Kalam, Actual Infinites, and Set Theory
In order to establish the second premise of the Kalam Cosmological Argument (“the universe began to exist”), William Lane Craig often argues against the idea of an “actual infinite” (i.e. an infinite amount of things/moments existing in reality). To do this, Craig shows that basic arithmetic cannot make sense of infinity, therefore supposing an “actual infinite” is absurd. Philosopher Colin Howson objects:
Lane Craig uses an argument that originates with Kant to ‘establish’ that time cannot be infinite in the past and still proceed into the future, on the ground that an actual infinite cannot exist because, among other reasons, if it did it would be impossible to add to it. But this claim is vitiated by the facts that (i) in contemporary set theory it is easy to show that there exists a sequence of infinite discrete ordered sets each with a greatest but no smallest member, each set extending its predecessor by an additional largest element; and (ii) the things in the domain of any consistent theory, as set theory is thought to be, are possible existents. Adducing similar observations, the distinguished philosopher of physics Michael Redhead concludes a review of Lane Craig’s argument with the remark that it, ‘seems a total muddle’. (Objecting to God, Pg. 92-93)
In other words, Howson argues that set theory makes sense of infinity in a way that arithmetic does not. And since set theory is consistent, then it is possible that “actual infinites” do exist.
Hi! You might be interested in watching what philosopher Quentin Smith says in this short interview:
http://www.closertotruth.com/video-profile/Arguing-God-from-First-Cause-Quentin-Smith-Part-1-of-2-/986
Smith explains that actual infinities are a part of multiple, well-confirmed scientific theories.
Welcome back. It’s been a long time. I was hoping you’d start up again. I have some fallacies that you might be interested in as well from several of his debates. Is this a good place to pass them on?
Gary
I’d say that an actual infinity of abstract (or ideal) objects (like those in set theory) is intuitively quite different from an actual infinity of concrete (or real) objects, such as physical beings or events.
I can clearly conceive the sets N, Z, etc. but I can hardly make sense of an infinite past.
Craig’s complaint about an infinite set is only that it remains “the same” when you add to it, subtract from it, divide it by two, etc. He uses Hilbert’s Hotel to make the point, as if Hilbert’s Hotel’s fame establishes its truth.
Infinity is not a quantity. It is the absence of maximum (or minimum) and is a concept, not a number. It’s used in physics and math in general as a “variable” for convenience (it’s easier than writing a very large number) but it does not represent an actual value, number, or quantity, and therefore you cannot add to it, subtract from it, etc. Math is able to deal with infinity quite easily, and without any paradoxes.
Claiming that it cannot actually exist because dividing it by half leaves it unchanged is like saying that you can’t, in actuality, have ZERO of something because if you did and you divided that in half, you’d still have zero, which is a problem. First, zero over two is not zero, and neither is infinity over two equal to infinity. In concept, if you don’t have anything, and you “give half of it away”, you’re still left with nothing, so it may sound like zero over two is zero, in the same way that Hilbert and Craig present “infinity plus one”. The concept is invalid. You can’t ACTUALLY give away half of your zero, nor can you ACTUALLY add one to infinity, because infinity is not a number. You CAN add one to any number.
There is no reason why zero cannot exist – it exists as a concept we all understand. The same applies to infinity (minus the ‘we all understand’ part) it’s a concept, and as such it can and does exist. There is no logical reason why time cannot be infinite in past and/or future.
Set theory is consistent but it is not complete. See Goedel for more on this concept.
I like Hilbert’s hotel, but
Grim reaper Parodox > Hilbert’s Hotel
http://alexanderpruss.blogspot.com/2008/01/grim-reaper-paradox.html
Paper using the Grim Reaper Paradox:
http://www.robkoons.net/media/83c9b25c56d629ffffff810fffffd524.pdf
Either way the KCA is sound, and while there are minor objections to it, the premises still stand strong.
I don’t know what the set theory is, but apart from this I don’t find Craig’s arguments convincing. The Argument from Hilbert Hotel shows a contradiction, but this contradiction does not arise only from the existence of an actual infinity. It comes from someone trying to count them. Therefor it is slippery slope.
The argument from a finit number growing to infinity is a straw man argument. An universe without beginning didn’t grow to infinity it was always infinite.