Cosmology Darwinism Philosophy Physics

An infinite past can’t save Darwin?

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Philosopher and photographer Laszlo Bencze shares with us a passage from Robert J. Spitzer on the impossibility of infinite past time. He explains,

If often happens that infinity is marshaled to prop up the notion that evolution can work via random mutations, no matter how heavily the odds are stacked against that possibility. If the finiteness of our universe limits the effectiveness of randomness in producing wonders, then infinity is offered as the handy solution. Our universe was preceded by an infinite number of other universes which rolled the dice an infinite number of times until finally our own time-bound universe happened to get it “just right.” An infinite number of universes of course entails infinite time, a concept tossed blithely into discussion as if it were no more problematic than booking a meal at a restaurant.

Here is one of several proofs that Spitzer offers to show the impossibility of infinite past time. I find it rather elegant:

Infinities within an aggregating succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for an aggregating succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. Now, if “infinity” is applied to an aggregating succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved or be actualized through an aggregating succession. Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to an aggregating succession. Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in an aggregating succession, “infinity” must be consider unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable). We are now ready to combine the two parts of our expression through our three common conceptual bases:

“Infinite Past Time”

“(The) unoccurrable (has) occurred.”
“(The) unachievable (has been) achieved.”
“(The) unactualizable (has been) actualized.”

Failures of human imagination may deceive one into thinking that the above analytical contradictions can be overcome, but further scrutiny reveals their inescapability. For example, it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved. However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved. This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.” It simply manifests a failure of our imagination. Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction. ( – New Proofs for the Existence of God, Robert J. Spitzer, p. 181 )

Readers? Thoughts?

See also: Arrow of time points to missing dark matter

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119 Replies to “An infinite past can’t save Darwin?

  1. 1
    daveS says:

    Bencze:

    Here is one of several proofs that Spitzer offers to show the impossibility of infinite past time.

    Is that really a “proof”? I don’t find it particularly convincing.

    Infinite past time is a strange concept, but in my view no stranger than finite past time, or an omnipotent being existing “outside of time”, etc.

  2. 2
    JDH says:

    I think this is really obvious if you think of it this way. One definition of infinity that I like is a kind of phenomenological definition. It goes like this:

    Think of the highest number you can imagine. There are still an infinite amount of numbers higher than that number.

    You notice the definition never speaks about where infinity actually lies on the number line. It just states that you can’t reach the point of infinity by any series of steps or any advancement by a finite number no matter how large.

    So don’t confuse the issue by thinking of the past. Let’s look at the infinite future. Can you go out to visit the “time” of the infinite future. NO. No such real time as we know it exists. Any “time” we thought was the infinite future would always have an infinity of time left before it reached the infinite future.

    Can we go to the infinite future and then get back to the present in a series of finite steps from the infinite future. NO. Because you can’t even start. You can’t reach from any present time to the point at infinity future, so you can NEVER get there.

    The point is that NO series of finite calendar days can make it back to the infinite past or get to the infinite future. In this regard there is no difference between future and past. Beings which exist in a world where time differences allow events to be discussed in a chronological order, can never get to either the infinite past or infinite future by the passing of time.

    So it really is a simple as this. Do you live in a world where events at a single point in space can be observed to occur in a time sequence – then there CANNOT be an infinite past. Time had to have a beginning.

  3. 3
    Jim Smith says:

    An infinite number of universes in sequence has the same problem as multiverse theories where you have an infinite number of universes “simultaneously”. You can explain anything by chance quantum fluctuations and therefore science looses all explanatory power. For example if there are an infinite number of universes it could be due to chance that our universe is 6000 years old and consists of just the sun and its planets with the rest being an illusion.

    If you invoke an infinite number of universes, or infinite time, to account for something improbable, you have no grounds on which to assert which improbable thing happened by chance.

  4. 4
    daveS says:

    JDH,

    It is (I assume) true that there is no “point of infinity” either in the future or the past.

    Furthermore, I assume you agree that there will be no “end” to time. People who go to heaven will exist there eternally.

    Why should there be a “start” to time, then? Why can’t time be unbounded in both directions?

  5. 5
    Mapou says:

    Infinite anything is nonsense, let alone an infinite past. Even Yahweh says “I am the first and the last”, which implies that there was a beginning.

    Infinity is illogical for a very simple reason. Nothing can be compared to infinity without introducing a contradiction. Why? It’s because any finite quantity is infinitely small compared to infinity, making the quantity both finite and infinitesimal at the same time. As simple as that.

  6. 6
    RexTugwell says:

    The above youtube video is very good. Bruce Gordon’s talk is especially amusing on the silliness of infinite past time and infinite universes. I wish I had the faith of the atheist; it’s truly profound.

    I highly recommend Spitzer’s video series (along with its study guide) From Nothing to Cosmos

  7. 7
    Laszlo says:

    As Spitzer makes clear, the only “prohibited infinity” is an actual infinity which means a completed set of things said to be infinite (including past time). Future time is not an actual infinity but a potential infinity. There is no prohibition against a potential infinity for the simple reason that it is not possible to reach the end. No matter how far time may progress, it will always be finite in duration. The future is always potential not actualized.

    As for infinite time in heaven, I think that is a highly misleading way to look at the next life. Eternity is not and cannot be the same thing as infinite time. It is something other than that, but the limitations of human thought cannot comprehend it.

  8. 8
    Mung says:

    Furthermore, I assume you agree that there will be no “end” to time. People who go to heaven will exist there eternally.

    God must just be a really, really old guy.

  9. 9
    daveS says:

    Spitzer, with some bolding added:

    No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable.

    Maybe I’m wrong, but it seems like he’s essentially assuming his conclusion here. If an infinite amount of time is available, then why wouldn’t an infinite amount [of an aggregating succession] be occurrable, achievable, or actualizable?

  10. 10
    Seversky says:

    Laszlo @ 7

    Since when has infinity been necessary to explain evolution?

    The future is always potential not actualized.

    When George Washington was appointed to command the American army in 1775, he did not know what would be the outcome of the Revolutionary War, let alone anything after. For him, it was all unactualized potential. Yet for us, what occurred in the 238 years between then and now is actualized history. So which is true, is that period unactualized potential or actualized history?

  11. 11
    Seversky says:

    Mung @ 8

    God must just be a really, really old guy.

    …or, like Peter Pan, He just never grows up.

  12. 12
    Aleta says:

    Mapou in 5 obviously doesn’t have a mathematical background in the concept of infinity.

  13. 13
    Mapou says:

    Aleta,

    Man, don’t attack my background. This is known as an ad-hominem argument. If you got an actual argument, let’s hear it. I suspect you don’t. But hey, just in case you do, come out with it and watch me shoot it down. LOL

  14. 14
    jimmontg says:

    Anyone hear of the mathematician David Hilbert and Hilbert’s Hotel with an infinite amount of rooms? When talking about an infinite amount of things all kinds of counter-intuitive things happen. Like if half of the infinite rooms were occupied you would still have an infinite amount of empty rooms and occupied rooms so like every other room would be empty. If 20 million people check out half the rooms are still full, but if everybody except 8 people checked out you would only have 8 people occupying 8 rooms. Got that? OK.

    There cannot be an infinite past in our spatial dimensions because there is only one dimension of time moving in one direction. If there were two dimensions of time, think of a two dimensional plane and we live on a single line going through it. Because of our restriction we cannot go back in time, but a Being that existed in 2D could be infinite and could see the beginning of our time line from start to finish if there is one, but that Being doesn’t have to set an end. Yet there has to be a beginning or we could never reach today on a single line of time because if you have crossed 99.9999% of infinity you still have an infinity to go.

    That is why theologians say God is outside of time and indeed created it as per Hawking/Penrose’s work on relativity when they figured out at the initial moment of the Creation time had a beginning as well as space itself. Nothing, or better to say not out of anything was the Universe created and YEC, OEC or EC , there is a Creation point and there has to be in one dimension of time or the past would not have been actualized to get to today.

    When God created Adam He said, “Let Us make man in Our image.” The very fact that we can even conceive of such mind bending concepts is part of that Image manifesting itself. The fact that we can apprehend the concepts doesn’t mean we can comprehend them in their fullness, as we are finite beings. This is an evidence for God’s existence argument.

    By the way, in the only Psalm attributed to Moses he declared “From Everlasting to Everlasting Thou art God.” So there is the answer for the “Who created God?” crowd.
    The Bible states that we cannot even imagine what God has in store for those that trust in Christ and his Saving Hands with the holes in Them. The God that Suffers and for those who don’t believe in God because of evil, you have to understand that this world is the best plan. How do I know that? God planned it from before the foundation of the world and God is the greatest conceivable Being and therefore His plan is the greatest conceivable plan, it cannot be otherwise even though there is evil.

    You must understand I speak from experience. I was raised in an abusive home and I got into a lot trouble and paid for it. I have suffered, I lost a son five years ago who was burned to death in a meaningless accident, but not to God was it meaningless and I never once was angry at Him as I am a sinner and have no right to accuse Him. I had kidney cancer and had one of my kidneys removed last Oct. I know what pain is and nothing was as bad as losing my 24 year old son. But I am just a creature and a sinful one at that.
    God is The Creator not me. God commands you to believe in His Son and I hope that somebody is praying for you because God will answer their prayers whether you like it or not. I also tend toward Calvinism too. LOL

  15. 15
    Mapou says:

    Infinities within an aggregating succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for an aggregating succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. Now, if “infinity” is applied to an aggregating succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved or be actualized through an aggregating succession. Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to an aggregating succession. Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in an aggregating succession, “infinity” must be consider unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable). We are now ready to combine the two parts of our expression through our three common conceptual bases:

    After reading this, my first thought was, why use so many words just to say that nobody can count to infinity? Infinity is obviously a stupid concept for propellerheads and pompous, insecure mathematicians.

  16. 16
    Brent says:

    Mapou,

    I agree with your sentiment at 15, but not with your stated application/conclusion in 5.

    If infinitude doesn’t exist in at least one sense, somewhere, or better put, some how, then nothing exists, period.

  17. 17
    Bob O'H says:

    No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable.

    In other news, I hear that Achilles still hasn’t overtaken the tortoise.

  18. 18
    kairosfocus says:

    Folks,

    RS is right, you cannot traverse a countable infinite in finite, discrete spaced steps. You can only be on the way and point onward.

    That is why our cosmos as a contingent being with processes of finite duration that lead out in causal chains cannot be of infinite duration. You can no more count down one by one from infinity than you can count up to it.

    Infinity minus one, infinity minus two . . . zero, 1, 2 makes no more sense than , 2, . . . infinity minus one, infinity.

    To get to root cause you have to switch order of being to necessary being, one so connected to the foundation of a world that without it there is no world. Things like the number 2 and the consequent relationship this and that which is not this — distinct identity and linked world partition W = {A | ~A}.

    Infinite regress is impossible and the attempt to draw being out of non-being is a failure on its face.

    Non-being has no causal powers so were there ever utter nothing such would forever obtain.

    If a world now is, a world root always was.

    And in wider context one capable of a cosmos fine tuned for life and with capability to be the IS that grounds OUGHT as we are responsibly free morally governed beings.

    There is (after centuries of debates) only one serious candidate for such, the inherently good creator God, a necessary and maximally great being worthy of loyalty and the reasonable responsible service of freely doing the good in accord with our evident nature.

    But, that is exactly what ever so many do not wish to face.

    KF

  19. 19
    kairosfocus says:

    Bob O’H: Surely you are familiar with L’Hospital’s rule and the like. Processes based on infinitesimals — I allude to non standard analysis that vindicates Newton et al — can under fsvourable circumstances terminate in finite times and spaces. But the limit is not guaranteed in general. That is distinct from finite separate successive steps of action that are like climbing a ladder rung by rung. You cannot traverse the countable infinite that way. And I am confident you know it so the posing of Zeno’s paradoxes is a red herring led out to a strawman. KF

  20. 20
    RexTugwell says:

    Infinity-of-the-gaps

    nothing more

  21. 21
    kairosfocus says:

    But we can drown you out through institutional and message dominance — until we go over the cliff through a march of folly.

  22. 22
    Aleta says:

    At 5, Mapau wrote,

    Infinity is illogical for a very simple reason. Nothing can be compared to infinity without introducing a contradiction. Why? It’s because any finite quantity is infinitely small compared to infinity, making the quantity both finite and infinitesimal at the same time. As simple as that.

    Mapou, your “logical” conclusion that infinity is illogical because it contains a contradiction is flawed because, among other things, you use “infinitesimal” in a colloquial way as meaning really small as opposed to a mathematically correct way.

    Second, there are all sorts of things about infinity that appear “illogical” to our common sense, but are mathematically true – “common sense” is not a very good guide in thinking about infinity.

    A couple of textbooks examples:

    An infinite set can contain an infinite subset: the set of all even numbers is obviously a subset of all integers (because there are also odd numbers), yet both the set of all even numbers and the set of all integers are infinite.

    The set of all even numbers and the set of all integers are the same size: we say they have the same cardinality. There are as many evens as there are integers, even though there are also the same number of integers that are not even.

    However, the same is not true of the integers compared to the reals: the integers are obviously an infinite subset of the reals, but there are more reals than there are integers.

    Last, infinitesimal doesn’t just mean “really small” in comparison to something else. It is a central concept in calculus, although it has been the subject of much discussion over the centuries.

    The basic idea is connected to that of limit: an infinitesimal is, informally, a non-zero “number” that is smaller than any number you can name, much as infinity is a “number” larger than any number you can name. But neither is really a number: more formally, they both refer what happens as you continue a processes.

    The classic story is of the crazy frog who start at one end of a room, and starts jumping to get to the other side. His first jump is 1/2 way across the room, but because he is tired his next jump is only 1/2 the remaining distance (so he is now 3/4 of the way); and his next is again 1/2 the remaining distance (so he is now 7/8 of the way), and so on.

    Does he ever get to the wall? No.

    How close does he get? Infinitesimally close. That is, no matter what finite distance you name (1 trillionth of the way), he will get closer than that; and if you name an even smaller number (1 quadrillionth) he will get closer than that.

    More formally, we say the limit of the series 1/2 + 1/4 + 1/8 + … = 1, but we can not say that 1/2 + 1/4 + 1/8 + … = 1.

    The even bigger idea – the idea that makes calculus possible, is that the ratio of two infinitesimals can have a finite, definite limit, even though the limit of each infinitesimal is zero. That is; 0/0 is undefined and indeterminate, but the limit of dy/dx, where dy and dx are infinitesimals each of whose limit is zero, is a definite number.

    Maybe you know all this, Mapou, and just choose to not accept these conclusions, and maybe you don’t. But some of the above is why it looked to me that you didn’t have much of a background in the topic.

  23. 23
    harry says:

    kairosfocus @21

    But we can drown you out through institutional and message dominance — until we go over the cliff through a march of folly.

    Exactly.

    Although I still entertain the hope that by the grace of God, for Whom nothing is impossible, the march of folly can be transformed into a march of Wisdom.

    What prevents that isn’t any lack of perfection in God’s plan to make it happen, or a deficiency in God’s power, which is limitless. So why don’t we see it happening? Because good Christians with the best of intentions throw themselves into their own good plans to make it happen. The problem is that their plans go forth with their power, which is nothing compared to that of the prince of this world.

    God’s plans go forth with His power. We need to beg God to show us our place in His plan. Of course, that will require us to leave our comfort zones and have the faith of those for whom Christ worked the miracles recorded in the Gospels, which most often have Him mentioning their faith as what brought forth the miraculous power of God.

    God’s plan will take us where we, on a human level, as Christ told Peter, would “rather not go.” (Jn 21:18) But God’s plan has more than enough power to reverse the march of folly. We just need to find our place in it and be willing to go where it takes us.

  24. 24
    daveS says:

    KF,

    RS is right, you cannot traverse a countable infinite in finite, discrete spaced steps. You can only be on the way and point onward.

    If the universe is infinitely old, we certainly can traverse a countable infinite. Think of a clock ticking once per second in that scenario. How many times has it ticked so far? Little-omega at least (I don’t think the symbol will display properly here).

    That was my point in post #9. It seems to me that this resolves the “analytical contradiction” than Spitzer constructs.

  25. 25
    Aleta says:

    Infinity, and other incarnations involving cardinality of infinite sets, are not numbers. You can’t count an infinite number of seconds. Saying the clock has ticked “little-omega” seconds isn’t meaningful, I don’t think.

  26. 26
    daveS says:

    Aleta,

    First, I should have said aleph-null instead of little-omega, my bad.

    But I dispute that you can’t count an infinite number of seconds. You can count an infinite number of natural numbers. How would seconds differ?

  27. 27
    Aleta says:

    You can’t count an infinite number of natural numbers either. The term “countable” is misleading: any set that can be put in a 1-1 correspondence with the natural numbers is countable, but there is a difference between countable finite sets and countable infinite sets. As Wikipedia says,

    In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish,, every element of the set is associated with a natural number.

    [My emphasis]

    So an infinite countable set may be countable in the mathematical sense, but that doesn’t mean we can actually count them. Infinity is a not a number: no matter how many numbers we counted, there are still more to count. If we can’t stop counting, I don’t think we can say we have in fact counted the numbers.

  28. 28
    Mung says:

    Infinity-of-the-gaps

    But the gaps ate infinitely small, unlike Darwinian gaps, which are gaping wounds.

  29. 29
    Mung says:

    The even bigger idea – the idea that makes calculus possible…

    In my experience, calculus is not possible.

  30. 30
    Aleta says:

    A trollish comment? Calculus works – the modern world would not exist without the mathematical tools brought to us through calculus.

    What do you mean that “calculus is not possible”?

  31. 31
    RexTugwell says:

    I got an A in calculus and still couldn’t wrap my head around it 🙂

  32. 32
    Mapou says:

    Aleta,

    I warned you that I would shoot your argument down. You wrote:

    An infinite set can contain an infinite subset

    There is no need to go further. You see, this is precisely the problem with all infinity mongers. They are what I call poofery artists or “pooferists”. Declaring “Let there be an infinite set” is no better than saying “abracadabra” and expect a fully formed rabbit to come out of a hat.

    The biggest problem with mathematicians is that they believe that numbers exist in the physical realm. I got bad news for you. You are all deluding yourselves.

    Immanuel Kant asked “If space exists, where is it?” I ask the same question about not just infinity but also about all numbers and other abstract mathematical concepts:

    If the number 2 exists, where is it?
    If distance exists, where is it?
    If volume exists, where is it?
    If circumference exists, where is it?

    My point is this. In the physical universe, there exist only particles and their properties. Everything else is either abstract (i.e., in the spiritual realm) or just plain poofery.

  33. 33
    daveS says:

    Aleta,

    Addressing your points in slightly scrambled order:

    Infinity is a not a number: no matter how many numbers we counted, there are still more to count.

    I would agree that “infinity” is not a number. However aleph-null, 1, 17 aleph-thirty-two, and other cardinal numbers are.

    You can’t count an infinite number of natural numbers either. The term “countable” is misleading: any set that can be put in a 1-1 correspondence with the natural numbers is countable, but there is a difference between countable finite sets and countable infinite sets.

    If we can’t stop counting, I don’t think we can say we have in fact counted the numbers.

    I appreciate the distinction between countable and “able to be counted”, if you will, but I think that relates to the very issue I’m raising. Who says that a countably infinite set is not “able to be counted”, given an infinite past?

    Referring back to the eternal ticking clock example, is the total number of ticks up to the present undefined? Is it finite? Or something else?

  34. 34
    Aleta says:

    Mapou says, “The biggest problem with mathematicians is that they believe that numbers exist in the physical realm.”

    That is wrong, and not relevant to this discussion. My guess is that virtually all mathematicians think of numbers and other elements of mathematics as abstract ideas, not as things that physically exist. I have no idea why you think mathematicians “believe that numbers exist in the physical realm”. I am discussing abstract ideas, not physical reality.

    I am discussing the idea of infinity as it is understood in mathematics. Since you dismiss mathematicians who have studied and use these ideas as “infinity mongers”, then there isn’t much for us to discuss.

  35. 35
    Aleta says:

    daveS asks, “Referring back to the eternal ticking clock example, is the total number of ticks up to the present undefined? Is it finite? Or something else?”

    It’s something else: it infinite, in the sense that it is bigger than any number that can be named. If you started counted backwards from now, you would never stop counting: it doesn’t make sense to say we can count the number of seconds if in fact we never could actually stop counting and declare a final amount.

  36. 36
    Mapou says:

    Aleta:

    there isn’t much for us to discuss.

    I agree. I guess I was wrong to think we were talking about the existence of infinity. Silly me.

    See you around, pooferist.

  37. 37
    Aleta says:

    Mapou, if you mean “the physical existence of infinity”, I was never talking about that, so yes, you were “wrong to think we were talking about the existence of infinity.”

    I don’t know where you’ll find someone who believes that that there is an infinite number of any thing in the physical world, though, so I’m not sure the pooferists you are arguing against actually exist.

  38. 38
    Mapou says:

    Aleta,

    Actually, the pooferists are all over the place in both the academic world and the Christian fundamentalist community. Albert Einstein was a pooferist because his theories assume the existence of continuity, i.e., infinite divisibility. Black hole preachers and other snake oil charlatans (e.g., Stephen Hawking) are all pooferists since black holes assume infinity. Materialists and Darwinists are constantly preaching to us about the existence of an infinite number of parallel universes and the like. Mathematicians are the most incorrigible pooferists of them all.

    The truth hurts. Live with it.

  39. 39
    Mung says:

    Mapou:

    My point is this. In the physical universe, there exist only particles and their properties. Everything else is either abstract (i.e., in the spiritual realm) or just plain poofery.

    Aleta:

    Mapou, if you mean “the physical existence of infinity”, I was never talking about that…

    A trollish comment?

  40. 40
    Aleta says:

    Mung, I know you’re getting me back for my comment, so I guess we’re even there.

    But, more substantially:

    a) what do you mean when you say calculus is impossible? Calculus works. What is impossible about it?

    b) do you agree with Mapou that “most mathematicians believe that numbers exist in the physical realm (see 32), or do you agree with me that very few mathematicians believe that, and that most mathematicians consider the concepts of mathematics, such as infinity and continuity, as abstractions.

  41. 41
    Mapou says:

    I should add that Einstein seemed to have realized late in his life that he was wrong about continuity. I guess it’s never too late to recant a lifetime of self-delusion. Not long before his death he wrote to his friend Besso,

    “I consider it quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics.” (From: “Subtle is the Lord” by Abraham Pais.)

  42. 42
    kairosfocus says:

    DS, the traversal of a countable infinity in finite discrete successive steps is impossible, which is what I spoke to. Steps is indicative of causal succession. Chains and ratios of infinitesimals etc is a different issue, as limit processes come in that may complete in finite spaces and times — forms like dy/dx, limit runs to 0/0 or inf/ inf etc. where the differing “rates” of approach can indeed yield finite answers, where however there is no simple general answer. KF

  43. 43
    kairosfocus says:

    Mapou and Aleta:

    Would nonstandard analysis be of help here:

    http://mathworld.wolfram.com/N.....lysis.html

    Note: http://www.cut-the-knot.org/Wh...../NSA.shtml

    Cf monograph:

    https://www.math.wisc.edu/~keisler/foundations.html

    KF

    PS: I found the definition of the contiunnum in terms of ability to interpolate a point between two neighbouring points helpful. I suggest on decimals A and B such that both are W + F = W.f1f2 . . . fn but then at some fn+1 they differ so we have an+1 and bn+1, we may then average these terms and interpolate within the two, accepting that the decimals are members of a common class, R.

  44. 44
    daveS says:

    KF,

    DS, the traversal of a countable infinity in finite discrete successive steps is impossible, which is what I spoke to. Steps is indicative of causal succession.

    But why is it impossible?

    Chains and ratios of infinitesimals etc is a different issue, as limit processes come in that may complete in finite spaces and times — forms like dy/dx, limit runs to 0/0 or inf/ inf etc. where the differing “rates” of approach can indeed yield finite answers, where however there is no simple general answer.

    I’m not referring to infinitesimals anywhere here. My example was an eternal ticking clock, with ticks separated by 1 second. Finite, discrete, successive steps throughout.

  45. 45
    kairosfocus says:

    DS,

    Count off from a beginning in steps:

    S1, s2, . . . sn.

    At any sn we reach, we can add another and each successive si will be finite, i.e. we are doing a physical analogue to counting.

    We will always be at some finite si, and will have infinitely many further steps to reach aleph_null or omega if you think in ordinal terms.

    Counting with finite discrete steps is inherently a potentially transfinite process not one that can successively reach the first transfinite.

    To see where the problems are with counting down, let us imagine, using A for aleph null:

    A, A-1, A-2 . . . A – n . . . 3, 2, 1, 0, . . .

    At any n, we have only taken a finite out and have still not left the domain of A.

    The problem is essentially the same.

    Supertask, cf Hilbert’s Hotel

    KF

  46. 46
    Aleta says:

    What I said. kf and I agree.

    And to dave S: my remarks about infinitesimals were a part of the discussion with Mapou. I know you are talking about discrete counting.

  47. 47
    daveS says:

    KF,

    DS,

    Count off from a beginning in steps:

    S1, s2, . . . sn.

    At any sn we reach, we can add another and each successive si will be finite, i.e. we are doing a physical analogue to counting.

    But in my eternal clock example, there is no beginning, no first tick. There is no S1, in your notation.

    To see where the problems are with counting down, let us imagine, using A for aleph null:

    A, A-1, A-2 . . . A – n . . . 3, 2, 1, 0, . . .

    At any n, we have only taken a finite out and have still not left the domain of A.

    True. This just means that you can step back in time any finite number of ticks and still the clock has ticked an infinite number of times before that.

  48. 48
    kairosfocus says:

    DS, You will see I began with a simple, readily understood case. In taking up the infinite step-down illustration, what I brought out is that removing any finite subset will do nothing. The idea that at any point there was an infinite past succession to that point is a mathematical suggestion not a physical one — and we are dealing with the physical case. Work flows, energy, trends to so called heat death, rise of entropy are all implicated. Once we have finite actual causally connected steps a world is running and you then face the limits imposed. An infinite past succession of steps is physically equivalent to, heat death has already occurred. There will be no free energy to drive such steps. No physical clock that has ticked infinitely many times and advanced the second hand accordingly will be feasible. KF

  49. 49
    daveS says:

    In taking up the infinite step-down illustration, what I brought out is that removing any finite subset will do nothing.

    Agreed.

    The idea that at any point there was an infinite past succession to that point is a mathematical suggestion not a physical one — and we are dealing with the physical case.

    Well, once you start considering physical evidence, then I agree an infinite past seems less likely.

    But Spitzer’s argument is based solely on the derivation of a supposed analytical contradiction, with no physics involved. Again, I could be wrong, but I don’t think it works.

  50. 50
    kairosfocus says:

    Cf Hilbert’s Hotel. Where, we are dealing with the cosmos.

  51. 51
    daveS says:

    What’s the relevance of Hilbert’s Hotel to what Spitzer says?

  52. 52
    kairosfocus says:

    DS You spoke of logical issues. The hotel shows at least some of them, whatever may o may not be right in the clip. KF

  53. 53
    Mung says:

    Aleta: …what do you mean when you say calculus is impossible?

    It was a joke. It was a comment about myself.

    Mung: In my experience, calculus is not possible.

  54. 54
    Aleta says:

    I see! 🙂 My apologies for my attitude.

    I really enjoyed teaching calculus to kids so that they could understand what it was about, both mathematically and philosophically, rather than seeing it as an arcane mumbo-jumbo set of techniques (although eventually one must learn the techniques). This is one reason why I’m interested in infinity: it’s critical to understanding what calculus is about and how it works.

  55. 55
    kairosfocus says:

    DS,

    The underlying notes for Spitzer seem to be here: http://magisgodwiki.org/index.php?title=Cosmology

    PDF fact sheet: http://www.magisreasonfaith.or.....tSheet.pdf

    In the talk I did not hear an argument regarding a beginning that was not a summary of physics [and he leads with a different entropy argument but it boils down to saying much the same — starlight drowned out], and that seems to be so in the notes too.

    Maybe I missed something, could you kindly outline what you are objecting to?

    Where is it in the video?

    KF

  56. 56
    daveS says:

    KF,
    It’s this text from the OP:

    Infinities within an aggregating succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for an aggregating succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable.

    That seems to me to be untrue if there actually does exist an infinite past.

  57. 57
    Mapou says:

    It is a lie that calculus (or anything else) uses infinity. If it did, it would be impossible to perform calculus computations on a finite machine (which all machines are). Calculus uses numbers and all numbers are finite and discrete.

  58. 58
    Aleta says:

    to Mapou: I can do calculus calculations with a piece of paper, using techniques developed using concepts of infinity.

    You have some strange beliefs, but they are not shared by the world of mathematicians, so I won’t bother with them.

    Although calling my statement that calculus uses infinity a lie, rather than a statement you disagree with, is a mistake of another kind.

  59. 59
    Mapou says:

    Aleta, a concept of infinity is not infinity. It is a lie that calculus uses infinity whether you agree or not. It is a very old lie that has retarded progress in science for centuries. Even brilliant men like Newton, Leibniz and Descartes believed in the lie. The universe is 100% discrete and finite.

    And I really don’t care what you or the world of mathematicians will not bother with. I’m an independent thinker, a rebel.

  60. 60
    Mapou says:

    Brent:

    Mapou,

    I agree with your sentiment at 15, but not with your stated application/conclusion in 5.

    If infinitude doesn’t exist in at least one sense, somewhere, or better put, some how, then nothing exists, period.

    If infinity exists, it does not exist in the physical universe, that’s for sure.

  61. 61
    Aleta says:

    Mapou, I have not been arguing that there is anything in the physical world that is either infinite in number, infinitely small, or infinitely small. I think I’ve made that clear.

    But it’s not a lie that “calculus uses infinity.” Math is an abstraction – a logical system – and within that system, infinity is a concept that is useful, and essential for the development of calculus.

    Furthermore, calculus, and math in general, can be applied to the real world even though there isn’t a perfect correspondence between it and the world. In this case, even though the infinitely small upon which calculus is based does not actually exist in the physical world, the components of the world are small enough that it is often easier – much easier – and accurate enough, to apply calculus than it is to treat those components with discrete math.

    An example is calculating compound interest. If you have $1000 in the bank at 10% interest for 10 years, it is easier to calculate the compound interest using the natural exponential function (which assumes instantaneous compounding) rather than the discrete exponential function: there is about a 37 cent difference in the two values, even though a day is a very large value in respect to infinitely small, and I could with some assurance do both calculations with sufficient accuracy using the scientific calculator on my computer.

    However, if you look other examples, such as heat loss which involves a huge number of molecules, a calculation using calculus will be vastly simpler (the discrete calculation would probably be impossible) and accurate beyond any practical limits.

    So the short summary is that even if infinity doesn’t actually exist in the real world, using math based on the concept of infinity does work well enough for a great many things.

  62. 62
    kairosfocus says:

    DS:

    I see, likely here http://magisgodwiki.org/index......st_Time.22

    To wit:

    Infinities within an aggregating succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for an aggregating succession [–> this implies a physical situation of finite sucessive steps] occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. [–> just made explicit] No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. [–> I would stress successive, finite] Now, if “infinity” is applied to an aggregating succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved or be actualized through an aggregating succession. Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to an aggregating succession. Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in an aggregating succession, “infinity” must be consider unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable). We are now ready to combine the two parts of our expression through our three common conceptual bases:

    “Infinite Past Time”

    “(The) unoccurrable (has) occurred.”
    “(The) unachievable (has been) achieved.”
    “(The) unactualizable (has been) actualized.”

    I believe no matter how fast was likely not intended to enfold the infinitesimal.

    Taken with that proviso (which is suggested by what I bolded), he is saying in effect what I have said.

    KF

  63. 63
    daveS says:

    KF,

    As I stated before, I’m not referring to anything infinitesimal here. Maybe I should have bolded only the “how long” part, and not “how fast”.

  64. 64
    kairosfocus says:

    PS: Notice how he goes on:

    Failures of human imagination may deceive one into thinking that the above analytical contradictions can be overcome, but further scrutiny reveals their inescapability. For example, it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved. However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved. This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.” It simply manifests a failure of our imagination. Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction.

    Another failure of our imagination arises out of thinking about relative progress in an historical succession. Our common sense might say that infinite past history is impossible because an infinity is innumerable, immeasurable, and unquantifiable, making the expression “an infinite number” an oxymoron. But then we get to thinking that infinite history seems plausible because each step relative to the other steps is quantifiable in its progression; each step is subject to relative numeration. Therefore, it seems like history can really achieve an infinite number of steps.

    However, as the above analysis reveals, this cannot be so because an infinity in a continuous succession must be unachievable or unactualizable as a whole (otherwise, it would be analytically indistinguishable from “finitude” in a continuous succession). Since, as has been said, past time must be achieved or actualized (otherwise it would be analytically indistinguishable from “present” and “future”), “infinite past time” must be an “achieved unachievable” or an “actualized unactualizable” (an intrinsic contradiction). Moreover, the expression “an infinite number” is also an intrinsic contradiction because “number” implies a definite quantity, whereas “infinity” implies innumerability (more than can ever be numbered). Therefore, infinite history and its characterization as “a completion of infinite time,” remains inescapably analytically contradictory.

    This intrinsic analytical contradiction reveals the problematic character of the very idea of “infinite past time.” It now remains for us to show the inapplicability of this problematic idea to our universe, and indeed, to any really possible changeable universe. This step will give ontological (“synthetic”) significance to the analytical contradiction by showing that the condition of the real world (i.e., our real universe, or any really possible changeable universe) will contradict (and therefore resist) the application of this problematic idea to it. The result will be that no real universe could have had infinite past time.

    Before we can proceed to this proof, we must first give an ontological explanation of real time[1], and then show that this real time must be intrinsic to any changeable universe, and then explain Hilbert’s distinction between actual and potential infinities so that it will be clear that “infinite past time” (as defined) must be an actual infinity which Hilbert shows to be inapplicable to any reality to which the axioms of finite mathematics can be applied. The ontological proof against an infinity of past time will follow from this.

  65. 65
    kairosfocus says:

    DS,

    no actual finite succession of steps in our world can be actually infinite. It is no more possible to physically count down from infinity as to count up to it.

    Abstract to a model ideal world without heat death or the like thermodynamic rundown as stars burn out and gradually fade away into cinders etc lurking and once finite duration steps are involved, we are back to the traversal of the transfinite in successive steps.

    Where we cannot reason properly, that we are here so we have achieved the infinite succession to get here.

    Take 0, 1, 2, 3 . . . A

    where A indicates order of Aleph null.

    Readily, we cannot traverse this.

    Simply multiply through by -1 and reverse the order:

    – A . . . -3, -2, -1, 0

    then concatenate with 1, 2, 3 . . . n, for now.

    Finite duration steps, we can readily go from 0, the singularity say to now.

    But all we did was to reverse sign, there is no more reason to imagine counting down from A-order to 0 in finite steps than counting up.

    However, when worldview commitments are in play, people will go p –> Q but I reject Q so I challenge P.

    I therefore leave it as a commitment implicit in certain views that we have actually successively traversed the infinite past to get here. How can you justify such a view, on the premise that every tub must stand on its own bottom.

    KF

    PS: I add, note onward discussion on Hilbert’s prohibition of type C infinities: http://magisgodwiki.org/index......Infinities

  66. 66
    daveS says:

    KF,

    The first bolded part is where he applies his alleged “analytical contradiction”, which is exactly what I am questioning. I’m not convinced that an infinite past creates an achieved unachievable.

    it will be clear that “infinite past time” (as defined) must be an actual infinity which Hilbert shows to be inapplicable to any reality to which the axioms of finite mathematics can be applied.

    Interesting. I’ll take a look at the Hilbert reference tomorrow if I get the time. Are you sure you want to go there, though? The trouble is apparently with C-infinities, which are described thusly:

    3) “C-infinity.” “Infinite” is sometimes used to signify “infinity actualized within an algorithmically finite structure.” Mathematicians such as Georg Cantor hypothesized a set with an actual infinite number of members (a Cantorian set) which would not be a set with an ever-increasing number of members or an algorithm which could generate a potential infinity of members. Examples might be an existing infinite number line, or an existing infinite spatial manifold, or the achievement of an infinite continuous succession of asymmetrical events (i.e., infinite past history).

    and:

    Thus, if C-infinities could really exist, there could be infinite space, infinity degrees Fahrenheit, infinite mass density, infinite physical force, and infinite past time.

    From your past writings, I take it you accept the existence of the set of real numbers, and so forth, which are of this type, right?

  67. 67
    kairosfocus says:

    DS, The reals are a mental not physical reality: contemplated eternally by the mind of root reality and accessible to our own. They are foundational to a world being instantiated, e.g. take 2 and twoness which is inextricably bound up in distinct identity and in thought. I should add, I have never advocated infinite past time (causal succession of existential, influencing instants linked to successive changes such as breakfast to we are hungry again, the meal having been digested); eternity is not time. That root, necessary being reality always was does not entail that our space-time domain is of infinite past duration, indeed the evidence is it has a finitely remote beginning and is contingent pointing to a begin-ner. KF

    PS: To get a simplistic analogy, contemplate how the north pole is north of all longitudes at all latitudes all at once so that the 24 hours are instantly present there.

  68. 68
    daveS says:

    KF,

    DS,

    no actual finite succession of steps in our world can be actually infinite.

    Agreed. But I’m not talking about a finite succession of steps.

    It is no more possible to physically count down from infinity as to count up to it.

    Of course an infinite past is not so popular now, but the eternal steady-state universe model was considered viable at one time. What would its age in seconds be?

    Simply multiply through by -1 and reverse the order:

    – A . . . -3, -2, -1, 0

    What is minus aleph-null? I don’t know anything about multiplying infinite cardinals by negative numbers.

  69. 69
    kairosfocus says:

    DS,

    We are talking about past infinite causal successions, which would entail finite discrete steps.

    I am aware a Steady State model was contemplated; that was in the context of imagining the observable cosmos broadly conceived to be the necessary being root of reality. For instance a steady influx of matter was envisioned. When I objected on energy conservation, I was told the rate would not be detectable on lab or solar system scale. That is, the mod is empirically substantially equivalent.

    Next, if you note what I did, I first set out a counting up, indicating by ellipsis that this extended beyond limits would be of ORDER — scale — Aleph null, i.e. a countable transfinite. Timed succession was implied.

    Then the reversal to count down temporally from a past beyond the singularity was represented. That is, a procedure of allegedly arriving at present n through 0 from an infinite past was symbolised. In effect think of a discrete number line extended to such an order, reflected in a plane mirror set normal to the line at 0.

    To represent that, use – signs and ignore the well known mirror reversal of shapes so a 3 would look like a curly E.

    Now, the challenge still obtains. If you will, contemplate the reversed line and step back, -1, -2, . . . you will never complete the traversal. By direct comparison, it is unattainable to step down from that order in finite successive steps either.

    If you do hold that to have been completed, kindly show us how so.

    KF

  70. 70
    daveS says:

    KF @67:

    I withdraw the second half of my post #66. I’ll take a look at the Hilbert reference when I can.

  71. 71
    GaryGaulin says:

    From:
    https://en.wikipedia.org/wiki/Cyclic_model

    A cyclic model (or oscillating model) is any of several cosmological models in which the universe follows infinite, or indefinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 theorized a universe following an eternal series of oscillations, each beginning with a big bang and ending with a big crunch; in the interim, the universe would expand for a period of time before the gravitational attraction of matter causes it to collapse back in and undergo a bounce.

  72. 72
    daveS says:

    KF,

    You keep talking about “finite steps”, which I find puzzling. Of course I’ve been talking about executing infinitely many steps all along.

    Could you rephrase the task you’re asking me to complete in more direct terms?

    If you’re asking me to count through all the natural numbers in increasing order, then count through them in reverse (in order), then I won’t be able to do that. Not in a countable number of steps anyway.

  73. 73
    mike1962 says:

    The future differs from the past:

    The past exists and is fixed

    The future does not exist and is not fixed

    See the difference?

  74. 74
    kairosfocus says:

    DS: Please, go find a flight of stairs and walk up then down. That should suffice to show finite, discrete, successive steps in sequence. KF

  75. 75
    Aleta says:

    I have been trying to understand Dave’s point, but I can’t, and I’ve forgotten/never really understood, what the point of contention is, but I’m interested.

    If now is zero, and you start walking backward into the past (using “walking” figuratively), you will never get to “negative infinity”, because infinity is not a “place to get to” but rather a shorthand for the concept that you can always keep walking: no matter where you are, you aren’t ever at infinity.

    This seems obvious, so I don’t know what other point Dave is trying to make. Want to try again, Dave?

  76. 76
    daveS says:

    KF,

    First, strike the last sentence of my post #72. I cannot do that task in any amount of steps.

    Let me address something which I should have noticed somewhere. There is no “counting down to 0 from aleph-null” in my example, which you seem to imply in several posts above.

    Here is what I have in mind:

    The most recent click occurs at t-zero

    The tick before that at t-minus-one

    Before that, at t-minus-two.

    And so forth. t-minus-3.15 x 10^7 occurred about 1 year ago.

    Every natural number is thus associated with a clock tick.

    There is no t-minus-aleph-null.

    Since I’m assuming an infinite past, all these times are well-defined.

    The clock never “started” ticking. It is eternal (although it could stop ticking now without changing my argument).

    What is the cardinality of the set of clicks before the present time? Aleph-null.

  77. 77
    daveS says:

    Aleta,

    I don’t think I really disagree with anything you said in post #75. Spitzer argues that a universe in which you can travel back arbitrarily far in time is impossible.

    I disagree, and don’t think his method of devising an “analytical contradiction” works, and this clock example was what I came up with to show why I think that. I’m having a hard time explaining the actual workings of this example, however.

  78. 78
    Aleta says:

    That all seems clear. Would it be reasonable to say that your point is that there is no starting point – no first moment of time, because the past is infinite? But that seems obvious also.

    Aleph-null is just a way of saying that a set of numbers can be mapped 1-1 with natural numbers, and obviously the negative integers can be so mapped, and thus have cardinality aleph null.

    And, as you say, since we are assuming the past is infinite, every negative integer has a clock tick – a one second moment of time, associated with it.

    Are you saying any more than these things?

  79. 79
    Aleta says:

    Dave, I wrote 78 before I read 77, so now I understand more.

    You write, “Aleta, “Spitzer argues that a universe in which you can travel back arbitrarily far in time is impossible.”

    There is a vast difference between discussing the pure mathematics of this, which is what I have been doing, and making claims about a universe and what may or not be possible in it. I don’t think we know enough, and ever will, about either time or universes to know whether our understanding about how our mathematical understandings of infinity as applied to our universe could be extrapolated to explain or describe possible other universes (if such exist), or realms that would not even be called universes.

    But those are much bigger, different questions. Again, I have just been discussing the math.

  80. 80
    daveS says:

    Aleta,

    I am also saying that this scenario is not incoherent (or at least that KF’s objections don’t take it down). Perhaps it also argues against Spitzer’s point. That’s about it.

    Re #79, which I just read: I agree. I am mainly interested in Spitzer’s particular argument here. I’m actually skeptical about an infinitely old universe overall, but who knows.

  81. 81
    Aleta says:

    Got it, Dave – thanks for the discussion.

  82. 82
    Brent says:

    Mapou @60,

    OK. I obviously didn’t understand your comment at 5. I really thought you meant to say that even God had a beginning.

    Otherwise, everyone would have been much better off taking your advice at 15 and just leaving the idea of an actual infinity alone.

  83. 83
    Brent says:

    daveS @80,

    . . . but who knows.

    From a very rough recollection of something Chesterton said, “The purpose of opening your mouth is to eventually close it on something solid.”

    If you haven’t closed your “mouth” on the solidity of the impossibility of an infinitely old universe, I hope you don’t eat in the same manner, or there will be no more daveS.

  84. 84
    daveS says:

    Brent,

    Well, I’m a layperson, so there’s a limit to how certain I can be on this matter.

  85. 85
    kairosfocus says:

    DS,

    no clock can credibly have ticked for a countably infinite number of times to date. Thermodynamics or cosmology alone in the context that an oscillator gates energy flow from a source in such a way as to have regular cycles, is enough for that. That was already settled above.

    Notice, too: once I realised fine distinctions were going to be important to you, I spoke also of starting at some point on a number line of the ORDER of Aleph null.

    If you will (and as Aleph null will be a point of contention), consider A to be a natural number so large that its reciprocal is an infinitesimal; i.e. it is a hyper-real with fractional part zero, and the resulting 1/A –> 0 but is not quite there. I assume you are willing to accept real numbers that can be specified as whole plus fraction with an extension of the place value system for representing the power series in the base to simply represent.

    Notice, I am in no wise specifying Aleph null as though it were a particular individual value on the number line. I give it as an order of scale. (One where it is countable in principle as the destination of a stepwise process that lists the naturals but is beyond the finite.)

    I trust that understanding a negative number line to be a vector reflection of the natural numbers should also be clear.

    Just as complex numbers will be a 90 degree anti-clockwise rotation using the i-operator [which, repeated twice effects just the reflection we want], and that the ijk system arises similarly.

    The reals, again, come out through defining whole and fractional parts and from that going to a continuum on the “any two neighbouring values can define another real value in between” principle.

    Coming back, the point is that it is obvious that at each step from 1, 2, . . . n, one has only gone through a finite process and so to continue thereafter to the order A, is going to constantly still lie unattainably far ahead — the ultimate chase after the end of the rainbow. The distance from 0 to A cannot be traversed by any finite discrete step by successive step process — an algorithm if you please.

    But by the mirror in 0 used, the span to be traversed in the other direction is the same.

    There is no good reason to imagine that one can start at some A and proceed by such a process to 0, on the strength of the first result.

    Setting aside Spitzer’s complex terms, that is effectively his point and beyond him, Hilbert’s via the famous hotel. Though, Hilbert showed that one would also destabilise the finite numbers by careless extrapolation — the point of the hotel. ( Cf here: http://world.mathigon.org/Infinity and: https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel Notice one can in principle introduce the step of sending guests in room n to room 2n, and putting in new guests into 2n +1, where n starts at 0 and repeat endlessly without ever failing to fit in a new set of guests. Also observe, this operation is an all at one go, not a stepwise deliverance of the change in succession.)

    I suspect Spitzer is correct to pin his attention on the way we can imagine but we are here so if we “must” have gone through an infinite past process to get here we can say at any point that is already accomplished. But that simply begs the question of finite discrete steps traversing a span that is countably infinite.

    And with our doubts on the table — we can all play the Cartesian trick of if I doubt or can construct a doubt I can dismiss, can you show us good reason to believe the supertask can be completed stepwise?

    As my old gramps used to say, every tub must stand on its own bottom.

    Remember, after finite time per astrophysics, the stars will burn out and onward in finite time they will radiate out heat to a point where the observed cosmos will be in heat death; there would be no effective heat gradient to drive a clock process. That it is not now in such a condition entails finitely remote past for the only actually observed physical cosmos.

    Darwin et al do not have an infinite past to play with in this cosmos, and it would be interesting for us to hear of others.

    KF

  86. 86
  87. 87
    EugeneS says:

    “New Proofs for the Existence of God” sounds dodgy to me. This is what I don’t like about it: IMO there cannot be any proofs for the existence of God. To believe or not is not a rational choice but primarily a moral choice of free will (the heart, not so much the mind). Otherwise, authors like this one would have done it once and for all. But it does not happen this way! There are considerations of all sorts, rational ones included. But no set of rational considerations necessarily entails faith. This is why, in fact, God praises human faith: it requires lots from a human being to believe. The Biblical concept of faith is so much different from the cold mental agreement with the existence of a God.

  88. 88
    daveS says:

    KF,

    DS,

    no clock can credibly have ticked for a countably infinite number of times to date. Thermodynamics or cosmology alone in the context that an oscillator gates energy flow from a source in such a way as to have regular cycles, is enough for that. That was already settled above.

    Gary Gaulin raised the issue of oscillating universes above, some of which would be consistent with an infinite past. We can replace the clock ticks with big bang/crunch cycles.

    For a theistic approach, the Creator could simply “mentally” note the passing of each second.

    Notice, too: once I realised fine distinctions were going to be important to you, I spoke also of starting at some point on a number line of the ORDER of Aleph null.

    If you will (and as Aleph null will be a point of contention), consider A to be a natural number so large that its reciprocal is an infinitesimal; i.e. it is a hyper-real with fractional part zero, and the resulting 1/A –> 0 but is not quite there. I assume you are willing to accept real numbers that can be specified as whole plus fraction with an extension of the place value system for representing the power series in the base to simply represent.

    Eh? There clearly is no such natural number A. Please, let’s just use the real aleph-null. And there is absolutely no need to bring in the hyperreals to this trivial example.

    I trust that understanding a negative number line to be a vector reflection of the natural numbers should also be clear.

    Just as complex numbers will be a 90 degree anti-clockwise rotation using the i-operator [which, repeated twice effects just the reflection we want], and that the ijk system arises similarly.

    Ok.

    The reals, again, come out through defining whole and fractional parts and from that going to a continuum on the “any two neighbouring values can define another real value in between” principle.

    Well… that might get you only to the rationals. But nevermind, we both know how to work with real numbers.

    Coming back, the point is that it is obvious that at each step from 1, 2, . . . n, one has only gone through a finite process and so to continue thereafter to the order A, is going to constantly still lie unattainably far ahead — the ultimate chase after the end of the rainbow. The distance from 0 to A cannot be traversed by any finite discrete step by successive step process — an algorithm if you please.

    Again this “order A” stuff. If you can rephrase this using the standard definitions, that would be great.

    ***

    I’ll have to pause there so I can get some work done. I really have no idea why this is so controversial.

  89. 89
    daveS says:

    KF,

    I’ll look at the rest of your post as I get time.

    Setting aside Spitzer’s complex terms, that is effectively his point and beyond him, Hilbert’s via the famous hotel. Though, Hilbert showed that one would also destabilise the finite numbers by careless extrapolation — the point of the hotel. ( Cf here: http://world.mathigon.org/Infinity and: https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel Notice one can in principle introduce the step of sending guests in room n to room 2n, and putting in new guests into 2n +1, where n starts at 0 and repeat endlessly without ever failing to fit in a new set of guests. Also observe, this operation is an all at one go, not a stepwise deliverance of the change in succession.)

    How does this “destabilize” the finite numbers (in standard terms, please)? Hilbert’s Hotel is a nice puzzle, but I don’t understand what you mean by that.

  90. 90
    daveS says:

    KF,

    Remember, after finite time per astrophysics, the stars will burn out and onward in finite time they will radiate out heat to a point where the observed cosmos will be in heat death; there would be no effective heat gradient to drive a clock process. That it is not now in such a condition entails finitely remote past for the only actually observed physical cosmos.

    As I stated above, physics considerations make an infinite past less likely, to me anyway. But you seem to think there is an elementary mathematical problem with this eternal clock example, which is what I’m addressing (and, ultimately, the Spitzer analytical contradiction argument).

    That’s about all I have to say on your post #85.

  91. 91
    kairosfocus says:

    DS, the oscillating universe does not get rid of finite past time as the entropy buildup locks out an infinite past. That is besides the challenge of getting a viable bounding model and the further issue that the density we observe points to unlimited expansion not recollapse, which as was highlighted far above, is a fine tuned point. KF

  92. 92
    kairosfocus says:

    DS, whole numbers are a subset of the real numbers. Perhaps, you are unaware of the hyper-reals, used in non standard analysis to give strict meaning to infinitesimals. I am simply specifying A to be of order such that its reciprocal 1/A will be infinitesimal, i.e. A is a countable transfinite, and will be of relevant order. KF

    PS: Go to Hilbert’s debates on the infinities.

  93. 93
    kairosfocus says:

    Dr Selensky, I am fully aware that worldviews are matters of comparative difficulties leading to hopefully reasonable faith. Further to this, was it Plantinga who spoke of using the implication structure to reduce knowledge to ignorance: P => Q but someone rejects Q having initially accepted the set of premises P. On rejecting Q s/he may then proceed to now suddenly no longer know P, i.e. we see a resort to modus tollens. So, there are no proofs beyond doubt of consequence, but the price paid in rejecting relevant aspects of P may be highly revealing. That is where the worldviews challenge really lies. Multiply by that we are inherently finite, fallible and morally struggling, and the challenge of reasonable faith and our responsibilities concerning what we accept as true and trustworthy come out. KF

  94. 94
    daveS says:

    KF,

    I’m aware of both the whole numbers and the hyperreals.

    In the hyperreals, no natural number has an infinitesimal reciprocal, which is what your construction requires.

    The infinitesimals are reciprocals of infinite numbers, not natural numbers.

    If you have an argument to make, please do so using standard constructions, otherwise we are going to get lost in the weeds.

  95. 95
    kairosfocus says:

    DS, take a whole number A of sufficient size that 1/A –> 0. That is big enough to make the point. Count up in steps you face a supertask, count down from it, the same. KF

  96. 96
    daveS says:

    KF,

    DS, take a whole number A of sufficient size that 1/A –> 0. KF

    Please define the meaning of 1/A -> 0 using standard terms.

    Then, can you give a specific example of such a whole number A?

  97. 97
    kairosfocus says:

    DS, this is now dancing around. It is you who need to show us cause to accept that for some whole number A [no fractional part] so large that 1/A –> 0, we can successfully traverse the span from A to 0 in finite discrete steps. A does not have to be the first, just . . . A . . . 2, 1, 0. KF

  98. 98
    daveS says:

    KF,

    I’m just asking you to demonstrate the task you assigned me:

    “DS, take a whole number A of sufficient size that 1/A –> 0. ”

    Can you show me how to do this?

    Edit from #97:

    It is you who need to show us cause to accept that for some whole number A [no fractional part] so large that 1/A –> 0, we can successfully traverse the span from A to 0 in finite discrete steps. KF

    Maybe if you define 1/A -> 0, I can do that. That is not standard notation.

  99. 99
    daveS says:

    KF,

    To address your question in #97, suppose A = 10^150. Is that sufficiently large? We can then traverse the sequence A, …, 2, 1, 0, in 10^150 steps. Which is finite.

  100. 100
    kairosfocus says:

    DS, we can look at it from the point that m –> 0, and 1/m = A, where it so happens that representing A as a decimal W.F for convenience F = 0, i.e. A is whole; e.g 1/0.001 = 1,000 and then drop m hard towards 0. Then study the descent in discrete steps that for argument has been going on long before A:

    . . . A . . . 2, 1, 0.

    The issue is to traverse A to 0, and it implies traversal of a number of steps that becomes beyond traverse as m approaches 0 and A increases to the transfinite.

    It matters not if aleph sub any arbitrarily high index or whatever lies left of A, A is of order that the cardinality of the number of steps to 0 is at least of the order aleph null.

    The steps onwards from A are a traversal that cannot be completed, Spitzers point.

    Now, as one who has advocated a down-count to 0 and an up-count to now, n, kindly explain to us how to do so:

    . . . A . . . 2, 1, 0, 1*, 2*, . . . n*

    * up-counting, with the singularity a reasonable zero point.

    KF

  101. 101
    daveS says:

    KF,

    Ok, so now we’re talking about an increasing sequence of A’s, not a fixed value. That’s not going to help your case.

    For example, the m’s could be defined by the sequence a_k = 1/k for all positive integers k. Then the associated A_k equals k.

    For each A_k, it takes k steps (finitely many) to traverse the sequence A_k, …, 2, 1, 0.

    The number of steps never is “of order aleph-null” (I’m having to guess what that means, admittedly).

    Disagree? Give me a value of k such that “A [A_k?] is of order that the cardinality of the number of steps to 0 is at least of the order aleph null.”

    If you decline to do so, I will step back for a while and give the onlookers a break from this bizarre thread. I’m astonished that we’re even having this discussion.

  102. 102
    kairosfocus says:

    DS, nope. Kindly note m –> 0, such that A will have the property that in effect we step, with a cut:

    A . . . 2, 1, 0, . . .

    And we make correspondence:

    A, (A-1), . . .

    0, 1, . . .

    As, the origin of a count is effectively arbitrary.

    Where, the span is such that the span from A to 0 will be transfinite, m being infinitesimally small.

    Which was said from the beginning.

    And we are no closer to you showing us how to count down to 0 having already completed transfinitely many steps.

    KF

  103. 103
    Mung says:

    DS, hint. You can’t do it. So don’t pretend as if you can.

  104. 104
    daveS says:

    Mung,

    DS, hint. You can’t do it. So don’t pretend as if you can.

    If this is humor, then I appreciate it! If not, can you tell me what KF is asking me to do in #102?

  105. 105
    kairosfocus says:

    DS:

    This:

    we are no closer to you showing us how to count down to 0 having already completed transfinitely many steps.

    In other words, counting down in discrete, finite, finite duration steps to the present from a transfinitely remote past — not merely asserting that at a given time such an order of prior steps was already completed.

    KF

  106. 106
    daveS says:

    KF,

    I never said I was able to do that. I am referring to a ticking clock, not a counting clock. In my post #76, I described how to associate a natural number with each tick for the purpose of counting the elements of the set of ticks, after an infinite number of ticks had already occurred.

  107. 107
    kairosfocus says:

    DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out. And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown. Grant you whatever before A, just now A down to 2, 1, 0. Where A = 1/m, m –> 0 i.e. is infinitesimal. KF

  108. 108
    daveS says:

    KF,

    DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out.

    There are oscillating universe models which are consistent with an infinite past, as I stated. Replace each tick with a big bang/crunch cycle.

    And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown.

    No. I am saying that Spitzer assumes that an infinite number of ticks cannot in principle occur as part of his argument. The burden is on him to prove that.

    A down to 2, 1, 0. Where A = 1/m, m –> 0 i.e. is infinitesimal. KF

    There are no infinitesimals in sight in your statement above. All the numbers are real and finite.

    Now that we’ve clarified the ticking vs. counting issue, do you still have mathematical (not physical) objections to the eternal ticking clock example? If so, I challenge you to take some time and write them out precisely (avoiding such concepts as “of order aleph-null” and abuse of the hyperreals). Maybe even post it as an OP.

  109. 109
    Mung says:

    I can’t make any sense of a timeless ticking clock. Is it like a ticking time bomb that can never go off?

  110. 110
    Mapou says:

    There is only change. Time is an abstract concept created by consciousness. The same can be said of space. Again, if space exists, where is it? If time exists, where is it?

    The entire 3D vista you think you see in front of you is actually an interpretation by your consciousness of a bunch of neurons firing in your visual cortex.

    The illusion of time and space is irrefutable proof that the human brain is inhabited by a spirit or conscious entity. Yep, the ghost in the machine and all that beautiful stuff.

  111. 111
    kairosfocus says:

    DS, the oscillating universe models have fallen to entropy rise challenges. Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points. KF

  112. 112
    kairosfocus says:

    DS,

    Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges.

    Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points.

    Beyond, Spitzer summarised arguments that the transfinite cannot be traversed in separate finite steps. He did not merely assume.

    Above, the exchange we have had is about precisely that.

    You have admitted that you are unable to show such a traverse, and are now adverting to oscillating models that have failed.

    I have taken time step by step to put the challenge in terms of completing the arrival at the present; in the face of many objections on your part.

    You have spoken of how at any specific point, already an infinite number of steps is complete. I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by in the end using some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes. The focal task then becomes traversing onward from A to 0, envisioned for the moment as the singularity, from which onward we go to now, n. Where as you objected to negatives [though how that was used was explained] I use asterisks to show the finite up-count since the singularity. Of course the lead ellipsis indicates that A is not the beginning of the steps we may identify and list as a succession, it is preceded by an arbitrarily and per your suggestion for argument even possibly transfinitely large and unending set of previous values:

    . . . A, . . . 2, 1, 0, 1*, 2*, . . . n*

    Such, of course was already outlined by way of making the way clear after successive objections.

    The start point for a count is arbitrary, so let us put the start at A and put it into correspondence with the naturals, i.e. this is in principle countable . . . as is implicit in stepwise succession as would happen with clock ticks, one providing the basis for the next as energy is gated from a source and as positive, precisely lagged feedback is applied:

    A, (A less1), . . .
    0, 1, . . .

    Given that the traverse from A to 0 is transfinite, the task here is comparable to counting up from 0 to a transfinite in finite successive steps, which is a supertask that is unattainable. (And I have taken the step of identifying A as a specific number a reciprocal of a number close to 0 [as the hyper reals approach takes to identify what an infinitesimal is, only in reverse], to avoid all sorts of issues on what does subtraction mean with a transfinite, Such will of course be of at least the order — scale if you will — of aleph-null from the origin at 0. I take it that we can accept the reasonableness of infinitesimals close to but not quite attaining to zero; such being foundational to a way to understand the Calculus.)

    For, once we count 0, 1, . . . n, we may always go on to n +1, etc in further steps, always being finite.

    The evidence is that traversing an infinite succession of finite discrete steps is a unattainable supertask, precisely as Spitzer sums up.

    The worldviews significance is this, that a contingent succession of being, with each being b_i subject to on/off enabling causal factors it must have in place for it to begin or continue to exist, must be a part of a chain of successive and in context finite discrete causes. This can be in principle enumerated and compared to the step-wise succession, e.g. of clock ticks on a clock.

    We then see that the traversal of an infinite succession of such beings is to be doubted, on grounds of needing to arrive at the singularity then onward up to us. From the singularity (for reference to current cosmology, actually any reasonable zero point would do equally) to us is explicable on a succession, but the problem is to arrive at 0.

    This may then b multiplied by the challenge that non-being, the genuine nothing, can have no causal powers. There is not space, time-point, energy, mass, arrangement, mind etc “there.” So were there ever utter nothing, such would forever obtain.

    We face then, the need for a necessary root of being to account for a world that now is.

    Necessary, so connected to the framework of a world that no world can be absent such. As an instance, 2 must exist in a world W where distinct identity, say A, exists: W = {A|~A}.

    A world now is, so something always was.

    Following, frankly, the line in the classic work, Rom 1:19 ff (which I find to be enormously suggestive of a frame for a reasonable faith worldview), this world is a world in which we find ourselves as self aware, responsibly free and rational individuals; contingent beings subject to moral government and intuitively sensing the need to respond appropriately to evident truth about ourselves and our circumstances in a going concern world.

    It is appropriate in such a context to ask, what sort of serious candidates — flying spaghetti monsters etc are patently contingent imaginary parodies that do not meet the criteria for necessary being and need not apply — can we see in making a worldview level choice?

    After centuries of debate, there is one serious candidate, by utter contrast with non-serious parodies, and by contrast with the challenge of traversing the transfinite etc.

    The bill to be filled looks extraordinarily like:

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

    This is Candidate A.

    Candidate B is: ___________ ?

    KF

  113. 113
    daveS says:

    KF,

    DS,

    Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges.

    Not all of them.

    As I also previously stated in post #88, the Creator could simply note the passing of each second. And I see no reason why the Creator could not arrange for a universe with an infinite past, intervening in the laws of physics where necessary.

    I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by in the end using some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes.

    There are no whole numbers which lie an infinite distance from 0.

    See the definition.

    At this point, you are conjuring up the logical equivalent of a square circle, which is frequently used as an example of contradiction in terms.

    You need to remove this fatal defect from your argument before I or anyone else take it seriously.

  114. 114
    daveS says:

    A qualification: As I stated in the new thread, by the set of whole numbers, I take it you mean {0, 1, 2, …}, as defined on the Wolfram site.

  115. 115
    Mung says:

    Meanwhile, a timeless ticking clock that never began ticking and will never end ticking is not a problem.

    Seems to me the ticks constitute an ordered sequence, each tick preceded by the previous one and followed by the next.

    Is there an isomorphism there to the natural numbers?

  116. 116
    daveS says:

    Mung,

    Is there an isomorphism there to the natural numbers?

    Yes, both are countably infinite, so they are isomorphic as sets. Order is not preserved, however.

  117. 117
    kairosfocus says:

    DS, I have continued in the new thread. I just note here that in fact there is a definition of a hyperinteger that can be seen as a whole but hyper-real number, i.e. it is arguable that my terminology is simplified but not inherently unreasonable, and we are still looking at your showing how to descend from an infinite past in discrete finite steps without running into the problem of counting down across a transfinite range to arrive at 0; my symbolising above is meant to find a way to focus this issue, and the issue exists even if my effort to put it in symbols fails. Cf: http://www.uncommondescent.com.....ent-596007 The context is that there is an extension of the Reals, cf. Keisler. Beyond, I see no reason to take an infinite oscillating physical cosmos tied to the one we observe as a serious alternative. KF

  118. 118
    daveS says:

    KF,

    Thanks, I’ll take it to the new thread.

  119. 119
    Laszlo says:

    I’m amazed at the quantity, length, and intensity of the responses to this post. Clearly infinity arouses passion like few other topics. I’m not going to add anything to the discussion but do wish to thank kairosfocus for his thoughtful and technical analyses.

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