Home > Mathematics > Kalam, Actual Infinites, and Set Theory

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.

Categories: Mathematics
  1. 02/06/2012 at 10:38 PM

    Hi! You might be interested in watching what philosopher Quentin Smith says in this short interview:

    Smith explains that actual infinities are a part of multiple, well-confirmed scientific theories.

  2. Gary
    02/08/2012 at 9:28 PM

    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?


  3. LauLuna
    03/29/2012 at 12:27 PM

    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.

  4. 05/08/2012 at 2:40 PM

    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.

  5. BAJ
    08/15/2012 at 10:08 AM

    Set theory is consistent but it is not complete. See Goedel for more on this concept.

  6. 10/11/2012 at 10:42 PM

    I like Hilbert’s hotel, but

    Grim reaper Parodox > Hilbert’s Hotel


    Paper using the Grim Reaper Paradox:


    Either way the KCA is sound, and while there are minor objections to it, the premises still stand strong.

  7. 12/04/2012 at 8:57 AM

    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.

  8. jogu
    05/24/2013 at 10:46 AM

    I don’t find WLC’s argument persuasive for reasons similar to the ones expressed above.

    I would just like to add that I don’t think WLC’s argument for the impossibility of an “actual infinite” centres as much around the impossibility of adding something to infinity as it does around the fact that “infinity minus infinity” is not mathematically well-defined in the arithmetic on the extended real line:


    (see the “Arithmetic operations” section)

    But what I find very ironic about the argument is that after WLC has “established” the impossibility of an actual infinite, he immediately (in his Kalam argument) turns to god as the ultimate cause of existence – a god who himself (herself?) is eternal and infinite in so many ways. So isn’t god the prime example of an “actually infinite” entity?

  9. David D
    05/27/2013 at 3:54 AM

    1) Wrong! Craig argues “not” an infinite amount of time or moments, but rather against an infinite number of actual events having occurred. Hilbert’s Hotel is a great example of how an infinite number of actual events leads to self contradictions, and self contradictions by definition are illogical, self refuting, and thus fallacious. aka absolutely false!

    2) It would follow logically that anything and everything that could possibly exist would exist if there were an actual infinite number of world/s and possibilities. When Craig talks about “possible worlds” he is referencing “possible realities” and not actual physical worlds such as an actual multi-verse or physical many worlds ensemble (such as Stephen Hawking envisions) But consider the implications of an infinite number of actual possibilities or events: It would follow logically that anything that could possibly exist, would exist : So if God (Contingent or Necessary) were even possible, it would follow logically that such a being does exist “contingently” is some possible and actual worlds and/or out of necessity in all possible and actual worlds.

    So rejecting Craig’s argument based on your presupposition that there is such a thing as an actual infinite number of possibilities and events; you are now in a unique position of explaining why God’s existence is impossible in all possible and all actual worlds?

    3) There is only two possibilities a) There is an actual beginning to the physical realm. b) The physical realm exist infinitely. If the physical realm has an actual beginning it follows logically that the cause of the physical realm having come into being is time-less, space-less, and immaterial. If “a” is true; Then Craig’s argument is sound! But lets assume that “b” is true: It would follow logically that out of an infinite number of possibilities that God would exist in some actual world as well as in some possible worlds contingently; and if by necessity it would follow logically that God would exist in all possible worlds as well as in all actual worlds. That is sound logic!

    4) Concepts are not physical realities! I can conceive a universe where bananas speak and rule their world by their vast super intelligence. Sounds silly? Well, that is imagined therefore a concept.

    Another example: The existence of God “IS” a concept! That is exactly what Craig argues: Actual infinite is a mere concept “imagination” but leads to self contradictions in reality or “actuality”: …. Actual infinite’s would have no beginning or end and could not be added to or subtracted from/// No beginning, no ending; but rather a beginning-less and endless series of infinite possibilities, excluding the possible existence of God in any possible world, right? To me, that is appealing to the “ABG theory”: “Anything But God”

    Lastly: What you have to show is how materialism can explain the beginning of the first physical “material” entity? It can’t be physical, or it would not be the origin of the first physical realm huh? It must transcend the physical and by definition be super-natural (Transcends any natural explanation) even If mere mindless energy. a) Why and how has this mindless energy existed infinitely and timelessly without beginning? b) How did physical reality come into existence from none-being ? To say that God is a better explanation than nothing does not take much of a leap of faith; for it is a rational and logical conclusion based on the evidence before us. There is absolutely no argument or evidence for why atheism is true! If so, lets hear the argument!

  10. 07/31/2013 at 4:22 AM

    What I find strange, regardless of the existence of paradoxes or not, is that people argue, that something must exist, because we speak in a certain way. In this case the arguments against actual infinities are just like ontological arguments. Our language evolved and is still evolving. We shouldn’t expect it to describe our whole world accuratly. Craig presupposes, that whatever it is it must be described in a way we already thought about.

  1. 07/13/2012 at 12:01 PM
  2. 07/13/2012 at 10:09 PM

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