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God as a necessary, maximally great, endless being vs. the challenge to an actual infinity

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In a recent thread, the Kalam Cosmological argument family was challenged on the issue: can an actual infinity exist? If not (presumably due to Hilbert’s Hotel-like absurdities), then God could not be an infinite being as such is impossible of being.

A thread of discussion developed, and I thought a summary intervention may be helpful. On further thought, perhaps it should be headlined:

_________________

KF, 12: >> I think several themes are worth highlighting.

It can be discussed that non-being, true nothingness cannot be a causal source. Were there ever utter nothing, such would therefore forever obtain. There would be no world.But, manifestly, there is a world.

So, we must ponder the logic of being, at least in a nutshell.

Candidates to being may be such that core characteristics central to identity stand in mutual contradiction such as those of a square circle. Such are impossible of being. And, we see principles of distinct identity necessarily embedded from the outset, especially that truths must be all so together so X and Y where Y = ~ X is not a possible state of affairs. If something could exist in a possible world were it actualised as a state of affairs, it is a possible being.

Of these, some Z can be in at least one possible world Wi, but not in another “neighbouring” one, Wj — contingent beings. The difference Wi – Wj will contain some unmet on-off enabling circumstance for Z . . . a necessary causal condition or factor for Z, say C. If Z is a fire, it requires heat, fuel, oxidiser and an uninterfered-with combustion chain reaction (cf. how Halon extinguishers work).

By contrast, we can see a being N that has no dependence on any such C, which will be in any and all possible worlds. That is, N is a necessary being and will be part of the common framework for any world W to exist. For example, distinct identity (A vs ~A) entails that two-ness and so also the endless set of naturals, must exist. [Beyond which lie the transfinites and the surreals as illustrated.] And, without necessary causal factors C, such has no beginning or end. Given that a world exists, at least one being N must be necessary. (Theists, classically hold that things like numbers are eternally contemplated by God, for instance.)

Any given case Ni is eternal, causeless, framework for any possible world, enabling and structuring it in some way. Notice, eternality not infinity, has been asserted, on the strength that for some W to be, some N must be as key to its framework. We readily see this for two-ness etc.

Such is strange to our ears, maybe, but that is a fault of our education not the logic.

Now too, our world is one of finite stage causal succession as we can see from succession of generations. But it is dubious for such to have existed to the infinitely — endlessly — remote past, as to succeed from some stage s_k to s_k+1, s_k+2 etc is equivalent to a counting succession 0,1,2 . . . which succeeds without limit but in an instance will be such that some later s_p to s_p+1, s_p+2 etc can again be matched 1:1 with 0,1,2 . . . thus showing that a transfinite span, credibly, cannot be traversed in finite-stage successive, cumulative steps. Thus if we are at a now, no S_k is transfinitely remote, even beyond say a big bang at 13.8 BYA. Our world W_a is credibly not some Ni, and has a beginning. It has a cause, a capable, sufficient one.

Where, we exist therein as responsibly and rationally free, morally governed creatures. This constrains the N_a that is at the world-root. For, post Hume et al, only at that level can the IS-OUGHT gap be soundly bridged. And we all know that after centuries of debate, only one serious candidate stands — just put up a viable alternative if you think you can. Good luck with that: ______ (Predictably, a fail.)

This is: the inherently good creator God, a necessary and maximally great being; worthy of loyalty and of the reasonable, responsible service of doing the good in accord with our evident nature.

But, what about, God is infinite?

I suggest, this first means that God is not externally limited or weakened so that he can be defeated or utterly frustrated in his purposes. Which, is among other things a way of saying that God is not evil, that being the privation, frustration or perversion of good capabilities out of their proper end.

God is also eternal and indestructible, as he has no dependence on external, on/off enabling causal factors. Thus, his being is without beginning or end, endless. This, being a characteristic of necessary being, which is required once a world is.

So, I think we need to reflect on what sense is meant when it is suggested that an actual infinity is impossible of being, and what are its strengths and limitations. For sure, an endless past of finite stage causal succession seems impossible and a physical, materially based infinite quantity is also dubious. But, the transfinite set of natural numbers and beyond the surreals great and small all seem necessary — framework to any world. Which in turn suggests mind capable of such a contemplation.

And more.>>

_________________

Food for thought. END

283 Replies to “God as a necessary, maximally great, endless being vs. the challenge to an actual infinity

  1. 1
    kairosfocus says:

    Is God a necessary (so, eternal), maximally great being without external limit to his being and power? How does this interact with the idea that no actual infinity is possible? (Does this, say, include that transfinite numbers and infinitesimals are impossible?) KF

  2. 2
    asauber says:

    kf,

    I think the simple answer is God is not subject to the rules of our universe that make an actual infinity an absurdity. We don’t really know what an infinity is/would be, anyway. So, I think, we are going beyond our capacities in this area. Just my 2 centavos.

    Andrew

  3. 3
    Origenes says:

    KF: Given that a world exists, at least one being N must be necessary.

    N is necessary for that world to exist. The world could not exist without N.

    This reminds me of ‘cogito ergo sum’, which I habitually cast into the incoherent form: “I doubt my existence, therefore I exist.” This draws attention to the impossibility for a non-existent entity to doubt its existence. In order for me to doubt my existence I must exist. The demand to explain my activity makes my existence “necessary.”

    Given that we seek understanding, the fact that a world exists, presupposes its cause — let’s call it the ‘First Cause’.

    How is logical that we stop there?

    Why not say: “Given that a First Cause exists, there must be cause for the First Cause?”
    We cannot say, in my opinion, “because that First Cause exists necessarily”. I also “necessarily” exist as an explanation for my activity (see above), but that necessity does not entail that my existence does not require an explanation.

    The only valid reason for stopping, in my opinion, is that by “world” we are referring to the reality as an entirety. So, the question is “what explains reality in its entirety?” And the answer: “The First Cause”.
    Given that we are talking about that which is foundational to all of reality, we can now see that it doesn’t make sense to ask “And what caused the First Cause?”
    Why doesn’t it make sense? Because there is no candidate available, since we referred to all of reality.

    If A causes all of reality, then there is nothing that can cause A.

  4. 4
    Dionisio says:

    KF,
    Thanks for writing this OP, a timely thought-provoking invitation to serious discussion on fundamental issues, right when we approach the traditional dates chosen for the celebration of a unique event that changed human history for good. Emmanuel became a reality, as it had been prophesied long before, and with it God’s immeasurable grace was poured over His sinfully rebellious creatures to save them from deserved damnation.
    But many have rejected the gracious offer, because this world prefers to sing Paul Anka’s song “My way” (Sinatra’s version was 75 weeks in the UK Top 40, a record still today). Along with John Lennon’s “Imagine” they are practically the hymn of Hades.

  5. 5
    daveS says:

    KF,

    I suggest, this first means that God is not externally limited or weakened so that he can be defeated or utterly frustrated in his purposes. Which, is among other things a way of saying that God is not evil, that being the privation, frustration or perversion of good capabilities out of their proper end.

    God is also eternal and indestructible, as he has no dependence on external, on/off enabling causal factors. Thus, his being is without beginning or end, endless. This, being a characteristic of necessary being, which is required once a world is.

    In my experience, when people say that God is “infinite”, they mean He has infinite powers, and that He can/could do anything outside the logically impossible.

    For example, let’s say the universe is finite in size, and consists of n fundamental particles. Then perhaps God could also have created a universe with 2n, 3n, 4n, and so on fundamental particles. In other words, in order to be infinite, God must have some power which is unbounded.

  6. 6
    Dick says:

    Origenes @3: “The only valid reason for stopping, in my opinion, is that by ‘world’ we are referring to the reality as an entirety.”

    Perhaps it’d be better to define ‘world’ as the entirety of contingent reality. If so, the cause of the world cannot be contingent since then it’d be part of the world and the cause of itself.

    The cause must therefore be non-contingent or necessary.

  7. 7
    Origenes says:

    Dick @6

    Dick: Perhaps it’d be better to define ‘world’ as the entirety of contingent reality.

    To be clear: the world is “contingent”, because it has a cause. And the cause is “necessary”, given that the world exists.

    Dick: If so, the cause of the world cannot be contingent since then it’d be part of the world and the cause of itself.

    So, the cause of the world cannot be ‘contingent’ (cannot be caused) because we have defined the cause as ‘necessary’?
    That would be assuming the conclusion, and that is exactly what I object to. As I have argued in #3, I am ‘necessary’ for my actions to exist, but that ‘state of being necessary’ does not exempt me from having an explanation. Similarly, the cause of a world X is not exempt from having an explanation, simply because it is the ‘necessary’ cause for that world.

  8. 8
    kairosfocus says:

    Origines, we are part of a world. Such a world cannot come from utter non-being nor from circular cause, nor from a chain of stages of cause-effect. This leaves necessary being as ground of reality, the issue is, of what character. Thence the further constraint that we deal with the morally governed and significantly free, just to be able to responsibly discuss. Where, necessary being will be framework to reality existing, thus any world, not just this one. KF

  9. 9
    john_a_designer says:

    The following is an argument I’ve developed for the existence of a transcendent, self-existent mind (God) which I think is a logical defeater for the idea that an infinite regress of “natural” causes is a good explanation for our existence.

    (1)Everything that begins to exist is contingent.

    (2)Anything that is contingent has an explanation for its existence. (It must be caused by something else.)

    (3)If the universe is contingent it has an explanation for its existence.

    (4)If the universe began to exist it must be contingent.

    (5)However, it is logically possible that something exists which is not contingent.

    (6)If such a being exists it exists necessarily. In other words, it is eternal or self-existent.

    (7)A necessary or self-existent being is not explained by anything else.

    Conclusion #1: Therefore, only a necessarily existing being could be the ultimate explanation for everything else.

    Conclusion #2: Therefore an infinite regress of contingent causes can never reach an ultimate explanation.

    Some implications:

    A necessary or self-existing being must have causal power. If it is the cause of the universe it must have sufficient power to cause the universe.

    It must have volitional intentionality. In other words, it does not need to create anything, it freely decides or chooses to create.

    It must be transcendent due to the fact that it is not contingent.

    It must have mind and intelligence as well as personal self-consciousness.
    This idea fits very nicely with a basic classical conception of God.

    In other words, if something is contingent then it cannot be necessary. If a necessary being exists it cannot be contingent. If it is even logically possible for the universe to be contingent how can we claim that it is necessary? In other words, if it is logically possible for the universe to be contingent then it cannot be logically necessary for it to be ontologically necessary. If a transcendent necessary being exists then it is logically necessary that it is ontologically necessary.

  10. 10
    kairosfocus says:

    Origines, kindly, consider whether distinct identity and so also two-ness had a beginning or can cease to be. Nope. We are so used to contingent beings that we find it hard to understand another mode of possible being is there, by the force of logic of being — and is exemplified in so familiar an entity as the number 2. Such beings have no dependence on on/off enabling causal factors [by contrast with a fire, a classic case of contingent being], and so a serious candidate to be like that will either be impossible of being or else will exist in the framework of any possible world. And this is not playing with definitions, we are looking at the nature of being per its logic and are considering cases in point. BTW, distinct identity [A vs ~ A] is the root of logic and of mathematics, which answers to why mathematics seems so unreasonably effective in studying the physical world around us. A possible world being in effect a reasonably complete coherent description of a way the or a world may be or is if actualised. A first, self-moved cause, may be contingent, us, or it may be necessary, presumably: God. Self-moved implicates reflexivity, especially acting on oneself by responsible rational freedom — such as ourselves. I choose to compose an OP and responsive comments, and actuate my fingers to do the typing, leading to a combox draft, then I submit and it is in the world through a mechanical chain of onward cause-effect bonds in cascade. So, it is not just necessary for this world, but necessary, period. KF

  11. 11
    Origenes says:

    KF @8

    KF: we are part of a world. Such a world cannot come from utter non-being nor from circular cause, nor from a chain of stages of cause-effect.

    Yet, the world can come from a cause A, which, in turn, is caused by X.

    KF: This leaves necessary being as ground of reality, the issue is, of what character.

    Not so fast! We must consider the possible scenario that our world is caused by A and that there is another world caused by B.

    KF: Where, necessary being will be framework to reality existing, thus any world, not just this one.

    Only if we consider the cause for all of reality do we arrive at one single cause.

  12. 12
    kairosfocus says:

    Origenes, sorry typo, an infinite successive chain of finite stage causes, as I discussed in my OP. The transfinite cannot be traversed in successive steps. The answer is finitely r5emote world root of necessary being character. Where such a being will be framework to any possible world, just as no world can exist without distinct identity thus two-ness. Such a being will be independent of on/off switch, enabling causal factors, it will not be like a fire. Such a mode of being is such that a serious candidate necessary being will either be impossible due to contradictory core characteristics or it will be in any possible world and actual in this one. It will be tied to the framework for a world to exist at all. Again, try the exercise of imagining a world where distinct identity does not exist, so two-ness does not exist, or where two-ness came into being, or ceases from being. You will find, not possible, any world Wi has characteristics that distinguish it in principle from not that world ~ Wi, two-ness is bound up in that distinct identity, and so are the core laws of logic . . . there can be no world where LOI, LNC and LEM do not obtain, nor the naturals and their directly connected extensions, which is very powerful indeed. Thus we see how such a being can be and is in a simple case. KF

  13. 13
    Origenes says:

    KF: The transfinite cannot be traversed in successive steps.

    I agree.

    KF: The answer is finitely remote world root of necessary being character.

    Something must be the foundation of reality.

    KF: Where such a being will be framework to any possible world, just as no world can exist without distinct identity thus two-ness.

    If you could create a world, then you would be the cause of that world. If we say that, in an ultimate sense, since the First Cause created you, the First Cause created the world you created, then we presuppose determinism and discount the possibility of you creating your world by your own personal free will. While it is the case that the First Cause is foundational to your existence, it is not necessarily the case that the First Cause is the only free actor.

    KF: Such a being will be independent of on/off switch, enabling causal factors, it will not be like a fire.

    I agree, such a being must be an indivisible unity.

    KF: Such a mode of being is such that a serious candidate necessary being will either be impossible due to contradictory core characteristics or it will be in any possible world and actual in this one.

    I don’t understand, by what logic must it be in any possible world?

    KF: It will be tied to the framework for a world to exist at all.

    There is no reality, and hence no world, without a First Cause. But the First Cause has chosen not to be the only actor, so, in principle, others can create a world as well.

    KF: Again, try the exercise of imagining a world where distinct identity does not exist, so two-ness does not exist, or where two-ness came into being, or ceases from being. You will find, not possible, any world Wi has characteristics that distinguish it in principle from not that world ~ Wi, two-ness is bound up in that distinct identity, and so are the core laws of logic . . . there can be no world where LOI, LNC and LEM do not obtain, nor the naturals and their directly connected extensions, which is very powerful indeed.

    I agree fully.

    KF: Thus we see how such a being can be and is in a simple case.

    There is a First Cause.

  14. 14
    kairosfocus says:

    Origenes, start with distinct identity and two-ness. Contrast a fire. The latter depends on enabling factors as illustrated in the OP, they are used to fight fires. By contrast, no possible world will be such that two-ness does not obtain, as distinct identity is bound up in what it means for a particular world to be. The difference between contingent and necessary possible beings, is there are worlds in which contingent beings would exist and “neighbouring” ones in which they won’t. The neighbour is one in which the switch is off — an enabling causal factor is in the blocking state. Think, spray Halon extinguisher at the base of a fire. But in the case of necessary beings, they exist in any possible world. This means they must be bound up in what it means for a world to exist, hence my framework language. Two-ness,as discussed, is a case in point. That is, this is not just an arbitrary definition plucked out of thin air. Let us start here, to be sure we have a clear view of what necessity of being implies. KF

  15. 15
    Origenes says:

    KF @

    I can easily agree that logic must hold in any possible world. Also, I agree that every world must have a cause. Thirdly I agree that reality, as a whole, must have a Foundation.
    What I do not see is that the necessity of logic takes you to the conclusion that the “Foundation” must create each world and/or must be in each world.
    Because every world is logical? By what reason is it that only the Foundation can produce logical worlds?

  16. 16
    kairosfocus says:

    Origenes, I am pointing out characteristics of necessary beings as a class and illustrating by way of distinct identity which happens to entail both the first principles or right reason and the natural numbers thence the extensions to the surreals, etc. These things do have causal influence, they constrain what can be e.g. through the logic of structure and quantity, mathematics. I am not saying that number and identity alone cause a world. The further point is, that necessity of being means that something of this character cannot not exist once there is any possible — coherent — world whatsoever. Such things are framework to any world existing. This is as distinct from contingent beings that need not exist in any possible world but will exist in some one world at least were it to be actualised, and would not exist in another. I used the further construction that the latter is a neighbouring world with an on/off enabling causal factor in the off position. Think, a fire. Now consider the possibility of candidate beings that have no dependence on any particular on/off factors. Now, perhaps the candidate is like a square circle, it has core proposed characteristics in mutual contradiction and cannot exist in any world whatsoever. But there is a case where a candidate necessary being is possible and exists therefore in at least one world, but as there are no factors that can turn it off, it will exist in any possible world. On inspection we see that such a being fills a bill: it has to be something that is framework to any world whatsoever existing. Distinct identity is a case in point, thus two-ness and numbers plus the principles of reason that are corollaries of distinct identity: LOI, LNC, LEM. In short, once there is any world, there must be a world-root that is unconditioned on any external on/off causal factors. The issue is what fills that bill, especially given that we find ourselves in an actual world and that just to reason together here, we are responsible and rationally free morally governed creatures. KF

  17. 17
    Origenes says:

    KF: In short, once there is any world, there must be a world-root that is unconditioned on any external on/off causal factors.

    Yes there must a foundation to reality and yes there must a cause for the world, but does it follow that they have to be same person? I have no objection of course, but I do not see the compelling logic.

    Can you please point out which premise you reject?
    1. The First Cause creates a free responsible rational independent being.
    2. The free being creates a world.
    3. The First Cause does not have to be involved in (2).

  18. 18
    polistra says:

    The great big problem with this whole argument is that you’re stuck on the unnecessary concept of particles. If you think of reality as particles, then you have the Hilbert thing.

    I doubt that any sophisticated religion thinks of God as built from particles, even if you want to think of “things” as Lego assemblages. But there’s no reason to limit this to the “spiritual” side of reality.

    When you start with waves, the concept of “finite” is impossible and meaningless. You can’t count the “pieces” of a wave because it’s continuous.

    Particles are an unnecessary and confusing and contradictory entity. Just think of everything, gods and nongods, as waves. Then the whole problem disappears, along with most other “contradictions” in “physics”.

  19. 19
  20. 20
    kairosfocus says:

    Origines, necessary being is not strictly about creators much less persons. Indeed, that is part of the point on two-ness and distinct identity — even a free being creating a world (actually, world is bigger than sub cosmos, it is about the overall system of reality) cannot build a world without distinct identity, and would be in the context of being a secondary creator or designer. Moreover, Demiurges are within, not beyond the world. There is a later inference to best explanation argument but that is later and tied to moral government of our rational amimality. KF

  21. 21
    Seversky says:

    There seems to be no getting away from an infinity of one sort or another, either an infinite multiverse or an eternal God or maybe they’re one and the same thing.

  22. 22
    mikeenders says:

    God is not infinite. he is necessarily bounded by who he is in timelesness. He cannot lie . He cannot change and he will never violate his own rules.

    Everything he does , thinks or says he actuates in his now. There is no past or future in his frame. He created time and process but is without it.

    Physical based infinities are paradoxical because they express endless increase or change. God does not and cannot increase his strength or even expend energy. It is complete authority with nothing against it. Complete never increases its set at complete for all time. God has no endless reservoir of energy he pulls from. Its just the same never increasing or decreasing or used up authority. You could beat God if he held back using his authority – Beat him in a hand wrestling match or you could even nail him to a cross.

    We see this misunderstanding of God and his power all the time but most glaringly with the old gotcha question from materialists – “can God create a rock so heavy he cannot lift.” As if God has the kind of material energy like humans that can be expended so he needs to increase it infinitely. he doesn’t. Its all authority. Nothing can disobey or resist and takes no energy as we know it. A stone of any size must obey regardless and with no increase in output from God. Like the centurion in the NT understood – giving a command to one or 200 soldiers is the same. no infinite power – just the same authority.

    God CREATED the concepts of power and energy we look at in the material world. This is where we miss the boat. material power is drawn from by a process (and time based) and a physical one. Authority is not process based and so there is no drawing from an infinite power.

    Now if materialism worked without contingent cause, no change and timelessness then it might work as an equal. It would NOT be paradoxical but then again if you have timelessness, no increase and no change materialism dies on the vine and is no longer materialism.

    Once you realize as Scriptures state – that God does not change I see no issue, God is not infinite in that sense of change or increase in ANYTHING he does in his now.

  23. 23
    kairosfocus says:

    Seversky, the set of possible worlds would be contemplated in the mind of God, eternally, just like the set of surreal numbers. Which highlights that there are more transfinite numbers, continuum numbers and so on in aggregate than the number of naturals. Thus the mind of God is infinite. KF

    PS: BTW, some think that we are in effect a species of thought in the mind of God. A sort of giant simulation. I actually think, some worlds are conceptual but not actual, and the best way to see our world is that God chose to implement it physically, though he sustains its every where and every when as well as every one by his power from moment to moment. Absent that enabling presence, poof, gone.

  24. 24
    kairosfocus says:

    ME, As you will see from the just above, I consider that it takes infinite mind to contemplate the panoply of infinite possibilities out there, infinite numbers, infinite possible worlds, and much more . . . note, omniscience. In addition, by being necessary and eternal, God is endless in existence. he is maximally great — possesses only great-making properties and such to their maximal degree, with no less-making properties. He is also every where and every when upholding reality from moment to moment and place to place by the word of his power. In him, we live and move and have our being. His limits are, first, those of necessary impossibility, e.g. God cannot create a square circle as such is impossible of being by virtue of containing inherently contradictory core characteristics. Likewise, he is bound by his character and promises/ word: it is impossible for the inherently good God to lie or to otherwise do what is wrong. He has granted a class of creatures the power to be responsibly and rationally free as en-souled, en-conscienced beings with minds of their own; which implies a voluntary restraint in his use of his power, the better to enable a world in which true love is possible — requiring actual freedom to choose (thus opening up a whole wold of possible goods that would not otherwise obtain). In the Judaeo-Christian tradition, this class embraces us, and possibly several races of what are commonly called angels. It may embrace other creatures in ways we do not understand, or have yet to encounter — who knows, even within our own galaxy there may be many other races capable of love and of other virtues as a result. KF

    PS: God as unchanging implies stable essential, core characteristics, so that God is utterly reliable. His word is given and it will not return to him void.

  25. 25
    kairosfocus says:

    Polistra, I long ago learned the concept, wavicles at the ultramicroscopic level. Light was a classic wave phenomenon, until cavity radiation forced the issue of facing the ultra-violet catastrophe by suggesting an energy penalty for the radiation field: E = hf. Then, five years later, Einstein’s Nobel Prize-winning solution of the photoelectric effect: light was always a particle, which he termed the photon. Even vibrations in materials are now regarded as phonons. Then, Thomson’s son discovered the wave properties of the electron — complementing his father’s discovery of that particle. Then, in recent times it has been demonstrated that an electron double slit interference experiment works still when we attenuate a beam down to a degree where only one electron at a time is in transit . . . so, wavicle. None of the above is seriously modified by this. If you imagine that superposition of possibilities and Schroedinger’s poor cat in a black box imply a breakdown of distinct identity, kindly consider that quantum phenomena and laws are studied and empirically tested through implicit reliance on distinct identity and linked distinct core characteristics. From scratch-marks on chalk boards as the math is set up to observing electron beam interference patterns, we rely on distinct identity. KF

  26. 26
    daveS says:

    ME, As you will see from the just above, I consider that it takes infinite mind to contemplate the panoply of infinite possibilities out there, infinite numbers, infinite possible worlds, and much more . . . note, omniscience.

    Yes, and as an example of His omniscience, God knows every digit in the decimal expansion of π. I don’t know how ME or anyone could deny that implies God is infinite.

  27. 27
    tgpeeler says:

    Concerning the possibility of an infinite regress being able to explain anything (granting for the moment that an infinite regress is even coherent). Let contingent (or possible) being = 0. Let actual being = 1. Contingent being (0) needs to be explained by prior being (1) since it is nothing before it becomes something (it is still only possible). But this means that somewhere in the regress there must be a 1. In other words, an infinite regress of 0s will always be 0. There will be no being without a 1 (necessary being) to terminate the regress. So even if I granted the possibility of an infinite regress it still explains nothing. An infinity of 0s still leaves us with 0. Since there must be a 1, it is by definition not contingent (1 is not equal to 0), thus it is non-contingent, or necessary. (One can take the essence/existence distinction to a being whose essence is existence with this rationale.)

    Perhaps a concrete example will make this clearer. Suppose I wanted to borrow an iPad from KF but he didn’t have one. Well, KF, being the generous helpful guy he is would offer to borrow one for me from someone else. But that person didn’t have one either. And so on… Could I ever have an iPad in my hands if no one in the regress didn’t have an iPad to give? No. An infinite number of people without iPads would be of no help to the one who needed an iPad. It is the same way with being. There must be a being in the regress who doesn’t get his being from a prior being. A first being.

    This may be a simpler way to say it: If there was never nothing there would still be nothing. But there is something. So there never was nothing. (Granted the linguistic games of talking about “nothing.”) That is, there has always been something. There is an eternal being. When one “runs the numbers” one gets to an uncaused, immaterial, necessary, immutable, and timeless being. Or God, as we Christians like to say.

  28. 28
    daveS says:

    tgpeeler,

    Isn’t there a difference between the two scenarios you describe?

    In the iPad example, no one has an iPad, so it’s impossible for you to get one that way.

    In the previous example, presumably each being in the chain can give being to the next. For example, a parent can give being to a child.

    I don’t see that the two scenarios are parallel at all.

  29. 29
    daveS says:

    PS to my #28:

    Here’s an illustration of the distinction I have in mind:

    1) Suppose you already have an iPad, and you tell me that KF gave it to you. KF got it from someone else, &c. ad infinitum.

    2) Suppose you want an iPad, and you ask KF. He asks someone else, &c. ad infinitum. (That’s what you desribed above).

    I don’t see that 1) is logically impossible, in any case.

  30. 30
    tgpeeler says:

    It’s the juxtaposition of the two scenarios that illustrates the point. Scenario 1. I don’t have an iPad. I ask an (infinite number of people) for an iPad. None of them have one. That explains why I don’t have one still. Scenario 2. I do have an iPad but the question now becomes how did I get it? Scenario 1 fails since if that were true I still wouldn’t have an iPad, but I do have one. So… in the second case, someone in the chain MUST HAVE had an iPad to give else I still wouldn’t have one. But I do have one. So the explanation is obvious.

    In the case of being, think of it the same way. My parents were 0 before they became 1s, and their parents, and so on. There has to be a Being in the regress that has being to give without getting it from another (contingent) being. No Being with being to give to every existent means no existents. But there are things. Blah blah blah.

  31. 31
    daveS says:

    tgpeeler,

    Thanks for the response.

    However, I don’t see that anyone in scenario 1 must have had an iPad without being given one; likewise, I don’t see that anyone in the chain of beings must have been a necessary being. Why can’t they all be contingent?

    I think it takes some additional metaphysical assumptions to rule this out.

  32. 32
    john_a_designer says:

    The 17th century philosopher G. W. Leibniz argued that an appeal infinite regress does not provide a sufficient reason to explain the universe or our existence. Notice this is not merely an appeal to intuition.

    For a sufficient reason for existence cannot be found merely in any one individual thing or even in the whole aggregate and series of things. Let us imagine the book on the Elements of Geometry to have been eternal, one copy always being made from another; then it is clear that though we can give a reason for the present book based on the preceding book from which it was copied, we can never arrive at a complete reason, no matter how many books we may assume in the past, for one can always wonder why such books should have existed at all times; why there should be books at all, and why they should be written in this way. What is true of books is true also of the different states of the world; every subsequent state is somehow copied from the preceding one (although according to certain laws of change). No matter how far we may have gone back to earlier states, therefore, we will never discover in them a full reason why there should be a world at all, and why it should be such as it is.3

    (3) Gottfried Wilhelm Leibniz, Philosophical Papers and Letters, ed. by Leroy E. Loemker (Kluwer Academic, 1989); p. 486.

    Using Leibniz’s argument above I think we can conceive of another plausible argument which argues against an infinite regress: an ontologically hierarchical regress rather than a temporal or sequential one. I call it the ontological ladder. The argument begins with the observation that even if we grant, for sake of argument, that an infinite regress of books being copied is possible, the books themselves are insufficient sustain the regress, because a book by itself cannot copy itself. To create a new book requires the existence of things like paper and ink (if it’s being copied by hand) as well the material to bind the book together. Of course, neither are paper and ink etc. sufficient to generate a copy of a book. We need either a conscious agent or a robot to copy the book. If it’s a conscious biological agent then he or she requires a cause as well as habitable world—an earth like planet with the right raw materials to sustain the copyist and provide the materials for book production and some kind technological society in which the copyist can learn to read and write and can manufacture the paper and ink etc. required to create new copies of the book.

    Even robots, which could theoretically build a copies of themselves could not simply be floating out in space. They also would need the raw materials to copy themselves and the books. So besides the books we needs a lot of other things that need to be explained. Furthermore, we know that people, robots, stars and planets are not the kind of things that have always existed, so we cannot claim that they are the kind of things that in and of themselves necessarily provide us with the complete reason or explanation as to why anything at all exists. But what does? If we continue to climb this ontological ladder, the argument goes, we seem to be moving towards some kind of higher being, a mind or intelligence, which is needed to explain not just the books but also the copying of the books, their material cause etc. Theists would argue only an eternally-existing transcendent mind (God), which avoids an infinite regress, is a sufficient explanation.

  33. 33
    tgpeeler says:

    I am not clear on what your objection is to this line of reasoning. In scenario 1 no one has an iPad. Thus I couldn’t possibly have one either.

    All beings cannot be contingent because all contingent beings at one time did not exist (this includes the universe, by the way) and it takes existence (an efficient cause) to bring something into existence. If you have no necessary existence then you have no existence at all. Think of it this way. Can you imagine typing on a non-existing keyboard or drinking non-existent water or eating non-existent food? Of course not. The principle we abstract from these (universal to all people) observations (evidence) is that existence precedes causation. No existence – no causation or creation. There has always been existence and it isn’t us or the universe. Perhaps this helps.

  34. 34
    daveS says:

    tgpeeler

    I am not clear on what your objection is to this line of reasoning. In scenario 1 no one has an iPad. Thus I couldn’t possibly have one either.

    Maybe the numbering of our scenarios conflict. In my scenario 1, there is an iPad that has been passed down through an “infinite regress”, so everyone has one.

    All beings cannot be contingent because all contingent beings at one time did not exist (this includes the universe, by the way) and it takes existence (an efficient cause) to bring something into existence. If you have no necessary existence then you have no existence at all.

    Well, I thought we were assuming that infinite regress was coherent, for the moment? In that case, I have in mind an infinite regress of contingent beings, each giving existence to its successor.

    There is no “bringing something into existence” from a vacuum. Every being is brought into existence by its predecessor.

  35. 35
    tgpeeler says:

    You can only imagine (if the infinite regress is assumed to be coherent) that you can have an infinite series of contingent beings but once you rationally dissect the idea you see that it is, in fact, incoherent. To say that you can have contingent beings without a necessary being is like saying you can have square circles. Once you understand what contingent means and what necessary means then you know.

  36. 36
    tgpeeler says:

    There has to be a first being. Here’s the argument for that. If there was no first being, there would be no subsequent beings. (Agreed? If not, you are claiming that nothing has the power to create.) But there are subsequent beings. Therefore, there WAS a first being. The first being can’t be preceded or caused by another being because then it would not be FIRST. But there is a FIRST being. An uncaused being. An eternal being. A necessary being. A changeless being. A timeless being. Do you see this?

  37. 37
    daveS says:

    tgpeeler,

    I think I understand contingent and necessary beings well enough. The contingent beings I’m referring to all depend on some other being for their existence, so they fit the definition.

    Unless you add some more premises (such as the principle of sufficient reason), I don’t see how you can show that the existence of contingent beings implies the existence of necessary beings.

  38. 38
    daveS says:

    tgpeeler,

    Cross-posted there. To answer your #36, no, I don’t see that your argument is valid (without adding more premises).

    Why couldn’t there be an “infinite regress” of beings? (I don’t like that term, but it’ll have to do, I guess). In that case, there would always be “subsequent” beings, but no first being.

  39. 39
    mikeenders says:

    “Yes, and as an example of His omniscience, God knows every digit in the decimal expansion of ?. I don’t know how ME or anyone could deny that implies God is infinite.”

    Its actually quite easy Dave…when you stop thinking of it in human terms. God does not access a knowledge of decimal expansion of anything. He determines what it is.

    there are no infinite possibilities “out there”. there are no possibilities till he determines them. KF is wrong. God can create a square circle if he wishes. there are no inherently contradictory core characteristics with it because God determines whats inherent to ANY construct.

    If tomorrow God determines that a square is a circle then that will be what it is (even if it requires several more dimensions). God is a Spirit – shape , size dimensions. angles (all which determine whats a circle and whats a square) are all his creations that he created by command where none existed before. So he solely determines the bounds of contradictions because he is the only true bound.

    Is very hard for us to grasp a being that completely exists to himself and from which ALL things flow including knowledge, time and physical dimensions. We are used to and surrounded by things that are not ours that we have to access or modify or that hinder us.

    So God is not sitting in heaven accessing an infinite storage of the expansion of any human mathematical term so he can give you an answer he draws from his memory. He is above and beyond that. he’ll tell you once what it is and that will be what it is.

    This is what the old theologians used to understand of God in the word Sovereign. Nothing exists outside of himself and his decree not even the placement of a decimal

    God IS limited……. by and to himself. he will never go outside of himself, think a different thought that is not not his, learn anything new. he will never access any more power than he already has because power does not exist that he does not now already hold. he cannot lie, he cannot become evil and he cannot change – all limits but great ones to have.

    To put it more succinctly God does not possess infinite power. He is power. The only power that ultimately exists. So its meaningless to talk about infinitely powerful when there is only one power and its defined by the authority of one entity. Its not meaningless to say all powerful because it highlights he has it all – no one else has any real power.

    God is all about authority and all authority is not infinite. its set at one unchangeable constant – all.

  40. 40
    asauber says:

    Why couldn’t there be an “infinite regress” of beings?

    Because context is removed. There’s no here or there in an infinity. There’s no now vs. then. There are no relationships because there’s no context anymore. There has to be finite context.

    If you try to insert finite things in an infinite context you are just mixing two incompatible ideas together. Illogical, the saying goes.

    Andrew

  41. 41
    EricMH says:

    If God is an actual infinity, are humans also actual infinities? The human mind can generate an unbounded amount of novel information, given unlimited resources. Does this denote an infinite source of information, or the creation of new information?

  42. 42
    Origenes says:

    KF @20

    Makes perfect sense. I think we are in agreement.

  43. 43
    asauber says:

    So, the way I see it, God made the finite for us to live in. A wondrous miracle to ponder this Advent, as He came to dwell with us in it, also.

    Andrew

  44. 44
    john_a_designer says:

    What I call the ontological ladder above @ #32 Ed Feser terms “an essential series of causes” (vs. “an accidental series of causes.”)

    I briefly discuss his argument (actually Aquinas’ argument) here:

    https://uncommondescent.com/philosophy/philosopher-ed-feser-offers-some-fun-richard-dawkins-vs-thomas-aquinas/#comment-643349

    Also see #5 on that thread.

    And then take a look at this interview where Feser very clearly and succinctly lays out his argument:

    https://www.youtube.com/watch?v=9R3BXJVjwKI

    He seems to do a much better job IMO when he is discussing his argument casually and extemporaneously.

  45. 45
    kairosfocus says:

    DS, a vacuum is not non-being. KF

    PS: The problems with infinite regress are pointed out above in brief and have been discussed many times. You cannot traverse a transfinite span in finite stage successive steps. The latter can never count up to the transfinite. An actually infinite past would require this, not just word games on at any point the infinite has already been traversed. As to the chain of present supportive causes, the point is, an infinite depth of the contingent does not amount to necessary being. Of course, one can blandly assert and cling to the opposite if s/he wishes.

  46. 46
    daveS says:

    KF,

    Yes, I was using “vacuum” as a figure of speech here, not using the term literally. I’ll be interested to see what tgpeeler says.

  47. 47
    Origenes says:

    DaveS

    The Leibniz quote in #32 by john_a_designer is about your scenario. Did you notice?
    In his example there are no IPads but, instead, books are copied. Leibniz’ conclusion:

    … we can never arrive at a complete reason, no matter how many books we may assume in the past, for one can always wonder why such books should have existed at all times; why there should be books at all, and why they should be written in this way.

  48. 48
    daveS says:

    Origenes,

    Yes, he’s invoking some form of the principle of sufficient reason there.

  49. 49
    kairosfocus says:

    DS,

    I think I should comment on the PSR. Take as a weak form, that of any entity or candidate entity A, we may freely inquire as to why it is, or is not, or is impossible. This is patently unobjectionable.

    Now, some answers.

    Some things are not as they are impossible of being, there being a mutual contradiction of core characteristics connected to A having a distinct identity as itself. There can be no square circles.

    Other things may be in some possible world, some possible, reasonably completely describable state of affairs. Such are possible beings.

    Of these, some could exist in some world say Wi but in a neighbouring one Wj, they would not, This is because, there is in Wj some on/off enabling causal factor for A that is in the off state, but in Wi, it and all other similarly enabling factors are on, and there is cumulatively a sufficient circumstance for A to be. A, e.g. a fire, is caused.

    Others, are such that there is no possible world in which they will not exist. Such as, distinct identity and thus two-ness. Such a case would be independent of on/off enabling causal factors, and in addition would have to be possible. Thus, we see that a necessary being is without cause and a serious candidate will either be impossible or else actual. That is best understood as being due to such necessary beings being part of the framework for any possible world to be.

    So, there will be a good and in my view sufficient reason behind anything that is, or is not, or cannot be.

    Simply trotting out PSR and by implication the objections that have been made, is not sufficient to set aside a matter. At least, once we have looked at the logic of being.

    KF

  50. 50
    daveS says:

    KF,

    I’m not “trotting out PSR” as an excuse to set aside the matter. I’m just saying that I think tgpeeler has to invoke it or something like it to arrive at a valid argument.

  51. 51
    john_a_designer says:

    Earlier at #9 I wrote:

    “If a necessary being exists it cannot be contingent. If it is even logically possible for the universe to be contingent how can we claim that it is necessary? In other words, if it is logically possible for the universe to be contingent then it cannot be logically necessary for it to be ontologically necessary. If a transcendent necessary being exists then it is logically necessary that it is ontologically necessary.”

    Or in terms of the PSR: If the universe is contingent (it had a beginning in space and time) then it lacks a sufficient reason for its own existence. Invoking the so-called multiverse does not solve this problem because a collection of contingent things, even a larger or infinite collection of contingent things is not sufficient to explain its own existence. That was Leibniz’s point.

    Also notice that for the atheist to argue that only contingent things exist he must smuggle in, or coopt, some theological concepts: infinite being, eternality and transcendence. For example, in invoking the multiverse is an attempt to bring in an outside transcendent cause. Furthermore, it’s a purely metaphysical argument that is neither self-evidently true nor scientifically provable. Even if we are someday in the far distant future we are able to create artificial wormholes, as Kip Thorne suggest in his book, Black Holes & Time Warps, which he argues could theoretically tunnel through into another universe, how would we know that we are really in another universe? How could we ever know that there are an infinite number of other universes? In other words, are we supposed to accept (believe) the idea of the multiverse on faith? So then it appears that the claim that atheistic naturalism/materialism is more reasonable than theism is just so much pretension and posturing.

    Furthermore, why are we as a species “hardwired” to ask these kind of questions? Why do we feel so compelled to explain our own existence?

  52. 52
    daveS says:

    JAD,

    Also notice that for the atheist to argue that only contingent things exist he must smuggle in, or coopt, some theological concepts: infinite being, eternality and transcendence.

    I for one, would not hesitate to shamelessly steal concepts from other worldviews, provided this doesn’t result in a glaring inconsistency.

    That said, who gave theists sole rights to “infinite being” and “eternality”? Even “transcendence”, for that matter?

  53. 53
    kairosfocus says:

    DS, pardon, but that is the typical way it is referred to — to dismiss. That’s why I took time to point to a weak and patently unobjectionable form that invites us to look at the logic of being. In that context JAD is correct to say existence of a contingent being is not self-explanatory as it depends on what I have termed on/off external enabling factors (as necessary causal factors is ambiguous). KF

  54. 54
    tgpeeler says:

    daveS – The premises in my argument are two. 1. Things exist. 2. Unless something exists, it cannot create or cause something. What more do I need?

    As far as who gave theists sole right to “infinite being” and “eternality” etc… I guess nobody did. We just use those ideas coherently.

  55. 55
    daveS says:

    tgpeeler,

    You can’t deduce much starting with those two premises. Here’s how I understand them:

    1) For some X, X exists.

    2) For every X, either X exists or X cannot cause or create anything.

    Both appear to me to be consistent with the existence of an infinite regress of contingent beings.

  56. 56
    Origenes says:

    DaveS @ 29

    1) Suppose you already have an iPad, and you tell me that KF gave it to you. KF got it from someone else, &c. ad infinitum.

    2) Suppose you want an iPad, and you ask KF. He asks someone else, &c. ad infinitum. (That’s what you desribed above).

    I don’t see that 1) is logically impossible, in any case.

    I take it that you do hold that 2) is logically impossible. If so, it is likely that you agree with KF’s

    The transfinite cannot be traversed in successive steps.

    The IPad in 2) would never come into existence. We would see an endless row of people, each requesting an IPad from the person before her/him. No one would have an IPad, everyone is waiting for someone else to give it.
    IOWs 2) is not an ‘IPad-producing-process.’ An IPad does not come into existence by a world solely inhabited by people who do not have one but want to get one from someone else.

    Let’s look at 1) again:

    1) Suppose you already have an iPad, and you tell me that KF gave it to you. KF got it from someone else, &c. ad infinitum.

    This is logically impossible, since you offer the scenario in 2) as an explanation for you having an IPad, but, as has been argued, 2) cannot be the explanation for you having an IPad.

  57. 57
    daveS says:

    Origenes,

    I actually do not agree with KF that the statement “the transfinite cannot be traversed in successive steps” has been shown to be true. That’s the issue at the center of all those “infinite past” threads we struggled through.

  58. 58
    kairosfocus says:

    DS, I pointed the disagreement. I suggest that if an iPad is infinitely remote, it can never be traversed to here and now in a succession of finite stage steps as it is precisely that, endlessly remote. KF

  59. 59
    Origenes says:

    DaveS @57,

    Do you agree with my analysis (see #56) that in your scenario 1) (see #29) you offer the scenario in 2) as an explanation for you having an IPad?

  60. 60
    daveS says:

    KF,

    I suggest that if an iPad is infinitely remote, it can never be traversed to here and now in a succession of finite stage steps as it is precisely that, endlessly remote.

    KF,

    No doubt you can guess what my response would be. 🙂

    And I can probably guess what your response to that will be.

  61. 61
    daveS says:

    Origenes,

    No, the two scenarios are very different. For one thing, everyone has an iPad in scenario 1, while no one does in scenario 2.

    More importantly, the traversal in #1 is moving forward in time, while the other one is moving backward. One is beginningless, one is endless. In one case, the infinite set traversed has order type ω*, while the other has order type ω.

    KF and I (among others) probably spent hundreds of posts debating the distinction between these two types of traversals.

    To sum up, my explanation in #1 is completely different than scenario #2.

  62. 62
    Origenes says:

    DaveS: No, the two scenarios are very different. For one thing, everyone has an iPad in scenario 1, while no one does in scenario 2.

    Oh, I misunderstood. I thought in scenario 1) a single IPad was in play. In #29 you are not making this clear.

  63. 63
    daveS says:

    Origenes,

    Sorry, you were right—there is just one iPad, but it’s being passed around to everyone.

  64. 64
    Origenes says:

    DaveS: More importantly, the traversal in #1 is moving forward in time, while the other one is moving backward. One is beginningless, one is endless.

    #1 seems to have a problem at the end (or in the now, if you will), because there is no one to give an I pad to (yet), so, unlike everyone else, that person has 2 IPads.
    #2 seems to be beginningless in the sense that there is no IPad coming into existence.
    – – – –
    edit: irrelevant because of #63

  65. 65
    tgpeeler says:

    daveS, my point is that an infinite regress of no one with an iPad means that I cannot have one. But (modus tollens) if I DO HAVE one then the explanation lies in the termination of the alleged infinite regress with someone who has one to pass forward. This seems very clear to me.

  66. 66
    daveS says:

    tgpeeler,

    Yes, that also seems clear to me, so perhaps we are talking about different things.

    My point was that there need not be a “first” person with an iPad. That is, why couldn’t everyone in the chain receive the iPad from someone else, and then pass it on to the next person?

  67. 67
    Origenes says:

    DaveS: My point was that there need not be a “first” person with an iPad. That is, why couldn’t everyone in the chain receive the iPad from someone else, and then pass it on to the next person?

    To be clear, there is only one IPad in play. And everyone at some point held this IPad. Right?

  68. 68
    daveS says:

    Yes, that’s right.

    Edit: Here’s a visual.

  69. 69
    tgpeeler says:

    Geez. I am beginning to doubt my command of the English language. The original point, if I remember correctly, was to deny the ability of an “infinite regress” to explain anything. In fact, I would deny the existence of any infinite regress because any series of events is finite by definition, but for the sake of argument, I am “playing along” with the idea that an infinite regress is possible.

    In my illustration, I tried to make the point that an infinite number of people without iPads (or whatever) could not explain why I (at the current end of the regress) had one. In order for the person next to me to be able to give me an iPad, he would have to have one first. And in order for him to have one, the person next to him would have to have one. And so on. If no one if the regress had an iPad to give then obviously I could not have one either. An infinite number of 0s does not = 1. BUT, if I did have an iPad, that means that someone in the regress had it FIRST. That means they didn’t have to get it from someone else because no one else had one. So the one with the iPad passes it forward and it becomes possible for me to have it. Note the regress is no longer “infinite” as it began with the one holding the iPad. I am sure this argument is demonstrative.

  70. 70
    tgpeeler says:

    I just saw the visual. This is not what I’m talking about. In fact, this illustrates my point. The only reason the last guy could have the football was that someone in the regress first had one to toss to him. If no one had a football, the guy at the end could never have had one. There is no such thing as an infinite regress, for one. It’s akin to saying square circle. And two, any appeal to an infinite regress as an explanation for anything fails because it’s an incoherent idea. Once the terms are defined it becomes obvious that “infinite regress” is nonsense.

  71. 71
    daveS says:

    tgpeeler,

    BUT, if I did have an iPad, that means that someone in the regress had it FIRST.

    But again, why must there be a first person to have the iPad? I see no argument to justify this conclusion.

    If infinite regresses are impossible, then of course that solves your problem, but my understanding is that we are, for the sake of argument, assuming they could exist.

    Under that assumption, why couldn’t everyone in the infinite chain possess the iPad briefly before passing it on to the next person?

  72. 72
    Origenes says:

    DaveS

    Suppose someone is standing outside the infinite regress holding an IPad. And suppose that no one in the infinite regress has an IPad.
    The person outside wants to give the IPad to someone inside the infinite regress, in order for you (at the end of the line, in the now) to have one, but he needs to give it to the ‘First Person’ in line. He must give the IPad to the ‘First Person’, because the requirement ‘everyone at some point held the IPad’ must be true.
    This person, outside the infinite regress, is in for an endless journey.
    IOWs the IPad never enters the infinite regress.

  73. 73
    tgpeeler says:

    daveS, “But again, why must there be a first person to have the iPad? I see no argument to justify this conclusion.”

    Because if there is no FIRST then no one has an iPad. Surely you can see this. If no one has one then no one has one. But I do have one. Ergo someone had one to give.

    Perhaps if you try to state your argument you will see the flaw.

    One clarification prompted by UB. I claim an infinite regress can’t exist because it’s a contradiction in terms. A regress is a series of events and there can be no such thing as an infinite series of events because a series can always have one more. We say “infinite regress” as if it is something, but it’s not. It’s no more a thing than a square circle is a thing. Something that is infinite has no potential to be more than it is. There is no adding to it. IF you are adding something to something else it’s not infinite.

  74. 74
    daveS says:

    Origenes,

    Certainly the outside person would not be able to give the iPad to the “first” person in the regress, since there is no such first person.

    I also agree with your last statement—in my scenario, the iPad never enters the regress, because it was never outside the regress to begin with.

  75. 75
    daveS says:

    tgpeeler,

    I think we’re still talking past each other on the iPad example. Perhaps my answer to Origenes above will help. In the scenario I described, I do not envision the iPad “entering” the infinite regress at all. It’s eternally being passed down through this chain of people, in a “beginningless” process. It wasn’t bought at an Apple store in 2010 and somehow inserted into the infinite regress.

    I don’t have much more to say about the iPads, but I do have a question/comment about your understanding of “infinite” and infinite regresses.

    I don’t know of any requirement that you cannot “add” anything to a collection which is already infinite. As an example, the set of integers is infinite, but you can add the number 1/2 to this set (and the result will be a new, infinite set).

  76. 76
    asauber says:

    “the set of integers is infinite”

    daveS,

    This is the infinite rut you are perpetually stuck in. A person can try to appeal to a mathematical infinity, but that is just a way of using your imagination.

    Numbers that relate to some kind of reality are finite.

    You think that because you can attempt to imagine some kind of infinity, it exists. There is nothing that can tell you it exists. Just like nothing can tell you a squared circle exists, even though you can try to imagine it.

    Andrew

  77. 77
    Origenes says:

    DaveS @

    DaveS: In the scenario I described, I do not envision the iPad “entering” the infinite regress at all. It’s eternally being passed down through this chain of people, in a “beginningless” process.

    There is an infinite chain of people eternally passing down one IPad.

    The problem does not go away. If all people in the infinite chain must have held the IPad, then there is no position for the IPad. And that’s the obvious problem: in order to be inside the chain the IPad needs to have a position.
    For any position inside the infinite regress goes that the IPad cannot be held by everyone prior to that position. Prior to any position, there is infinite number of people, so it cannot be the case that everyone held the IPad. Every IPad actually passed down the chain started ‘illegally’.

  78. 78
    vividbleau says:

    tgp

    “Because if there is no FIRST then no one has an iPad. “

    Im sure you recognize that it is Daves position that there need not be a first perhaps that is why you might feel you are talking past one another.

    Daves

    Im not clear how you can add anything to an infinite, if you were able to add something to an infinite it seems to me that means it wasn’t infinite. Using your example of adding 1/2 would that not be the “current” last number in the chain? If its the last “current” number then does not the number chain end at that point until another number is added? How can an infinite have an end?

    Vivid

  79. 79
    tgpeeler says:

    Yeah, I’m through on this thread. Best shot. If daveS can’t see it well then he can’t.

  80. 80
    tgpeeler says:

    I would be interested to see daveS make some kind of argument for his position, whatever that is, rather than just say “gee I don’t get what you are saying.”

  81. 81
    vividbleau says:

    Tgpeeler

    Eternal I Pads.

    Vivid

  82. 82
    daveS says:

    Origenes,

    The iPad does have a position, it just varies over time as it is passed from one person to the next.

  83. 83
    daveS says:

    vividbleau,

    I’m using the term “infinite” here only with reference to the number of people involved in the chain. I’m not saying the chain of people is unbounded in every possible sense, or something like that.

    We have an assumed chain of people, passing an iPad between one another. It’s easy to understand what a finite chain would look like: It would have some nonnegative integer number of members (5, 17, 1001, for example). The chain we’re discussing is not like that. Its length exceeds any finite integer value. It’s ok for it to have an end, and we could always add additional people to that end.

  84. 84
    Origenes says:

    DaveS @

    DaveS: The iPad does have a position, it just varies over time as it is passed from one person to the next.

    Yes, that is your claim. I know. According to you, the IPad can be inside the infinite regress and can be held by one of its members at time t.
    However, as I have argued (see #77), the IPad cannot have a position (cannot be held or passed on by any of its members) in the infinite regress AND meet the requirement: ‘this IPad is held by all preceding members in line’.

  85. 85
    daveS says:

    Origenes,

    It is true that prior to any position in the chain, there are infinitely many other positions/people.

    Why is it impossible for all of them to have had the iPad?

  86. 86
    vividbleau says:

    Daves

    “Its length exceeds any finite integer value”

    Am I to understand that you are talking about its “current” finite integer value? That its current length exceeds any finite integer value?

    Vivid

  87. 87
    asauber says:

    “The chain we’re discussing…”

    Yes, the chain we are discussing is literally beyond imagination. We can’t imagine in infinite, therefore this chain is illogical.

    Andrew

  88. 88
    Origenes says:

    DaveS: It is true that prior to any position in the chain, there are infinitely many other positions/people.
    Why is it impossible for all of them to have had the iPad?

    Perhaps one could say that for each member the IPad is ‘infinitely remote.’ For each member it can never be the case that the IPad is at a finite distance.

    Surely, an IPad infinitely remote from each member is incoherent. What it means is that, in fact, the IPad is not inside the infinite regress. An infinite regress cannot house an IPad which is held by all. There is still someone, holding the IPad outside the infinite regress, set on an infinite journey in search for the First Person in order to pass on the IPad.

  89. 89
    kairosfocus says:

    Folks, the point of claiming an infinite chain is to have thereby a necessary being, one independent of separate enabling causal factor. So, the first issue is, is the chain as a whole independent? (A chain of people doesn’t work as NB, as every member by definition is dependent; undermines aspects of the analogy, but it can be used with care.) Next, we see the issue of succession in finite stages and traversal of a countable infinity. The point here, is that if every member of the chain has had the iPad AND the chain is transfinitely long, there must be members that are transfinitely remote and once held the iPad. This is because, the material actually infinite chain constitutes a line of people — no people, no chain at that point. Let one such transfinitely remote person be P, and the next P+1, then P+2 etc. Where, for all we know, there is an infinity of further members beyond P, P-1, P-2 etc, that does not affect the logic of P, but it cuts off side tracks on oh you have proposed a first case. It is easy to see that progression here is equivalent to counting like 0,1,2 etc. Now go forward to some finitely remote point from P, P+K. Here, P+K+1, P+K+2 etc succeed. now, further this is the same as 0,1,2 etc. No matter how large P+K is, you have never traversed the transfinite, the endless span ahead of P+K is just as “long” as that beyond P. Now, let the now person be N, who has the iPad. We readily see he never received it from P or P+K as there is not a powerful enough transfer mechanism. So if N has the iPad, it is certain that it has only been handed on a finite number of times from some F, a first stage. This is the basic point I have repeatedly made regarding a claimed infinite past of a temporal-causal, successive cosmos, where we are at a now stage. Positing the dubious as though as we can construct a mathematical model it can be physically instantiated is an error. An infinite past has to have been made up of stages that were once the actual present and were causally succeeded by the next stage. Such faces the traversal of a countable transfinite endless span to reach now, which cannot be done in steps. Math models always deliver the infinite span all at once as a stipulation. The actual past was finite and had a beginning at some F. Thus, the whole series is contingent. We must look elsewhere for a necessary being world root. Which takes us back to the OP. KF

  90. 90
    daveS says:

    vividbleau,

    I wouldn’t use the word “current”. I’m looking at the chain from the point in time where tgpeeler (or whoever is the last person in the chain) receives the iPad and the whole process concludes. So the chain has stopped “growing” at that point. At that point, its length is larger than any finite integer value. Of course, that means that at any prior point, its length also was larger than any finite integer value.

  91. 91
    daveS says:

    Origenes,

    Judging by your #88, there still must be a problem in communication. In my scenario, the iPad is always within a finite distance of every member of the chain.

  92. 92
    daveS says:

    KF,

    Just one comment on your #89: I’m not aware that any of us is claiming that this infinite chain of contingent beings would be a necessary being. I’m certainly not.

  93. 93
    kairosfocus says:

    DS, I spoke to the wider context i/l/o the OP. KF

  94. 94
    JSmith says:

    I am having a hard time reading this infinite iPad regression and not laughing. It reminds me of the stupid argument of how many angels can dance on the head of a pin.

    Infinity is a mental exercise that has very few real world applications or implications. The difference between two seconds (or feet or litres) can be broken up into an infinite number of partitions. But that doesn’t mean that these individual partitions are measurable. BIPM has hundreds of people attempting to make these measurements. The philosophical implications are non-existent.

  95. 95
    Origenes says:

    DaveS: In my scenario, the iPad is always within a finite distance of every member of the chain.

    How?

  96. 96
    daveS says:

    Simply put, because every person in the chain is within finite distance of every other person in the chain, and the iPad is in possession of exactly one person in the chain at a time.

    Imagine all the people in the chain lined up single file in order, where the iPad is being passed to the right. In this case, the rightmost two people are KF and tgpeeler.

    Now the people in the chain pair up one-to-one with the nonpositive integers {…, -3, -2, -1, 0}. The distance between any two people is the absolute value of the difference between their corresponding numbers.

    Any two nonpositive integers are separated by a finite distance, hence any two people in the chain are separated by a finite number of steps.

    Therefore, the iPad is always within a finite distance of anyone else in the chain.

  97. 97
    Origenes says:

    DaveS: Simply put, because every person in the chain is within finite distance of every other person in the chain …

    The chain stretches infinitely into the past, so, if the IPad is in the chain, and everyone held it, the IPad was once in the infinite past.

    How does it get from there to here in the now?

    You are telling me that the IPad, somewhere in the infinite past, is within a finite distance from us, here in the now, but I don’t see that at all.
    To me, the IPad is not only infinitely far away in the past, it’s also infinitely far away from the infinite past, and, to go even further ….. and so on.

    The IPad cannot possibly enter the infinite chain and be held by all prior to the entry point. There cannot possibly be a start of the passing on of the IPad. The IPad cannot possibly be somewhere in the chain and have been held by all previous ppl, because ‘all previous ppl’ is endless. Nothing can satisfy infinity. You cannot finish infinity. The IPad cannot be here because it an infinity of ppl would never be finished passing it down to here. A process which involves an infinity of ppl cannot finish. Put another way, an infinity cannot be finite.
    You cannot traverse infinity.
    Again, an infinite regress cannot house an IPad held by all.

  98. 98
    daveS says:

    Origenes,

    The scenario I am describing must of course include an infinite past, but there are no points in that past infinitely remote from the present.

    All that is required for the past to be infinite is that there be points arbitrarily (finitely) remote from the present (and that’s what you will find if you look up some of the papers on this subject).

    Here’s one way to see why:

    a) The past is finite if and only if there exists some positive integer N such that all points in the past occurred less than N years ago. For example, empirical evidence suggests that all points in our past occurred less than 15 billion years ago. Therefore we (according to scientific findings) likely live in a universe with a finite past.

    b) Now negate both ‘sides’ of the above proposition: The past is infinite if and only if there is no positive integer N such that all points in the past occurred less than N years ago. That is equivalent to the proposition stating that for every positive integer N, there exists a point in the past which occurred more than N years ago.

    Conclusion: An infinite past requires only that there exist points arbitrarily remote (but still finitely remote) from the present.

  99. 99
    Origenes says:

    DaveS: The scenario I am describing must of course include an infinite past, but there are no points in that past infinitely remote from the present.

    Indeed, there are no actual points in the past infinitely remote from the present. Every time we choose (actualize) a number, it will be finitely removed from the present. Sure.
    However, there is a huge distinction between the infinite and a actual (finite) number. The infinite is infinitely far removed from the now. Obviously I cannot offer an actual (finite) number, because there is none.
    If there is an infinite number of people involved with the IPad, then it cannot reach the now, because an infinite number of ppl cannot finish/complete/finalize what they are doing.

  100. 100
    vividbleau says:

    daveS

    “I wouldn’t use the word “current”. I’m looking at the chain from the point in time where tgpeeler (or whoever is the last person in the chain) receives the iPad and the whole process concludes. So the chain has stopped “growing” at that point. At that point, its length is larger than any finite integer value. Of course, that means that at any prior point, its length also was larger than any finite integer value.”

    Why wouldn’t you use the word current? The last person in the chain is the current person that has has the iPad no? So the process concludes and the chain has stopped growing until what?

    If I am understanding you correctly we have an infinite past but a finite future.

    I am still unclear has to how you can add anything to an infinite nor have I seen any kind of explanation from you as to how that can be.

    Vivid

  101. 101
    kairosfocus says:

    DS, let me add, that what was on the table 100 or so years ago was a steady-state, eternal physical universe. This sort of context would enable a chain of people to hand over the iPad, i.e. a going concern world. Go back much beyond that and you will see discussions of life being eternal in the domain of reality too — and not just with ethical theism. Currently, there is a common discussion of forms of multiverse. Every one of these is addressing in some way, perhaps implicitly, the issue of a world root that is without beginning. We can pause on the genuine nothing case and quickly note that if there were ever utter nothing, such non-being can have no causal capability so such would forever obtain. When some physicists relabel quantum foams and the like, they are not genuinely appealing to non-being. So, we can see how the existence of a world in one form or another requires something that is infinite in duration. Something, of one form or another. KF

  102. 102
    kairosfocus says:

    JS, by speaking of a chain of discrete members and a finite hand-over process, the discussion is beyond the continuum of infinitesimal elements. As L’Hospital’s rule tells us (as an easy way to see), the latter can complete in finite time or attain a finite result. The former forces the issue of finite step cumulative process facing a transfinite traverse. That, is a supertask, a futility. KF

  103. 103
    Origenes says:

    DaveS

    Let’s assume that passing on the IPad takes one second each and every time, without exception.
    How do we explain that, in the now, person A holds the IPad and not person B, who stands next in line? The only two reasons I can think of are that in the infinite past the hand-over process didn’t start one second earlier than it apparently started or didn’t start with a person 1 position closer to the now.

    Are there other explanations?

    If there are not, we arrive at the same problem: the hand-over process cannot have a start, because every entry implies ignoring the people before the entry point.

  104. 104
    asauber says:

    We can only conclude that daveS is an Infinitismalist. But was he always one? haha I kill myself.

    Andrew

  105. 105
    daveS says:

    KF,

    Yes, I more or less agree with what you’re saying in #101.

  106. 106
    daveS says:

    vividbleau,

    I guess I was confused with how you meant “current”; with your further explanation, it looks correct. At the moment that tgpeeler receives the iPad, the length of the chain exceeds any finite integer value.

  107. 107
    daveS says:

    Origenes,

    If there is an infinite number of people involved with the IPad, then it cannot reach the now, because an infinite number of ppl cannot finish/complete/finalize what they are doing.

    Well, they do have an infinite amount of time to complete the process (plus the process is beginningless rather than endless), so it’s not clear to me they won’t be able to finish.

    Let’s assume that passing on the IPad takes one second each and every time, without exception.
    How do we explain that, in the now, person A holds the IPad and not person B, who stands next in line? The only two reasons I can think of are that in the infinite past the hand-over process didn’t start one second earlier than it apparently started or didn’t start with a person 1 position closer to the now.

    Are there other explanations?

    If there are not, we arrive at the same problem: the hand-over process cannot have a start, because every entry implies ignoring the people before the entry point.

    There definitely is no start to all this. The question of how two different instances of the process could end at different times (say off by 1 second) is a little tricky.

    First, let me clarify what I am saying: I am claiming that you can’t prove that this process is impossible using math and/or logic. I’m not claiming it is possible, or that I can explain why it happened or what would have caused it, had it occurred.

    So, if in one instance person A has the iPad one second earlier than person B, then all I can say is that for all n > 0, the person n positions to the left of A also had the iPad one second before the person n positions to the left of B. In other words, the infinite “tails” to the left of A and B are out of sync by 1 second.

    I suggested in the original threads on this that a God who exists outside of time and who experiences all instants of our time “simultaneously” could arrange for this to happen. That’s just a suggestion, however, not part of my actual claim.

  108. 108
    Origenes says:

    DaveS: So, if in one instance person A has the iPad one second earlier than person B, then all I can say is that for all n > 0, the person n positions to the left of A also had the iPad one second before the person n positions to the left of B. In other words, the infinite “tails” to the left of A and B are out of sync by 1 second.

    The question is: how do you explain that fact?
    How do you explain the difference between the scenario where, in the now, person A holds the IPad and the scenario where, in the (same) now, person B holds the IPad?
    I would argue that both scenarios are possible. Why would one of them be true? What other than a different start can explain that one of the scenarios is actual?

    DaveS: I suggested in the original threads on this that a God who exists outside of time and who experiences all instants of our time “simultaneously” could arrange for this to happen. That’s just a suggestion, however, not part of my actual claim.

    How can God cause this to happen? How does he do that?

  109. 109
    daveS says:

    Origenes,

    I’m explicitly stating in #107 that I don’t have an explanation (I also don’t know precisely what events even have explanations, but that’s a much bigger question). I don’t know how a God would cause any of this to happen either, that’s why I called the above a mere suggestion.

    All I’m claiming is that the iPad scenario(s) cannot be shown to be impossible using mathematics and logic. Others disagree, and those people are who my argument is directed toward.

  110. 110
    daveS says:

    vividbleau,

    To answer your question about adding anything to an infinite, I’ll give another set example.

    The set {…, -3, -2, -1, 0} is infinite.

    We can add the element 1 to this set, obtaining:

    {…, -3, -2, -1, 0, 1}

    which is also infinite.

    That’s the sense in which I say it’s possible to add to an infinite.

  111. 111
    Origenes says:

    DaveS: I’m explicitly stating in #107 that I don’t have an explanation …

    Not true. You do have an explanation Dave. You happen not to like it, but it explains things perfectly — a specific start of the hand-over process.

    DaveS: All I’m claiming is that the iPad scenario(s) cannot be shown to be impossible using mathematics and logic.

    That’s not how reasoning works, Dave. You cannot claim that a square triangle is possible, and brush off questions like “how can something have 3 and 4 sides at the same time?” with “I don’t have an explanation, but that does not mean that it is impossible.”

    I have argued that the hand-over of the IPad cannot have a start and be held by all. And I have argued that only a specific start can explain why, in the now, person A has the IPad and not person B.

    If you do not have an alternative explanation, then a start is necessary and your claim has been falsified.

  112. 112
    daveS says:

    Origenes,

    I meant that I don’t have an explanation for why the two infinite regression scenarios are out of sync by 1 position.

    If the regression is finite, of course there’s a mundane explanation.

    If you do not have an alternative explanation, then a start is necessary and your claim has been falsified.

    Nonsense. Just because some particular person does not have an explanation for some event does not mean the event did not occur or is impossible. Recall that my claim is simply that you cannot prove with just mathematics or logic that these infinite regressions are impossible.

  113. 113
    asauber says:

    “We can add the element 1 to this set, obtaining:

    {…, -3, -2, -1, 0, 1}

    which is also infinite.”

    daveS, this is another illustration that you are not maintaining logical consistency. Adding something to a set that remains same after you add to it is not logical. If infinite is the result of adding something to the infinite, you haven’t added anything. It’s just nonsense.

    You are just rearranging symbols. It doesn’t mean anything though.

    Andrew

  114. 114
    daveS says:

    Origenes,

    PS to this in my #112:

    Recall that my claim is simply that you cannot prove with just mathematics or logic that these ifinite regressions are impossible.

    The reason I’m claiming this is that when these threads first started, people were proposing mathematical/logical arguments (without premises such as PSR) that allegedly showed an infinite past is impossible. I believe they all have errors, which led me to conjecture that there aren’t any such arguments which are sound.

    If you do present one, then you will falsify my claim.

  115. 115
    Origenes says:

    DaveS @
    My argument is logical. Okay, maybe in another form it rings a bell:

    Suppose we can travel back in time and follow the IPad into the infinite past; witnessing each hand-over. Obviously, this journey towards the past would never end, because the past is infinite. So, it will never be the case that we can rightfully say: all members of the infinite chain held the IPad.

    This means that there is no stepwise path from the now to the ‘limit’ of the past.

    But ONLY if there is a stepwise path to the limit of the past can it be so that all members of the chain are involved in the infinite hand-over of the IPad.

    If there is no path from A to B, then there is no path from B to A.

  116. 116
    vividbleau says:

    Dave’s re 110

    So the infinite set ENDS at 1 ? What kind of infinite has an end?

    Vivid

  117. 117
    daveS says:

    Origenes,

    You are right that we could never verify, by travelling back in time, that every member of the chain had the iPad at one point. But that’s not the issue I’m debating. I am only concerned with whether one can prove that it’s mathematically impossible for that to have happened.

    Now to address the issue of moving backward or forward in time: I mentioned this earlier when I wrote of traversing an ordered set of type ω vs. one of type ω*. The traversals are very different. I agree that you can’t traverse the chain “backwards”, but the forward traversal is a different animal. This is not like reasoning about traveling between two towns, where you can get from town A to town B if and only if you can get from B to A.

  118. 118
    daveS says:

    vividbleau,

    This infinite set does have a greatest element, so an “end” in that sense. The salient point is that it has infinitely many elements. Clearly it doesn’t have a finite number of elements, right?

  119. 119
    vividbleau says:

    daveS

    Well it doesn’t contain all the possible elements does it?

    Vivid

  120. 120
    daveS says:

    vividbleau,

    Probably not, although it’s not clear what “all the possible elements” would be. All integers? All rational numbers?

    In any case, you can’t express every element in that set in a list that terminates, so it’s not finite.

  121. 121
    Origenes says:

    DaveS: You are right that we could never verify, by traveling back in time, that every member of the chain had the iPad at one point.

    True, but there is more to it. There is an infinite movement towards the past (an infinite receding) which cannot be (logically and mathematically) combined with the IPad moving into the opposite direction and not having a starting point.
    I will come back to this.

    DaveS: The traversals are very different. I agree that you can’t traverse the chain “backwards”, but the forward traversal is a different animal.

    Where lies the difference? If we have a time machine and can travel towards the infinite future, witnessing future hand-overs of the IPad along the way, we, also, never reach the ‘limit’ of the future. So, it seems, I can make the same argument.

  122. 122
    tgpeeler says:

    The PSR follows from the universal observation that existence precedes causation. That premise is used in the argument.

  123. 123
    Origenes says:

    DaveS @117

    O: Suppose we can travel back in time and follow the IPad into the infinite past; witnessing each hand-over. Obviously, this journey towards the past would never end, because the past is infinite. So, it will never be the case that we can rightfully say: all members of the infinite chain held the IPad.

    DaveS: You are right that we could never verify, by traveling back in time, that every member of the chain had the iPad at one point.

    Traveling back in time, witnessing each IPad hand-over, we take the exact same path as the IPad. Just like us, the IPad is set on an infinite journey, at the exact same pace, along the exact same path.
    It follows that if we cannot witness that all members of the chain held the IPad, then the same is true for the IPad itself.
    IOWs if it is true that “we could never verify, by traveling back in time, that every member of the chain had the iPad at one point”, then it also true that the IPad itself cannot undergo that every member of the chain had it at one point.

  124. 124
    kairosfocus says:

    DS

    I suggest, once we have a real sequence, relabelling members makes no difference. Where, qualification for membership is that the token (here, iPad) has passed through or into your hands. So your:

    { . . ., -2, -1, 0}

    Is not essentially different from:

    { . . . p, p+1, p+2 . . . -2, -1, 0, . . . , n-1, n}

    Now, I make a difference, based on a claim to reality. If the sequence left is transfinitely long, it requires that transfinitely remote members passed the token, T, to the next recipient, so that we may interpret p as a transfinitely remote member, and further have T pass to the last two, just now.

    But we have seen above that an ellipsis of endlessness cannot be traversed in finite successive steps, as at any k we may go k+1, k+2 etc with endlessness still in “front.” So we have no basis to go from a transfinitely remote p to 0, thence to n.

    There is no transfinitely remote p once we reach n.

    Reaching n is sufficient to rule out a transfinitely remote p once the stepwise succession is in force.

    There is some first point F, at finite remove such that we actually have:

    {F: . . . p, p+1, p+2 . . . -2, -1, 0, . . . , n-1, n}

    We may continue forward, endlessly for all we know, but at no n will the number of terms become transfinite.

    Or, back to omega, using w. The first transfinite ordinal is not the direct incremental successor to any particular finite ordinal. Instead it reflects a qualitative leap to recognise that a countable but endless set has a new type of quantity, its cardinality aleph null, which is transfinite. So, w is the qualitatively different successor to endlessness, not the immediate stepwise successor to some particular finite value, say v, such that passing a token one further step right will attain:

    v, +1 –> w.

    So, we have no warrant to speak of an endlessly remote actual past, given the relevant logic of structure and quantity. Nor will saying that at any particular p we already have traversed the transfinite succession to its left help, as that simply begs the question. Notice, we always deliver endlessly successive sets all at once as a conceived whole in our mathematical reasoning, as we cannot exhaust such a set by succession. Hence the use of what I have called ellipses of endlessness.

    In ontological and cosmological terms, this brings us to how a going concern world in the now as a time-token has passed to us causally in succession of finite stages or eras, credibly implies a definite beginning. Which points to a finitely remote world root, and onward to that root in at least a key part being necessary being. (I here accept a point my son has argued with me on, that such an entity could in principle sustain other entities by eternally sustained choice or whatever.)

    We really do need to recognise a necessary being world root, and to see that our being under moral government is likewise a constraint on possibilities. Namely, we need to bridge the IS-OUGHT gap in the root. Even our reasoning in this thread depends critically on our being governed by duties of care to truth and to reasoning soundly towards it.

    Hence, the significance of ethical theism as an answer to this.

    KF

  125. 125
    daveS says:

    Origenes,

    Traveling back in time, witnessing each IPad hand-over, we take the exact same path as the IPad. Just like us, the IPad is set on an infinite journey, at the exact same pace, along the exact same path.
    It follows that if we cannot witness that all members of the chain held the IPad, then the same is true for the IPad itself.
    IOWs if it is true that “we could never verify, by traveling back in time, that every member of the chain had the iPad at one point”, then it also true that the IPad itself cannot undergo that every member of the chain had it at one point.

    The differences between the two are:

    1) The iPad’s journey has an endpoint, and it’s always a finite distance from that endpoint.

    2) There is no endpoint to the journey of the person traveling backward in time, and there are always infinitely many steps left in the journey.

  126. 126
    asauber says:

    2) There is no endpoint to the journey of the person traveling backward in time, and there are always infinitely many steps left in the journey.

    This is pure assertion. There is nothing that indicates this is possible.

    Andrew

  127. 127
    asauber says:

    And to get poetic about it, just because you can imagine a pot o’ gold at the end of the rainbow, doesn’t mean it’s possible.

    Andrew

  128. 128
    daveS says:

    KF,

    Now, I make a difference, based on a claim to reality. If the sequence left is transfinitely long, it requires that transfinitely remote members passed the token, T, to the next recipient, so that we may interpret p as a transfinitely remote member, and further have T pass to the last two, just now.

    Uuuhhm….

    Let’s say this is a picture of the head of the queue. The guy on the left will be the last person to receive the iPad. Let’s call him John. The line extends to the right without end.

    You’re saying that there are people in this line infinitely far from John. By the well-ordering property of the positive integers, there is a particular infinitely remote person who is closest to John. Let’s say his name is Ishaan.

    Notice how these people are all standing very close to each other (some even touching).

    That means that the person standing in front of Ishaan (call him Li Qiang) is only finitely remote from John. So we have two people, literally touching each other, one of whom is infinitely distant from John, and one of whom is finitely distant from John.

  129. 129
    Origenes says:

    DaveS:
    The differences between the two are:

    1) The iPad’s journey has an endpoint, and it’s always a finite distance from that endpoint.

    2) There is no endpoint to the journey of the person traveling backward in time, and there are always infinitely many steps left in the journey.

    (1) is completely wrong. When we travel back in time, we witness the IPad journey towards the infinite past. The IPad’s journey towards the infinite past has no endpoint. And the IPad is never at a finite distance from that (non-existent) endpoint.

  130. 130
    daveS says:

    Origenes,

    “When we travel back in time”, we embark on journey 2).

    1) Is the journey the iPad travels along in normal time order.

  131. 131
    Origenes says:

    DaveS @130

    “When we travel back in time”, we embark on journey 2).

    Indeed. And so does the IPad.

    1) Is the journey the iPad travels along in normal time order.

    Indeed. So?

  132. 132
    daveS says:

    Indeed. So?

    So:

    1) The iPad’s journey (moving forward in time) has an endpoint, and it’s always a finite distance from that endpoint.

    2) There is no endpoint to the journey of the person traveling backward in time, and there are always infinitely many steps left in the journey.

  133. 133
    Origenes says:

    DaveS @132

    So? How is that relevant to my argument?

    The IPad’s journey backwards in time has no endpoint, which means that not all members in the past could have held the IPad (see #123).

  134. 134
    daveS says:

    Origenes,

    The IPad’s journey backwards in time has no endpoint

    It didn’t travel back in time. You need to show that not all members of the chain could have held the iPad as it moved forward in time.

  135. 135
    Origenes says:

    DaveS @134

    If X didn’t travel back in time, we are not allowed to study its history? Is that a new rule?

  136. 136
    tgpeeler says:

    Geeeeez. It’s not this hard daveS. If we take infinite, which I do, to mean pure actuality with no potential, then ANY talk of sequences or regresses is absurd on the face of it. Consider this, an “infinite” regress would have no potential to have more individuals (seconds or causes) but clearly, if we are talking about actual seconds or causes then we can always add one more. Ergo, it is not “infinite.” The illustration I used was to posit an infinite regress, point out the implication of such a thing is absurd, and demonstrate that, in fact, there is no such thing as an infinite regress in reality (this universe).

  137. 137
    daveS says:

    tgpeeler,

    In that case, you and I are using different definitions of “infinite”. I believe Origenes, KF, and I are all using the same definition, however.

  138. 138
    asauber says:

    different definitions of “infinite”

    There’s no scientific definition of “infinite”. It’s a fantasy with a label on it.

    Andrew

  139. 139
    vividbleau says:

    daveS re 137

    I did not know the definition of “infinite” was up for grabs? What definition are you KF and Origenes using?

    Vivid

  140. 140
    daveS says:

    vividbleau,

    KF and I had used definition 4 on this page in the past. Origenes seems to be using the same one.

  141. 141
    kairosfocus says:

    DS 128, I noted that membership in the chain comes from having had or just received the token. If that chain is from the transfinite, there will thus be a transfinitely remote member P, that handed to P+1 etc until 0, 1, 2 . . . n-1, finally n. But the traverse from P to 0 cannot be done in finitely many steps, and there is no particular natural number whose +1 successor is w. That is the order type of the naturals as a whole is w, we cannot count them out. That, I already noted on. The scenario of neighbours one finitely remote the next infinitely, does not obtain. You do not traverse to the transfinite in successive +1 increments. You look at the set as a whole, recognise its endless character then understand this is itself a new type of quantity, and define order type of that set as w. KF

    PS: If at EVERY stage, the token was finitely removed from n, now, it was NEVER transfinitely removed.

    PS: Infinite is beyond the finite, endless, or of order type at least that of the natural successive counting numbers. For any counting number k, regardless of how large, we can always use +1 to go to k+1, k+2 etc, which is readily put in 1:1 correspondence with 0,1, 2 etc that is we see endlessness in action, we can always go on stepwise as though we had just begun, we cannot exhaust counting numbers by a “long enough” count. We zoom back out, notice the pattern of endless succession and see the implied new type of quantity, transfiniteness or a countable infinite in this case.

  142. 142
    daveS says:

    KF,

    The scenario of neighbours one finitely remote the next infinitely, does not obtain.

    Can you point out specifically the flaw in my argument in #128 which shows it does obtain?

  143. 143
    Origenes says:

    DaveS: The iPad’s journey (moving forward in time) has an endpoint, and it’s always a finite distance from that endpoint.

    You are saying that the distance between the iPad at any position N in the past and the now is finite. And I take it that you say this because in order for the iPad to reach the now, this distance needs to be finite.

    There is a problem:

    If the distance from the iPad to the now is always finite, then we run into a problem with an infinite past. Given that infinity exceeds any finite number, it follows that there is always a part of the past where the iPad has not been. Because the distance between the iPad and the now is always a finite number N, and because there is always an infinity (in the direction of the past) before each N, it cannot be the case that the iPad has been at all positions.

    For any N goes: infinity – N = infinity

    In order for your claim to be true the outcome should be ‘zero.’

    Therefore, on any position N, the iPad does not cover the infinite past.

  144. 144
    daveS says:

    Origenes,

    I don’t follow where the “infinity – N” comes from. What does it represent?

  145. 145
    Origenes says:

    DaveS @144

    Infinity = all the positions between the infinite past and the now.
    N = the number representing any position that the iPad ‘visited’ in the past.

    (note that N is a finite number, because it represents the distance between the iPad and the now, which is, per your own claim, always finite.)

    For any (finite) position N goes that infinity is larger, which can be expressed as:
    infinity – N = infinity

    If your claim would be true, the iPad traveled through the entire past, then we would expect:
    infinity – N = zero.
    IOWs N would encapsulate the entire past.

    But that is obviously not the case.

  146. 146
    kairosfocus says:

    DS, on track record you are doing a for sake of argument. I have long since pointed out here and elsewhere, that endlessness onward is an integral part of the definition of the natural counting numbers. As a result the sense in which w is the order type of the naturals is NOT the same as implying there is some particular finite counting number v such that

    v, +1 –> w.

    Instead, w is recognition of a new type of quantity due specifically to the endlessness implied by the ellipsis in the like of:

    N = {0,1,2 . . . }

    That onward endlessness is the reason why only a finitely remote p* can succeed stepwise to attain n:

    { F: . . . p*, P*+1, P*+2 . . . -2, -1, 0, 1, 2 . . . n-1, n}

    The alternative, with p transfinitely remote from 0:

    { . . . p, p+1, P+2 . . . -2, -1, 0, 1, 2 . . . n-1, n}

    is infeasible of actual stepwise completion as the endless span intervenes as was already pointed out.

    Where, to be a member, every value must have once held the Token T that shifts right in +1 steps, or has just received it. So, if the series has no finitely remote first stage F, then T would have had to have originally been transfinitely remote. Or else “infinite” becomes meaningless. For if EVERY value to the endless left is finitely remote, then there is no transfinitely remote member to the left, there is no transfinite past.

    Yes, I am aware of claims that every member of N is finite but N is transfinite. With all due respect to the Mathematical thinkers who have so spoken, this is a contradiction in terms, regardless of who says it.

    What is more correct, is clearly this: that the set N can be indicated as an endless succession, and every particular value we can state or symbolise as say k, will be finite and will be endlessly exceeded by applying +1 increments: k, k+1, k+2 etc. Thus, the endlessness property beyond any given finite is a part of the structure of the naturals. And that is crucial to the consideration here.

    KF

  147. 147
    Origenes says:

    Infinity : the total distance between the (infinite) past and the now.
    N : the variable that represents the distance between the iPad and the now, as it occurred in the past. For each N goes that N is a finite number, per DaveS’ claim (see #143).

    If the iPad would have visited the entire past (if each member of the infinite chain held the iPad at some point in the past), then we would expect: infinity – N = zero. IOWs the distance between the iPad and the now would at some point have been so large that it spanned infinity. N would ‘cover’ the total distance between the (infinite) past and the now, hence: infinity – N = zero.

    However, since N is always a finite number and infinity is always larger than any finite number, this cannot be the case:

    For any N goes: infinity – N = infinity.

  148. 148
    kairosfocus says:

    Origines, your point is sensible but the phrasing may need reworking as subtraction becomes problematic when transfinite sets are involved. Perhaps, it would be helpful to say, a countable finite number n will be a bounded — thus finite — subset to a set that can be matched 1:1 with the counting naturals or some isomorphism thereof. If n has a fractional part, it will be bound in the same way and the next highest counting number will be bound likewise. That is, you can count on as though you had started at 0,1,2 etc. This implies that no finite specific value can be “next door” to an actual transfinite value. The endlessness is a key structural part of the naturals, and that needs to be reckoned with. KF

  149. 149
    kairosfocus says:

    TGP, pure actuality with no onward potential of course is a perspective on being infinite, i.e. such a thing has exceeded limits. Having onward potential implies boundedness. Applied to number-type sets, the set of counting naturals will exceed any proposed limit, but any finite number, k, will be bounded as we can do k, +1 -> k+1, etc. We also need to recognise a potential infinity as a way of saying that something if pursued would become unbounded, usually to detriment. For instance, a chain of warrant: A as B, as C . . . where the ellipsis implies no limit. This then leads to the challenge, that finite and bounded creatures such as us cannot exhaust such. therefore chains of warrant must be bounded, and bounded without begging the question. Which, is a huge issue. KF

  150. 150
    daveS says:

    Origenes,

    I guess I still don’t see how your argument works. Here’s what I understand you can say about N:

    N of course depends on time, and counts down through the nonnegative integers as time elapses and the iPad approaches the end of the line.

    At some point, N will be 17, and one “transfer” later, N will be 16.

    At any point, N plus the number of positions “behind” the iPad must sum up to infinity (aleph null, precisely). But since the number of positions behind the iPad is always aleph null, then that means aleph null + N = aleph null. Subtracting N from both sides (which is not problematic) results in aleph null = aleph null – N. This agrees with one of your equations, but I don’t know if there are derived the same way.

    When N = 0, that is when the iPad has finally arrived at the end of the queue, then we have aleph null = aleph null – 0 = aleph null, as expected.

    I’m not seeing any problem yet.

  151. 151
    kairosfocus says:

    DS, the token cannot traverse a transfinite span and will never arrive. This is tied to there being the stipulation that for an actual chain, every member must have either passed the token on or have just now received it. So, if any member p+ is always at finite remove, no member is at transfinite remove. In this context, were there some p at transfinite remove, passing the token onward gives p, p+1, P+2 , etc which sequence can never cumulate at any particular value — required to pass form one to the next — to having traversed a transfinite span beyond p. It will never arrive. KF

  152. 152
    daveS says:

    KF,

    But can you point out precisely where my argument in #128 goes wrong? If you are correct about no neighbors in the queue existing, one infinitely remote, and one finitely remote, then there must be a flaw in my argument.

    I’ll repeat it here with a few edits for clarity:

    Assume we have an infinite queue with John at its head. Further assume that some members of the queue are at infinite distance from John. In diagrammatic form:

    … *__*__*__*__*__*__ John

    where the asterisks represent members of the queue. Note that the iPad will be traveling left to right.

    1) Premise: Any nonempty collection of members of this queue must contain a particular member which is farthest to the right (referring to the above diagram).

    2) By assumption, the collection of members of the queue at infinite distance from John is nonempty, therefore there is a particular member, Ishaan, who is farthest to the right among all members infinitely distant from John.

    … Ishaan … * __*__*__ John

    3) Clearly Ishaan is not John, so Ishaan has a neighbor immediately in front of him, say Li Qiang.

    … Ishaan __ Li Qiang … * __*__*__ John

    4) Because Li Qiang is farther to the right than Ishaan, he is only finitely distant from John.

    5) Therefore Ishaan and Li Qiang are neigbors, one of whom is infinitely distant from John, while the other is finitely distant from John.

  153. 153
    Origenes says:

    KF @148

    KF: … your point is sensible but the phrasing may need reworking as subtraction becomes problematic when transfinite sets are involved.

    Point taken. I was aware of the fact that (2 x infinity) – infinity = ? is problematic. However, I did not know that infinity – N = infinity when N is a finite number, is also problematic.

    KF: Perhaps, it would be helpful to say, a countable finite number n will be a bounded — thus finite — subset to a set that can be matched 1:1 with the counting naturals or some isomorphism thereof.

    Indeed, that is all I need to make my argument: N is a subset of infinity or, put differently:

    For any N goes: infinity > N (infinity is larger than N)

    It follows that:
    infinity – N > 0
    This is all I need to show that there must be positions in the past that N does not cover.

  154. 154
    daveS says:

    Origenes,

    Actually infinity – N = infinity is just fine.

  155. 155
    asauber says:

    infinitely distant from

    This is a confusion.

    Distance is finite.

    The question is, daveS, why do you insist on clinging to confusions like this?

    Andrew

  156. 156
    kairosfocus says:

    DS, done several times. KF

  157. 157
    daveS says:

    KF,

    Could you remind me please, which step is wrong?

  158. 158
    kairosfocus says:

    Ds, try 146, and before that 124. KF

  159. 159
    daveS says:

    KF,

    #124 was written before my #128! I’m asking what the problem with that argument is. I’ve even numbered the steps for your convenience in #152. Which of the 5 listed steps is/are incorrect??

  160. 160
    kairosfocus says:

    DS, yes 124 is before 128. In short the problem was anticipated. When you put it up notwithstanding, I pointed to the solution again. KF

  161. 161
    daveS says:

    Ok. I’ve asked the same question four times—I guess I’ll move on now.

  162. 162
    kairosfocus says:

    Ds, that’s also how many times you passed over your answer, which pivots as you know on there being no neighbouring finite v that increments directly to w. As stated several times, w is a successor to the naturals, where endlessness captured by the ellipsis is pivotal and is a structural part of the set. It is therefore recognised that endless countability is a new type of quantity that is the order type of the naturals, N. This is what then feeds the point I have repeatedly highlighted now and before, that any particular finite k, however large will have successors k+1, k+2 etc which means you could put K on in correspondence with 0,1,2 . . . thus you cannot exhaust the endlessness in counting by finite successive stages. Which is where the problem with an infinite token passing comes in. For, only cases that have had the token or just got it count. So, if there were an infinite chain, the transfinite would have to be traversed stepwise to reach n-1, n, the now terminus. And, this also requires transfinitely remote members to have had and passed the token. But as such a case P does not have a strong enough means to reach n, there cannot have been a transfinite p. All of this has been repeatedly explained but studiously glided over, by your count four times. Your move may be rhetorically clever but in the end it shows that you do not have a means of doing more than suggesting that at any given stage the transfinite token pass has already happened. That is little more than begging the question at stake. KF

  163. 163
    daveS says:

    KF,

    The answer to my question would be some subset of {1, 2, 3, 4, 5}, the set of symbols I used to enumerate my steps. If you don’t know which steps are problematic, then that’s perfectly fine, just say so. Or, if you think some steps are definitely ok while others are dubious, you could specify which.

  164. 164
    Origenes says:

    DaveS: I’m not seeing any problem yet.

    We have two sets:

    A: All positions of the chain from the infinite past to the now:
    [infinity ….. -2 , -1 , 0]
    B: All positions that the iPad entered in the past to the now: [N ….. -2 , -1 , 0]

    DaveS’ claim: “the iPad entered all positions of the chain from the infinite past to the now.”

    If the iPad entered all positions from the infinite past to the now, then A – B = 0.

    However, we know that N is a finite number. And we also know that infinity is larger than any finite number. Because A is infinite and B is finite, it follows:

    A – B = A

    Conclusion: DaveS’ claim is false.

  165. 165
    daveS says:

    All:

    I’m about to head out for some Christmas season travel, so I’ll be semi-offline for a few days. I might be able to sneak in some posts if I get good access to WiFi here and there. Feliz Navidad.

    Origenes:

    My assumption is that the positions the iPad has entered is the set {…, -3, -2,-1, 0}, that is all positions with nonpositive index/coordinate. There is no lower bound N on this set of positions. It’s exactly the same as the set of all positions in the chain.

  166. 166
    kairosfocus says:

    DS, enjoy your Christmas. Meanwhile, the situation still stands that you have not solved the passage from the transfinitely remote problem, and we have reason to hold that no stepwise, finite stage, cumulative process can do so. Saying or suggesting that at any given stage s, the actual transfinite passage has already been done, does not answer the problem. And, I have again pointed out that there is no neighouring finite v in the counting numbers such that v, +1 –> w . . . there is no one step increment of mere degree from finite to transfinite. Endlessness is inherently structural to the naturals [that is a crucial point captured and symbolised by the innocuous-seeming but extremely powerful ellipsis, . . . ], and this is the exact reason why we cannot traverse it stepwise. Remember, N = {0,1,2, . . . } Where w is order type as identifying the scale of the endless succession as a new type of quantity, qualitatively different from finite ones precisely because it embeds endlessness of counting succession. We need a more powerful operation than steps of finite size to bypass a transfinite traverse or to leap over it. Sort of, like taking reciprocals of infinitesimals — indeed, [0,1] captures a dual of the whole number space including transfinites once we bring in infinitesimals as neighbours of 0 and use reciprocals 1/m –> M, with M > w, where too, trivially any natural value n is dual to 1/n, with w > n. Or in this case, living in a world created by or at least rooted in a necessary being first cause at finite remove. Such a being is necessitated by our being here as a going concern world that is clearly contingent. In such a case, no finite stage stepwise process will be transfinite and there is no need to suggest or imply an infinite past finite stage causally cumulative succession of any kind. KF

  167. 167
    Origenes says:

    DaveS: My assumption is that the positions the iPad has entered is the set {…, -3, -2,-1, 0}, that is all positions with nonpositive index/coordinate. There is no lower bound N on this set of positions. It’s exactly the same as the set of all positions in the chain.

    No, that’s not possible, because you have stated the following (emphasis added):

    DaveS: The iPad’s journey (moving forward in time) has an endpoint, and it’s always a finite distance from that endpoint.

    The position of the iPad in the past, relative to the now, is always finite, this means that the set of positions that the iPad entered in the past up to the now can only be a finite set. Never can it be a an infinite set.

    So, again, we have two sets (see #164), one finite and one infinite — the latter is larger. Unless you are willing to argue that there is no difference between an infinite set and a finite set, your claim is refuted.

  168. 168
    kairosfocus says:

    Origenes, DS’s underlying point is that the naturals, of infinite number, are all finitely removed from 0. This is the context for my pointing out so insistently the endlessness of the set as a whole as a key structural feature. Consequently, as each step imposes a +1 increment, the total remove must be equally endless. Where at ANY finite k, no matter how large, you can go on k, k+1, k+2, etc, which can be “slid” to 1:1 correspondence with 0,1,2 etc equally endlessly. Indeed, in previous discussions I used the picture of pink and blue marked tapes endless to the right to make the point. So, there literally is no last natural v such that v, +1 –> w, omega, the first transfinite. Instead, by reckoning endless succession as integral to the naturals, we recognise that w is successor to the endlessness, its order type in more technical terminology. Now, scroll up to the OP and notice the diagram of the surreals. Do you see how the number line is flipped upwards and bent to the right like a bow? The negative integers are a mirror image of the counting numbers. This is where I now point out that to have traversed the chain from a negative transfinite labelled member, a transfinite traverse would have to have been crossed. That cannot be done stepwise, the token just cannot cross endlessness in +1 steps. That it is at n-1, n now means it has only traversed finitely many steps. And indeed you are right to point out that if every negatively labelled member of the chain is finitely remote, the traverse indicated is never infinite. Talk about infinitely many finite and bounded negative integers falls into a mare’s nest. Let us trust we can now return from this tangent. KF

  169. 169
    Origenes says:

    KF @168

    KF: Origenes, DS’s underlying point is that the naturals, of infinite number, are all finitely removed from 0. This is the context for my pointing out so insistently the endlessness of the set as a whole as a key structural feature.

    I believe I do the same when I point out that there are two types of sets, one infinite set and one finite set. Of course, a finite set is a set that has a finite number of elements.
    I should, perhaps, explicate an assumption that I make during my reasoning: the past is settled. Given that the past is settled, at the moment the iPad arrives in the now, the number of positions that preceded its arrival is a fixed number. Once in the now, that number is not up for change. Given that the past is settled, in the now there can be no extra positions added to that number.
    Can that number be “infinity”? No, that number is finite, which is guaranteed by the fact that, per DaveS’claim, the iPad in the past is “always a finite distance” from the now and due to fact that no more positions are added, since the past is settled.
    So, we have a finite set of positions that the iPad entered between the past and the now.

    I do not see why it is any more complicated than that.

  170. 170
    daveS says:

    Origenes,

    How many elements does the set {…, -3, -2, -1, 0} have? Infinitely many (aleph null). Are any infinitely distant from 0? No, every integer can be obtained by adding 1 (or -1) finitely many times to 0.

  171. 171
    daveS says:

    PS to #170:

    I see KF already brought this point up. Of course the past is settled and no additional positions can be added to it after the fact. I don’t believe that creates any problems for my position, however.

  172. 172
    kairosfocus says:

    DS, no, we may reach finite values stepwise from 0, but we may not exhaust the naturals that way as for every k or – k we may reach, we may count on endlessly from there. We may place count on from K in correspondence to count from 0, this is structural. Just so, no finite stage cumulative process can span endlessness. KF

  173. 173
    daveS says:

    KF,

    By definition, every natural number can be reached by applying the successor operation a finite number of times to zero. Any number not reachable in this way is not a natural number.

  174. 174
    daveS says:

    KF,

    PS to #173:

    In my #170, I mean that given any integer k, it lies finitely many steps from 0. I’m not saying you can construct the entire set Z in finitely many steps.

  175. 175
    kairosfocus says:

    DS, your challenge is that to have a token T coming from an endless past to now, you have to do the equivalent of spanning and exhausting the set of naturals in a stepwise, +1 per stage, cumulative process. The symmetry between the natural counting numbers and the negative integers — a simple transform converts the one to the other (and cf. the Surreals as illustrated in the OP) — implies just that supertask: spanning endlessness in successive +1 increments. The means hath not the power as at every k it arrives at, k+1, k+2 etc continue endlessly and in 1:1 correspondence with the counting numbers from 0 on. We may even view this as an informal mathematical induction . . . though strictly, the succession onwards is implicit in that procedure. The token passed to some p (transfinitely remote) would always have that endlessness still before it. And for the claim that the token has come from an actually transfinite chain, some p that once held the token then passed it +1 to p+1 etc must be transfinitely remote. As a consequence, if a passed token has just arrived at n-1, n from a chain of members that passed it in succession, we may freely infer that the chain is not transfinitely long. KF

    PS: Transform, in simple representation:

    {0,1,2 . . . } x(-1) –> {0, -1, -2, . . . }

    PPS: We must also recall that properties of individual cases and properties of the whole may but need not differ — for a wall to be a red brick wall, every brick must be red, but the mass of the whole is greater than that of any given brick. Yes, any specific k can be exceeded k+1, k+2 etc. Put that in correspondence with 0, 0+1, 0+2 etc and we see that no particular integer we state has spanned endlessness. The endlessness, however is material. It is a structural component of the set. Extending to Z, this is double-sided. Succession to w and -w is by recognition that countable endlessness has a new type of quantity, symbolised by its order type w.

  176. 176
    Origenes says:

    DaveS @75: In the scenario I described, I do not envision the iPad “entering” the infinite regress at all. It’s eternally being passed down through this chain of people, in a “beginningless” process.

    I would like Dave to elaborate on his vision, because I don’t understand how it could work.

    For any position, between the infinite past and the now, goes: towards the past are infinite positions. However, Dave claims that the iPad only goes towards the now only after it entered all positions in the past.
    But how does the iPad accomplish that? I see two options and both fail:

    1) by first going towards the past, but that would be an endless journey.
    2) by starting from the first position of the past, but there is no first position of the past.

    DaveS: Of course the past is settled and no additional positions can be added to it after the fact.

    So, a beginningless infinite process has come to an end.
    🙂

  177. 177
    Origenes says:

    KF: … supertask: spanning endlessness in successive +1 increments. The means hath not the power as at every k it arrives at, k+1, k+2 etc continue endlessly and in 1:1 correspondence with the counting numbers from 0 on.

    Indeed, there is no progress on a relevant level. It is as if incremental steps take place on a micro-level, while infinity takes place on a macro-level, which is insensitive to the first.
    It does not matter how many steps you take, for any position, between the infinite past and the now, goes: towards the past are infinite positions.
    How did progress take place? How did the iPad traverse?

  178. 178
    daveS says:

    KF,

    The sets {…, -3, -2, -1, 0} and {0, 1, 2, 3, …} are not isomorphic as ordered sets, agreed?

    Are we also agreed that for each natural number k, k is finite?

  179. 179
    daveS says:

    Origenes,

    More precisely, I claim that the iPad never entered the chain—it has always been in it. You are right that if I claimed it did enter the chain at some time, then my proposal would be impossible.

    I also don’t claim to have an explanation for how this was all accomplished—I’m merely asserting that it did, and asking for anyone to prove that it’s logically/mathematically impossible.

  180. 180
    Origenes says:

    DaveS: More precisely, I claim that the iPad never entered the chain—it has always been in it. You are right that if I claimed it did enter the chain at some time, then my proposal would be impossible.

    Okay, let’s suppose that the iPad was always in the chain and, in the past, always moved towards the now. If it ever moved towards the past it would never return.

    But here is another problem: for any position in the past it goes that there are infinite positions to the ‘left’.

    So, isn’t it necessarily the case, that whenever the iPad moves to the right (towards the now), it does not enter positions on the left?
    IOWs if the iPad was always in the chain and always moved to the ‘right’, there were always positions on the left it did not enter.
    Always having been inside the chain, doesn’t seem to alter the fact that moving to the right implies always infinite positions on the left.
    If an infinite can be traversed by incremental steps (which is highly doubtful) then one requirement seems to be that one takes steps towards that infinite and not in the opposite direction.

    = = =
    BTW I do think I have provided logically/mathematical prove that the positions constitute a finite set.

  181. 181
    daveS says:

    Origenes,

    Yes, for every position, there are infinitely other positions to the left. And I am assuming it never moves to the right.

    But that doesn’t imply that the iPad missed some positions (at least I haven’t seen any proof of this).

    If the iPad is currently 100 steps from the end, then 900 seconds ago, it was 1000 steps from the end, for example.

  182. 182
    kairosfocus says:

    DS, your argument keeps implying that at any given stage transfinitely many token-passes have already happened. For instance, see your: “for every position, there are infinitely other positions to the left.” This consistently begs the question at stake. KF

  183. 183
    daveS says:

    Oops—correction to the second sentence: I am assuming it [the iPad] never moves to the left. Rather, it always moves toward the right, toward the (one) end.

  184. 184
    daveS says:

    KF,

    To your #182: I am assuming such (explicitly, so no question-begging), and asking you to show me how this results in a contradiction.

    An analogy: I could claim that there are only finitely many primes, and ask you to prove this claim is false by demonstrating that it leads to a contradiction.

    Note that I am not trying to prove the iPad completed this traversal, so it’s impossible for me to beg the question.

  185. 185
    kairosfocus says:

    DS, the assumption begs the question of stepwise traversal of the transfinite. We have put that back on the table and have shown that it is inherently deeply problematic. KF

  186. 186
    daveS says:

    KF, et al,

    Since my challenges in #128 and #152 didn’t lead to much response, I’ll post something which I had planned to bring up if that discussion had developed further.

    We are talking about this infinite chain of people who are passing an iPad toward the person at the end of the chain (John). It has been claimed at various junctures that if this chain is infinite, then there must be people in it who are at infinite remove from John (for example, Ishaan and everyone behind him). Needless to say, I disagree.

    I hesitate to bring up another model of this scenario since that has led to confusion in the past, but I think this one might be helpful.

    The people in the chain pair up one-to-one with the digits in the decimal expansion of π. Let’s ignore the “3” to the left of the decimal point. John pairs up with the “1” in the tenths position, the next person pairs up with “4” in the hundredths position, and so on. Notice this association gives us an isomorphism of linearly ordered sets in a natural way.

    A couple of facts/observations:

    (1) The decimal expansion of π has infinite length, obviously, because π is irrational. Err, right? Does anyone think that the decimal expansion of π is actually finite because the digits are each finitely distant from the decimal point? I sincerely doubt it. Does anyone think that there are digits in the decimal expansion infinitely distant from the decimal point? I doubt this also.

    (2) I claim it’s clear that given any nonempty collection of (the positions of) these digits, there is a single position closest to the decimal point. For example, consider the positions of all the digits which equal zero. The zero closest to the decimal point is in the 32nd position. You can even get fancier and look for the positions of strings of digits. The first occurrence of “12345” begins at position 49702. Interestingly enough, this string of digits occurs 2018 times in the first 200 million digits of pi. This supports my premise (1) in #152.

  187. 187
    kairosfocus says:

    DS you full well know that by begging the question of the infinite traverse you leave on the table a finite descent to n. KF

  188. 188
    daveS says:

    KF,

    If, by “begging the question” you mean stating in all-caps, flashing neon letters that I’m assuming up front that the traversal has taken place, yes.

    What about the digits of π illustration? Do you agree that the expansion has infinite length, yet each digit is at finite distance (number of steps) from the decimal point?

  189. 189
    kairosfocus says:

    DS, the traversal of a transfinite span is precisely the issue at stake. By assuming that you beg the question. KF

  190. 190
    daveS says:

    KF,

    Ok, I will happily withdraw my assumption which you term “begging the question”, and ask you to prove this traversal is impossible directly. I think that has been your strategy all along, anyway.

    I would ask you to please address the π example first, however, since I believe it is relevant to the argument you would make regarding the traversal itself.

  191. 191
    kairosfocus says:

    DS, I have long since shown the problem i/l/o what an infinite actual chain and past of token passing requires. For members must have once held T or have just now received it. So, if the actual chain is transfinitely long, it means transfinitely removed members must have existed. Thence, they would have had to successively pass T +1 at a time to -2, -1, 0, 1, 2 . . . and on to n-1, n. This means some transfinitely remote P passed to P+1, P+2 etc, crossing a transfinite span to reach -2,-1,0 etc. But that means spanning endlessness in steps and we already say that after any finitely large k from P, P+k, we can redefine as K, k+1, K+2 etc and match 1:1 from P on. Or, from a new span, 0,1,2 on. Always, endless succession will still be ahead at any K in finite reach from P. Where of course for all we care there is onward endlessness to the left of P. From P we do not have a powerful enough means to get to -2 etc. So, the issue is, infeasible supertask. Further to this, inferring that we are always after the transfinite has been bridged so we are always within finite reach of -2 etc, begs the question. KF

  192. 192
    daveS says:

    KF,

    So, if the actual chain is transfinitely long, it means transfinitely removed members must have existed.

    Let’s assume this is the case for the moment.

    Then exactly one of the following must be true:

    A) Some digits in the decimal expansion of π are infinitely remote from the decimal point.

    B) The decimal expansion of π has finite length.

    Which is it?

  193. 193
    Origenes says:

    DaveS

    DaveS: But that doesn’t imply that the iPad missed some positions (at least I haven’t seen any proof of this).

    If the iPad always moves to the right, then, given that for every position there are infinitely positions to the left, it is necessarily the case that the iPad missed positions.
    Why? Since there is no ‘First Left.’

    Because there are infinite positions to the left, there are only two options available to anyone who wants to enter them:

    1. Go to the left.
    2. Go to the right, but depart from the ‘First left.’

    (1) Is a journey with no return. (2) Is not possible, since there is no ‘First left’.

  194. 194
    kairosfocus says:

    DS, to be a member of the chain requires that one has passed the token or just received it. If transfinitely long, that requires actual members. As for expanding pi, it is endless, there is no last digit. We cannot exhaust how far it goes because we are finite, we can only point onward. KF

  195. 195
    daveS says:

    KF,

    To the extent I understand your #194, I don’t know that I disagree with any of it.

    Going back to my #192, which is true, A or B?

  196. 196
    kairosfocus says:

    Ds we cannot fully expand pi, and the digits in principle continue endlessly, there being no last reachable digit. There is an implied power series and its endlessness is a structural part of what it is, of course coupled with convergence so beyond some partial sum, onward partial sums will be within a delta neighbourhood of the limit, never mind how small (but finite) we set delta at. KF

  197. 197
    daveS says:

    KF,

    Obviously we cannot fully expand π in decimal form, but each of its decimal digits exists necessarily. I suspect you would agree that God knows all decimal digits of π.

    Furthermore, this sequence of digits pairs up one-to-one with the people in the line passing the iPad around.

    If there is someone in the queue infinitely distant from John, then there is a digit in the decimal expansion infinitely far from the decimal point.

    Therefore the question is, which statement is true, A or B? Perhaps consider how God would answer this question, since He can comprehend the entire infinite collection of digits.

  198. 198
    daveS says:

    Origenes,

    It seems you are still thinking of the iPad entering the entire chain at some time, so that some particular person will be first to get it (and then they have to decide whether to pass it to the left or the right).

    That is not what is happening. The assumption is that each person in the line receives/received the iPad from the person standing behind him. There is no person who received the iPad first.

  199. 199
    Origenes says:

    DaveS @198

    DaveS: It seems you are still thinking of the iPad entering the entire chain at some time, so that some particular person will be first to get it (and then they have to decide whether to pass it to the left or the right).

    I do not think of iPad entering the chain at some time.

    DaveS: That is not what is happening. The assumption is that each person in the line receives/received the iPad from the person standing behind him. There is no person who received the iPad first.

    I know. Stop worrying.
    Again, my reasoning does not start by assuming that the iPad must enter the chain at some time. Instead, it goes like this:
    Because there are infinite positions to the left, there are only two options available for the iPad to visit them all:

    1. Go to the left.
    2. Go to the right, but depart from the ‘First left.’
    (1) Is a journey with no return. (2) Is not possible, since there is no ‘First left’.

    Both options fail, ergo, the iPad misses positions on the left.

  200. 200
    daveS says:

    Origenes,

    I’m honestly not sure that you’re serious, but if you are correct, the iPad could only be held by finitely many people in the line. Therefore, it would be possible in principle to describe the path of the iPad completely (say by listing the numbers corresponding to those who held it).

    Would you please do that for an example path?

  201. 201
    Origenes says:

    DaveS @200

    DaveS: … if you are correct, the iPad could only be held by finitely many people in the line.

    If the iPad only goes to the right, then only finitely many people could have held it. I agree.

    DaveS: Therefore, it would be possible in principle to describe the path of the iPad completely (say by listing the numbers corresponding to those who held it).
    Would you please do that for an example path?

    I don’t think it’s possible. And if it were possible I would probably not interested. For me it is enough to refute your incoherent concept by logic.

  202. 202
    daveS says:

    Origenes,

    It definitely does only go to the right (and incidentally cannot skip anyone in line).

    The only way this could happen, assuming it misses some positions, is if some particular person had the iPad first.

  203. 203
    kairosfocus says:

    DS, there you go on the assumptions. Assuming the Token’s journey never had a beginning implies leftward endlessness already completed at any particular value. The possibility of such endlessness is what is at stake. This is where the criteria for membership in the chain and their consequences help us. For leftward endlessness of the chain to hold, there must be members at every actual stage. Were this to include members endlessly remote leftwards, then the +1, stepwise passing process has no power to actually traverse endlessness to reach within a finite range of n. So, if the token is passing up to n, no member of the actual chain is transfinitely remote. We may therefore conclude it had a beginning leftward. This is linked to the case that if every member is finitely remote, the whole chain — unlike an ellipsis of endlessness [which points to what we cannot attain in steps] — is finite to the left. KF

  204. 204
    Origenes says:

    DaveS @202

    For any position, going rightward is not a method to visit positions on the left. If all positions on the left have been visited, it follows that the iPad must have (already) visited the ‘First Left.’ If the iPad, going rightward, did not visit the First Left, then it never will.

    However, there is no First Left, since the left is infinite.
    – – – – – –
    1. The iPad visited all positions on the infinite left.
    2. The iPad always goes rightward.

    Therefore, from (1) and (2)

    3. The iPad must have visited the First Left, in order to visit all positions on the left.
    4. There is no First Left.

    Therefore, from (3) and (4)

    5. The iPad did not visit all positions on the left.

  205. 205
    daveS says:

    KF,

    It’s an assumption intended to assist Origenes in a reductio ad absurdum only. That’s how reductio arguments work. Please don’t pretend otherwise.

    In the conversation between you and me, that assumption has been withdrawn, as you apparently prefer direct proofs.

    Edit: I infer from the rest of your post that you choose option A, which I described earlier. There are digits in the decimal expansion of π infinitely far from the decimal point.

  206. 206
    daveS says:

    Origenes,

    You are correct that there is no “first left”, but then I don’t know what you mean when you use that term several times in your argument. Can you rephrase the argument with no use of “first left”?

  207. 207
    Origenes says:

    DaveS @206

    Sure, no problem:

    – – – –
    1. The iPad visited all positions on the infinite left.
    2. The iPad always goes rightward.

    Therefore, from (1) and (2)

    3. The iPad must have visited the farthest point of the left, in order to visit all positions on the left.
    4. There is no farthest point of the left.

    Therefore, from (3) and (4)

    5. The iPad did not visit all positions on the left.

  208. 208
    daveS says:

    Origenes,

    mfw

    I think we still have a problem …

  209. 209
    Origenes says:

    DaveS,

    Answer me this:

    Going rightward, one has to visit the farthest point of the left in order to visit all positions.

    Agree? If so, we do not have a problem.

  210. 210
    daveS says:

    Origenes,

    No, I disagree, because there is no farthest point of the left.

  211. 211
    Origenes says:

    DaveS: No, I disagree, because there is no farthest point of the left.

    That is not an answer to my question, which is irrespective whether there is a farthest point or not.

    Here it is again:

    Going rightward, one has to visit the farthest point of the left in order to visit all positions.

    Agree?

    – – –
    edit:

    Consider the set {1, 2, 3 …, 9}. Suppose one travels this set rightward (towards the 9), which position must be visited in order to visit them all?

    The farthest to the left perhaps?

  212. 212
    daveS says:

    Origenes,

    I (still) disagree.

    Suppose I said “One has to visit the man holding the square circle in order to visit all positions”. Do you agree with that? (Hopefully not)

    The issue is that neither “the farthest point of the left” nor “the man holding the square circle” has a referent. You can’t require that the iPad visit positions which literally do not exist.

  213. 213
    Origenes says:

    DaveS @212

    Suppose that I claim that a triangle is, in fact, a square. To show me wrong you would, perhaps, say:

    for a triangle to be a square it needs to have a ‘Fourth Corner.’

    “What do you mean by this ‘fourth corner’?”, I answer. “The ‘Fourth Corner’ has no referent. You can’t require that the triangle has something which literally does not exist.”

    What would you tell me to do with my claim?

  214. 214
    daveS says:

    Origenes,

    You can require that a square have a fourth corner. If it has three but not four corners, it’s a triangle and not a square.

    This is not really analogous to the situation under discussion, btw.

    Edit: Reading this post again, the best answer is simply that I would not say “for a triangle to be a square, X must be the case” in any form. A triangle cannot be a square, period.

  215. 215
    daveS says:

    Just saw this:

    Consider the set {1, 2, 3 …, 9}. Suppose one travels this set rightward (towards the 9), which position must be visited in order to visit them all?

    The farthest to the left perhaps?

    Yes.

  216. 216
    Origenes says:

    DaveS: You can require that a square have a fourth corner.

    Similarly, I require for a set — that has been traversed to the right and has been visited at all positions — that it has a farthest point on the left.

  217. 217
    daveS says:

    Origenes,

    You can require whatever you want; adding this extra condition just means you are using a different definition than pretty much everyone else, so communication becomes difficult.

  218. 218
    Origenes says:

    DaveS,

    O: Consider the set {1, 2, 3 …, 9}. Suppose one travels this set rightward (towards the 9), which position must be visited in order to visit them all?

    The farthest to the left perhaps?

    DaveS: Yes.

    For ‘going rightward’ & ‘visit all positions’, it is logically required to visit position (1) — the farthest position on the left. There is no other method to fulfill both “only going rightward” and “visit all positions”, than to visit the position “the farthest on the left”.

    O: I require for a set — that has been traversed only to the right and has been visited at all positions — that it has a farthest point on the left.

    DaveS: You can require whatever you want; adding this extra condition just means you are using a different definition than pretty much everyone else, so communication becomes difficult.

    Adding an “extra condition”? I am pointing to a logical requirement! Using a “different definition”? Of what?

  219. 219
    kairosfocus says:

    DS, the meaning of being in the chain imples that now token holder plus all to L-ward must have held T. The problem is not there, it lies in adding that as the process is increment +1 in succession the L-ward extension can be transfinite in extent AND T reaches n at some point, now. For if L-ward is transfinite and every member must hold the T, stepwise increment has to span the transfinite. But we know as shown already that it is too weak to do that. KF

    PS: the idea of transfinitely many digits of pi is an abstraction that is in principle delivered all at once by the meaning of whatever power series or series is used to deliver pi. That is utterly distinct from +1 succession r-ward to span a transfinite expanse from L-ward.

  220. 220
    daveS says:

    Origenes,

    You have observed that for finite sets (where there is an element “farthest to the left”), a traversal must visit that element farthest to the left.

    You haven’t proved that this requirement also holds for sets with no element farthest to the left (one can’t, obviously). You have drawn a false analogy, where the concept “farthest item on the left”, which does exist in the finite case, has no counterpart in the case of the iPad example.

    Regarding definitions, according to #216, it appears that you are simply choosing to interpret the statement “a set has been traversed to the right and visited at all positions” as meaning (in part) that the set has an element farthest to the left.

    Based on your next post, it looks like you are really saying that “the set has an element farthest to the left” is a logical consequence of the statement, so I withdraw my complaint about definitions.

  221. 221
    daveS says:

    KF,

    PS: the idea of transfinitely many digits of pi is an abstraction that is in principle delivered all at once by the meaning of whatever power series or series is used to deliver pi. That is utterly distinct from +1 succession r-ward to span a transfinite expanse from L-ward.

    Yes, this discussion about π is distinct from the traversal question, but the issues are related.

    Each digit in the expansion has a well-defined distance from the decimal point. The question is whether any of these digits are indeed infinitely remote from the decimal point.

    Based on your recent post, your answer is “yes”. Is that correct?

  222. 222
    Origenes says:

    DaveS

    To be clear, my reasoning goes as follows:

    A: I have a set of positions which has been (1) traversed solely rightward & (2) has been visited at all positions.

    B: If that is true, then your set must have a farthest position to the left, which has been visited. Because for any set, that is the only way to accomplish both (1) and (2).

    A: Uh … my set has no farthest position to the left, because on the left my set is infinite.

    B: Ah! Then you do NOT “have a set of positions which has been (1) traversed solely rightward & (2) has been visited at all positions.”

  223. 223
    kairosfocus says:

    The relevant power series in 10^-n has no final term n_final with n_final equals “infinity,” it embeds endlessness multiplied by convergence. The ellipsis is really significant.

  224. 224
    Origenes says:

    DaveS: You have observed that for finite sets (where there is an element “farthest to the left”), a traversal must visit that element farthest to the left.

    Not necessarily, “There must be a limit to the left that has been visited” is a logical conclusion that can be drawn even if one has never seen a specific set.
    – – – – – –

    “I have come from afar and followed the path of an arrow to your blessed city, Sire. And, noble King, I can be of service to you, because I have seen everything from afar to here.”
    “Then there must be a boundary to afar which you have seen, stranger,” spoketh the wise King.

  225. 225
    daveS says:

    Origenes,

    It’s an assertion which holds true under certain circumstances, but you haven’t shown it follows logically from the premises I laid out (which you concisely summarized in the first (A) of #222).

  226. 226
    Origenes says:

    DaveS,

    Traveling to the right and visit every position can only take place if travel space has a certain form. The travel space must have a farthest border on the left and all positions must be aligned from left to right (between that farthest border and the now), so, that, by going rightward, all positions can be visited.

    Now you tell me that your infinite set has no border on the left. Well, this tells us that your travel space has not what it takes for a good ol’ rightward traveler who wants to visit all positions. However, don’t you worry, these old-timers are tough as nails and used to set-backs.

  227. 227
    daveS says:

    Origenes,

    Is your first paragraph above any more than bare assertion? It looks as if you’re just stating what needs to be demonstrated.

  228. 228
    Origenes says:

    DaveS: Is your first paragraph above any more than bare assertion? It looks as if you’re just stating what needs to be demonstrated.

    Let me try to dig deeper:

    First, it is clear that there are certain requirements to the travel space. For instance, if one travels only rightward & visits all positions, then the positions must be arranged such that moving rightward coincides with (or ‘results in’) visiting all positions. If we have a travel space with positions all over the place and in every direction, then moving rightward cannot do the job of visiting all positions. We need all positions neatly lined-up left to right.

    Okay, about the boundary:

    If one travels only rightward & visits all positions, then all lined up positions on the left must have been already visited or they never will. This means that, at any point, traveling rightward & visit all, cannot take place if there are still unvisited positions on the left. And, since we never go leftward, this implies that at some point our rightward traveler must visit a position with no positions on the left at all: the ‘First Left’, a.k.a. ‘the farthest point on the left’, a.k.a. ‘the boundary.’

  229. 229
    daveS says:

    Origenes,

    Up until the last sentence, everything you have written follows from the original premises I stated. How do you prove this last statement?

    And, since we never go leftward, this implies that at some point our rightward traveler must visit a position with no positions on the left at all: the ‘First Left’, a.k.a. ‘the farthest point on the left’, a.k.a. ‘the boundary.’

  230. 230
    Origenes says:

    DaveS @

    This may be it:

    Because it is the case that the traveler only visits positions going rightward, he must have visited a position from where all other positions are on its right. IOWs from where there are no positions on the left — boundary.

    If it is never the case that the traveler visits a position from where all other positions are on the right, then there are necessarily unvisited positions.

  231. 231
    kairosfocus says:

    DS, do you see how you are implicitly begging the question of traversal of the infinite, again? KF

  232. 232
    daveS says:

    KF,

    No, I am not attempting to prove anything, so I cannot be begging the question. I have pointed this out many times.

  233. 233
    daveS says:

    Origenes,

    This suffers from the same problem as the argument in #222. How do you get from the stated premises to the conclusion that there exists some position in the iPad’s trajectory where all other positions lie to its right?

    You can certainly make some existential statements, for example, given any position in the chain, there is a unique position to its immediate left. That holds since we have assumed the chain has the same order properties as the nonpositive integers.

    But you haven’t derived the fact that this leftmost position must exist, starting with the original premises.

  234. 234
    Origenes says:

    DaveS #233

    DaveS: This suffers from the same problem as the argument in #222. How do you get from the stated premises to the conclusion that there exists some position in the iPad’s trajectory where all other positions lie to its right?

    What? To be clear, I am not arguing that your particular set contains such a position.

    DaveS: But you haven’t derived the fact that this leftmost position must exist, starting with the original premises.

    I agree. But never did I attempt to prove that in your set this particular position exists. I am truly surprised that you think I did.

    My argument is simply this:

    1. The traveler only visits positions going rightward.

    Therefore

    2. Only positions which are located on the traveler’s right are being visited.
    3. The traveler visits all positions.

    Therefore, from (2) and (3).

    4. At some position all other positions must be located on the right (and none on the left).

    – – –
    If it is never the case that the traveler visits a position from where all other positions are on the right, then there are necessarily unvisited positions.

  235. 235
    daveS says:

    Origenes,

    I would add this clarification to your #2:

    ____

    2′: Only positions which are/were located on the traveler’s right at some moment are being visited.

    ____

    The collection of “positions which are located on the traveler’s right” varies with time, right?

    Choose any position, say n steps from the end, where n is a positive integer.

    Then that position was to the right of the traveler n + 1 seconds ago, hence it was visited.

  236. 236
    Origenes says:

    DaveS @235

    With a slight modification, I agree with your clarification. My revised argument goes:

    – – – –
    1. The traveler only visits positions going rightward.

    Therefore

    2. Only positions which, at some point in time, are located on the traveler’s right are being visited.
    3. The traveler visits all positions.

    Therefore, from (2) and (3).

    4. At some point in time, at some position, all other positions must be located on the right (and none on the left).

    – – –
    If it is never the case that the traveler visits a position from where all other positions are on the right, then there are necessarily unvisited positions.
    – – – –
    – – – –

    DaveS: The collection of “positions which are located on the traveler’s right” varies with time, right?

    Right.

    DaveS:
    Choose any position, say n steps from the end, where n is a positive integer.
    Then that position was to the right of the traveler n + 1 seconds ago, hence it was visited.

    At one point in time n + 1 must have been on the right of the traveler, otherwise n + 1 was never visited.
    But this is a general rule (!): For all visited positions goes that they must have been on the right side of the traveler.
    And, of course, we also know that ALL positions have been visited. Moreover we know that they are neatly lined-up left to right.
    So, at some point in time, at some position, all other positions must be located on the right (and none on the left).

  237. 237
    daveS says:

    Origenes,

    From your (2) and (3) you can certainly conclude that each position must have been, at some moment depending on the position, to the right of the traveler. (2) & (3) -> (4) is a non sequitur.

  238. 238
    Origenes says:

    DaveS: (2) & (3) -> (4) is a non sequitur.

    On the ground of you saying so?
    – – – –
    edit:
    Bonus:

    1. Only positions once on the right of the traveler are visited.
    2. If, at all times, there are positions to the left of the traveler, then there are unvisited positions.
    3. The traveler visits all positions.
    4. At some point in time there were no positions to the left of the traveler.

  239. 239
    daveS says:

    Origenes,

    Yes. 😛

    Well, if you want to formalize your argument symbolically so that we can check its structure purely mechanically, then you can prove me wrong.

    Edit:

    2. If, at all times, there are positions to the left of the traveler, then these are not visited.

    Where does this come from?

  240. 240
    Origenes says:

    DaveS @239

    DaveS

    2. If, at all times, there are positions to the left of the traveler, then these are not visited.

    Where does this come from?

    I think you know: Only positions on the right are visited. So, positions which are, at all times, on the left of the traveler have, obviously, no chance of ever being visited.
    – – –

    DaveS: Yes. 🙂 Well, if you want to formalize your argument symbolically so that we can check its structure purely mechanically, then you can prove me wrong.

    I think I leave the mathematics to your capable hands 🙂

  241. 241
    daveS says:

    Origenes,

    (1) If, at all times, there are positions to the left of the traveler, then these are not visited.

    (2) So, positions which are, at all times, on the left of the traveler have, obviously, no chance of ever being visited.

    These are very different, just as the following are different:

    (1) For every time, there exist positions which are on the left of the traveler.

    (2) There exist positions which are, for every time, on the left of the traveler.

    (1) is consistent with the original premises, while (2) is not.

    (Edited: 1 & 2 were reversed).

  242. 242
    Origenes says:

    DaveS,

    At the moment the following formulation of the argument seems most succinct to me:

    1. Only positions once on the right of the traveler are visited.

    Therefore,

    2. If, at all times, there are positions to the left of the traveler, then there are unvisited positions.
    3. The traveler visits all positions.

    Therefore, from (2) and (3)

    4. At some point in time there were no positions to the left of the traveler.

    For some obscure intuitive reason, I hold that the alternative formulation (below) might be inferior:

    1. The traveler only visits positions going rightward.

    Therefore

    2. Only positions which, at some point in time, are located on the traveler’s right are being visited.
    3. The traveler visits all positions.

    Therefore, from (2) and (3).

    4. At some point in time, at some position, all other positions must be located on the right (and none on the left).

  243. 243
    daveS says:

    Origenes,

    In the succinct version, you would need this to make the argument work:

    “If there are positions, which are at all times to the left of the traveler, then there are unvisited positions.”

  244. 244
    Origenes says:

    DaveS @243

    I agree, that is better. So, again, for clarity:
    – – – – –

    1. Only positions once on the right of the traveler are visited.

    Therefore,

    2. If there are positions, which are at all times to the left of the traveler, then there are unvisited positions.
    3. The traveler visits all positions.

    Therefore, from (2) and (3)

    4. At some point in time there were no positions to the left of the traveler.

  245. 245
    daveS says:

    Origenes,

    Now the problem is that there aren’t any such positions. If you disagree, name one by giving its distance (in number of steps) from the front of the line.

  246. 246
    Origenes says:

    DaveS: Now the problem is that there aren’t any such positions.

    If that is the case, then “at some point in time there were no positions to the left of the traveler.”

    There is no other explanation.

    For instance, because the positions are lined-up left to right, it cannot be the case that the traveler is busy traveling rightward, while their are still unvisited positions to the left, and later in time there are not. So, at some point in time, the traveler is visiting a position with no positions on the left and all other positions on the right.

    Again, no other explanation. But if you think you can provide one, do let me know.

    If you disagree, name one by giving its distance (in number of steps) from the front of the line.

    Wait a minute my argument shows that your premises are incoherent … — they cannot both be true — and now you expect this incoherent ‘thing’ to produce exact numbers?

  247. 247
    daveS says:

    Origenes,

    Ok, I misunderstood your reasoning.

    So, (2) and (3) from #244 tell you that it is not the case that “there are positions which are at all times to the left of the traveler”. That means that given any position, it was once to the right of (or in the same place as) the traveler.

    That’s completely consistent with the premises.

  248. 248
    Origenes says:

    DaveS @247

    DaveS: So, (2) and (3) from #244 tell you that it is not the case that “there are positions which are at all times to the left of the traveler”.

    Exactly.

    DaveS: That means that given any position, it was once to the right of (or in the same place as) the traveler.
    That’s completely consistent with the premises.

    Indeed. But, in fact, I can get that from “the rightward traveler has visited all positions” alone.
    It is, of course, the step towards (4) “at some point in time there were no positions to the left of the traveler.” that is crucial. Maybe I can break this up further:

    – – – – –
    A. Every visited position was once to the right of (or in the same place as) the traveler.
    B. All positions are visited.
    C. The positions are aligned left to right.
    Therefore, from (A), (B) and (C)
    D. All positions were, at a certain point in time, all to the right of (or in the same place as) the traveler.
    Therefore, from (D)
    E. “At some point in time there were no positions to the left of the traveler.”

  249. 249
    daveS says:

    Origenes,

    I agree that each position was, at some time depending on the position, to the right of the traveler, but how do we know they all were, simultaneously?

  250. 250
    kairosfocus says:

    Ds, you are posing a scenario. I am pointing out that it embeds a dubious assumption that decides the matter without proper warrant. KF

  251. 251
    daveS says:

    KF,

    I have presented a collection of premises. Origenes is working to show that those premises entail a contradiction. That is all.

  252. 252
    Origenes says:

    DaveS

    I agree that each position was, at some time depending on the position, to the right of the traveler, but how do we know they all were, simultaneously?

    Multiple reasons. The fact that the positions are aligned is obviously important. From this it follows that, if it was never the case that all positions were simultaneously on the right of (or at the same place as) the traveler, then it was *somehow* always the case that there are visited positions, without an explanation for them being visited.
    IOWs if it is never the case that all positions are ‘in front’ (or at the same place) of the traveler — all for him to visit — then he was inserted somewhere in the chain with *magically* visited positions ‘behind’ him.

    The traveler must have all positions in front of him, because he must visit them all.
    – – – – –
    At times this is getting so obvious, that it becomes confusing in the sense that I lose track of what the problem is.

  253. 253
    daveS says:

    Origenes,

    Recall that I never claimed that I could explain how all this happened, or even if there _was_ an explanation. I’m just asking whether these premises entail a logical or mathematical contradiction, and I am convinced they do not. Once you start asking about explanations, things get more complicated.

  254. 254
    Origenes says:

    DaveS.

    Is there a mathematical/logical contradiction in “positions are visited, not by the traveler, but somehow *magically*”? If there is not, then your premises are irrefutable.

  255. 255
    daveS says:

    Origenes,

    I don’t know what it means for positions to be visited “magically” but not by the traveler. As far as I am concerned, they are either visited by the traveler or not, and that’s it.

  256. 256
    Origenes says:

    DaveS
    Okay, let me redo #252:

    DaveS: I agree that each position was, at some time depending on the position, to the right of the traveler, but how do we know they all were, simultaneously?

    For one thing, if it was never the case that all positions were simultaneously on the right of (or at the same place as) the traveler, — if the traveler never had all positions before him in time — then some positions must have been always on the left of him. Positions which are always on his left are, by definition, never visited by the rightward traveler, who (obviously) only visits positions on his right.

  257. 257
    daveS says:

    Origenes,

    — if the traveler never had all positions before him in time — then some positions must have been always on the left of him. Positions which are always on his left are, by definition, never visited by the rightward traveler, who (obviously) only visits positions on his right.

    There would always be positions on his left, but there are no particular positions that would always be on his left.

  258. 258
    Origenes says:

    DaveS: There would always be positions on his left, but there are no particular positions that would always be on his left.

    If there are always positions on his left, then, at no point in time, the rightward traveler can visit all positions.
    Why? Because our rightward traveler cannot visit positions which are on his left.

    At no point in time he is in a position to visit them all.

  259. 259
    daveS says:

    Because our rightward traveler cannot visit positions which are on his left.

    But they were on his right in the past. Each position was on his right in the past, despite the fact that there are always infinitely many positions on his left.

  260. 260
    Origenes says:

    DaveS: But they were on his right in the past.

    Okay, but then I want to talk about that time in the past. The time before he visits them.

    DaveS: Each position was on his right in the past, despite the fact that there are always infinitely many positions on his left.

    That is incoherent. If there are “always infinitely many positions on his left”, then at no point in time can the rightward traveler visit/access them all.

    IOWs, at any point in time, every position the traveler visits has infinite neighboring positions on its left, which he cannot access.

    If there is no point in time that the traveler has all aligned positions in front of him — if there is no point in time where he can look to the right and see them all — then at no point in time is he in a position to visit them all.

  261. 261
    daveS says:

    Origenes,

    I just don’t see anything incoherent about my statement in #259, or how it would prevent the traveler from visiting all positions. If we were supposing that there must be a “first visitation”, where the traveler visits a leftmost position, then I would agree with you, but that’s not the case here.

  262. 262
    Origenes says:

    DaveS,

    DaveS #259: Each position was on his right in the past, despite the fact that there are always infinitely many positions on his left.

    O: That is incoherent. …

    If there is no point in time that the traveler has all aligned positions in front of him — if there is no point in time where he can look to the right and see them all — then at no point in time is he in a position to visit them all.

    DaveS #261: I just don’t see anything incoherent about my statement in #259, or how it would prevent the traveler from visiting all positions.

    Please explain why, according to you, it is unnecessary for the traveler to be in a position to visit all positions. Logic informs us that if the traveler was never in such a position, then he could not have visited all positions.

    Perhaps you can point out which step you disagree with.

    1. The rightward traveler can only visit positions situated on his right.
    2. The rightward traveler cannot visit positions situated on his left.
    3. “At every point in time, there are infinitely many positions on the traveler’s left.”

    Therefore, from (2) and (3)

    4. At every point in time, there are (infinitely many) positions that the traveler cannot visit.

    Therefore,

    5. It cannot be the case that the traveler visited every position.

  263. 263
    daveS says:

    Origenes,

    We need to qualify (1), (2), and (4), to take into account that the “positions on his right” depend on time. Suppose the traveler is at position P at time t_P. Then:

    1. The rightward traveler can only visit positions to the right of P at times later than t_P.

    2. The rightward traveler cannot visit positions situated to the left of P at or after t_P.

    3. At every point in time, there are infinitely many positions on the traveler’s left. (No changes needed here).

    From (2) and(3):

    4. At every point in time t_P, there are infinitely many positions that the traveler cannot visit after time t_P.

  264. 264
    Origenes says:

    DaveS: 4. At every point in time t_P, there are infinitely many positions that the traveler cannot visit after time t_P.

    Indeed, at every point in time t_P, there are (infinitely many) positions that the traveler cannot visit (at that moment in time).

    But that’s all I need to make the point:

    During his entire travel there are always positions inaccessible to the traveler. If there are always positions inaccessible, then logic informs us that one cannot have accessed them all.

  265. 265
    daveS says:

    Origenes,

    But there are no positions that were always inaccessible to the traveler. Each position was/is accessible at some moment.

  266. 266
    Origenes says:

    DaveS: But there are no positions that were always inaccessible to the traveler. Each position was/is accessible at some moment.

    In #245 you wrote (tongue in cheek): “.. name one by giving its distance (in number of steps) from the front of the line” and here you implicitly do something similar. Indeed, you are right, I cannot name specific positions that were always inaccessible to the traveler. However, for my argument it is not required to point out specific positions and show that they are inaccessible. And that is very convenient for me, since it cannot be done.

    All I need to do is point out that at every point in time there are inaccessible positions — a general observation without specifics. During his entire travel there are always positions inaccessible to the traveler. If there are always positions inaccessible, then logic informs us that one cannot have accessed them all.

  267. 267
    daveS says:

    Origenes,

    Actually it was not tongue in cheek, although I wasn’t clear on your reasoning when I wrote it.

    Edited: Do there exist specific positions which were never accessible?

  268. 268
    Origenes says:

    DaveS: Do there exist specific positions which were never accessible?

    Irrelevant to the issue at hand. My argument is based on the insight that, given a strict rightward traveler, for every position goes that there is only access to the right and no access to the left of the traveler. This ‘limited access’ (LA) is a problem which must be solved in order to have access to all positions.

    However, as we have seen, LA exists on every position at each time. Although LA differs in its specifics at each position, it never differs in its general form (which can be described as: ‘no access to positions on the left’).
    So, the general form of LA, is a problem which exists independent from specifics and therefore specific positions are irrelevant to the problem. To be clear, the problem of LA cannot be solved by specific positions.

  269. 269
    daveS says:

    Origenes,

    Irrelevant to the issue at hand.

    Please answer the question.

  270. 270
    Origenes says:

    DaveS,

    as I already stated in #266, I cannot name specific positions that were always inaccessible to the traveler. But I am no mathematician.

    Edit: however, for every position goes limited access, so they must exist. Perhaps there is a term for them.

  271. 271
    asauber says:

    Do there exist specific positions which were never accessible?

    When something is strictly imaginary (like the attempt to utilize something called infinity), things that don’t or can’t exist appear to be possible.

    The question is wrong. The question is “Do there exist specific positions in your imagination which were never accessible?”

    Depends on what you like to imagine.

    Andrew

  272. 272
    Origenes says:

    DaveS: But they were on his right in the past. Each position was on his right in the past, despite the fact that there are always infinitely many positions on his left.

    Each position was on the right of the traveler in the past ….

    So, at one point in time, the traveler looked to the right and saw all and each positions (except for the one he visited at that same point in time)?

    – – – –
    A.Sauber,

    The concepts ‘potential’ and ‘actual’ may play an important role in this story.

  273. 273
    daveS says:

    Origenes,

    To answer your #272 first: No.

    Suppose you just took a slow 10-minute walk. Then each of your feet would have been off the ground at various times, but it’s likely (or at least possible) that there was no time when both feet were off the ground.

  274. 274
    daveS says:

    Origenes,

    To answer #272, “always inaccessible” means “always to the left of the iPad”.

    Choose any position, and it must be some nonnegative integer number n steps from the front of the line. The iPad would have been to its left n + 1 seconds ago.

    Therefore no position is always inaccessible.

  275. 275
    Origenes says:

    DaveS: Choose any position, and it must be some nonnegative integer number n steps from the front of the line. The iPad would have been to its left n + 1 seconds ago.
    Therefore no position is always inaccessible.

    Cannot be true. Choose any position and there is a inaccessible position on its left. At every point in time there are inaccessible positions — a general observation without specifics. During his entire travel there are always positions inaccessible to the traveler. If there are always positions inaccessible, then logic informs us that one cannot have accessed them all.

  276. 276
    daveS says:

    Origenes,

    This is one of those times when I really doubt you are being serious…

  277. 277
    Origenes says:

    DaveS

    For any point in time goes that I am infinitely serious.

    BTW if you are correct and there is always an infinite number of visited positions on the left of the traveler, then there was once an infinite number of unvisited positions on the right of the traveler.

    So we have an infinite number of positions on both sides of the traveler. It’s getting weirder.

    An infinite number of positions on the right causes many problems for your position.

  278. 278
    daveS says:

    Origenes,

    BTW if you are correct and there is always an infinite number of visited positions on the left of the traveler, then there was once an infinite number of unvisited positions on the right of the traveler.

    0___0

    It’s a lot easier to post these fallacious arguments than it is to refute them. I don’t think I can keep up.

  279. 279
    Origenes says:

    DaveS

    O: BTW if you are correct and there is always an infinite number of visited positions on the left of the traveler, then there was once an infinite number of unvisited positions on the right of the traveler.

    DaveS:
    0___0

    It’s a lot easier to post these fallacious arguments than it is to refute them. I don’t think I can keep up.

    Unwarranted response.

    Let’s break it down:
    1.

    O: there is always an infinite number of visited positions on the left of the traveler …

    DaveS: … there are always infinitely many positions on his left.

    2.

    O: … then there was once an infinite number of unvisited positions on the right of the traveler.

    DaveS: Each position was on his right in the past …

    So, infinite positions on the left, which were all on the travelar’s right in the past.

    How is anything I’ve said “fallacious”?

  280. 280
    daveS says:

    Origenes,

    Thanks, but I’m going to let someone else field this one.

  281. 281
    jdk says:

    Again this topic??? I feel for you, Dave.

    “Infinite distance” is not a well defined mathematical term. The set of all integers is infinite, but no two integers are an “infinite distance” apart, for what that’s worth.

  282. 282
    daveS says:

    Hey jdk, good to see you.

    Yes, this topic is like a perpetual motion machine.

  283. 283
    Absolutely says:

    The original question may be answered already but if not there is No question that the whole argument is a classic Category Mistake.

    Infinite in relation to God is not literal.

    I think it’s extremely important when discussing natural theology or the God of the Bible to guard ourselves from being overly pedantic. Although God and the nature of reality is obviously deep serious stuff… the answers are not complicated at all and certainly are easily accessible to all.
    We all actually understand infinity well in reality and many are quite intuitive, if math was in our ciriculum, in recognizing infinity on paper is just a tool & never meant to apply to the real world. So running into the barstool atheist or even the shockingly biased physicist turned philosopher syndrome rampant today shouldn’t require more than a minute to dispense of their utter nonsense and quite frankly, outright lies in many of them, regarding infinite regression.

    It’s been settled to the point that it can be asserted & if an atheist wants to fight it with hypothetical Sets just let him go off & argue with deep fried poop.

    The very presence of a “debate” only serves to illustrate someone’s gone crazy & has begun to babble 🙂 (BTW, I don’t mean the tailend of what’s played here in the comments…sounds like fun)

    Anyway… Yeah

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