In the beginning ...

[thestoat]

Ah, I see, so you believe maths is wrong and you are right.

Just because you do not understand the maths does not make it incorrect or nonsense.

(As to confounding finite and infinite numbers, you were the one who wrote that we can "approximate" a finite number to an infinite number. If there is a problem, it is yours, not mine.)

Yes, I did write that, and once again (for the last time because repeating myself is becoming boring) the approximations hold true when a large number may be considered infinity to aid in simple calculations. This sort of thing happens a great deal in physics and mathematics. I have given real examples where it is appropriate to do so. Infinity is not a real number, yet you persist in treating in so.

[Crackpot]

Is so times infinity!

[Andrew D]

And so we come to the last refuge: "I can't explain it, but I believe it, so that settles it."

[Crackpot]

Glad you can finally admit that.

[thestoat]

And so we come to the last refuge: "I can't explain it, but I believe it, so that settles it."

Not really since I don't think you do believe it. It seems to me a little like the imaginary number i (or j if you are an engineer) where i = square root of minus 1. Doesn't make sense when thinking about apples in a basket but it is hugely important in physics and engineering. People don't understand TVs, computers, etc, but their understanding is not required to make the concept useful.

[MajGenl.Meade]

Here's the problem (again):

If a thing does not happen given infinite time in which to happen, then it is an impossible thing

That is a leap, not a logical progression. "Impossible" means that it cannot happen. But you keep substituting "does not" for "cannot".

How do you justify that leap?

Well Andrew if you take a single sentence from any entire syllogism, then you can always say the same thing.

All rabbits have four legs
A "Smith's Red Rock" is a rabbit
Therefore a Smith's Red Rock has four legs

To which your brilliant answer is: "That's a leap there - that a Smith's Red Rock has four legs. How do you justify that leap?"

When can a possible thing happen if it does not happen in inifnite time? The answer is "never" - in which case it is identical to an impossible thing, which cannot happen. Which means it was not a possible thing in the first place. You keep treating infinity as if it had a time limit - in which case many possible things might not happen (they run out of time).

If you view infinity as a set and the earth as another set. "It's possible there are pink lions on earth". Every single cm. of the earth is subjected to scientific scrutiny and no pink lions show up. Now you can argue until you are blue in the face that there may still be pink lions on earth - because they are "possible". But the set has been exhausted and there were none. I conclude that pink lions were not possible but were impossible. The set of infinity has been exhausted (yes I know) but you maintain that the possible pink lion just might not show up in the set but still remains possible. Where? When?

Leaping lizardes (and rabbits)

Cheers
Meade

[thestoat]

Meade ...

I suggest "It's possible there are pink lions on earth"

But ... after infinite time, if they don't turn up, then I'd say it was impossible for them to be pink ...

[Crackpot]

Is that "are" or "could be"?

[thestoat]

lol - good point CrackPot, could be... depening on what you smoke

[Andrew D]

Glad you can finally admit that.

Care to cite examples, Crackpot?

Or perhaps we should revisit the omnipotence paradox ....

[Andrew D]

General Meade, you wrote:

The non-believer (admit it or not) must argue that time has no beginning -- that the universe (or call it what you will) has infinitely existed in one form or another -- a singularity, an eternal oscillation from one state to another. It ignores the inconvenient fact that in infinite time (and they mean an infinite regression into the "past"), all that could possibly happen must have happened. In which case we would not be here at all by now.

Now you ask:

When can a possible thing happen if it does not happen in inifnite time?

With respect to an infinite past, the answer seems straightforward enough: A possible thing which has not happened in an infinite past could still happen in the future (infinite or finite) or, for that matter, at this very moment.

With respect to an infinite future -- which is apparently not required by your description of what a non-believer "must argue" -- the question becomes "when does the moment occur at which a possible thing, if it has not occurred, become a thing which cannot occur?" The answer must be "at the end of the infinite future time," because at any moment at which the infinite future time has not yet ended, there must be more time remaining in which the possible thing could occur.

But the infinite future time cannot have an end; that is what makes it infinite. So the moment at which a possible thing, if it has not occurred, becomes a thing which cannot occur never arrives. And that means that the possible thing remains possible, because the time at which it becomes impossible never comes.

If the past is infinite, a possible thing which has not yet occurred could occur now or in the future, so it need not have already occurred. If the future is infinite, a possible thing which has not yet occurred could occur at some later time in that infinite future, which is to say that it could occur sometime between now and forever. In either case, the possible thing has not and will not become impossible.

For purposes of discussion, I am willing to concede that if an infinite future ever ended, a possible thing would become impossible when the infinite future ended. But given that an infinite future, by definition, never ends, I do not see how that advances the analysis.

[Crackpot]

Glad you can finally admit that.

Care to cite examples, Crackpot?

Or perhaps we should revisit the omnipotence paradox ....

What you think you actually proved any anything in that "debate" other than you can prove anything when you insist on "facts" that haven't been proven?

I've long since given up trying to convince you of anything on the subject of paradoxes (Of which the concept of infinity contains many) after realizing a long time ago that despite your rather brilliant legal mind it is a subject you just plain don't get. I'm the same way with medicine I just don't get it. But unlike you I fully realize it.

The best I can do is recommend some reading on the off chance that you can find the book:

Aha! Gotcha: Paradoxes to puzzle and delight By Martin Gardener.

It's a book that my dad got my brother and I when we were kids. I first read it in my Tweens and could only grasp a small amount of but through the Years Picked up more and more. Perhaps it could help you too.

[MajGenl.Meade]

Thanks Andrew. It appears crackpot has identified a real problem; that the paradox of infinity is at the moment eluding your fine analysis.

A recap. I do not believe that an actual infinity of moments can exist. Therefore, you and I agree that “not all possible things” have happened and that they in fact may not happen. They may remain “possible” until the end of time – after which they are no longer possible. OK so far?

You surely agree that in the “set of infinite moments” any thing which does not occur in that set must be a thing that cannot occur? There is no other time outside the set of infinite moments in which this thing can occur. It must occur within the set or never occur at all.

The paradox of infinite time (no beginning and no end) is that the number of moments prior to THIS moment is equal to the number of moments AFTER this moment. That is, an infinite number of moments has already happened. In fact, a set of infinite moments has already happened, which any future infinite moments you choose to posit cannot increase.

Since we know that any possible thing which does not occur inside the set of infinite moments cannot happen and is therefore impossible, we must conclude that if the universe has “always existed” and therefore there is an infinite regress of time, all possible things have already happened including our extinction.

Since we are not extinct, this proves that an actual infinite regress of moments (time) is impossible, for not all possible things have happened yet. We agree on that.

We disagree because…. I’m not sure why. You deny that a non-believer must posit an infinite number of moments (always existing universe) but then you put forward the ping-pong universes in a cycle of expansion and contraction which had no beginning as if that’s an alternative. It isn’t. It relies upon the eternally existing (in the past) universe which means infinite regress.

Now – which is it? Did the universe have a beginning – a time when nothing was and then something was? Or has it always existed – in which case you do argue for an infinite regress just as I stated would have to be the case

Regards
Meade

[Big RR]

Or did time, as we understand it, begin when the universe began? We are constrained to this universe for our understanding, and time is reckoned based on processes within this universe. that other universes may have existed before, or that we may become something different later, is immaterial--we are constrained to the universe at hand.

[thestoat]

There are a couple of other things to ponder here. Certainly time is a human concept and did not exist before the big bang, implying we have so far only a finite amount of time under our belts (though of course previous universes may have existed before the big bang with similar times, giving an effective "infinite time" before now).

We could speculate our universe is infinite. I don' know, but I think this idea doesn't hold much water since it has been measured to be expanding, and something of infinite size cannot expand - can it?

But, assuming our universe is finite in size, what if there are an infinite number of universes? In that case, there will be others with humans on identical planets. In one of those universes, I exist and am rich (not this one, unfortunately). In another, humans exist and they all worship a god. In another, humans exist and worship no god whatsoever. In others, humans have indeed become extinct. Etc.

Or maybe there are other universes here, in out own but removed from us along some extra dimensional plane. There could be infinite universes along the 5th, 6th or 7th dimensions. (In fact, I believe that infinity to the power 3 is infinity, but infinity to the power infinity is a higher order of infinity, but my memory here is rusty).

The mind boggles...

[thestoat]

Aha! Gotcha: Paradoxes to puzzle and delight By Martin Gardener.

Thanks for that tip Crackpot ... I've just ordered it. Love that sort of stuff

[Andrew D]

Glad you can finally admit that.

Care to cite examples, Crackpot?

Or perhaps we should revisit the omnipotence paradox ....

What you think you actually proved any anything in that "debate" other than you can prove anything when you insist on "facts" that haven't been proven?

Actually, I was thinking of the crowning achievement of your argument -- when you linked, supposedly for my "edification," an article which refuted your position. Brilliantly done.

[Andrew D]

A recap. I do not believe that an actual infinity of moments can exist. Therefore, you and I agree that “not all possible things” have happened and that they in fact may not happen. They may remain “possible” until the end of time – after which they are no longer possible. OK so far?

At the moment, I see no reason to dispute that.

You surely agree that in the “set of infinite moments” any thing which does not occur in that set must be a thing that cannot occur? There is no other time outside the set of infinite moments in which this thing can occur. It must occur within the set or never occur at all.

I see a problem here: A hypothetically infinite set of moments[1] does not necessarily contain all moments. Consider the infinite set of all odd integers. That set does not contain any even integers. Nonetheless, it is infinite. Likewise, of course, the set of all even integers is also infinite, even though it does not contain any odd integers. Thus, there are integers not contained within each infinite set of integers: The even integers are not contained in the infinite set of all odd integers, and the odd integers are not contained in the infinite set of all even integers.

Now consider the hypothetically infinite set of all past moments. That set does not contain any future moments. (Nor does it contain the present moment, assuming that there is such a thing.) Nonetheless, it is hypothetically infinite. And whatever reasons there might be for concluding that such a set of past moments cannot actually be infinite, the fact it does not contain all moments (i.e., it does not include any future moments) is surely not among them: If a set of integers can be infinite despite not containing all integers, is there any reason why a set of moments cannot be infinite despite not containing all moments? And the same goes for the hypothetically infinite set of all future moments: It does not contain any past moments, but it is still infinite.

The hypothetically infinite set of past moments does not contain all moments, just as the infinite set of all odd integers does not contain all integers. And the hypothetically infinite set of future moments does not contain all moments, just as the infinite of all even integers does not contain all integers. So does it not follow that a possible event which has not occurred in the hypothetically infinite set of past moments could still occur in the hypothetically infinite set of future moments? If not, why not?

The paradox of infinite time (no beginning and no end) is that the number of moments prior to THIS moment is equal to the number of moments AFTER this moment. That is, an infinite number of moments has already happened. In fact, a set of infinite moments has already happened, which any future infinite moments you choose to posit cannot increase.

I do not see that it matters whether the infinite number of moments in the hypothetically infinite set of all past moments is or is not equal to the infinite number of moments in the hypothetically infinite set of all moments, past (and present?) and future.[2] (Or whether the infinite number of moments in the hypothetically infinite set of all future moments is or is not equal to the infinite number of moments in the hypothetically infinite set of all moments.)

If the infinite number of integers in the set of all odd integers is the same as the infinite number of integers in the set of all integers, there are integers (the even ones) which the set of all odd integers does not contain. And if the infinite number of integers in the set of all odd integers is not the same as the infinite number of integers in the set of all integers, there are still integers (the even ones) which the set of all odd integers does not contain. (And the same goes for the set of all even integers vis-a-vis the set of all integers: Regardless of whether the infinite number of integers in the former is or is not the same as the infinite number of integers in the latter, there are integers (the odd ones) which the set of all even integers does not contain.)

And that, it seems to me, is what matters: The hypothetically infinite set of all past moments, despite its infinitude, does not contain all moments, because it does not contain any future moments. Therefore, a possible even which has not occurred in the infinite set of all past moments is not necessarily impossible, because it could still occur in one of the future moments which is not contained within the hypothetically infinite set of all past moments.

Since we know that any possible thing which does not occur inside the set of infinite moments cannot happen and is therefore impossible, we must conclude that if the universe has “always existed” and therefore there is an infinite regress of time, all possible things have already happened including our extinction.

But as far as I can tell, we do not know that any possible thing which has not occurred within the hypothetically infinite set of past moments cannot happen within the hypothetically infinite (or, for that matter, even within a finite) set of future moments. Saying that (a) because a possible thing has not occurred within the hypothetically infinite set of all past moments, it cannot occur within the set of all moments, past (and present?) and future, seems to me like saying that (b) because even integers do not occur within the infinite set of all odd integers, they cannot occur within the set of all integers, odd and even.[3]

Or is there something about infinite sets of moments that is qualitatively different from infinite sets of integers? Is there some reason why a set of integers can be infinite without containing all integers, but a set of moments cannot be infinite without containing all moments?

Since we are not extinct, this proves that an actual infinite regress of moments (time) is impossible, for not all possible things have happened yet. We agree on that.

I agree that not all possible things have happened yet. Again, however, I do not see how a possible thing's not having occurred in the hypothetically infinite set of all past moments precludes its occurring in the set of future moments which the hypothetically infinite set of all past moments does not contain. Even given an infinite set of all past moments, there are still moments out there (the future ones) in which the possible thing could occur.

You deny that a non-believer must posit an infinite number of moments (always existing universe) but then you put forward the ping-pong universes in a cycle of expansion and contraction which had no beginning as if that’s an alternative. It isn’t. It relies upon the eternally existing (in the past) universe which means infinite regress.

Now – which is it? Did the universe have a beginning – a time when nothing was and then something was? Or has it always existed – in which case you do argue for an infinite regress just as I stated would have to be the case

I am not sure whether a non-believer must posit an infinite number of past moments. What interests me is your (and thestoat's, but he declines to attempt to explain it in English[4]) assertion that an infinite number of past moments is not possible.

As I wrote before, I did not put forward the hypothesis of a cyclical universe as way of dealing with the general proposition that an infinite number of past moments is impossible. I put it forward as a way of dealing with the much more specific proposition that entropy makes the infinite (in time) existence of our universe is impossible. But it is not the only conceivable alternative to that specific proposition. Perhaps as the universe changes, the operation of entropy will change or cease. Perhaps the theory of entropy will, like so many theories before it, turn out to be wrong.

Did God have a beginning -- a time when God was not and then God was? Or has God always existed, in which case you are arguing for an infinite regress just as much as I am?

Or, if God can be uncaused, why cannot the universe be uncaused?

-------------------------

1. I assume that we agree that what is at issue is the hypothetical infinitude of the number of moments contained in the set(s), not any infinitude of each moment.

2. I assume that this is what you are driving at. The proposition that the number of moments in the hypothetically infinite set of past moments is equal to the number of moments in the hypothetically infinite set of future moments seems to me, at least at this moment, unexceptionable: Neither of those sets contains any moments also contained in the other set.

3. If you are saying that an event which does not occur in either the hypothetically infinite set of past moments or the hypothetically infinite set of future moments (or in the present moment?) is an impossible event, then I see no reason to dispute that proposition. If it is true, the impossibility of the event depends on the exhaustion of the hypothetically infinite set of all future moments: As long as there are still future moments out there, the event remains possible. And because the set of all future moments is hypothetically infinite, there are always future moments out there. So the theoretical impossibility of the event never occurs. An event which remains possible forever is not, it seems to me, an impossible event.

4. In the case of approximations, thestoat goes even further. Confronted with a discussion of the matter which states that although surely and almost surely are equivalent in questions of ordinary statistical probabilities but are importantly different in questions of infinitudes, he omits the qualification that they are equivalent only in questions of ordinary statistical probabilities, ignores the statement that they are importantly different in questions of infinitudes, and asserts that a finite number can be approximated to an infinite number. That sort of thing leaves me at a loss as to how to respond other than to point out his tactic.

[thestoat]

4. In the case of approximations, thestoat goes even further. Confronted with a discussion of the matter which states that although surely and almost surely are equivalent in questions of ordinary statistical probabilities but are importantly different in questions of infinitudes, he omits the qualification that they are equivalent only in questions of ordinary statistical probabilities, ignores the statement that they are importantly different in questions of infinitudes, and asserts that a finite number can be approximated to an infinite number. That sort of thing leaves me at a loss as to how to respond other than to point out his tactic.

All you need to do, Andrew, is simply say "I simply do not understand any of the maths behind infinity" rather than harping on about how maths is wrong and you are right. Each of your complaints above have been answered by me several times (the most recent leaving me state I am bored by repeating myself). If you don't understand - say so - I find it pointless to keep repeating a question that has already been answered several times. Repeat a question and you'll get a repeat of the answer - unless you say tht you don't understand the answer you are given.

[Crackpot]

Aha! Gotcha: Paradoxes to puzzle and delight By Martin Gardener.

Thanks for that tip Crackpot ... I've just ordered it. Love that sort of stuff

It's possible it may be beneath you (you seem to have a firm grasp on most of the concepts within) but it is an entertaining read none the less.