I'm involved in a discussion on an Islamic website regarding the existence of God. My Muslim opponent is leaning heavily on the Kalam Cosmological Argument, part of which depends on the assertion that an actual infinite (specifically an infinite temporal past) cannot exist. He bases this assertion on thought experiments like Hilbert's Hotel.

My response is chiefly that similar arguments can be made against something like the existence of an infinite future, which must be possible according to Islam. It seems to me that the seeming absurdities only demonstrate that conventional mathematics cannot use operations like addition and subtraction to deal with an actual infinite, but that does not mean the actual infinite itself does not exist. It only meants that, if it did exist, we do not have the math to perform certain operation with it, or that it does not lend itself to those operations at all.

Does this argument make sense from a mathematical perspective? Have any mathematicians proposed arguments that demonstrate that an actual infinite could exist?

For my purposes, it is not necessary to demonstrate that an actual infinite can exist. I need only show that it remains a question that has not been settled.

Oh, and try and keep answers at the level someone with no math beyond high school can understand, if that's possible.

TIA.

Consider the past as negative numbers, the present as zero and the future as positive numbers.

ughaibu wrote:Consider the past as negative numbers, the present as zero and the future as positive numbers.

The standard response to that is that the future is only a potential infinite; one can keep adding future time to it infinitely, but it is never a complete infinite set. Whereas an infinite past consists of an actual infinite number of events that have occurred.

One interesting approach to the problem is taken by Wes Morriston:

http://colorado.academia.edu/WesleyMorr ... l_Infinite

There, he proposes a thought experiment in which God commands a pair of angels to take turns praising God for the rest of eternity. Morriston goes on to demonstrate that such a scenario meets the criteria for an "actual infinite," as set by theists who use the KCA to argue for the existence of God. Yet it is possible to construct arguments similar to Hilbert's Hotel to demonstrate that such a scenario (which must be possible under conventional theism) is also absurd.

However, I'd like to have some actual mathematical arguments to offer, rather than just counter with a different flavour to theological wibble-woo.

Shrunk wrote:Does this argument make sense from a mathematical perspective?

No. Craig's arguments have absolutely no mathematics in them, and in at least one case, he arrogantly asserts that he has better insight into these matters than mathematicians do. All Craig has is wibble, and his assertion that mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions is straight-up bullshit. There is no contradiction.

Axiom systems for classical mathematics typically assert that an actual infinite exists axiomatically. The mathematics works out consistently. Hilbert's Hotel is not a counterintuitive result (the observation made in the thought-experiment can in fact serve as a definition for the actual infinite). If you want a really counterintuitive result, look up the Banach Tarksi paradox: you can decompose a ball into six bits, move them around, and produce two balls the same size as the first.

I grant that reasoning about the actual infinite smells funky. Craig doesn't give any indication that he knows why though.

Evolving wrote:

Can an actual infinite exist?

- is this in fact a mathematical question? If it is, then clearly it can and does exist, axiomatically explicitly (as VazScep correctly states), and it is easy to name some examples that are not merely "potential infinites" (whatever that means) - how about the decimal places in pi, for instance, or the number of fractions between 2 and 3.

Or is it a physical question, i.e. is there a thing (a physical entity) of which there are, in reality, infinitely many? That question is not so easy to answer with "yes".

But is it any easier to answer with a "No"? The theistic argument depends on that being the case.

Can you give us your opponent's argument that Hilbert's Hotel rules out an infinite past in detail? It's hard to examine an argument we can't see.

An argument cannot hold when built around the unknown. As for infinity , dealt with already :)

twistor59 wrote:About the only useful things that can be uttered about physical infinity are contained here:

logical bob wrote:Can you give us your opponent's argument that Hilbert's Hotel rules out an infinite past in detail? It's hard to examine an argument we can't see.

This is probably the argument Shrunk is talking about:




Anyway, is the Hilbert's paradox even a real paradox? As far as I know it's a veridical paradox. (In this page, veridical paradoxes are defined as paradoxes which are seeming absurdities that are nevertheless true because they are perfectly logical.)
Just because something is counter-intuitive doesn't necessarily mean it's a paradox.
.

murshid wrote:

logical bob wrote:Can you give us your opponent's argument that Hilbert's Hotel rules out an infinite past in detail? It's hard to examine an argument we can't see.

This is probably the argument Shrunk is talking about:

I was hoping there'd be more because that's so lame. Also Shrunk is talking to Muslims, who probably aren't drawing directly on Craig.

Anyway, for what it's worth, Craig would seem to be saying
1. There can't be an actually infinite set
2. Therefore there are only finitely many past events
3. Therefore the universe had a beginning

This does seem to commit him to
1. There can't be an actually infinite set
2. Therefore there are only finitely many numbers
3. Therefore there is a largest number

It would seem to rule out an endless number of future events as well. Does Craig address this? It seems Shrunk's opponent does as he apparently says future events make up an ever expanding finite set, a potential rather than rather an actual infinity. Numbers can be seen as ever expanding in the same way. It seems Shrunk's opponent is at least doing better than Craig, whih is why I asked about the specifics.

As VazScep has already said, the axioms in set theory don't provide an actual infinity unless you invoke one by including the axiom of infinity specifically for that purpose. Neither do they preclude one. Actual infinities generate some results which are counterintuitve the first time you see them but no paradoxes at all.

Shrunk wrote:
However, I'd like to have some actual mathematical arguments to offer, rather than just counter with a different flavour to theological wibble-woo.

Craig is not impressed by purely mathematical arguments because he doesn't argue against the logical consistency of the mathematical concept of an actual infinity but against the metaphysical possibility of an actual infinity in concrete, spatiotemporal reality. Actually infinite sets of abstract mathematical objects don't trouble him anyway because he is a mathematical antirealist/antiplatonist. For him, the realm of mathematics is a fictional realm. If things such as numbers don't (really) exist, there aren't (really) any infinite sets of them either.
Craig simply undermines the argument that actual infinities are metaphysically possible because there are actual infinities in the abstract realm of mathematics by denying the reality of mathematical objects such as numbers.

http://plato.stanford.edu/entries/platonism-mathematics

http://plato.stanford.edu/entries/ficti ... athematics

Teuton wrote:Craig simply undermines the argument that actual infinities are metaphysically possible because there are actual infinities in the abstract realm of mathematics by denying the reality of mathematical objects such as numbers.

Yes, but it's not at all clear that if the universe had no beginning the set of past events would "have reality" any more than the set of numbers.

Yes, an action of something real is an actual infinite. You reading this is an actual infinite. Everything that happens, Has happened and shall always remain as something which happened. Now, can we move on to something else.

All alleged absurdities regarding actual infinities have to do with the point that the word "more" becomes ambiguous in the realm of the infinite:

– There are "more" natural numbers than odd natural numbers in the sense that the set of odd natural numbers is a (proper) subset of the set of natural numbers.

– There aren't "more" natural numbers than odd natural numbers in the sense that the set of odd natural numbers and the set of natural numbers are equinumerous, i.e. have the same cardinality, aleph_0.

Craig equivocates between these two senses. By doing so, he can claim that an absurd or inconsistent situation arises where infinite wholes both are and aren't greater than their proper parts.

Teuton wrote:All alleged absurdities regarding actual infinities have to do with the point that the word "more" becomes ambiguous in the realm of the infinite:

– There are "more" natural numbers than odd natural numbers in the sense that the set of odd natural numbers is a (proper) subset of the set of natural numbers.

– There aren't "more" natural numbers than odd natural numbers in the sense that the set of odd natural numbers and the set of natural numbers are equinumerous, i.e. have the same cardinality, aleph_0.

Craig equivocates between these two senses. By doing so, he can claim that an absurd or inconsistent situation arises where infinite wholes both are and aren't greater than their proper parts.

Thats because he doesnt add them up till infinite, if he did then he shall see that it was he who was being absurd.

Teuton wrote:
Craig equivocates between these two senses. By doing so, he can claim that an absurd or inconsistent situation arises where infinite wholes both are and aren't greater than their proper parts.

Craig's argumentation collapses once one acknowledges the fact that, e.g., the natural numbers are "more copious" than the odd natural numbers without being more numerous than them. This is odd but not absurd.

Shrunk wrote:The standard response to that is that the future is only a potential infinite; one can keep adding future time to it infinitely, but it is never a complete infinite set. Whereas an infinite past consists of an actual infinite number of events that have occurred.

Shrunk, thinking it over, this might be the key point. The difference between an actual and a potential infinite doesn't rest with "how many there are" but with whether we can get hold of them all at once. There's a form of maths called constructivism in which, for reasons of philosophical preference, there are no infinite sets. Constructivists don't say there are only finitely many numbers, they just say you can't capture them all in a set.

Your opponent seems happy that to say there doesn't have to be a last event in the future, it's just that we can't capture all future events in a set. He seems to feel that because past events have already happened this somehow means that they can be captured in a set, giving rise to the dread actual infinity if there wasn't a first event. This is the move in the argument that doesn't follow. Nothing stipulates that it must possible to form a set of all events that have happened. The past can be treated as a potential infinity just as much as the future, or the integers.

I stress that I'm not a constructivist and see no problem using the theory of actual infinities. But if you did come to the discussion with only constructivist maths and no actual infinities it does not follow that there must be a first event.

murshid wrote:

logical bob wrote:Can you give us your opponent's argument that Hilbert's Hotel rules out an infinite past in detail? It's hard to examine an argument we can't see.

This is probably the argument Shrunk is talking about:

That's basically it. The argument is that, since it is possible to generate conclusions that seem absurd from the assumption of an actual infinite, then an actual infinite cannot exist. My position is that the absurdities arise from attempting to treat infinity as an ordinary number (pardon my non-technical term). That could simply indicate that an actual infinite exists, but cannot be defined or described by conventional arithmetic. In the same way, trying to describe quantum phenomena using conventional Newtonian physics generates results that seem absurd, but that does not mean quantum mechanics is false.

If you want to read the whole discussion, it starts here as a tangent on a discussion of evolution. The person I am debating with is shikran, who enters at post #374.

http://www.ummah.com/forum/showthread.p ... ion/page10