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.

  11. Jonathan E. Kiser
    02/18/2015 at 10:56 PM

    David D
    You ask what proof there is that Atheism is true? How about you use the same logic and ask why in any possible world could Theism be true? Atheism is a statement that no God was needed for all things to exist. You have to make a lot of assumptions first to argue for the existence of any type of God much less the God of the Bible. You are straining the limitations of understanding to have an God is Always the Answer theory…it is actually lazy to suppose a God/gods made all that is…why can’t you just say..”I do not know!” Because you do not know! The advantage of a Scientific Approach, aka an Atheistic viewpoint by nature, is there is always room for improving upon data. I admire your intellectual attempt to confirm WLC argument…but it is missing the point big time.

    • David
      09/29/2015 at 12:48 AM

      Not making any assumptions!

      The best science based on the evidence us that the universe (physical reality) came into being. If the physical realm came into existence it follows logically that the cause is 1) an abtract object with out physical properties. Or 2) nothing…. I simply choose to believe that something is a better explanation than your (assumptions of nothing)

      Without God there would be no objective right or wrong .. True or false… Correct or incorrect possibilities. There is no logical reason to believe that a randon unguided series of meaningless and unintentional events is the best expanding to such objective truths such as 2+2=4. There is no reason or evidence that a blind random process would prefer or create such necessary truths.. If 2+2=4 then order + logic + objective truth equals God.

      Seriously, explain to us logicaly how nothing gave rise to a brain capable of understanding objective truths? What if believing a lie or an illusory perception of reality better helps us survive? Does not naturalism teach us then that we would be better to believe the lie? Athiesm cannot be explained logically, for without God materialism and naturalism leads to a world of losy unguided with no hope of understanding the true nature of reality… I have good reason to believe that ( God exist) is as true as 2+2=4

  12. RBC
    08/13/2015 at 7:53 AM

    If Infinity made sense to Physicists, perhaps we would not be searching so diligently for a unification theory. The 2nd premise of Lane’s argument is highly plausible. As a Deist, Theist or Agnostic, you really shouldn’t be upset by his argument. There’s no logical inconsistency with his argument and no reason for his argument to be upsetting to your belief system. Science corroborates his argument, yet that doesn’t mean he has proven God’s existence with his argument. He has however proven that belief in God is as plausible as the unknown naturalistic mechanism required if the condition of no God is reality. Inflationary hypothesis is more of a stab in the dark than belief in God. His argument would perhaps be upsetting to Secular Humanist, as it violates their religious/philosophical tenets. Lawrence Krauss has been debunked, this argument which has been around since Plato and Aristotle has not.

  13. 09/28/2015 at 9:57 AM

    His premises are entirely based on intuition, so I’m not sure why anyone accepts that as compelling. Sound, sure. But it’s sound to say “all swans are white”, and this doesn’t make it a compelling statement.

    Everything that begins to exist has a cause – NOTHING in our experience ever begins to exist (things only change state). So what is this statement based on? Intuition which is irrelevant to the premise. Change (of state) usually has a cause, but not necessarily. Quantum events leading to observable result (such as radioactive decay) do not appear to have causes.

    The universe began to exist – This could be true, or it could be false. Induction from observation only takes us to within a fraction of a second AFTER a potential beginning. Craig’s conclusion is not based on such observations, but on his assertion that there must have been a beginning because you can’t count time forever into the past. Not only is this false, it’s irrelevant. So while the universe may have had a beginning, neither current science nor anything else Craig offers leads us to accept this is compelling.

    Even if his premises were based on compelling evidence, the entire argument is self-defeating because he changes the use of the word “universe” mid-sentence. “The universe had a beginning” either means that OUR universe had a beginning (but other things preceded it, such as other universes, or whatever you want to name “God”), or it means that everything that exists now began at this “beginning” (and nothing preceded it, not even “God”). Craig’s preferred conclusion (God preceded and created “everything that exists now”) is not supported by either possibility.

    So we’re left with the usual special pleading:

    1. everything that exists today began to exist at one point, except for God, who exists today, but existed before this point.

    2. No thing can have existed infinitely into the past because I can’t count to infinity, but God existed infinitely into the past.

    • David
      09/29/2015 at 1:37 AM

      So you have existed infinitely and didn’t come into existence at conception of birth, but rather existed infinitely and chances states? Could you explain to us on what foundation you make such assumption? Your conscious bring didn’t begin to exist in time but rather just exist infinitly? I guess that makes you God! The problem is, without God there is no good reason to believe that anything you say is necessarily true, because you can’t show that objective truth has any real benefit to any creature including so called intellectual secular humanist . Again if a lie helped you survive your (god) random unguided natural selection world prefer the lie to truth and the illusion to reality… Convince us that a blind unguided natural process would prefer objective truth. . or any truth for that matter…. Athiesm leads to self processed ignorance of the nature of any objective truth… Yet I have good reason to believe that the world is ordered to account for such objective truths as 2+2=4…. If God does not exist then I have no reason to accept your argument,: in fact I wouldn’t have any reason to accept that an illogical unguided series of random events could produce a mind capable of believing anything…. Since I believe that objective truths exist, I have good reason to believe that God exist….. Athiesm had no good argument to even assume that it nay be true…. For if it did it would follow logically that I couldn’t be certain that logic was logic.. truth, truth, or fact was fact. I reject atheism on the grounds that it leads to self contradictons as well as holding to an uncertain world view concerning the true nature of reality.

  14. 09/29/2015 at 10:26 AM

    OK let me respond to your points one by one.

    “I” didn’t exist infinitely, but “I” am just a combination of things that existed before “I” did. Those things didn’t come into existence when I was born, they were joined together to form a new, emergent state. A brick is made of atoms, and when you make a brick in the brick factory, you combine these atoms into what is now a new brick. But none of the atoms are new – they didn’t begin to exist. Atoms are a combination of previously existing quarks, which are a combination of basic units of energy. Energy isn’t created or destroyed, ever! other than, possibly, when all energy began to exist, if that ever happened – the big bang theory does NOT say anything began to exist. The foundation of my statement is the very well established, highly tested modern physics.

    Objective truth doesn’t need to have any benefit to anyone – that’s what “objective” means – “it’s true whether you like it or not.”

    Natural selection isn’t random. Think about it – “only what can, survives” is not the same as “X survives, and/or Y survives, randomly.” I think you can see yourself that “only what can, survives” is an unguided objective fact (which you called objective truth).

    Atheism is the claim that theism is an unjustified claim. I don’t see how this leads to a “self processed ignorance of the nature of any objective truth”… The only way to become aware of objective facts is by the testing of observations, and the testing of inductions based on observations. No other way of selecting opinions (hypotheses) comes close in its ability to weed out opinions/beliefs/delusions from facts. This has nothing to do with atheism, other than the fact that theism consistently fails this test, justifying atheism.

    God is not needed in order to explain why 2+2=4. 2+2 can only be 4. Even God cannot make 2+2 be anything else, because 2, and 4, are descriptions of quantities, and quantity is a basic characteristic that falls under the law of non-contradiction (A is not not-A). This law is basic (a first principle) – it cannot be otherwise, which means it does not require someone to create it. Which means it necessarily precedes any law giver. Think about it – if this law is true (A is not not-A) then it’s true. And if it’s false, then it’s true! so it’s always true. No God needed. No God can override it (because if he can, then he can’t).

    I believe objective facts exist, because observation leads to this conclusion. But no observation leads to the conclusion that God exists.

    “I reject atheism on the grounds that it leads to self contradictons [sic] as well as holding to an uncertain world view concerning the true nature of reality.” I’d like to hear how atheism leads to self-contradiction. If you’ve attempted to explain this above, I didn’t get it, so please explain it at a lower-reading level for my benefit. Also, no worldview is certain. You cannot logically prove that we are not in the matrix right now. But you can make judgments based on justification – we have no reason to suggest we are in a matrix right now, and that’s as close as we can get to certainty. I have no valid, compelling reason to believe God exists. That isn’t certainty, but it doesn’t have to be certainty. If you believe God exists, are you certain? if so, then you should be able to prove God exists. If you can’t, how can you say you’re certain?

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

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