# Fun with the hyperreal numbers (and with the idea of an infinite actual past)

The hyperreals are an extension of the real number line that brings to bear a reciprocal relationship between the very large and the very small. By so introducing extensions to the real number continuum, it forms a base for an infinitesimals approach to the calculus and makes sense of a lot of the tricks used by early pioneers of Calculus from Leibniz and Newton to Euler and beyond. (Though, it is clear in retrospect that they missed a lot of the pathologies that are now part of the far more cautious approaches of today.)

And yes, here is a case where Wikipedia does some good (likely, in a context where there are few basement trolls capable of making a mess):

Let’s zoom in on the graphic, which illustrates the “hyperreal microscope” of *R:

Let us note the definitional relationship between the infinitesimal and the hyper-large:

1/ε  =  ω/1

Where also, a common principle used is that ε is so small that ε^2 = 0. Where, to see what that is suggesting consider that (1/10)^2 = 1/100, and (1/10^50)^2 = (1/10^100), i.e. squaring drastically reduces the scale of a very small number.

This is all quite interesting, and has been used to rehabilitate some of Euler’s work, e.g. here.

(This is also quite relevant to some of the “Math tricks” used by Physicists and Engineers. The reference to the hyperreals may be a way to rehabilitate some seemingly dubious tricks.)

The principle that ω is a number greater than any finite sum 1 + 1 + 1 + . . . + 1 implies that it is of order type at least comparable to the first transfinite ordinal. The inclusion of the further numbers such as ω/2 indicates a reference to the surreals, and something like root-7 times ω indicates an onward transfinite continuum. I do not at this juncture specifically identify this ω with the familiar first transfinite. (Perhaps someone cares to clarify?)

So, we may at least highlight the surreals, where the vertical bars indicate continua — note the place for “infinitesimals”:

All of this is interesting in itself, as numbers are the tools of ever so much analysis and we here enrich appreciation of our favourite tool-box. (I confess, this weekend was more spent with dynamic-stochastic general equilibria, linked rational expectations, questions on modern theories of growth and human capital, etc. All, with Garrison’s Austrian approach to macroeconomics lurking, and blending in issues of saturation and stagnation at points along the PPF as well as what happens to shocked economies with low investor confidence . . . as in, 20+ years on from devastating volcanic eruptions. This stuff was the oh, what about light exploration as a relief.)

But all of this converges on something which has come up for strong, sustained exchanges several times here at UD. Namely, the suggestion of an actual infinite causal-temporal past of the [wider?] cosmos. For, if ω is such that no finite succession from 0, 1, 2 via 1 + 1 + 1 . . . + 1 can reach it, then counting down — notice, the ladder-like succession of steps (and how the surreals extend this to construct continua and to go into transfinite ordinals) — from it in finite succession [or by symmetry counting algebraically upwards from – ω] may reach to something like ω/2 [or – ω/2] but it will be futile for getting to a finite reach of a zero-point.

In short, we can see here a reason to hold that there was no actually transfinite causal-temporal succession of states that have managed to reach the present. Nor will it do to posit that at any given past time p that can finitely succeed to now, infinity past was already traversed. That begs the question of HOW.

This also surfaces a logic of being point.

Namely, that non-being has no causal powers, so that if ever there were utter nothing, such would have forever obtained. Thus also, circular causation is forbidden as this would imply that the not yet existent acted as a cause. Thus, we either have an infinite succession of contingent beings as the world-root or else there is a necessary being at world root. That is, an entity such that it is utterly unlike a fire, which has several required external, enabling “on/off” causal factors for it to begin or be sustained:

Where this goes, is that a necessary being is framework for any possible world to exist. So, in any world, it would be. It neither began nor can it cease from being. For instance, in reality we may consider a world W, which must have a distinct identity. So, too, we may consider some distinct thing in W, A that contributes to its identity. Then, we may look at W = {A|~A}. This shows two unities, i.e. two-ness. No world is possible without two-ness, and beyond, the panoply of numbers.

This is what gives bite to Berlinsky’s remark that was just raised here at UD:

>>There is no argument against religion that is not also an argument against mathematics . . . .

Mathematicians are capable of grasping a world of objects that lies beyond space and time ….

… Come again …

DB: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.

… And this is something that you, a secular Jew, believe? …

DB: What a question! . . .  I have no religious convictions and no religious beliefs. What I do believe is that theology is no more an impossible achievement than mathematics. The same rational standards apply. Does the system make sense; does it explain something? Are there deep principles at work. Is it productive? >>

So, now, we see from the hyperreals augmented by ideas of causal-temporal succession, that it is hard to defend the notion of a transfinite actual past of contingent beings leading up to now. This points to there being an actual beginning of the world and that this traces to a finitely remote necessary being world root.

That’s enough for a UD Sunday reflection! END

PS: As it is being claimed or implied that no serious thinker thinks like that, I add a clip, just for record:

## 98 Replies to “Fun with the hyperreal numbers (and with the idea of an infinite actual past)”

1. 1
kairosfocus says:

Sunday reflections: Fun with the hyperreal numbers — implications for the origin of the world in a finitely remote necessary being world-root and a side-light on Berlinsky’s point that “There is no argument against religion that is not also an argument against mathematics.” KF

2. 2
daveS says:

KF,

For me, one of the most interesting things about ‘the’ hyperreal numbers is that they have the same cardinality as the reals.

For, if ω is such that no finite succession from 0, 1, 2 via 1 + 1 + 1 . . . + 1 can reach it, then counting down — notice, the ladder-like succession of steps (and how the surreals extend this to construct continua and to go into transfinite ordinals) — from it in finite succession [or by symmetry counting algebraically upwards from – ?] may reach to something like ω/2 [or – ω/2] but it will be futile for getting to a finite reach of a zero-point.

Certainly this is true—you cannot count up from 0 to ω in finitely many steps, nor can you count down from ω to 0 in finitely many steps.

In other words, if you start at 0 (ω) and count up (down), you will not reach ω (0) at any point in the future.

I believe that if Ben Waters, William Lane Craig, or David Snoke were to enter the conversation, the first thing they would ask is “What is this ω doing in this discussion? There is no such time coordinate in our conception of infinite past“.*

Of course I know what your response will be (and my response to your response, etc.)

Nor will it do to posit that at any given past time p that can finitely succeed to now, infinity past was already traversed. That begs the question of HOW.

It obviously leads to the question of ‘how’, but does not beg any questions.

*Although some (including Quentin Smith, IIRC) have considered various models of time with exotic topologies.

3. 3
kairosfocus says:

DS, indeed any finite segment of the line has continuity cardinality and in the last discussion I noted how the counting numbers have duals per 1/n as a dust in [0,1], mostly near 0. Yes, the idea of a continuum begins to look stranger and stranger the more closely you look at it — an infinitely dense set of points constituting a line such that a cut at a given point entails no gap looking on either side. Going on, if there were an infinite actual past, then that means that every point along that past — I usually count in finite stages had to have once been now then was succeeded, down to now. That is the reason I have consistently rejected the proposal that at any p as described, the past infinite traverse was already done. And yes, that proposal begs the question at stake. I also find the notion of a transfinite actual past with only finitely remote past moments or stages, questionable. Going back to the math side, I find the hyperreal line interesting in and of itself. KF

4. 4
daveS says:

KF,

It probably isn’t worthwhile debating the meaning of “beg the question” much further, but I will clip this from wikipedia:

To beg the question is to assume the truth of the conclusion of an argument in the premises in order for the conclusion to follow.

The proposal in question is not an argument, so it’s literally impossible for it to ‘beg the question’.

Continuing:

Many modern English speakers use beg the question to mean “bear the question”, “suggest the question,” “raise the question”, “invite the question”, “evade the question”, or even “ignore the question”, and follow that phrase with the question, for example: “I weigh 120 kg and have severely clogged arteries, which begs the question: why have I not started exercising?” In philosophical, logical, grammatical, and legal contexts, some commenters believe that such usage is mistaken, or at best, unclear.

Back to the post, and the substantial matter:

I also find the notion of a transfinite actual past with only finitely remote past moments or stages, questionable.

That’s fine that you find it questionable, but that is unquestionably what people such as WLC and David Snoke are talking about, so I suggest we address that issue and set the ωs aside.

5. 5
johnnyb says:

“That’s fine that you find it questionable, but that is unquestionably what people such as WLC and David Snoke are talking about”

I was under the impression that WLC was against the infinite past, and used a near-identical argument as KF to argue against it.

6. 6
daveS says:

johnnyb,

He does indeed argue against an infinite past, but he does not assume that an infinite past entails the existence of these “ω-points” infinitely remote from the present.

7. 7
johnnyb says:

Interestingly, the transfinite numbers actually *remove* a lot of the paradoxes associated with actualized infinities, including those of the remote past (though not all of them, such as the one related here). Prior to Cantor, the concepts of ordinality and cardinality were combined, so that every omega (w) wound up being the same, leading to contradictions.

By distinguishing between ordinal and cardinal numbers, we can have *different* infinities within the same cardinality. However, the ordinals are essentially arrangement-dependent. The way I like to think about it is to think about a darts target, but instead of a circle, they are infinitely stretched bands. The center is the thinnest, and the other ones are wider. There are actually the same *quantity* of points in each line, but the thinner ones are harder to hit. Thus, the particular transfinite ordinal is based on relative ease of throwing a dart and hitting a particular band.

To my mind w isn’t so much of a number, as a unit. It’s a starting point for thinking about transfinity. Thus, the transfinites are relative to one another, not absolute (i.e., there is not a “first” transfinite number). w is just kind of an arbitrary yardstick for consideration of them.

8. 8
kairosfocus says:

DS,

I think we can all agree that for a past time p to be the actual past, it had to have once been the present but gave rise to successive stages down to now.

Now, the core issue is what it means for the causal-temporal past in toto to have been transfinite. Where, as we have addressed finite stages, counting stages backwards or forwards necessarily implies accumulating duration.

In that context, to assert or imply that for any actual past stage p, the cumulative time down to the present will be finite, also that prior to p there was an already completed further past that is infinite is to state a conclusion that requires warrant.

I suggest, that warrant will not be adequate.

First, an actually infinite past implies completion of an infinite succession of stages that were once the present but have now been succeeded. Thus, some stages of the actual past, per this claim, must be transfinitely remote. Let q be one such stage, per the argument. We then face the implication of stepwise, finite stage succession that spans a transfinite gap.

That is not possible, stepwise advance is not powerful enough of either a mathematical or a physical operation to do that. Math, being taken as the logic of structure and quantity.

What we can trivially warrant is that if any past stage p as discussed is finitely removed in steps, it is also finitely removed in duration.

Bridging to p from some q is likewise challenged by the endless span. And if one instead asserts there was no q, all stages are at finite remove then that directly leads to the conclusion: a finite causal-temporal past.

This puts the immediate questions on the table.

KF

9. 9
daveS says:

KF,

In that context, to assert or imply that for any actual past stage p, the cumulative time down to the present will be finite, also that prior to p there was an already completed further past that is infinite is to state a conclusion that requires warrant.

It is being proposed (by WLC, Snoke, et al) that for any actual past stage p, the cumulative time down to the present is finite (among other things).

It’s merely a proposal submitted for discussion, as in “let’s examine whether this proposal, along with the proposal that the past is infinite, leads to a contradiction”. I don’t see the need for any warrant here. We can talk about any number of hypothetical propositions, regardless of whether we have warrant.

First, an actually infinite past implies completion of an infinite succession of stages that were once the present but have now been succeeded. Thus, some stages of the actual past, per this claim, must be transfinitely remote.

I have yet to see any compelling evidence for this. WLC is a pretty sharp guy. Are you curious why he, who arguably should be the world’s leading expert on this topic, does not come to this conclusion?

10. 10
kairosfocus says:

DS, what does it mean to have an actually infinite past? KF

11. 11
daveS says:

KF,

It means that the past is not finite.

And a finite past is a past for which there exists a positive integer N such that every instant in time occurred within the last N years.

You can also rephrase this in terms of a sequence of stages, as we have occasionally.

12. 12
kairosfocus says:

DS, in short there would have to be past times or stages Q such that no finite past stage P would be just as remote. For, I take it the past constitutes stages such that they were once the present but have been superseded by successor stages down to now. The challenge of endlessly remote past stages Q remains. KF

13. 13
daveS says:

KF,

DS, in short there would have to be past times or stages Q such that no finite past stage P would be just as remote.

Eh? Just as remote as what? I don’t think I understand.

Edit: If you’re saying that there must exist a stage Q such that no finite past stage P is just as remote as Q, then no, I don’t believe that follows.

14. 14
kairosfocus says:

DS, no finitely remote stage P [stage count from now to P = N, N a natural number] will be as remote as Q. That is Q is transfinitely remote. KF

15. 15
daveS says:

KF,

Ok, based on #14, no, that definitely does not follow.

16. 16
kairosfocus says:

DS, if you are asserting, at least as a case to be considered:

It means that the past is not finite.

And a finite past is a past for which there exists a positive integer N such that every instant in time occurred within the last N years.

I suggest that in this context, “not finite” implies that there is no integer N such that the stage count to ANY past point P will be less than N. Endless, in short.

I would then note that the past implies all and only those states, S such that they were once the present but have been succeeded by a chain of subsequent stages to the present.

So, the actuality criterion binds our consideration.

I suggest that if there are actually past moments such that no integer value stage count equals or exceeds them, that is tantamount to there being actual past stages Q that are transfinitely remote.

No, there is not an open admission.

But the point is, we are not here dealing with oh you can always add another integer to any given integer K so they go on endlessly; then, we assign an order type to that sort of entity, omega and identify it as order type of the naturals.

No, we require ACTUAL past moments in a successive chain to now.

So, if the chain is not finite, it has links that are transfinitely remote.

Where, step by step succession cannot traverse endlessness.

KF

17. 17
daveS says:

KF,

I suggest that in this context, “not finite” implies that there is no integer N such that the stage count to ANY past point P will be less than N. Endless, in short.

I take it this means that there does not exist an integer N such that for every stage P, P occurred less than N stages ago. If, so, I agree.

I would then note that the past implies all and only those states, S such that they were once the present but have been succeeded by a chain of subsequent stages to the present.

Yes. I would use the words “consists of” instead of “implies”, perhaps.

So, the actuality criterion binds our consideration.

I guess so? I don’t know what “non-actual” past stages or a “non-actual” past would mean, however. How can there exist stages which have already occurred, are in the past, yet are “non-actual”?

I suggest that if there are actually past moments such that no integer value stage count equals or exceeds them, that is tantamount to there being actual past stages Q that are transfinitely remote.

But there aren’t any such moments/stages. At least not in the conception of WLC, David Snoke, and others.

18. 18
kairosfocus says:

DS, the issue is the logic of structure and quantity i/l/o what is implied by what is said. The past is the set S, where there is a way to count the chain. Then, there is no N in the counting numbers that is sufficient to exceed and bound the number of links to any P in S. This leads to the conclusion that there are members of S like Q where the span of steps would exceed ANY natural number, N. This then runs right into the problem that stepwise, finite stage succession cannot span endlessness. KF

19. 19
daveS says:

KF,

Then, there is no N in the counting numbers that is sufficient to exceed and bound the number of links to any P in S. This leads to the conclusion that there are members of S like Q where the span of steps would exceed ANY natural number, N.

How?

Would you mind explaining this step in detail, symbolically if possible?

I’m asking because this looks like an erroneous version of DeMorgan’s laws. Which I’m sure you are aware of, so I’m at a loss to understand why you would make this sort of elementary error.

20. 20
kairosfocus says:

DS,

The point hinges not on De Morgan on NAND or NOR of composite statements, but on the meaning of the [actual] past and the further meaning of your:

It means that the past is not finite.

And a finite past is a past for which there exists a positive integer N such that every instant in time occurred within the last N years

For the actual past S to not be finite, then: there [is not ANY] positive integer N such that every instant in time occurred within the last N years.

P is some particular, finite past event. One that is such that some N will exceed the count to P.

Think of it, now, as a subscript we may tag all or any such events we count back to with. As the past is a successive chain, we may count the links back from now and tag every counted link with a p subscript, like a check-mark. The claimed infinite past implies that there would be some Q such that it will be beyond ANY N, that is we cannot find a way to comprehend the set of the past by count. It is endless.

However, distance or depth in time or count by finite stages to any past event or stage R, will be specific to that event. It is not an abstract set, for an event R to be a member it must have once been the now and to have since been succeeded by other events in a chain of finite stages down to today.

So, if S contains all and only finitely remote past events, there would be no transfinite past. Recall, no R will be in S unless it was an actual past stage, and holds a specific place in the chain.

That is, if you claim S is in aggregate transfinite but has in it no Q which is itself remote beyond any natural number, then there is a contradiction of meanings.

For, transfinite is not here just an aggregate property but must obtain for particular stages.

In short, the claim that there is an endlessness of count in aggregate does not confer infinite remoteness to any particular member R. But, to have a transfinite past we need to have such events that are beyond count so their remoteness in causal-temporal succession cannot be exceeded by ANY natural number N. There is — for want of that transfer from the aggregate to the particular member — therefore no R in S that may be tagged Q. That is, the past only contains finitely remote members of S and is finite.

The future is POTENTIALLY infinite, continuing endlessly, but at any moment it is as yet unrealised. The future is for us an inductive projection from the past to now on the further assumption that nothing will cause the succession process to cease. But at any “now” the chain of events or stages so far is finite and open to further additions.

Of course a discussion like this is not the sort of thing that one could have with the general public as it pivots on several abstruse issues such as what it would mean to be finite or infinite, etc. This is dialectic.

But, that does not mean it is unimportant.

For, the consequence of a world of causal-temporal succession is we face an Agrippa-like trilemma. Circular causal root can be ruled out as it entails a future, unformed event or stage causing its predecessor. Non-being having no causal power, that is impossible. Likewise, that the world should spring forth from utter non-being full-orbed and operating, is likewise impossible. So, we are looking at an infinite contingent chain in causal-temporal succession or a finitely remote world-root that comes from necessary being.

It turns out that atheistical systems such as evolutionary materialism are locked into implying an infinite past of causally connected stages. But such is deeply questionable for reasons as described.

This means the best option on the table is that the past is finite, rooted in a necessary being world-root.

KF

21. 21
daveS says:

KF,

I will go through and interpret your argument. Please point out any errors.

P is some particular, finite past event. One that is such that some N will exceed the count to P.

Ok. Let’s say we choose P to be the event of Neil Armstrong first setting foot on the Moon, and think of P as part of a particular very long chain of events stretching into the past. P is of course a member of many such chains, but we will fix one.

Think of it, now, as a subscript we may tag all or any such events we count back to with.

As the past is a successive chain, we may count the links back from now and tag every counted link with a p subscript, like a check-mark.

Ok, I think that’s clear enough.

The claimed infinite past implies that there would be some Q such that it will be beyond ANY N, that is we cannot find a way to comprehend the set of the past by count. It is endless.

Eh? But that’s exactly what you’re supposed to be demonstrating, namely that there is a Q, such that for every positive integer M, Q occurred more than M stages before the present. Why must such a Q exist?

There is another problem here, which has come up frequently in other threads, that of the distinction between ontological questions vs. epistemological questions.

Who really cares if a human observer, working backwards through this chain of events after they have occurred, has difficulty counting/comprehending the chain at some point? The events in the chain exist, regardless of our finite intellect. Not that this is really an issue—we have an ample supply of counting numbers in our arsenal.

22. 22
kairosfocus says:

DS, my point is that the past and time or stage count to particular events or stages is specific. The idea that we get to deal with infinity by implication of a set that is endlessly continued so the aggregate is infinite does not work. The past is fixed, it happened, it is not adding in the remote zone. KF

23. 23
daveS says:

KF,

DS, my point is that the past and time or stage count to particular events or stages is specific. The idea that we get to deal with infinity by implication of a set that is endlessly continued so the aggregate is infinite does not work.The past is fixed, it happened, it is not adding in the remote zone.

True, but AFAIK, no one is suggesting otherwise.

This person moving backward in the chain is observing something that already exists, and which is not changing.

That’s been my presumption all along.

24. 24
kairosfocus says:

PS: Back, I should add, the set is structured, formed by a dynamic, time-forward causally connected process. That process imposes stepwise, finite stage succession.

25. 25
kairosfocus says:

DS, given that each stage has its own time locus and is at a specific link in the successive chain, the property of duration since some member of S, R is a property of R. As a result, if some P is finitely remote we can count back to it through its successors. Where also, as duration attaches to each member, if the past-stages set is infinite in the past, it must contain members Q that have the property of being themselves infinitely remote in time from now. Otherwise, there simply was no such infinitely remote actual past. But the dynamic of stepwise succession from finite stage to finite stage does not have the capability to actually span endlessnes. So, if we are here after a past, that past will contain only members P and no members Q. The past events are finite and the overall set of events is also finite. KF

26. 26
daveS says:

KF,

Where also, as duration attaches to each member, if the past-stages set is infinite in the past, it must contain members Q that have the property of being themselves infinitely remote in time from now. Otherwise, there simply was no such infinitely remote actual past.

Well, I have no interest in getting into a debate over semantics. I will point out once again that the above is in conflict with the usage of the term “infinite past” in the philosophical literature.

I am interested in the issue of “infinite past”, as understood by WLC, Quentin Smith, etc., which involves no assumption of “ω-points”. I will use the notation IP(−ω) to denote this concept, and perhaps IP(+ω) for your version.

But the dynamic of stepwise succession from finite stage to finite stage does not have the capability to actually span endlessness.

Edit x 2: This is in fact irrelevant. The traversal in question is actually beginningless, not endless.

27. 27
kairosfocus says:

DS,

Last point first.

Remember, kindly, the arrow and trajectory of time: forward. So if there were a beginningless, infinitely remote temporal-causal sequence of past stages to now, it would have had to span from the transfinitely remote past to now.

So, that a stepwise, finite stage succession cannot span a transfinite range to REACH the present, that is material. And IIRC that has been one of WLC’s concerns among others.

But names are not the key issue, substance is.

What I have pointed out is that for any R in S, which has a particular place in the sequence, duration to now is a property of the event given lapsed time. That is, date x to now is driven by when date X was. 1930 to now is 87 – 88 years ago.

The aggregate set of past events only exhibits a duration because of the specific durations to now of its members R.

So, if the claim is, a transfinite past with no beginning, then any event R has an onward endlessly remote onward chain of events. But that does not change the point that duration (t_now – t_R) is set by when R happened. Set t_now = 0 by convenience and WLOG, so we see duration = – t_R.

So now, if you claim time without beginning, then necessarily transfinite past duration.

Further to such, that entails cases R that carry the transfinite duration. So were there an actual infinite, beginningless past there would be cases Q such that t_R(Q) is itself of transfinite scale. It matters not that beyond any given Q there are more stages in an onward transfinite chain, the focal point is Q.

The point I have made is that duration since a transfinitely remote Q would have to proceed in stepwise finite stages to now. But that is the problem, such a dynamic does not have the power to span such a traverse.

So, we are warranted to infer a finite past such that all past moments, stages etc are only finitely remote from now and the number of such stages is itself countably finite.

KF

28. 28
kairosfocus says:

PS: I find in the Sanford Enc Phil:

>>6.2 Impossibility of an Actual Infinite

In defense of premise 2, Craig develops both a priori and a posteriori arguments. His primary a priori argument is

5 An actual infinite cannot exist.
6 A beginningless temporal series of events is an actual infinite.
7 Therefore, a beginningless temporal series of events cannot exist.

Since (7) follows validly, if (5) and (6) are true the argument is sound. In defense of premise (5), he defines an actual infinite as a determinate totality that occurs when a part of a system can be put into a one-to-one correspondence with the entire system (Craig and Sinclair 2009: 104). Craig argues that if actual infinites that neither increase nor decrease in the number of members they contain were to exist in reality, we would have rather absurd consequences. >>

And also . . .

>> 6.3 Successive Addition Cannot Form an Actual Infinite

Craig’s second argument addresses this very point.

8 The temporal series of events is a collection formed by successive addition.
9 A collection formed by successive synthesis is not an actual infinite.
10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103).

The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite (Craig and Sinclair 2009: 118).

It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future. Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.>>

So, there actually are similar arguments in WLC. Though, he focusses on the overall set. I have pointed out that the members form the set and duration of the whole depends on duration since its members were the present.

But that is not the primary issue.

I also note that I speak to finite stages in succession precisely because this sets to one side the “Zeno paradox” distraction. The focal matter is a succession of finite stages.

Likewise, you will observe that above I do not speak of Q as a beginning but a point in the set S, for all we care for the sake of argument for the moment, there could be an onward even more remote transfinite chain, much like we may see in the hyperreals above in the OP.

The focal matter is that duration attaches to the specific event once we accept now as a zero point [or a similar convention such as the singularity, which I used earlier].

The span onward cannot be bridged in succession by stepwise addition of finite stages, so cannot reach to now.

29. 29
daveS says:

KF,

Remember, kindly, the arrow and trajectory of time: forward. So if there were a beginningless, infinitely remote temporal-causal sequence of past stages to now, it would have had to span from the transfinitely remote past to now.

Yes, so the traversal is beginningless, not endless. That’s exactly my point. It’s important to be precise about the order type of the collection of stages being traversed.

Further to such, that entails cases R that carry the transfinite duration. So were there an actual infinite, beginningless past there would be cases Q such that t_R(Q) is itself of transfinite scale. It matters not that beyond any given Q there are more stages in an onward transfinite chain, the focal point is Q.

Where is this Q coming from?

If you are talking about a scenario where an infinite past is assumed to include such Q’s, then of course they exist.

But I’m strictly speaking of what I called IP(−ω), consistent with WLC, Quentin Smith, etc. In their conception, you don’t get this Q for free.

It sounds like you only want to talk about IP(+ω), in which case you’re talking about something different than me, WLC, Smith, etc.

30. 30
kairosfocus says:

DS,

What I put up stands on its own merits, and I have shown that very similar thought is being addressed.

Maybe it is time for you to explain to us how in a world where duration since an event is specific to that event, you propose to have a beginningless and infinite past without events that are a transfinite duration away.
===

Recall, duration since attaches to the dynamically connected finite stages, not to the conceptual container, the set of the actual past.

In this context it seems more and more distractive to bring up an onward further endless [from the perspective of counting back up the chain of causes that is perfectly legitimate] stages in the chain. Yes, that only means that there are many cases of further durations to address.

Duration or count to now attaches to the specific link in the chain, not to the conceptual container. To suggest an infinite duration as there are onward more remote links does not eliminate the point that for a duration you need now this end and then that end. So if you claim transfinite duration you imply events Q with transfinite separation in stages from now.

And that pivots on what duration means.

KF

31. 31
kairosfocus says:

F/N: Kindly see the clip from J P Moreland, Ed. that has been appended to the OP. KF

32. 32
daveS says:

KF,

Duration or count to now attaches to the specific link in the chain, not to the conceptual container. To suggest an infinite duration as there are onward more remote links does not eliminate the point that for a duration you need now this end and then that end. So if you claim transfinite duration you imply events Q with transfinite separation in stages from now.

When WLC and Quentin Smith speak of an infinite past, they mean a past in which given any positive integer N, there exist two stages/time coordinates in the past separated by at least N steps/time units. There need not exist any two stages/time coordinates separated by infinite distance.

This is similar to a discussion of a spatially infinite universe, with infinite “diameter”. In that case, it just means that there are physical points separated by arbitrarily large distances. There need not be any particular points separated by an infinite distance.

33. 33
daveS says:

KF,

F/N: Kindly see the clip from J P Moreland, Ed. that has been appended to the OP.

Yes, that is one example of the very exotic topologies that Quentin Smith, I believe, has also written about. There’s nothing “unserious” about considering those possibilities, to be sure.

Edit: I’ve skimmed a few of Moreland’s arguments just now, and they are not very good, to say the least.

34. 34
daveS says:

PS to my #33:

Perhaps a bit hasty. He has generated some new ideas and discussion around the “infinite past” issue, in any case.

35. 35
kairosfocus says:

DS,

Let me start with the point Berlinski made, which this was primarily a back-up for. Mathematics is inseparable from modern technical thought, and is an abstract mentally perceived or even intuited realm that interacts causally with the observable reality. This of course echoes Wigner’s amazement at the effectiveness of mathematics in the physical sciences. I have long thought that an extension of the definition I was taught long since answers aptly: mathematics is (and studies) the logic of structure and quantity.

So, it is in fact highly relevant to take Wiki as a witness against known ideological interest and to highlight the place of the hyper-reals [and so too the infinitesimals], which were relevant to the emergence of Calculus from the days of Archimedes to the C18. There being challenges, the epsilon-delta limits approach was substituted across C19, but from the 1940’s – 60’s the non-standard analysis approach rehabilitated the use of infinitesimals.

A realm of abstract ideas that are explored on logic-guided intuitions and formulated into schemes of thought turns out to be foundational to modern society. That should give any card-carrying materialist serious pause.

Next, was it two years ago, I was thinking along the lines that are now confirmed to be reasonable. I well recall the fairly sharp dismissiveness with which I was greeted. But there we have it, use of reciprocals as a catapult function between the hyper-reals and the infinitesimals, the use of an essentially extended reals line with a continuum. That then brings back into play the substantial concerns and issues I raised then and have in some measure continued to raise since.

It also calibrates the credibility [or, actually, want of it] of the dismissiveness in response.

In that light, I am not particularly suprised to see that you were first trying the tactic of isolating and dismissing my remarks as idiosyncratic and dubious musings, Above, you were confidently trying to slice off and dismiss. In that context, I highlighted Stanford Enc of Phil and now Moreland, to see that there is in fact a serious discussion along the lines of the logic I have put up. In response you try to still distance WLC et al and dismiss the most closely parallel to my arguments.

By now, that attitude and approach are sadly unsurprising.

Let me pick back up SEP and WLC’s core points [cf. 28 above and the onward link], for a moment:

5 An actual infinite cannot exist.

[He has many reasons for that, which are often hotly contended but on the whole I think he has a serious point . . . at minimum he has shown that modern atheism effectively is reduced to a highly contentious assumption about the past, which is very different from its boast that it can be seen as a default for the intelligent person]

6 A beginningless temporal series of events is an actual infinite.

[Note this]

7 Therefore, a beginningless temporal series of events cannot exist.

Since (7) follows validly, if (5) and (6) are true the argument is sound . . . .

>> 6.3 Successive Addition Cannot Form an Actual Infinite

Craig’s second argument addresses this very point.

8 The temporal series of events is a collection formed by successive addition.

9 A collection formed by successive synthesis is not an actual infinite.

10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103).

The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite (Craig and Sinclair 2009: 118).

It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future.

[–> relative to the distant past in view. This is also the context in which I have consistently spoken of finite stage causally successive stages]

Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.

In short these arguments are indeed quite similar to what I have put on the table, and those of J P Moreland, though compressed are even closer. So, the implied oh you are ranting on idiosyncratically and can be dismissed rhetorical tactic fails.

Now, we can return to the substantial matter.

I recently had to compose a Crypt-stone summary for my Dad. In identifying the duration of his life, I posed two dates, just as was done for my grandparents, aunts, uncles and so forth in the same family burial plot. (The motto on the stone is, “study to shew thyself approved unto God” which was underscored in every Bible in the house and was inscribed in the only Bible he ever actually gave to me. It is apt to his life and a lesson to the nation.) The duration, as simple logic tells, is date of death, less date of birth, though of course it is usually given as a forward direction, X – Y, the duration being the dash, a compressed timeline.

Duration, from the actual (as opposed to imagined or assumed or theoretical) past to the present is similarly a timeline of stages or events. As we may here involve going beyond the big bang singularity, using something like years is probably not the best, so I have emphasised a stages-count; which is more than good enough for our purposes. A hypothesised beginningless past would have a past infinity of stages.

We set, WLOG, now as zero (previously, I used the Singularity). So the duration since the past event at X is 0 – X = – X, where X is a negative number. This, you have not challenged effectively. It highlights, that duration attaches to the specific beginning and end events of a duration. And since we are dealing with actual events, at specific points in the chain of successive causal-temporal stages to now, I have every right to point out that duration is primarily since a specific event, not a property of the collecting container, the set S of the past series of stages.

So, I have singled out that we can denote an event in general as R, and the duration since can be seen as -R_t, for want of better symbols for the moment. So, for a finitely remote event, P, we can directly see that time since is – P_t. What you obviously wish to put on the table is that the set of all events elaborates events of type P, into the past infinity, such that no natural number N can bound the value of t, therefore you have a past infinity of finitely remote events in finite stages. This is a way to imply dismissal of the force of the point that from the far past zone, to claim that it is beyond any natural counting number N (or integer) from us, one is implying that some values of t are hyperreal. Thus, you have your cake and eat it: only finite points in the past, but collectively an infinity.

I don’t think that works.

First, I repeat, duration does not attach tot he set as a whole, but it is since particular events or stages in the set. So, to claim a past infinity of beginninglessnes, you imply that here are particular actual events or stages Q in S such that – Q_t is hyperreal. I have sometimes put it that if every event in the past is finitely remote, that directly implies that the past is finite.

So, we have here another really strong and questionable worldview commitment of the modern atheist. An infinite past containing only finitely remote events, such that somehow mysteriously, through the magic of the ellipsis of the past infinite, we only designate finitely remote past events but infer an infinity of no0-beginning for the set as a whole.

I repeat, duration is primarily since the particular event to now. That is not a controversial, idiosyncratic, readily dismissed claim.

So, directly, if one claims a beginningless past of finite scale stages in a causal-temporal succession down to now, then one IMPLIES — whether or not one wishes to ACKNOWLEDGE — actual past events that are transfinitely remote.

There is no escape into the magical power of the three dot ellipsis.

So, events Q are on the table such that duration since Q — regardless of onward infinite sequence — that is -Q_t, is transfinite, hyperreal for convenience, i.e. its reciprocal q is all but vanishingly close to zero, smaller than any real number, and such that q^2 ~ 0. That seems to me, a reasonable definition of such a transfinite.

The problem then directly surfaces as is also shown by WLC and JPM et al: to pass from Q to Now, let’s denote now as 0, requires stepwise, finite stage causal-temporal succession across a span of transfinite character. Traversing the infinite in finite stage successive steps.

I have long since shown how this is a futility as once we have taken any k steps, no matter how large, we can effectively start again as though we had taken none — think here of the blue vs pink punch-tape example I used previously:

0 –> 1 –> 2 –> 3 . . .

k + 0 –> k + 1 –> k + 2 –> k + 3 . . .

That is, one to one correspondence, the proper subset is in 1:1 correspondence and we cannot traverse the transfinite span in the ellipsis. (Going to the OP for a moment, we could put the ellipses in between the two omegas (+/-) and the zero point. You cannot go to either or from either to zero in finite stage steps.

The issue is serious.

Now, the SEP continues, citing a key objector who comes up early in Google searches, long before WLC himself does in his own voice (which is telling in itself):

Morriston (2003: 290) critiques Craig’s thesis that forming the infinite series of events by successive addition presupposes a beginning point. He asks,

Why couldn’t there have been an infinite series of years in which there was no first year? It’s true that in such a series we never “arrive” at infinity, but surely that is only because infinity is, so to speak, “always already there”. At every point in such a series, infinitely many years have already passed by.

Why do we need to “arrive at infinity?” But Craig’s point is not that we cannot arrive at infinity in the past, but that we could not traverse the infinite to arrive at the present moment. Why this moment rather than another? But maybe, Morriston replies, that is just the way it is; “the past just is the series of events that have already happened”. To require a reason for the series of past events arriving at now is to appeal to the principle of sufficient reason, which he deems both suspect and inappropriate for Craig to invoke (Morriston 2003: 293). Furthermore, he argues, Craig’s argument mistakenly presupposes two independent series—a series of events and a series of segments of time they occupy—such that one can ask about how the former is mapped onto the latter, whereas in fact the two series are not independent. Craig (2010) replies that it is not a matter of sufficient reason, but that Morriston simply has not paid sufficient attention to the distinction between past and present tenses, on which potential and actual infinites are founded. A finite series that has the potential for further members, as with future events beginning with now, is actually finite and only potentially infinite. But a beginningless series of past events cannot add new members; it is actually, not potentially, infinite. There is a relevant distinction between the two series.

In short, it is clear that the objector wishes to have the traversal of the infinite in effect set aside as already accomplished at all times we may specify.

In my mind, as I noted above, that already begs the question. The issue is how we get to finite reach of 0, not that from some finitely remote P we may access 0 and use the power of the past ellipsis to duck the point at stake.

I am also aware of the dismissive attitude to the PSR in various forms. However, there is a relevant weak form that one may inquire of anything that is [or is not etc] why is that so and then proceed with inquiry. This is self-evident as one may then directly engage the matter. Which is what we have done above. We find that stages in the temporal-causal domain of interest are (unsurprisingly) causally successive, one stage receding to the immediate past as it generates a successor and often leaves traces of once having been the actual present. Then as successive stages pass, it is increasingly distant in the past as the days of Rome now are or those of Pharaoh in Egypt.

So, it is entirely appropriate to address that dynamic of stepwise stage by stage succession and its implications.

In short, the issue is still on the table: transfinite duration in the past implies that there are members of S that are of the character of Q not P. This, as duration is since event R not lost in the magic of ellipsis. So, he decisive point is that one cannot span a transfinite traverse in whatever direction in stepwise succession, though as seen in the OP and for the past two years IIRC, one may catapult over it through the power of the reciprocal function which allows us to leap from infinitesimal to hyperreal and back again.

I conclude, for cause, that there is no good reason to infer that one may take as a default view that there has been a beginningless, thus transfinite, past. further, that we are only warranted to see that S contains elements of finite remoteness, P only. Third, the assertion or assumption of a beginningless past implies that there are events of character Q, as duration since attaches to the particular event not the magical power of the ellipsis and the brackets used to denote a set as a conceptual object. but there is no power in stepwise finite stage succession to actually span the transfinite range implied, so we are warranted in the conclusion that spanning from Q to 0 is an infeasible supertask.

There was an actual past, and we are warranted to conclude that it was finite, i.e. there is some counting number N that does exceed the valid number of past events/stages, whatever that may be.

The physical world has credibly had a beginning, even if we go beyond the big bang to suggest [without actual observations] that there are prior stages. And indeed that is also the message of accumulating entropy, that the past has not been long enough to reach to a deteriorated, degraded cosmic state.

A complex, fine tuned, coherently functionally specific world with a beginning points to its being the product of intelligently directed configuration. this raises the further issue of a capable designer of such a world.

And yes, that puts the Kalam cosmological argument back on the table. SEP:

Craig formulates the kal?m cosmological argument this way (in Craig and Smith 1993: chap. 1):

1: Everything that begins to exist has a cause of its existence.

2: The universe began to exist.

3: Therefore, the universe has a cause of its existence.

4: Since no scientific explanation (in terms of physical laws) can provide a causal account of the origin (very beginning) of the universe, the cause must be personal (explanation is given in terms of a personal agent).

This argument has been the subject of much recent debate . . .

KF

36. 36
daveS says:

KF,

I’ll respond more fully to this later, but for now, let me clarify that I’m not trying to “isolate” or “dismiss” your line of reasoning, but rather simply to draw some critical distinctions between the models of an infinite past that are being discussed.

WLC and others are talking almost exclusively about IP(−ω), while you sometimes are talking about IP(+ω). At certain junctures we need to be very explicit about what is on the table.

However, on a personal level, at times during this debate I have unquestionably been overly snarky, or a “jerk”, and for that I do apologize.

37. 37
kairosfocus says:

DS, appreciated. Let’s see how we can move forward. I await your onward comment. KF

38. 38
daveS says:

KF,

I’ll have to respond in small bites.

We set, WLOG, now as zero (previously, I used the Singularity). So the duration since the past event at X is 0 – X = – X, where X is a negative number. This, you have not challenged effectively.

Definitely not. I believe that it is correct.

It highlights, that duration attaches to the specific beginning and end events of a duration. And since we are dealing with actual events, at specific points in the chain of successive causal-temporal stages to now, I have every right to point out that duration is primarily since a specific event, not a property of the collecting container, the set S of the past series of stages.

How can we resolve this?

I suggest looking at the Moreland passage above.

He speaks of a beginningless infinite past. I can’t view the preceding page which that quote is taken from, but I presume he allows that a sequence of stages through an infinite past could have order type ω* (as opposed to ω* + ω*, from the quoted passage). Certainly other writers do.

Now the set of nonpositive integers, which does have order type ω*, does not contain any elements infinitely remote from 0.

So no particular elements of this set are infinitely remote from 0, yet Moreland (likely) regards the set as a whole as having infinite “duration”.

From my reading, I believe that’s consistent with what others have in mind. An infinite past simply must contain arbitrarily long intervals. Just as a spatially infinite universe must contain arbitrarily long distances.

39. 39
kairosfocus says:

DS, I am speaking to a stage or event in the chain, and I am pointing out that durations since are tied to those events primarily. Event R is such that duration since R to now = – R_t. And as the set is really a conceptual shell, there is no such thing actually as duration tied to the set as such but rather to the events in it and the dynamics that connect them in a successive chain. So, if one suggests a transfinite duration, there need to be events such that the duration since them is transfinite. And actually Moreland’s point in the excerpted footnote [and that is all of it there above] is that the duration from the claimed beginningless past to now would involve an event that has receded to the endlessly remote past. He objects that for such to have allegedly happened would imply traversal of a transfinite span in successive finite stage steps, which is impossible. But we do not need to go quoting JPM or WLC etc, the matter is laid out above. KF

40. 40
daveS says:

KF,

I think that leaves us at an impasse, then.

Correction to part of my #38:

From my reading, I believe that’s consistent with what others have in mind. An infinite past simply must contain arbitrarily long finite intervals. Just as a spatially infinite universe must contain arbitrarily long finite distances.

41. 41
daveS says:

PS to my #38/40:

The most common “application” of these debates is to arguments such as the Kalam Cosmological Argument, which seek to show at some point that the universe had a beginning, some finite number of years/stages ago.

They do this by showing that the past is finite.

But if there existed sequences of stages with order type ω*, with all stages occurring finitely many steps from the present, then the universe could not have a beginning.

Therefore, presumably “the past is finite” must be inconsistent with the existence of such ω*-type sequences, hence these sequences could only exist in case the past is infinite.

42. 42
Scotoma says:

Hyperreal numbers is just a rename for abstract or imaginary numbers. While they have their use in mathematics, they have no value in reality without absurdity. About as silly as supposing that an infinite number of moments can be divided (for example) by 2 and have a result of two finite sets. There cannot be an infinite succession of moments prior to this moment otherwise this moment could never arrive, since there is no such thing as an infinite number of moments + more moments. If there were an infinite number of moments prior to this moment, then all the energy of the universe would have been converted to heat by now and the universe would have reached maximum entropy and be cold, dark and dead. Yet, here we are, and the moments keep on coming. Trying to correlate imaginary “hyperreal” numbers to reality is to make much ado about nothing.

43. 43
kairosfocus says:

Scotoma, yes, it is arguable that had the physical cosmos been infinitely old, it should have faced in effect so-called heat death. KF

PS: The hyperreals lend support to the infinitesimals, which for over 2,0000 years have proved their worth.

44. 44
kairosfocus says:

DS,

I again point to the actuality principle. The past is actual and comprises a chain that is causally successive.

Each past stage was once the actual present and causally gave rise to its immediate successor.

This proceeded in a chain down to now. We can collect the chain as a structured, dynamically connected whole, S. In it is a succession of finite stage links, typical value R.

Each such R occurred at some time (or stage count back from now) R_t. By setting now to zero, the duration since R is

Dur_R = 0 – R_t = – R_t

This cannot be controversial, and every R is a finite stage, that is we are not dealing with an ever finer succession which may converge on a limit. Zeno’s paradox is irrelevant.

All of this connects to the claim, here, of an allegedly infinite actual past, a physical, causal-temporal succession to date that is claimed to have no beginning.

Were that so, the cumulative chain would be infinite, as there is no basis for its convergence.

But as duration since attaches to the individual member R, we can then distinguish a sub-type P that is finitely remote from us and — what is controversial because of implications — the sub-type Q that is transfinitely remote. This last implying that the reciprocal of its duration since is an infinitesimal as described in the OP. For a case of sub-type P the reciprocal would be a finitely small fraction.

Were there such an event as Q, we face the challenge of transfinite, cumulative, finite stage succession amounting to a transfinite value. That is set by the dynamics of succession demanded by the actuality principle.

We know that such a stepwise succession can only ever amount to a finite value, as at each finite stage k it can be seen as starting over again and put in 1:1 correspondence with the original chain. Both with endlessness still ahead. Finite stage succession has no power to stepwise span a transfinite range. In sequence and series analysis we only illustrate certain stages and then pull back to look at the whole at once, assessing an infinite summation’s trend. In fact, we use the trick that for a truly convergent series or sequence, beyond a given point, the onward sequence of partial sums or the terms of the sequence will be within a given small neighbourhood of there it would end in an infinite limit. We never actually do the infinite summing up. We cannot.

So now, we proceed to see that a hypothetical Q that proceeds to trigger a transfinite chain to now is an infeasible supertask.

The only credible members of S are of type P.

I point out immediately that duration since any — repeat, ANY — P will attach to P and will be – P_t, a finite value. So, as there are no members of the chain of type Q, the cumulative duration is finite, and that is not a matter of a convergence as every stage P is finite.

Now, the arguments I have seen from you and others try to posit a case that once we have any definite P that is at some definite duration to now, the beginningless chain is BEFORE P. So, we can never access a first member. And by appealing to the dismissals of the principle of sufficient reason the implication is that such is a causeless, unintelligible brute fact. (And people talk dismissively about blind faith?)

I say, no.

First, I am unwilling to surrender the intelligibility of the world and its major features without very good reason. I suggest that a weak-form PSR is patently valid: of any thing X that is (or is/ may be not) we may ask why and proceed to investigate. This does not assume a conclusion. But it can be confident, i/l/o the logic of being: possible vs impossible, contingent vs necessary. Cause, connected to contingency. For this, consider two “neighbouring” possible worlds, A and A’. In A X is, but in A’, which by distinct identity will differ in some relevant core characteristic c, X is not. We can then plausibly argue that c is or is connected to an enabling, on/off causal factor for X to be.

So now, we come back to our main discussion.

The issue is whether we can have a beginningless, causally successive temporal sequence of stages R such that the past is cumulatively infinite but all members of S are of sub-type P. To which, I answer no, as duration since is dynamically tied to the specific actual member R. So, if the chain as a whole is allegedly transfinite and beginningless, that directly implies that in the chain there are specific stages of the actual past that were once the present but which are now infinitely remote. And no, this is not to say that such events were the beginning of the chain, they are just within it. For all we care, there are onward members that may continue onward without limit, for argument.

Of course, if you ask the man in the Clapham bus stop what it would mean to have such an infinite past, he would instantly agree that it would involve specific stages of sub-type Q.

Is he wrong?

I think, not.

The way of proceeding where for instance we argue that the natural counting numbers succeed from zero and any member we can reach will have an endless onward chain of successors so that any member k that we identify is finite misleads us. For, the key property is that the chain is endless so the indefinite onward unspecified extension is material in seeing that the set is infinite. We then recognise a new order type omega as a type of quantity in itself.

By looking at that seemingly innocuous three dot ellipsis, we can see that it is telling us to shift perspective from the von Neumann-like succession to pondering the whole set all at once, recognising its divergent character. This is how we technically conceive the infinite nature of the naturals without noticing fully the lack of power of finite stepwise succession to deliver the actual endlessness. Or, reversing to look at the negative integers, beginninglessness. The right/left succession from 0 is just a matter of mirror reflection.

I think our challenge is conceptual and dynamical.

The actual past cannot be handled by infinite span ellipsis.

It proceeded stage by stage to get here, now, and duration since primarily attaches to the individual member R as -R_t.

So, if the whole is infinite in the past, it involves members of subtype P and of subtype Q. But the latter is not possible.

So, we are warranted to talk only of a finite actual past, which therefore is bounded by some N in the naturals such that

N exceeds max{-R_t)

There credibly was a finitely remote beginning to the world’s timeline. not just yes the Big Bang, but whatever was a material entity before it.

We face the implication of boundedness on spatio-temporal domain by an initial point, a point of causation. And onward, we find reason to suggest a different order of reality and a world-root necessary being of that order. (For convenience, we may use the traditional term, eternity.)

And yes, this is also connected to the point that, had the past proceeded without beginning, we should face entropy maximmisation. Or else, continual creation, much as the long since abandoned steady state view held.

KF

45. 45
kairosfocus says:

DS,

I wish to look at the LHS hyperreals line as a proxy for a beginningless timeline to now, 0.

In such a case we can count leftward in steps, presuming each is a stage in the past accumulating to the present. A successive L-ward count can always add one more.

So counting, we cannot span to a transfinite value, for obvious reasons. But be reversing direction, we cannot span a transfinite traverse in the R-ward (future-ward) direction.

So now, we face: but at any actual point on the line we reckon R, there is already a L-ward, beginningless, thus transfinite span. The infinity is already completed.

How so?

Oh — and following objectors, that would be to invoke PSR which is deprecated.

No, there is a dynamic which shows that the stages proceed causally, in finite stage steps, thus cumulatively and successively R-ward. We do have a dynamic that shows that one sufficient reason for a stage is that it causally succeeds another. And in the suggested case of a beginning, that it proceeds from the act of a world-root. Where the presumed beginningless process does not access this case.

So, we are left to the presumption that somehow the transfinite has already been spanned, without explanation. This patently begs the questions at stake. And no, a suggested infinite convergent succession of in the end infinitesimal stages such as with Zeno’s paradox does not answer. The cases are not comparable.

The point remains, that the past is actual, stage by stage to now, and we consider cumulatively causal finite stages; years for convenience. Duration since a stage, R, attaches to that stage, – R_t. Thus, to claim a beginningless, infinite- in- the- past world is to imply a world with specific events or stages such that duration since such a stage of sub-type Q will have been transfinite. That is, for such a stage, the duration since, – Q_t will exceed any natural counting number. It will be of a transfinite magnitude comparable to omega on the hyperreals line cf. OP.

Now, the dynamic process that creates the future, R-ward in the cause-to-effect direction, is immediately successive stage to the next, as we are familiar from the Calendar. There is no mechanism that delivers a chunk or a large span of time all at once.

The problem, of course, is that the dynamic at work does not have power to span the transfinite.

The presumption of an already completed past infinity begs that question. It is required to show it possible on a relevant dynamic, or else it is a physically groundless assertion.

Overall, we are only warranted to discuss stages of sub-type P, finitely remote. And to suggest, oh enough of those will be transfinite, fails to address that the dynamic is stepwise successive, on finite stages.

This exercise is then highly relevant to elucidating what we mean when we speak of the past and alternative models of how the past of origins gives rise to the present; to reflecting on cosmology. Where, if we have reason to believe that a claimed mechanism is infeasible, that is highly relevant.

Such also helps us clarify our understanding of infinity, numbers and processes that manipulate the infinite.

The point is, we face a causal form Agrippa trilemma; where origin from utter non-being (which hath not causal capability) is patently absurd. Circular cause is ruled out as it requires the future to cause the past, a case of cause from utter non-being. The above gives reason to set aside beginningless infinite physical world. This leaves, a finitely remote beginning from an adequately capable world-root.

A highly significant conclusion.

KF

46. 46
kairosfocus says:

F/N: “arbitrarily long but finite intervals” seems to be by definition, finite and thus containing a finite number of steps. To go to the transfinite we have to pull back and look at the set as a whole, the function of that three dot ellipsis: . . . So, no, if you mean an interval from 0 that is greater in size than any arbitrarily high n in N, that is any finite n in N, we have arrived at the point where the ellipsis is pivotal. The ellipsis means, we recognise that no stepwise process we undertake that is tied to 0 can span what we are dealing with, as at any k we can go to k+1, k+2 etc and be as good as having just started at 0. So, we summarise in a seemingly simple symbol. That’s math. We are dealing with physics, where each stage R gives rise to the next, and then the next etc. Going R-wards, causally. That successive process simply cannot span or traverse a transfinite span. Where a past without beginning will have to be such a transfinite span. KF

PS: I think Moreland has in mind something like a case Q with a leftward transfinite descent to Q, then a rightward transfinite succession down to now.

47. 47
daveS says:

KF,

It’s clear I’m not going to be able to keep up. I will respond to a few selected points.

Regarding the PSR: I have only claimed that IP(−ω) creates no mathematical/logical problems. Certainly if you invoke some form of the PSR, and demand an explanation for a hypothetical infinite chain of events, there likely will be issues. I have speculated that a god, existing outside of time, and capable of viewing all points in our time “at once” could solve these issues, but at the moment, my interest in that particular line of argument is at a lull.

So, if the chain as a whole is allegedly transfinite and beginningless, that directly implies that in the chain there are specific stages of the actual past that were once the present but which are now infinitely remote.

I don’t see how this follows. At this stage I think it’s fair to ask for a formal argument, with numbered steps, which is obviously valid, like the Kalam argument presented in post #35.

48. 48
daveS says:

KF,

Regarding #46: When we use Gauss’ Law to calculate the electrical field generated by an “infinite” charged wire, in what sense is the wire “infinitely long”?

Are there particular pairs of atoms in the wire which are infinitely far from each other? Or are we simply saying that there are pairs of atoms at arbitrarily large distances from one another? (You may answer this both as someone trained in physics as well as the man on the Clapham bus).

49. 49
kairosfocus says:

DS,

No the problem I have pointed to is inherent, it cannot be simply set aside as of no interest or significance, poof gone through inattention. So, the issue is not whether oh a set as a whole is transfinite can substitute for in this case, the set is built up dynamically so the dynamics have to be accounted for.

As for PSR weak sense: as I pointed out, we ALREADY have in hand a dynamical explanation of stages in the process of temporal succession.

Do you have another?

No, I don’t think so, apart from God creates and sustains all times at once presumably from a base in eternity. That is not what evolutionary materialistic cosmology is prepared to concede.

On A vs B theories of time, I suggest that simultaneously sustained times can only make sense from eternity and such support.

The physics of an infinite linear charge or an infinite transmission line etc are examples of the calculus at work, on increments. In these cases, the integration to achieve resultant action or effect at a point of interest, is a summation to a limit or a summation of infinitesimals, you pick your choice. A ladder of networks of infinite succession may be more to the point, but the result is, summation of a series with an ellipsis and convergence issues in play.

No actual successive impact out to a real infinity is in view, such are mathematical models. Lumped parameter, circuit theory models also imply scale less than 1/10 wavelength of EM waves for the frequencies of interest and several other subtleties.

The analysis abstracts from the real cases as the analysis would then collapse into hopeless complexity for most practical things. So no there is no actual implication of atoms at arbitrarily large transfinite distances. Just as point particle models in Newtonian dynamics do not actually mean to imply infinite density.

Our problem is whether you really mean an infinite past or simply a very large remote past. In either case, duration since R is attached to R. This means that the set which is in effect a conceptual bag that catches up the members, does not have in it a duration bigger than that possessed by some at least of its members. If every member period is only finitely remote in time then the whole is necessarily finitely remote in time. Which implies a first stage, a beginning.

If you argue no beginning, you imply that for any specific n in N, there are members at stages of further remove from now than n. That is, almost by definition, transfinite. Where since the stages are each finite, the span of time from the beginningless past to now is beyond any n in N, i.e. it is transfinite — even ignoring for the moment that S cannot have a span beyond its members.

So, we have every right to challenge that you suggest a timeline much like:

Beginningless so transfinitely remote range, where on actuality,q is a member:

. . . q –> q+1 –> q+2 –> . . .

procceeding stepwise to a finite range down to now:

–> p –> p+1 . . . –> 0

The first two ellipses are transfinite in scope.

Now, I infer from the line of your arguments that the accumulation:

. . . p-2 –> p-1 –> p –> p+1 . . . –> 2 –> 1 –> 0

does not requite any members of subtype q to attain transfinite, beginningless character.

I disagree as the structured set S:

{. . . p-2 –> p-1 –> p –> p+1 . . . –> 2 –> 1 –> 0}

can only have duration as far back as that of its members.

So if S is transfinite, members of subtype q are present.

The challenge being that stepwise, finite stage succession cannot traverse a transfinite span. The reciprocal function between infinitesimals and hyperreals does have such power, but that is not relevant to the dynamics at work.

KF

50. 50
daveS says:

KF,

No, I don’t think so, apart from God creates and sustains all times at once presumably from a base in eternity. That is not what evolutionary materialistic cosmology is prepared to concede.

WLC, Moreland, and others are not arguing from an evolutionary materialistic perspective, yet they still conclude that an infinite past is impossible. I’m just saying that even in their worldview, their arguments fail.

The physics of an infinite linear charge or an infinite transmission line etc are examples of the calculus at work, on increments. In these cases, the integration to achieve resultant action or effect at a point of interest, is a summation to a limit or a summation of infinitesimals, you pick your choice. A ladder of networks of infinite succession may be more to the point, but the result is, summation of a series with an ellipsis and convergence issues in play.

I will pose the question once more: in the hypothetical infinite charged wire, do we assume that there are particular pairs of atoms at infinite distance from one another?

51. 51
daveS says:

PS:

Here’s what I’m looking for in terms of an argument:

1) Assume we have a beginningless sequence of stages:

. . . p-2 –> p-1 –> p –> p+1 . . . –> 2 –> 1 –> 0

where there is no first/leftmost stage in the sequence.

2) ??

3) Therefore, there exists a stage q in the sequence such that q is infinitely stages before/to the left of 0.

52. 52
kairosfocus says:

DS, already given, just not acknowledged. KF

53. 53
kairosfocus says:

PS: Let me clip 49:

the structured set S:

{. . . p-2 –> p-1 –> p –> p+1 . . . –> 2 –> 1 –> 0}

can only have duration as far back as that of its members.

So if S is transfinite, members of subtype q are present.

The challenge being that stepwise, finite stage succession cannot traverse a transfinite span. The reciprocal function between infinitesimals and hyperreals does have such power, but that is not relevant to the dynamics at work.

54. 54
daveS says:

Hm. This means that a wire has finite length provided every pair of atoms in that wire is separated by a finite distance.

55. 55
kairosfocus says:

DS, the model of a wire, as already pointed out is an idealised simplification not a reality claim. The claimed infinite past is an actuality claim. That is why I pointed out that duration only extends so far as duration since members. An infinite past implies infinitely distant stages in time. But such have no mechanism to account for succession down to today as stepwise succession cannot span the transfinite. And BTW, a B-theory, all at once timespan sustained from a common point in eternity would not be going through such a succession as such as actual causal dynamic — that’s a talking-point fail on your part. And more, but this is enough for now. KF

56. 56
daveS says:

KF,

DS, the model of a wire, as already pointed out is an idealised simplification not a reality claim.

Not sure about that, but my point is that it’s one more example of usage of the word “infinite” consistent with mine. A wire in which pairs of atoms are separated by arbitrarily large finite distances is infinite, according to the physicists.

Can you find any published examples where the author unequivocally asserts that infinite lengths/time intervals/whatever must have parts separated by infinite “distances”? I have yet to find any, except for a 40-some year old paper, and that claim was disputed immediately.

And BTW, a B-theory, all at once timespan sustained from a common point in eternity would not be going through such a succession as such as actual causal dynamic — that’s a talking-point fail on your part.

I don’t know what this is about, but no matter, if we can’t agree on the meaning of “infinite past”, or even what a wire of infinite length would be, then we are not prepared to communicate in any depth on such questions.

57. 57
kairosfocus says:

DS, the infinite wire case is an idealisation, that is not up for debate. I am amazed that you are trying to squeeze a reality claim about atomic separations out of it. An infinite past claim by contrast is a reality claim, that the actual past extends in duration beyond any natural number, and is therefore infinite. That is a matter of the meaning of the claim, not citations of papers. Besides, we already had a round of citations, and I am by no means impressed with how you responded. I point out that the ellipsis to the left denotes an infinite extension. I point out again that no collection of successive stages of time will have a duration to now that exceeds that of actual members. So, if infinite past is claimed, it implies actual past stages transfinitely remote from us in time. The problem is, stepwise succession of stages cannot traverse a transfinite span. That gives a pretty good reason to infer that such implied stages did not exist and that the past is finite. KF

58. 58
daveS says:

Well, in any case, I’m not inclined to budge on this matter, as I believe my understanding of the meanings of “infinite” in this context is correct.

59. 59
critical rationalist says:

Well, in any case, I’m not inclined to budge on this matter, as I believe my understanding of the meanings of “infinite” in this context is correct

Why argue over the definition of words when they are ultimately undefined? Definitions rely on other words, which rely on other words, etc., which leads to an infinite regress.

As such, words should be shortcuts for ideas, which only need be defined to the extent that we can talk about them in the context of a problem.

IOW, we should be willing to accept the terms used by others when having a discussion because discussions are about ideas, not definitions.

For example, if you start out with the definition of knowledge as justified, true belief, the discussion will go nowhere because there is no where to go.

60. 60
jdk says:

Given a line of infinite length, there are no two points that are “an infinite distance” apart. As always, I’m with Dave on this (because he’s right! 🙂 )

61. 61
daveS says:

CR,

If I understand your post correctly, I generally agree. I’m usually willing to use whatever definitions anyone proposes, as long as they are stated clearly.

But there is a fair amount of writing on this subject, and it seems pointless not to comply with the definitions that have been established through usage over the years. Especially when we refer to these writings from time to time.

There has been so much needless confusion in these threads, it’s mind-boggling. Of course, I’m always one of the first to respond when a new thread comes up. Just like Charlie Brown is always ready to practice place kicking with Lucy.

jdk,

Thanks for the support 🙂

62. 62
kairosfocus says:

DS, could you kindly summarise what an infinite past implies about past stages of the world? Specifically, in your view, explain how an infinite past does NOT imply that past actual moments have now receded beyond the finite, i/l/o the lack of power of stepwise succession in that regard. Kindly, explain how the set of actual past stages could have an aggregate duration that is different from the duration since its particular members. KF

63. 63
kairosfocus says:

JDK,

a line of infinite length is not a physically instantiated reality in an actual space, it is a concept.

That is why for instance, assessment of the E-field at a point due to a line of charge viewed as endless (or more importantly for SI Unit of current definition, the B field due to a line of current elements) is about an idealised and reasonably calculable case, not an actual one. Likewise, classic point particles etc are ideals used in Newtonian analysis. That is this is of long standing.

Such an infinite line as proposed is such that it corresponds to the reals, for which endlessness on either side of 0 is material.

That is, for any specified point along the line, there is more beyond without limit. Thus to specify or even symbolise particular points will always yield a finite separation as the beyond implies a bound, but that does not tell the whole story that is expressed in the ellipsis in many contexts or by pointing arrows in the case of lines. Indeed, it is endless continuation which is the heart of our understanding of the infinitude of the reals or the integers.

In this context, I point to the OP above to the Hyperreals and infinitesimals as giving an alternative way to conceptualise using the catapult power of the reciprocal function thus yielding a continuum that enfolds a span of numbers that when — using w for omega and e for epsilon — we take 1/e we have w, and the converse. Where e^2 ~ 0. That is there is a cloud of all but zero numbers that allows us to create a conception that is relevant.

I add: if you will, pivot discussion on two operations, + and x with identity elements 0 and 1 used as start-points. From 0 by successive addition we see unlimited incrementation beyond, leading to a view of the infinite. Namely, we see an endless successiveness and infer to a new order of quantity, order type of such endless succession, a transfinite result. Of course I have in mind here the von Neumann succession, and we can fill in a continuum by using in effect power series based on 10^-n or the like base, decimal fractions etc continued endlessly. I find this leads to a gap, where we put that ellipsis of endlessness.

However, the movement instead down the interval (0,1] from 1 leads to a different but complementary and enriching result, using multiplicative inverses thus reciprocals and projecting a continuum beyond 1. That model if you will then attains to a point as we reach e near to zero in a continuum such that e^2 ~ 0, that 1/e = w, a hyperreal. This is the Robinson-type discussion. Where I further add, the counting numbers are picked up from the succession 1/1, 1/2, 1/3, 1/4 etc as we sink towards 0, in a way that naturally supports a continuum-based view. We then simply multiply through by -1 to get the negative side. It makes sense in this context to conceive of a continuum that involves ellipsis yes when we look at the “beyond the reals” aspect — reals here being in effect also the continuum that fills in the stepwise ladder of the counting numbers. That ellipsis is where the issues of interest come as we begin to look at descent from the claimed infinite past in finite stage steps. So, no there is no reason to make a middle wall of unbridgeable partition, a result that the surreals approach also allows us to see. BTW, this also shows us how the cardinality of the finite span continuum (0,1] — a proper subset — is in correspondence with the wider reals beyond 1 and it makes for a natural extension to hyperreals.

By contrast, the real past of our world is actual, whatever it was.

Time is a feature of our world and it proceeds by a known dynamic, this is not mysterious: as one finite stage causally transitions into the next, conveniently we can speak of years.

That sort of finite stage cumulative succession gives a distinctive forward direction and it does not have power to span the transfinite. Going forward there may well be the potential infinite but looking back, there is a fundamental asymmetry of a character that entails a finite succession so far. And no, I think we can see that those who wish to suggest that the past lacks actuality (an objection that has in fact been made) are missing something fundamental about reality.

Similarly, the Zeno paradox discussion is about non-finite, infinitesimal succession yielding a finite limit. And even that may well be an idealisation — is time, in the fine, a continuum in a world where energy-time uncertainty applies giving a limit to resolution that is material to our understanding of forces and fields, including virtual particles?

Those who suggest a physical, causal-temporal succession without a beginning, imply an infinite past and need to put on the table an adequate dynamic. Suggesting that oh, for any remote past point which can access now stepwise there is an onward further beginningless past does not answer the question.

Nor is the PSR objection reasonable: a known causal-temporal dynamic is on the table.

That dynamic points to the actuality of the past and the impotence to deliver an actual infinite succession, where the time-span to the remote past cannot exceed actual durations since actual past stages of the world. So, to posit no beyond any particular past stage p there is onward endlessness in the past . . . i.e. beginninglessness . . . leads to the need for an adequate mechanism that allows causal temporal succession of transfinite nature.

It is in that context that I have pointed to the principle of actuality, such that were there an actual transfinite past there would be transfinitely remote stages of the past. The issue is not whether one can conceive in the abstract of a timeline that is infinite in the past, but that it has to have actuality that answers to the nature and limitations of causal-temporal succession.

No such dynamic has been offered, just a concept that we can have an infinite where any two particular points we can specify or symbolise and access stepwise — no catapults allowed — have a finite span. That is not in dispute. The issue is to address infinite succession.

The evidence so far is that no such dynamic is on the table.

Absent such, I remain content to conclude that we are only warranted to speak of a finite actual past of finite stages that have cumulatively acted step by step to arrive at now.

KF

64. 64
kairosfocus says:

CR, to assert that the terms of language are ultimately undefined is to abandon meaning, truth, thought, reason and communication. Further to this, a discussion on what we mean in context of accurate description and understanding of empirically examined reality is fundamental. Thus, there is no hall of distorted mirrors in a world of words about words about words: the plumb-line aptly illustrates what is straight and upright, testing what we have erected — and your attempted objection long since failed. Just so, the self-evident undeniable truths you stoutly resist serve as plumb-lines starting with the first principles of right reason you must implicitly rely on to express text in English even as you try to undermine such. I also note that no-one here at UD of consequence has sought to defend the justified true belief conception of knowledge, given the Gettier examples and the like as well as the manifest weak form sense of knowledge that is commonly used. As you know or should long since have acknowledged, we have spoken to warranted, credibly true (and reliable) beliefs, with that warrant amounting to degrees of certainty that range up to utter certainty for a relatively few self-evident truths. KF

65. 65
daveS says:

KF,

I will address this part of your question:

Kindly, explain how the set of actual past stages could have an aggregate duration that is different from the duration since its particular members.

Recall the relevant definition of infinite:

extending beyond, lying beyond, or being greater than any preassigned finite value however large.

I submit that the following principle is obviously true:

The duration of the past is at least as great as the duration between the present and any particular point in time (or stage) in the past.

Suppose we have a sequence of points in time or stages of the following form:

. . . p_3 -> p_2 -> p_1 -> p_0

where the durations of the intervals between the p_k and p_0 are all finite, but are arbitrarily large, e.g., where each such interval has duration k seconds/years/stages.

Then according to the obviously true principle above, together with Merriam-Webster, the duration of the past is infinite.

66. 66
kairosfocus says:

DS, pardon but that is not an explanation, it asserts what is to be explained per relevant dynamics. The duration since a given past stage R is -R_t, setting now = 0. there is no past beyond actual stages [taken as having finite durations in themselves] and each stage causally gives rise to its immediate finite stage successor, down to now. This creates the sequence from a finitely remote stage to now with no problem. The challenge to any claimed infinite actual past is that the past will not have duration beyond actual stages which were once the present, and stepwise succession cannot bridge the transfinite. If the past was actual and comprises a succession of stages beyond any assignable specific finite number, then it had to have had actual stages that were transfinitely remote. Such stages could not give rise to a stepwise succession to 0, per the infeasible supertask involved. Suggesting that beyond any stage R there were endlessly more in a beginningless sequence only draws out the problem it does not solve it. In short the definition that turns on an unlimited l-wards extension of finitely remote stages R, only manages to entail that for each specified R there are further l-wards stages without limit. That is no particular specified or symbolised stage has attained to the actual transfinite, the operative point is in the l-ward ellipsis which points onward to the implicit transfinite. KF

67. 67
daveS says:

KF,

It explains how “the set of actual past stages could have an aggregate duration that is different from the duration since its particular members”.

The “aggregate duration” is infinite, yet the duration since each particular member is finite.

68. 68
kairosfocus says:

PS: Merriam-Webster:

Definition of infinite

1 : extending indefinitely : endless

infinite space

2 : immeasurably or inconceivably great or extensive : inexhaustible

infinite patience

3 : subject to no limitation or external determination

4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large

infinite number of positive numbers

b : extending to infinity

infinite plane surface

c : characterized by an infinite number of elements or terms

an infinite set

an infinite series

It seems the key point of meaningfulness is in a blend of 1, 2 and 4a:

1 : extending indefinitely : endless . . . . 2 : immeasurably or inconceivably great or extensive : inexhaustible . . . . 4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large

Notice, the key operative terms: “endless . . . . inexhaustible . . . . extending beyond, lying beyond, or being greater than any preassigned finite value however large.”

Those seem to be central to the problem.

69. 69
daveS says:

KF,

Yes, in my illustration, the duration of the past is greater than any preassigned finite value however large.

70. 70
kairosfocus says:

DS, the key issue is actuality. Surrounding the sequence of successive, cumulative, causally active stages with a conceptual bracket does not add any duration- to- now whatsoever that is not already present in the particular members. Thus, again, the key is what I spoke to two years ago as the ellipsis of endlessness [or I add: limitlessness or beginninglessness]. The infinity proper lies in the extension beyond any given finite member, not in the particular finite members themselves, by definition, such are exceeded, e.g by adding 1, 2, 3 . . . with that telling ellipsis there yet again. Let me add, further, that the very description of a member that it is finite implies that up to here cannot exhaust the membership, it continues beyond without limit if actually infinite. Where, there is no dynamic of stepwise succession that can traverse a transfinite span. KF

71. 71
daveS says:

KF,

Do you therefore agree with my post #65?

72. 72
kairosfocus says:

DS, I have pointed out my concerns already. KF

73. 73
daveS says:

KF,

Until I get a clear yes/no answer to my #71, I’m not willing to continue to the more complex aspects of this discussion. It’s essential that we each know what definitions the other person is using, for obvious reasons.

74. 74
kairosfocus says:

DS,

Let us follow up that definition. Can we with profit pre-assign a transfinite number and view it as a reasonable extension to the naturals and reals?

I believe so, let me clip from 63 above:

an infinite line as proposed is such that it corresponds to the reals, for which endlessness on either side of 0 is material.

That is, for any specified point along the line, there is more beyond without limit. Thus to specify or even symbolise particular points will always yield a finite separation as the beyond implies a bound, but that does not tell the whole story that is expressed in the ellipsis in many contexts or by pointing arrows in the case of lines. Indeed, it is endless continuation which is the heart of our understanding of the infinitude of the reals or the integers.

In this context, I point to the OP above to the Hyperreals and infinitesimals as giving an alternative way to conceptualise using the catapult power of the reciprocal function thus yielding a continuum that enfolds a span of numbers that when — using w for omega and e for epsilon — we take 1/e we have w, and the converse. Where e^2 ~ 0 [ADDING: , so also [2e]^2 ~ 4 x 0 –> 0, i.e. we can see a stepping down in this zone to 1/2e = 1/2 x 1/e = w/2 etc and will recognise that e is a yardstick for a zone that also extends above it significantly.] That is there is a cloud of all but zero numbers that allows us to create a conception that is relevant.

I add: if you will, pivot discussion on two operations, + and x with identity elements 0 and 1 used as start-points. From 0 by successive addition we see unlimited incrementation beyond, leading to a view of the infinite. Namely, we see an endless successiveness and infer to a new order of quantity, order type of such endless succession, a transfinite result. Of course I have in mind here the von Neumann succession, and we can fill in a continuum by using in effect power series based on 10^-n or the like base, decimal fractions etc continued endlessly. I find this leads to a gap, where we put that ellipsis of endlessness.

However, the movement instead down the interval (0,1] from 1 leads to a different but complementary and enriching result, using multiplicative inverses thus reciprocals and projecting a continuum beyond 1. That model if you will then attains to a point as we reach e near to zero in a continuum such that e^2 ~ 0, that 1/e = w, a hyperreal. This is the Robinson-type discussion. Where I further add, the counting numbers are picked up from the succession 1/1, 1/2, 1/3, 1/4 etc as we sink towards 0, in a way that naturally supports a continuum-based view. We then simply multiply through by -1 to get the negative side. It makes sense in this context to conceive of a continuum that involves ellipsis yes when we look at the “beyond the reals” aspect — reals here being in effect also the continuum that fills in the stepwise ladder of the counting numbers. That ellipsis is where the issues of interest come as we begin to look at descent from the claimed infinite past in finite stage steps. So, no there is no reason to make a middle wall of unbridgeable partition, a result that the surreals approach also allows us to see. BTW, this also shows us how the cardinality of the finite span continuum (0,1] — a proper subset — is in correspondence with the wider reals beyond 1 and it makes for a natural extension to hyperreals.

Clearly (0,1] is a continuum, and the relationship of multiplicative inverse leads to the hyperbolic function that takes reciprocals. There is no reason why we should not see in this way a continuum extending to ever higher values of reals as the number being inverted sinks ever closer to zero. It makes sense to see a zone that is all but zero such that e^2 ~ 0. The multiplicative inverses of these members of the interval, points on the restricted line from 0 to 1, have every right to be regarded as numbers, numbers beyond any finite value that we can assign.

Transfinites.

The more or less familiar hyperreals.

With, the rehabilitated infinitesimals as the bridge-anchor to the continuum of numbers in (0,1].

(Note SEP here.)

We can then conceptually explore this richer terrain accessed by a more powerful set of operations than +.

Here, we can see w appearing and extending beyond in a further continuum, one that is somehow connected to the natural numbers and reals. The ellipsis appears beyond any definable finite, indicating that we have passed a traverse that successive finite stage addition cannot traverse. We can speak to the onward surreals but it is enough to say, subset.

With aid of the ellipsis, we can step down from w, as the OP illustrates, and as the surreals also illustrate. Of course coming back down in finite stage steps will not bridge to the finite span from 0.

With this broadened picture I think we can more clearly see my concerns.

We can sensibly represent some Q as being comparable to w, and see that -Q_t would be a transfinite. We can see why stepping down in increments from a transfinite to get within finite span of 0 would be futile. Where, actually, stage succession is like pushing a plate on top of a sinking, push-down stack. Can we get from Q to 0 from the l-ward side by a stacking process? No.

So now to bridge to the claimed infinite past, is this just a claim to finites beyond counting?

I think that concept is incoherent.

If we are in the beyond counting and bounding by successor range we are in the hyperreal transfinite domain.

And that brings out the concern on oh we already load up a beginningless past that is infinite l-ward at any given R that can count down to 0, now.

KF

75. 75
kairosfocus says:

DS, I will not play the yes/no interrogator lockup game. I have repeatedly and with significant discussions pointed out why I am unhappy with the constructions that have been put. I am currently expanding on the concepts involved as just now. I have reason to hold my views and have my concerns, they are not just irrational or stupid, incoherent, stubborn rants — the force of some dismissive language above is all too plain. By dint of right of reply, I already have reason to know that my intuitions and explorations of was it two years ago were given inappropriate short shrift, and I think I have a right to draw them out in these current reflections. Maybe I will see that there are needed changes, but so far it looks to me like the power of the ellipsis beyond limits needs to be clarified. It looks a lot like, a second approach to the transfinite through reciprocals is fruitful and shows ways to fill in what is not communicated by successive addition. It also bridges to issues of coherence in the Calculus in terms of learning and forming concepts. KF

76. 76
jdk says:

When I wrote, “Given a line of infinite length, there are no two points that are “an infinite distance” apart.”,

kf replied

a line of infinite length is not a physically instantiated reality in an actual space, it is a concept.

This is true. There is nothing that we know of that is of infinite length in any actual space. When talking about infinity, I don’t see how we can be doing anything other than talking about concepts.

Also, kf writes,

Thus to specify or even symbolise particular points will always yield a finite separation

I take it therefore that kf agrees with my original statement.

All the rest of the concepts he offers about hyperreals etc. are additional thoughts and topics, but they don’t change the fact that there are no two points that are “an infinite distance” apart.”

77. 77
daveS says:

KF,

I’m not trying to “lock” anyone up here, certainly.

However, you made a request in your post #62, which I responded to in #65.

Isn’t it fair to ask you whether my post #65 is a satisfactory response to your request?

I simply cannot tell from your #70.

78. 78
daveS says:

KF,

If we are in the beyond counting and bounding by successor range we are in the hyperreal transfinite domain.

But we aren’t in the “beyond counting and bounding by successor range”. All the proposed points in time/stages are finitely many years/seconds/steps from the present.

79. 79
kairosfocus says:

JDK, it seems you keep missing the issue. I am not defending that our actual world timeline can have an infinite, beginningless past. I am showing that such a claim implies that there are actual past stages that were once the present but have been succeeded stage by finite stage to now be transfinitely remote. I have put on the table ways to represent that situation, and we can see that they are inconsistent with the dynamic of stepwise, finite stage succession. The idea that oh we can assert that at any past stage R, there always were limitless further past stages (think, years for convenience) — notice, the telling beyond limits — amounts to saying that we have a cumulative transfinite past, as the set of past events cannot have a duration beyond the particular events in it. If it is transfinite, there are stages in it that are at transfinite remove. The set only adds conceptual braces around the actual past stages, it does not manufacture additional duration out of the mere act of collection. Last, that any actual separation of any two distinct stages in time will be finite is another way of saying the past is finite. KF

80. 80
jdk says:

No, kf, I’m not “missing the issue”: I am just not interesting in, nor addressing, the issues your are discussing about time and the nature of the past.

I am merely saying that, to use your words, “any actual separation of any two distinct stages in time will be finite”, and I agree with you about that, and you agree with me that “there are no two points that are “an infinite distance” apart.”

That’s all.

However, that also means that the phrase “transfinitely remote” in respect to points on a number line is meaningless.

81. 81
kairosfocus says:

JDK,

transfinite separation on a number line is actually meaningful, cf. OP. More relevant to our concerns, if any two stages of the actual past will be finitely separated, then we have a fairly good reason to hold that it, as a whole, is therefore finite.

Just set one as now [ at 0], the duration since any other stage R, -R_t will be finite.

Conceptually collecting the causal-temporal sequence of stages into a set cannot add any further duration to them.

But of course, what is being pushed in is the ellipsis, indicating that beyond any particular R, there are limitlessly more prior stages, beyond any particular natural number, regardless of how large. Thus, we are invited to accept that there has been an infinite causal-temporal succession of past stages (years, for convenience), all of which are somehow also finitely remote from now.

Further, we are invited to accept that the limitless succession is always already accomplished when we contemplate a given point or stage, so that succession from some specific past point R to now is of no consequence.

My first concern is that no causal-temporal stepwise, finite stage succession can traverse an actually transfinite number of stages, for fairly obvious reasons. After k stages we may go on to k+1, k+2, etc. So, we are as good as at the beginning, 0.

But, it will be suggested: at any particular stage, there is no beginning, the limitless succession has already happened.

Already, a problem, the big event is off stage, invisible, unobservable and inexplicable. Yet, it is the key to the whole.

Big questions lie unanswered, even unanswerable on the scheme of thought.

There is even a suggestion, to ask for an intelligible, reasonably empirically or analytically anchored explanation is to resort to the much deprecated principle of sufficient reason. However, to inquire into why is not to presume an answer, though we have a known dynamic of succession at work and given any present state — it needs not be initial — a transfinite traverse of stages cannot be completed, only a finite number.

Recall, the actual past is such that every past state was once the present, then was succeeded by the next and so forth. So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is infinite, beyond any finite span of stages.

There are fairly obvious tensions here.

But of course, circular cause and appearance from nothing are appeals to cause from non-being. A finitely remote beginning requires a necessary being world root, which to many is unacceptable for essentially worldview reasons. That leaves the above or something much like it as the view to be stoutly held.

That set of alternatives speaks.

KF

82. 82
jdk says:

kf writes,

JDK, transfinite separation on a number line is actually meaningful, cf. OP

Not if we are just talking about the reals (or the integers) on a number line.

kf writes,

More relevant to our concerns …

,

but he means “More relevant to my concerns …”, as I am not interested in the issue of an infinite past.

83. 83
daveS says:

KF,

Last, that any actual separation of any two distinct stages in time will be finite is another way of saying the past is finite.

I say this is false.

Now how do we resolve the dispute?

84. 84
jdk says:

Hi Dave: my apologies for busting in, as the main topic here, between you and kf, is whether an infinite past is possible. (I think that’s the topic.) The fact that the the distance between any two points is finite appears to be accepted by both of you. So I’ll bow out, and let kf think about your 83.

85. 85
daveS says:

jdk,

I’m not the thread owner, but these related points are of interest to me as well. I think there is essentially no chance of agreement on the specific question in the OP, but the most interesting parts of these threads has been the side discussions that arise.

86. 86
jdk says:

So Dave. What do you think of this formulation:

Let the number line be a model for time, with an arbitrary point P0 the present, and the future in the positive direction and the past in the negative.

For convenience sake, consider the integers only (with the reals to be understood as “filling in” the rest of the line), so that there is a discrete distance of 1 unit between “moments”.

For any positive point F in the future, F + 1 is even further in the future. Therefore, the future is infinite in the sense of being unbounded or unlimited, even though every point in the future is a finite distance from the present. There is no point in the future that we can’t eventually “get to” from the present, even though there is no limit to how far in the future we might go.

Likewise, for any negative point P in the past, P – 1 is even further in the past. Therefore, the past is infinite in the sense of being unbounded or unlimited, even though every point in the past is a finite distance from the present. There is no point in the past from which we can’t “have gotten to” the present from, even though there is no limit to how far back in the past we might go.

The confusion here is that “infinite” is not a place on the number line, or a property of a particular point: you can’t be “infinitely far away” from the present. “Infinite” refers to a property of a sequence of numbers, stating that further numbers can be generated without end.

87. 87
kairosfocus says:

DS, I made several responses in the range 60 – 7 and am surprised that you do not find them clear enough. I summarise, that the thing to be explained is the claimed infinite past, i/l/o the context of a known step by step dynamic that cannot span an infinite range. I find that you are in effect saying that the infinite span of steps — the main thing to be explained — has already happened at any given R leaves the key question on the table as though it has no significance. Recall, the actual past is such that every past state was once the present, then was succeeded by the next and so forth. This also means that the set of the past events and stages cannot have duration beyond that of these events or stages. Putting them inside a set of braces adds nothing, it is the ellipsis of limitlessness that is key and what it here implies is an assumed completed infinite succession already completed. So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here. You may believe that an infinite succession of the actual past has already occurred at any specific point, that that infinite succession was somehow spanned step by step, that every stage in that infinity is only finitely remote from us but the whole is beyond all limits in the past — where finite directly implies within a limit, and so forth, but I think something is wrong with the cluster of claims. KF

88. 88
kairosfocus says:

JDK,

on the claim of an actualised past infinity, I am by no means convinced that we are speaking of the reals only, that needs to be shown — especially as the duration since the past is a duration since specific actual events or stages. You may not be interested in the claimed infinite past but that claim is interested in you, on the roots of the world around us.

Your projection of a future as infinite fails to reckon with the difference between a potential and an actual infinite. That is, from any given now, the future is open-ended but cumulative. Likewise, you have simply restated that the past is infinite, begging the question at stake, especially given the dynamic at work, stepwise succession of — for convenience — years. The key thing that lends limitless character to the negative number line is the ellipsis or the pointing arrow, indicating as a concept limitless continuation . . . no-one has argued that there is a single point of transition to the infinite n + 1 = “infinity,” it is the property of limitlessness that leads to recognition of a new type of quantity, the transfinite.

However, the analysis on taking reciprocals from (0,1] as the values tend to 0 above goes beyond that which additive succession gives us, allowing us to see how there is a reasonable continuity of range though of course there is that ellipsis there to indicate there is a span of transfinite character involved. That is, the hyperreals perspective allows us to see that as we tend to an e such that e^2 ~ 0 we are in a relevant range that is transfinite for 1/e = w/1, though of course there is continuity right down to zero on the interval implying a fundamental unity to the whole process.

By contrast successive addition only attains the potential infinite, as it has weaker power. The ellipsis gathers up “the rest” in one conceptual leap.

And as the negative reals are in effect the mirror image if we know that we cannot span up to the actual infinite by successive addition, there is reason to see that claiming to descend form the same scale is not going to be feasible, where claiming oh it has already happened at any given R seems very evasive.

But stepwise succession of transfinite character is precisely the thing to be grounded on a reasonable dynamic of causal-temporal succession, where there is reason to see that such a succession cannot span a transfinite.

I note, circular causation in a chain implies that a future, non existent entity causes itself, and that descent from infinity in steps is questionable, leaving the only serious option, a finitely remote world-root. This last is also supported by the point that the entropy of an infinitely old world should have long since reached to utter degradation.

KF

89. 89
daveS says:

jdk,

I do agree with everything you said in #86. I think that’s the “standard” model of time that practically everyone uses, and that while timepoints separated by infinite distances are conceivable, they are not physically realistic.

90. 90
daveS says:

KF,

I find that you are in effect saying that the infinite span of steps — the main thing to be explained — has already happened at any given R leaves the key question on the table as though it has no significance.

I’m not claiming it has no significance. Keep in mind I’m not trying to “explain” how all this came about. I’m in effect asking whether you can show all this is impossible.

This also means that the set of the past events and stages cannot have duration beyond that of these events or stages.

I don’t agree with this principle.

How would you apply it to a set of past events where there are events at arbitrary finite distance from the present? That is, where there are events 1 step before present (BP), 100 steps BP, 10^10 steps BP, 1 googol^googol steps BP, and in fact at any positive integer number of steps BP.

What is the duration of this particular past? It’s clearly not the duration between any particular event and the present. But if it’s finite, it must have some definite value, correct? What is that value, and how do you make that choice?

So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here.

That is an accurate description of the scenario, of course, but I am not detecting any tension.

91. 91
daveS says:

PS:

So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here.

I will point out that even if we call such a past “finite”, in such a scenario the universe would not have begun finitely many years/stages ago. Therefore in order to get the Kalam Cosmological argument to work, one would need to show this sort of “finite” past is impossible.

92. 92
kairosfocus says:

DS,

I have been busy, let me turn to something key.

I see you in 90:

{KF:] This also means that the set of the past events and stages cannot have duration beyond that of these events or stages.

[DS:] I don’t agree with this principle.

How would you apply it to a set of past events where there are events at arbitrary finite distance from the present?

It is central to our considerations to recognise that we are dealing with the real world, not a mathematical thought space. Collecting the chain of past events or stages into a set by in effect putting conceptual braces around them CANNOT change the actual stages of the past. And duration to now for a past stage R is – R_t. That is, duration since attaches to the actual stage of the past.

As a direct consequence, the duration of the actual past as a whole similarly attaches to the chain of events.

Surely, it is possible for that chain to be finite, in which case the overall duration of the chain is Max ( – R_t), i.e. the duration since the first event. And if the chain is infinite, for argument, then it is because particular events have been so far superseded by the next then the next etc that they are now remote beyond any arbitrarily high stage count.

I have tried to capture such an event, Q, by using models of the number line that use hyperreals tied to infinitesimals, using reciprocals and the hyperbolic function to catapult to such a range. This, because you have spoken to beginninglessness again and again. For example:

Beginningless so transfinitely remote range, where on actuality, q is a member:

. . . q –> q+1 –> q+2 –> . . .

procceeding stepwise to a finite range down to now:

–> p –> p+1 . . . –> 0

Reducing and putting braces to highlight the set of the actual past, S:

S = {. . . q –> q+1 –> q+2 –> . . . –> p –> p+1 . . . –> 0}

This, you have objected to, in effect trying to impose that the values are all finite but go beyond any arbitrarily high specific finite value:

S = {. . . q –> q+1 –> q+2 –> . . . –> p –> p+1 . . . –> 0}

Ironically, it is exactly the issue of finitude of stepwise R-ward succession that has been central to my point: e.g. stepwise cumulative succession of finite steps from a reference point 0 may amount to an arbitrarily large value k, but such a value will then be bounded and exceeded by k+1, k+2 etc.

Thus, I have emphasised the impotence of stepwise succession to actually span a transfinite range, to actually reach the infinite. In that context I have consistently pointed to the ellipsis or with lines the pointing arrowhead. Sometimes, “etc.” fills this role.

Stepwise succession may point beyond itself to the potentially infinite, but it cannot span to that zone that is so easily reached by the hyperbolic catapult from infinitesimals in the continuuous interval (0,1] . . . where continuity is key and helps us appreciate that e such that e^2 ~ 0 is in the line in the near-neighbourhood of 0 where taking reciprocals will fail at 0 itself. This is how we reconcile speaking of hyperreals beyond any finite counting number k attainable by stepwise succession from 0.

Now, time acts causally and cumulatively, R-ward in the sequence. In speaking of beginninglessness, you are focussing on an ellipsis L-ward, and wish to confine it to speak only of an indefinite, limitless L-ward extension of predecessor finite values.

The root challenge here is tied to how we get the L-ward side of a number line, multiply through the R-ward side by – 1.

We thus can reflect on how a succession from some j to j+1 etc and onwards to k can be exceeded by k+1, k+2 etc on. That is, stepwise succession cannot actually span the limitless, beyond any arbitrarily high but finite span [k – j]. Where, multiplying through by – 1 so to speak, or merely reversing direction cannot change the underlying dynamic of only attaining to a finite span of succession.

In this context, playing around with ellipses that are now finite then transfinite then finite again to suit points becomes fatally equivocating. For, there is no clear dynamical means to span a limitless range. And, it therefore makes sense to speak of some q in that far past zone which is so far succeeded that it is limitlessly, transfinitely remote.

Such a q cannot be succeeded stepwise to 0, now.

That is clear.

What is a lot less clear is whether the murky zone l-ward ellipsis of the reduced succession you keep on truncating to is a way to have q without acknowledging it.

Likewise, we do not get to impose what is convenient to our worldview preferences by default on logical possibility in our view is enough to effectively infer truth or actuality. Comparative difficulties on factual adequacy, coherence and explanatory power apply.

Just what is a beginningless, beyond any specific n in N span in the past if it does not contain any actually transfinite members of the class w etc?

Does that make a material difference?

It seems to me extraordinarily like a have your cake and eat it rhetorical situation.

Especially, where the same causal dynamics by which the present stage becomes the next have lurking in them the inexorable degradation of energy availability to do work. And, for feasibility, all materially relevant conditions must be fulfilled. Indeed, entropy has been spoken of as time’s arrow. An actually beginningless, ultraremote past for a cosmos that as a whole constitutes an isolated thermodynamic system will — in succeeding itself stepwise to now — have so degraded its available energy by now that the sort of life we enjoy would not be possible. IIRC 10^25 s is a yardstick of such time, about 100 mn times the duration since the big bang. Large but definitely finite and well within 10^100 s.

Of course as Sears and Salinger remarked in my foundational thermodynamics textbook, that the physical cosmos as a whole is isolated is an open question. That is, a quasi-infinite physical world would be contingent on an external enabling entity. One, extraordinarily like God, if one is interested in cosmological arguments. (My actual focus is on the physical import of the temporal-causal successiveness we note, as we look back to origins.)

A world of actual physical reality makes the suggestion of a beginningless past highly questionable, on grounds tied to causal-temporal successiveness. Even, if one can eliminate q definitively and avoid effectively putting it back in under another symbol, j.

Back on that, it seems to me that stage R taken as a pure counting number r, will span all ordinals in the L-ward sense. That leaves very much open that some would be beyond w. With the hyperbolic catapult and -1 multiplier mirrors in play, there is no inherent reason that I see to cut off cases where r is in a zone where the yardstick

-(1/e) = – (w/1)

does not apply.

That is, number line and REAL number line are different, the latter being a subset. Where a function based on the interval (0,1] is able to generate the wider span.

On reflection, the restriction to finitely large values is implicitly shaped by the physical consideration of actual stepwise succession of stages. But, physical considerations cannot be cherry-picked, feasibility requires that the implied energy dissipation in causal-temporal succession also applies. So, an actually transfinite span — murky or not — cannot be actually bridged and were such to happen, energy in the cosmos would have so degraded that life such as we experience would be physically impossible.

More to the point, an oscillating cosmos type model would dissipate after ~ 100 bounces and the evidence seems to be that we are having accelerating expansion rather than the slowing down that marks moving to an extreme of an oscillation. A budding-off or bubbling model would run out of available energy. Multiverses popping up out of utter non-being are not plausible. Circular causation is a species of popping out of nothing.

All of this leaves as best explanation, finitely remote origin from a necessary being world-root. (Indeed, we are actually debating the nature of such a NB, and my note on energy dissipation is a way of saying the physical cosmos — even if extended to a beginningless past — is radically contingent so not a plausible NB.)

KF

93. 93
daveS says:

KF,

And if the chain is infinite, for argument, then it is because particular events have been so far superseded by the next then the next etc that they are now remote beyond any arbitrarily high stage count.

If by this, you mean just that there is no maximum separation between the present and past events/stages, yes, I agree. If you mean there is a particular past event/stage separated from the present by infinitely many steps, then I would disagree.

What is a lot less clear is whether the murky zone l-ward ellipsis of the reduced succession you keep on truncating to is a way to have q without acknowledging it.

Likewise, we do not get to impose what is convenient to our worldview preferences by default on logical possibility in our view is enough to effectively infer truth or actuality. Comparative difficulties on factual adequacy, coherence and explanatory power apply.

From my perspective, you are the one adding these infinitely distant q-points to the standard model of an infinite past without justification. The standard model which theists such as WLC and David Snoke use, incidentally.

In order for me to take this model with q-points seriously, you would need to provide a rigorous proof that they must exist, that is, that IP(−ω) implies IP(+ω). I do acknowledge that you have attempted arguments in this direction, but AFAICS, they have been non-sequiturs.

Just what is a beginningless, beyond any specific n in N span in the past if it does not contain any actually transfinite members of the class w etc?

Does that make a material difference?

It seems to be, for both of us.

You have argued strenuously that it is nonsensical to call such a past “infinite”.

I have pointed out that it would be most convenient to use definitions consistent with that of WLC and the rest of the published literature on the subject, so we can at least be clear on what those authors are saying.

On reflection, the restriction to finitely large values is implicitly shaped by the physical consideration of actual stepwise succession of stages. But, physical considerations cannot be cherry-picked, feasibility requires that the implied energy dissipation in causal-temporal succession also applies. So, an actually transfinite span — murky or not — cannot be actually bridged and were such to happen, energy in the cosmos would have so degraded that life such as we experience would be physically impossible.

Recall that I am essentially working in a theistic perspective here, as is WLC. Ultimately, I am challenging his Kalam argument.

Now recall some common theistic assumptions: God created the universe, God exists in some timeless realm and can view all of our time at once, and very importantly, God intervenes occasionally our universe.

Surely under those assumptions, God could step in every few billion years and “turn back the entropy clock”.

94. 94
kairosfocus says:

DS,

what I have argued is that there is good reason to doubt claims that are sometimes made for a beginninless physical cosmos. Those reasons start with what is REALLY meant by such a claim.

If the past counted by stages [e.g. years] extends limitlessly beyond any natural number n in N, then that is a claim to a past infinity. As the past is not a concept but actuality that has been succeeded by subsequent stages though causal processes, then it is natural to ask what would such an infinitely remote actual stage look like.

The natural response would be to model such, and the hyperreals line gives us a way. The dynamic of succession indicates that a stepwise incremental process cannot span from such a past to now.

The answer is, no, what is meant is that every actual past value is finite but the overall chain extends limitlessly beyond any finite counting number n.

That is at least close to borderline incoherent.

The response is, putting conceptual braces around the chain does not give it a duration that extends beyond that of actual past stages. The above is perilously close to implying such stages of order w.

No, they are all finite but extend beyond any particular finite counting number n, no matter how large.

On one side, that suggests at first an indefinitely large but finite duration. But no, it must go beyond any such ordinal . . . and a successive count to the past is a chain of ordinals. The order type of that chain is of course w, a transfinite ordinal.

But then, such a suggestion is before actual proofs:

0,1, 2 … w . . .

The ellipsis of limitlessness in the context of counting numbers points to that order type.

But, maybe it is just a recognition of endlessness of succession appended to the naturals for convenience?

The other approach, the hyperreals unifies the two, no it is not simply recognition of a new class of quantity, it is legitimate to see the number line extended to hyperreals.

So, I say, values q of order w exist as numbers, the question is, can they exist as past times. If no, then the ellipsis needs to be explained in terms of what its order type means as applied to the actual temporal order. Leaving that vague as every case is finite but the cases go on and on beyond any n in N sounds less than satisfactory.

It sounds suspiciously close to exploiting the grey zone between concepts to have one’s cake and eat it.

And that is before the thermodynamic limitations come in play.

Where, if your intent is to address the Kalam cosmological argument (in which, oddly I have little interest) then invoking God as stabilising thermodynamics seems to yield the wider point. On cosmological reasoning I would suggest that the existence of a contingent world suggests dependence on a higher order of being.

I don’t think your objection is a serious one.

Back to WLC, let me again note from the SEP summary, clipping 35 above, also addressed to you:

5 An actual infinite cannot exist.

[He has many reasons for that, which are often hotly contended but on the whole I think he has a serious point . . . at minimum he has shown that modern atheism effectively is reduced to a highly contentious assumption about the past, which is very different from its boast that it can be seen as a default for the intelligent person]

6 A beginningless temporal series of events is an actual infinite.

[Note this]

7 Therefore, a beginningless temporal series of events cannot exist.

Since (7) follows validly, if (5) and (6) are true the argument is sound . . . .

>> 6.3 Successive Addition Cannot Form an Actual Infinite

Craig’s second argument addresses this very point.

8 The temporal series of events is a collection formed by successive addition.

9 A collection formed by successive synthesis is not an actual infinite.

10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103).

The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite (Craig and Sinclair 2009: 118).

It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future.

[–> relative to the distant past in view. This is also the context in which I have consistently spoken of finite stage causally successive stages]

Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.

In short WLC is raising arguments that should sound at least vaguely familiar.

If one explicitly argues that the l-ward ellipsis points to actual past moments q or order type w or leaves the matter in the vague terms of oh the l-ward succession is beginningless and has members that exceed any specific n in N no matter how large, it seems that you are trying to traverse the transfinite in finite stage steps.

A futility.

KF

95. 95
daveS says:

KF,

May I suggest you contact David Snoke or WLC and present your ideas to one of them? At this point I don’t have anything more to say.

96. 96
kairosfocus says:

DS, I thank you for the discussion. However, it is still a question: does or does not a beginningless past with members of the actual past being beyond any particular stated n in N, imply a traversal of the transfinite? If it does, I suggest the transfinite traversal done stepwise is questionable, and I would say credibly a futility. Also, to suggest that for any particular R in the past the traversal from the beginningless past can be regarded as already complete, seems to me to be a skirting of the precise point at stake. So, while many do argue such a past, and while it is obviously so hard to discuss it with any reasonable hope of a resolution that I can see why many would not address it at all, that question is there on the table. KF

97. 97
daveS says:

KF,

Thanks to you as well.

98. 98
kairosfocus says:

F/N: Perhaps, it is time to draw up some conclusions, and make statements of appreciation.

First, deep appreciation must be extended to DS, JDK and other interlocutors. DS has been particularly helpful over two years, indeed it is he who drew my attention to the surreals as a means of seeing the grand picture of numbers great and small. Despite disagreements and divergence of views, that has been important.

Now, what of the state on the merits? Of what significance is all of this stuff on abstruse topics linked to numbers?

Perhaps, we can start with an extended form of the definition of Mathematics I was taught in M100 by a famous prof: mathematics is [the study of] the logic of structure and quantity.

So, if we are to properly, truthfully understand the world, we should study that logic, the issues of structure and the nature of quantities involved with “numbers great and small.” The surreals are important, and the hyperreals (thus also, infinitesimals).

That search for truthful, adequate understanding has been my main motive, a motive that I recommend to one and all.

Closely linked is the educational issue: how can we bring more and more to a better understanding of these things? (Let me disclose that a bit less than two years ago I was a guest on a local radio show on education matters, and these general issues were focal. I found it particularly interesting to see the public’s reaction to my tying-in music with Mathematics and Physics. That may be a hint on reaching out to people where they are.)

Next, I find that the concept of the hyperreals is truly significant. Especially, by way of the thought-exercise of allowing a point of interest to follow the locus of sinking from 1 towards 0 along the open continuum- interval (0,1] and taking the reciprocal, 1/x.

This generates the line of the reals beyond 1 as a continuum also. Along that line we see 1/1, 1/2, 1/3 etc generating the marker-points that come up from von Neumann-style succession or successive addition by 1’s. Let’s use von Neumann as his approach shows how abstract and powerful numbers are in light of (i/l/o) set concepts. To do so, start with the empty set and denote what are called order types, extending to omega, which for convenience we can represent as w:

{} –> 0
{0} –> 1
{0,1} –> 2
{0,1,2} –> 3
. . .
{0,1,2,3 . . . } –> w

We see here how the set that collects nothing has cardinality (“size”) zero, and let that define our start point for quantity. Zero is an entity with distinct identity. It is something by contrast with nothing — and yes the triple first principles of right reason are embedded here: world = {A|~A} where A is itself i/l/o core characteristics that mark its distinct identity and ~A is another entity distinct from A, the rest of the world and what is in it. So too we may see the corollaries, LNC and LOI: no x in World is A AND ~A, any x in World will be A X-OR ~A, in one or the other, not both, not “nowhere.”

But this already gives us one, and two.

Further distinct identities follow in a chain, 3, 4, etc. The three-dot ellipsis — how I wish it were standard to use four dots for limitless extension — then indicates an endless ordered succession. This is complementary to

1 + 1 + 1 + . . .

We then find that we have countable numbers in endless succession, that endlessness allowing us to recognise a new type of quantity, the order type of the natural counting numbers, omega [here, w].

The set of the naturals is deemed infinite, by way of that ongoing endlessness of succession. In this context, w is the first transfinite ordinal number; denoting the order-type of the naturals.

Back to the 1/x on the continuous, open interval (0,1]. And by multiplying through by -1, we have the negative number line.

Here, we can see the filling-in between the naturals [and integers], which decimal representation allows us to complete by using ever finer decimal fractions without limit. This, we also see in the inner branches of the surreals diagram. The line of the reals is the continuum towards w. The idea of the surreals is to squeeze in on specific values from above and below like the closing jaws of a vice.

Now, we know that 1/x, x = 0 is forbidden, as there are all sorts of resulting problems. However, on the brink of 0, very interesting things happen. Let e be such that e^2 ~ 0, a yardstick infinitesimal. Where [2e]^2 ~ 4 x 0 = 0. Now we hold by selection of the “appropriate” value of e:

1/e = w/1

so, too

1/2e = 1/2 x 1/e = w/2

We unify, and extend the number line to the hyperreals. This goes beyond the somewhat unsatisfactory picture given by whole number succession, that w is the order type of the naturals.

Of course, the point is that (0,1] is a continuum, so the numbers line is here unified for the very small and the very large. With, the span beyond w also a continuum as we can keep sliding in in the all but zero range without actually hitting 0.

And we now use the property that e is not a FIRST infinitesimal but a YARDSTICK one to see that the range of numbers naturally assigns to something like w/2 and the like. As a consequence we can make sense of the idea that infinitesimals are smaller than any real, and hyperreals are larger than any real. That is, the ellipsis marks that veiled grey zone in which finite results yield to infinitesimals on the brink of 0, that are tied by the catapult hyperbolic function 1/x to hyperreals beyond any finite real. Where, of course, the natural counting numbers are the whole number mileposts along the way.

We have built up a significant set of concepts which allow us to better address structures and quantities.

Let me add that if we look at planar space and locate the numbers line as the o-x axis, we may then suggest an anticlockwise, right angle rotating operator i such that i*x gives us a perpendicular axis. Apply i twice:

i*i*x = – x

We just defined i as the square root of -1. We also define a quantification of planar space, where the orthogonal [= perpendicular] o-y axis is the i*x axis. This actually gives us a way to define a new class of numbers, complex numbers:

z = a + i*b

By watching our i’s carefully, we now have a way to handle quantities with both magnitude and direction, algebraically. Complex numbers are one algebraic way to handle vectors, with powerful results all over mathematics. And, we may extend to three-axis vectors using the ijk unit vectors scheme. (And BTW, that is one reason Physicists use j for the sqrt – 1, it helps further unify.)

Infinitesimals, of course are one gateway to calculus. In effect every real number is surrounded by a cloud of neighbouring infinitesimals. This gives a fairly straightforward interpretation to many Calculus operations. The C19 delta-epsilon, limits based approach came up to handle issues triggered by naive use of infinitesimals. Just as, naive set theory has had to be refined. In pure Mathematics, much effort has to be put in in working around truly pathological cases.

I think this picture is a useful exercise for its own sake.

It also helps us to clarify what is at stake in discussing what it means to claim that the [quasi-]physical cosmos has had a beginningless, thus infinite, past. That is, the logic of structure and quantity includes the bridge between the finite and the transfinite, which then applies to the question of a world or a world-root that was always there. The logic of being applied to the being and origin of the physical cosmos, is not without reference to the logic of structure and quantity.

Truth is unified, coherent, as all truths must hold together in a world, and if truth A is the denial of truth B that would not obtain.

Now, the source of our world cannot be utter non-being, as such hath not causal powers. Were there ever utter nothing, such would forever obtain. That a world in which we exist as self-aware, conscious en-conscienced, thinking and understanding creatures self-evidently exists as a going concern, SOMETHING always was. The debate in cosmology is the nature of that world-source.

Further to this, the world is spatial and temporal. Moving beyond Zeno’s paradoxes, we can consider the going-concern world as a structured set succession of finite-duration stages — years, for convenience — down to now, S:

S = { . . . –> p –> p+1 . . . –> 0}

Now is ever moving on from one stage to the next, there is a cumulative, causal-temporal order to the physical world.

Now, our observed world is generally held to have come from a “singularity” some 13.85 BYA, the big bang. A finitely remote origin. But, whence the “bang”? Not from utter non-being. Down that road lies a great discussion populated with fluctuating quantum foams, oscillating worlds, budding sub-cosmi, multiverses, strings and branes etc.

However, we may look from a different angle, Agrippa’s trilemma in causal form on going concern world-source:

[a] finitely remote cause by a necessary being world-root,

[b] infinite causal regress of a beginningless [quasi-]physical wider cosmos,

[c] self-causation in a circle at the root.

We can address c first. If there is circular cause, the non-existent future reaches back to cause its past. This boils down to appeal to cause from non-being. Ruled out.

Option a would work, but is often deprecated as appeal to the unobserved, god of gaps, violation of Occam’s razor, etc. However, the issue is, there is of necessity a root that was always there, we do not pull a world out of a non-existent hat.

As for necessary being, the von Neumann chain and the linked logic show that distinct identity thus two-ness [and beyond the panoply of numbers] will always be there. It is possible, as actual. It is there in any possible world, not least because such a possible world holds its own distinct identity. It never began, it cannot cease from being, it is independent of external enabling causal factors. Necessary being, though strange to our thoughts perhaps, is real.

Option c is favoured by many and has been the occasion for several exchanges along lines much as above.

The pivotal issue, of course, is what lurks in that grey area indicated by the ellipsis. The answer is, the bridge to the transfinite.

So, if we consider the proposed beginningless causal-temporal world, we find the idea that first, the past is actual. That is, any past stage R was once the present, a duration – R_t from now, where now is regarded as 0 for convenience and past times are negative:

Duration since R:

D(R) = 0 – R_t = – R_t

The beginningless past claim, then, is that for any specific R, and for any specific n in the naturals (N), there will be actual onward past times that are before R, beyond any specific n. That is, limitless in the past beyond R.

Such is a strong claim indeed.

Does it imply that there are past times that are remote of order w?

I have symbolised that situation as in effect S:

{ . . . q –> q+1 –> . . . p-1 –> p –> p+1 . . . 2, 1, 0}

where the first and second ellipses are transfinite in span.

This expansion of the ellipsis beyond p has been denied:

{ . . . q –> q+1 –> . . . p-1 –> p –> p+1 . . . 2, 1, 0}

In particular, it has been held that all specific vales leftwards [L-wards, contrast R-wards] of p are also finite but extend without limit.

Part of the reason, doubtless, is that a stepwise finite stage cumulative process cannot traverse a transfinite span. For, if it attains any stage k, we can treat k as if it were the beginning again, k, k+1, k+2 etc and 1:1 match process from the original point regarded as 0. That is part of how we understand the process to be transfinite, beyond limit. And with that stricture, no stepwise process will span an actual infinite traverse, So, it is held, there is no actually transfinitely remote actual past point. The infinity lies in the beginninglessness, beyond limits nature of the succession. Where all is neatly coherent, it is not impossible and cannot be dismissed; implying, adequate and sufficiently satisfactory explanation.

Another claim was that q smuggles in a beginning. Obviously, no, it too is preceded by a L-ward ellipsis.

So, is that where we must stand? I don’t think so.

The problem lurks in the grey area, we are obviously embarked in the zone where the finite transitions to the transfinite. Where, the implicit pivot of the argument is non-traversal of the transfinite. But, a successive stepwise process that extends L-wards beyond any n in N is clearly of transfinite character. Just what is it that lies “beyond any n in N” other than the transfinite? But if we focus on finitude of specific numbers R or n, we may miss that character.

So, it seems that whether or not one will accept that there is an implied actual past time of character q, one implies successive, stepwise traversal of a transfinite span.

Where, such a stepwise process does not have that power. That is a point by W L Craig, too, as summarised by SEP (it is not just an idiosyncratic, readily dismissed notion):

5 An actual infinite cannot exist.

[He has many reasons for that, which are often hotly contended but on the whole I think he has a serious point . . . at minimum he has shown that modern atheism effectively is reduced to a highly contentious assumption about the past, which is very different from its boast that it can be seen as a default for the intelligent person]

6 A beginningless temporal series of events is an actual infinite.

[Note this]

7 Therefore, a beginningless temporal series of events cannot exist.

Since (7) follows validly, if (5) and (6) are true the argument is sound . . . .

>> 6.3 Successive Addition Cannot Form an Actual Infinite

Craig’s second argument addresses this very point.

8 The temporal series of events is a collection formed by successive addition.

9 A collection formed by successive synthesis is not an actual infinite.

10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103).

The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite [–> in such a stepwise fashion] (Craig and Sinclair 2009: 118).

It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future.

[–> relative to the distant past in view. This is also the context in which I have consistently spoken of finite stage causally successive stages]

Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.

Beyond, lurks time’s arrow; entropy. Part of the driving-force of causal-temporal succession is that there are rich concentrations of energy that support processes of change. But that means the sources are gradually degraded and as they are finite, we look at what has been called heat death. Such is a condition where the degradation of energy has attained a point where the energy sources — stars are especially in view — have been used up.

That we are not in this degrades state implies, then, a finite date; at least i/l/o understanding the cosmos as a whole as an isolated system. Of course, it was suggested above, that perhaps there is an external non-physical source able to sustain the world “forever” through adding fresh energy. This is in effect continuous creation by an entity that would recognisably be God.

And that is a shocker for confident evolutionary materialism advocates.

Is there a physically beginningless world coeval with God who sustains it from without?

To such a model, we can note that one may speculate as one pleases, the evidence we can observe prunes speculations by highlighting evidence of a beginning rooted in back-projection of an expanding world with background microwave energy consistent with the about 14 BY origin. Likewise, we would expect to see a huge number of degraded stars, including cooled down white dwarfs. These are not seen with any numbers to support a beginningless world working much as what we observe.

In all of this light, the best explanation on the table is finitely remote origin of our world, tracing to a necessary being world-root.

Beyond, we may discuss implications of such a world having in it conscious, minded creatures able to seek out explanations of an intelligible world, who find ourselves under moral government.

Such is an onward exercise and the road to it lies through the AI issue, including the further exploration of memristors.

DV, another day.

KF