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Posted by on Mar 25, 2013 in Cosmological argument, Infinity, Mathematics, Time, William Lane Craig | 46 comments

Infinity minus infinity

William Lane Craig on the infinite – Part 2 of ∞


This is the second of a series of posts on William Lane Craig’s treatment of the infinite.  It’s a long one, but there are a lot of important topics to cover and points to make, some of which I have not seen anywhere else (especially the ideas in Section 4).  So please get comfortable before reading on!

In the first post of the series, Infinite Dreams, I explored Craig’s motivation for rejecting the existence of actual infinite collections, and outlined the form of his argument against them.  To briefly summarise:

  • Craig mainly wants to argue that the past series of events could not be infinite in order to give philosophical support to the second premise of the Kalam Cosmological Argument: The universe began to exist.
  • To do this, Craig attempts to argue that no collection could be infinite.
  • And Craig attempts to show that no collection could be infinite by explaining why he thinks certain collections could not be infinite.

In the first post, I highlighted exactly why this kind of reasoning is unsuccessful: even if it was granted that the handful of examples Craig considered were indeed problematic, this doesn’t mean that all examples would be problematic.

However, there is an even bigger problem for Craig.  In fact, none of the examples he considers lead to logical absurdities at all, and I intend to explain why this is the case in the next couple of posts in this series.  Naturally, this doesn’t lead to the conclusion that there do exist infinite collections.  But my purpose here is simply to show that Craig’s arguments against the existence of infinite collections are flawed.

Craig considers a fair few examples, so it is difficult to know where to start.  However, in his article Reply to Smith: On the Finitude of the Past, Craig says:

[M]y strongest arguments in favour of the impossibility of the existence of an actual infinite [are] those based on inverse operations performed with transfinite numbers.

I don’t know if Craig still considers these to be his strongest arguments against infinite collections, but he has used these arguments in a lot of his recent debates, so I think it’s fair to suppose he still thinks they are strong arguments.  In any case, I think it is probably fitting to deal with these arguments first.

1.  Craig’s case against the infinite

The following excerpt is from Craig’s debate with Prof Peter Millican (Oxford):

Have you ever asked yourself where the universe came from?  Why anything at all exists?  Well typically, atheists have said that the universe is just eternal and uncaused.  But there are good reasons, both philosophical and scientific, which call that assumption into question.

Philosophically, the idea of an infinite past is very problematic.  If the universe never had a beginning, that means that the number of past events in the history of the universe is infinite.  But the real existence of an actually infinite number of things leads to metaphysical absurdities.

For example, suppose you had an infinite number of coins, numbered 1, 2, 3, and so on to infinity, and I took away all the odd numbered coins.  How many coins would you have left?  Well, you’d still have all the even numbered coins, or an infinity of coins.  So infinity minus infinity is infinity.

But now suppose instead that I took away all of the coins numbered greater than 3.  Now how many coins would you have left?  Three!  So infinity minus infinity is three.

In each case, I took away an identical number of coins from an identical number of coins, and came up with contradictory results.  In fact, you can subtract infinity from infinity and get any answer from zero to infinity.

For this reason, inverse operations like subtraction and division are simply prohibited in transfinite arithmetic.  But in the real world, no such convention has any sway.  Obviously you can give away whatever coins you wish.  This, and many other examples, suggest that infinity is just an idea in your mind, not something that exists in reality.

But that entails that past events, since they’re not just ideas but are real, must be finite in number.  Therefore, the series of past events cannot go back forever.  Rather, the universe must have begun to exist.

Watch Craig say this here.

Before critiquing Craig’s reasoning, it is worth pointing out that even if his concerns about infinite coin collections were ‘on the money’, this would say nothing about the possibility or impossibility of an infinite past.  Unless Craig thinks it is possible to “take away” a (finite or infinite) number of moments from the past, his complaints about infinite coin collections (which are all based on supposed problems with taking away coins) do not translate directly to complaints about an infinite series of past events.  If he tried to counter this by saying that we can imagine “taking away” past events, then there would be problems with an infinite future as well, since we could equally well imagine “taking away” future events.

It should also be noted that Craig’s comments here seem to be at odds with his views on time.  Craig is a presentist.  He defends what is known as an A-Theory of time.  Craig summarises his views on time, and contrasts them with B-Theory models, in an interesting (and very accessible) video, The Nature of Time.  As stated in that video, his basic position is as follows:

The moments of time are ordered by past, present and future, and these are real and objective aspects of reality.  The past is gone; it no longer exists.  The present is real.  The future has not yet come to be, and is not real.  And so the future is not “out there”, ahead of us, down the line, waiting for us to arrive; the future is pure potentiality.  Only the present is real.

The reader will of course have noticed that Craig says here that “[t]he past is gone; it no longer exists”, but says above that “past events … are real” when arguing against an infinite past.  Since Craig’s problem is with infinite collections of real things, not with infinite collections of non-existent things, it seems that an infinite past should be no problem for him, even if it was granted that actual infinite collections were impossible.  If he has a problem with an infinite past, then he may as well have a problem with an infinite future since, on his view, the number of future events is also infinite.  (Craig believes in a never-ending afterlife and therefore, presumably, that the number of future events is greater than any finite number.  Anyone who says that the number of future events is zero will be quickly proved wrong when the second hand creeps forward another tick.)  Attempting to avoid this problem by claiming that future events are not real will not help, because neither are past events real on Craig’s view, so it seems that there could only be a problem with an infinite past if there was also a problem with an infinite future.  (Christian philosopher Wes Morriston makes a similar point in his article Beginningless Past, Endless Future, and the Actual Infinite.  Craig’s response to this article, as well as Morriston’s response to Craig’s response, can be downloaded from Morriston’s webpage.)  Craig does actually address the point I am making here in an online article, Is a Beginningless Past Actually Infinite?, but it is my view that Craig’s response is unsatisfactory.  Although it would take us too far afield to give a detailed critique of Craig’s article at the moment, I do intend to come back to it in a future post in this series.

But these preliminary considerations are not central to my argument here, so let’s put them aside for now and move on to the topic of subtraction.

2.  Subtraction as the inverse of addition

When you learnt arithmetic back in primary/elementary school, you probably did many worksheets with exercises like these:

  • 2 + 3 = __
  • 5 + 7 = __
  • 3 + 1 = __

As you got older, you would have seen more advanced exercises such as:

  • 1 + __ = 5
  • __ + 6 = 9
  • 3 + __ = 8

Each of these exercises has a unique solution.  For example, the answer to the problem 1 + __ = 5 is uniquely determined by the numbers 1 and 5.  We generally denote the answer by 5 – 1 and, in fact, we define 5 – 1 to be “the solution to the equation 1 + __ = 5″.  If two different people solved the equation 1 + __ = 5 in two different (but valid) ways, they would always get the same answer, so there is no ambiguity in defining 5 – 1 in this way.  So the second set of exercises could be rewritten as:

  • 5 – 1 = __
  • 9 – 6 = __
  • 8 – 3 = __

This is one way to define the subtraction operation: as the inverse operation of addition.

Now, if you have some intuition with infinity, you would probably agree with the following sums:

  • ∞ + 1 = ∞
  • ∞ + 2 = ∞
  • ∞ + ∞ = ∞

All of these (and many more) mean that the equation

  • ∞ + __ = ∞

does not have a unique solution.  There are several values that could correctly fill the blank.  In fact, there are infinitely many values that could correctly fill the blank: 0, 1, 2, 3, 4, … , ∞ all work fine.  Since there is no unique solution, we cannot unambiguously define ∞ – ∞ to be “the solution to the equation ∞ + __ = ∞”.

But is this a problem?

It actually isn’t, although Craig thinks it is, as we’ll see later.  As Millican pointed out in his debate with Craig, there are other operations that run into similar “problems”.  For example, when you learned multiplication, you would have worked on exercises like these:

  • 2 × 5 = __
  • 3 × 9 = __
  • 4 × 6 = __

And then you would have moved on to more advanced ones like:

  • 3 × __ = 21
  • __ × 8 = 32
  • 2 × __ = 18

Again, there are unique answers to each of these problems.  And this allows us to define 21 ÷ 3 to be “the solution to the equation 3 × __ = 21″.  And these three exercises could be rewritten as:

  • 21 ÷ 3 = __
  • 32 ÷ 8 = __
  • 18 ÷ 2 = __

But what happens when we do multiplication with zero?  Well, zero multiplied by any number is still zero:

  • 0 × 0 = 0
  • 0 × 1 = 0
  • 0 × 6 = 0

And these serve to show that the equation

  • 0 × __ = 0

does not have a unique solution.  Because of this, it is impossible to define 0 ÷ 0.  But does this mean that we should be dubious about the number 0?  Should we suppose that it is impossible for someone to have no coins?  I certainly don’t think so!  Amazingly, we’ll see that Craig does have some kind of problem with the number 0, though the reasons he gives are baffling to say the least.

Before we come to examine Craig’s response to these ideas, it is worth noting that anyone who has studied mathematics at university level will have seen a few more examples of this phenomenon: the squaring operation, multiplication of matrices or in certain rings.  These are all operations that are not invertible in general.  But it doesn’t mean that there are irreparable problems with arithmetic in these settings.  It simply means that these operations are not invertible in general.  No more, and no less.  This would only be a problem if you had a predetermined view that every operation must be invertible.

As it happens, my area of mathematical expertise is Semigroup Theory, which is the study of algebraic systems with an operation that is generally not invertible (this can be compared to Group Theory, the study of algebraic systems with an invertible operation).  A few of my most recent papers concern infinite semigroups – I have a feeling Craig would be appalled if he read them!

3.  Craig’s comeback

Here is what Craig has to say in response to all of this:

Dr Millican says that infinity minus infinity is not defined in transfinite arithmetic – there’s more than one solution to the equation [ie, the equation ∞ + __ = ∞].  And that is precisely the problem when you try to translate this into the real world.  You can slap the hand of the mathematician who tries to subtract infinity from infinity, but you can’t stop someone from taking away a certain number of coins.  And the contradiction is that you have identical quantities, you subtract identical quantities, and you come up with non-identical results.  And it needs to be understood that infinity in this case is not that sideways lazy 8, it’s the number aleph null, which is a number.  It is the cardinal number of infinity.  And it would be identical minus identicals yields non-identicals, and I submit in reality that’s absurd.

As for zero, I think zero is very problematic, frankly.  I mean, suppose someone said “There’s an elephant in the quad” and I said “Well, I don’t see any elephant in the quad”, and he says “Well, there is an elephant in the quad and its number is zero”.  Well, I think that’s very problematic.  I don’t think there is such a thing as zero.  It means just the absence of something.

Watch Craig say this here.

Now there’s a lot to say about this speech from Craig.  I’ll deal with the obvious things first, and then get to the heart of Craig’s misunderstanding.

Lazy 8.  First, Craig’s comment about the symbol ∞, or the “sideways lazy 8″, is neither here nor there.  Mathematicians use the symbol ∞ for a variety of different purposes, and one of them is to denote the cardinal number “aleph null”, or \aleph_0, which is also sometimes denoted ω (the lower case Greek letter omega), although the ω symbol is usually only used when referring to \aleph_0 as an ordinal number rather than a cardinal number.  Aleph null is not “the cardinal number of infinity” – in fact, this expression doesn’t make any sense at all.  Rather, aleph null is the cardinal number of the set of natural numbers.  Interestingly, there are other infinite cardinals:  \aleph_1, \aleph_2, and so on.  In fact,  \aleph_1 is a bigger infinity than \aleph_0, and \aleph_2 is bigger still than  \aleph_1.  For any infinity, there is an even bigger infinity.  Yes, this means there are infinitely many different kinds of infinities!  I’m surprised that Craig never complains about this!

So, even though Craig might be upset about it, I’m just going to use the symbol ∞ to mean the cardinality of the natural numbers.  Mathematicians typically do this when they are not talking about several kinds of infinity at once.

Zero.  Next, I hope it is abundantly clear to everyone reading that Craig’s comments about zero are hopelessly incoherent.  I happen to agree that the conversation Craig presents is problematic.  But this only illustrates a problem with the kind of person that might say such things, not a problem with the number 0.  I don’t really know what Craig was thinking, here.  Perhaps he meant his hypothetical person to say that “the number of elephants in the quad is zero”, rather than “there is an elephant in the quad and its number is zero”.  But this wouldn’t be a problem at all.  I’m quite happy for someone to tell me the number of elephants in the quad of the University of Birmingham is zero.  Unless there was an actual elephant there, this would be a true statement.  As it happens, the number of elephants sitting in my lap as I type this is also zero.  Thank goodness for that!  But if Craig really has a problem with the number zero, then subtraction becomes problematic even when restricted to whole numbers.  If Craig gave all his coins away, how many would he have left?  Zero.  We know that

  • 5 – 5 = 0
  • 7 – 7 = 0
  • 9 – 9 = 0.

And all of this is perfectly consistent.  There is no problem whatsoever in me saying that I have 0 daughters, and 0 degrees in theology.

In the end, Craig’s comments about zero are completely nonsensical, but they’re not central to his argument, so I’ll let them go.  I’m prepared to accept that this was an off-the-cuff statement from Craig, and that he might have said something more coherent if he had a bit more time to think about his words.  So let’s move on to the more important problem.

4.  Subtraction as “taking away”

Another way to define subtraction is via the operation of “taking away”.  Sometimes “5 – 3″ is read as “5 take away 3″.  And, to help you calculate 5 – 3 when you were learning, your teacher might have said something like:

  • If you had 5 apples, and I took away 3 of them, how many would you have left?

You might have actually done an experiment like this with apples several times, and seen that the answer is always 2.  In fact, it doesn’t matter which 3 apples you take away: you will always have 2 left.  But there’s more.  If you started with 5 bananas and took any 3 of them away, you’d have 2 left.  And this illustrates a general point.  If you start with a collection of 5 objects of any kind, and take any three of them away, then you will always have 2 left.

All of this seems completely unremarkable.  But it illustrates a couple of basic theorems in set theory.  To state them, I’ll first need to introduce a tiny bit of notation.

Let A be a set (ie, a collection of objects).  The objects contained in A are called the “elements” of A.  We write |A| for the number of elements in A.  For example, if A is the set consisting of all your fingers, then |A| = 10 (probably).

A set is called a “subset” of if each element of B is an element of A.  We write A to indicate that is a subset of A.  For example, if A is the set consisting of all your fingers, and if B is the set consisting of all the fingers on your right hand, then ⊆ A.  Essentially, ⊆ means that A is a set that extends B, in the sense that it contains everything that B contains, and possibly more.

If  A, then the “complement of B in A” is defined to be the set of all elements from that are not in B.  We write for the complement of B in A.  Continuing with the above example, if A is the set consisting of all your fingers, and if B is the set consisting of all the fingers on your right hand, then is the set of all the fingers on your left hand.

And now we can state a couple of theorems of set theory:

Theorem 1.  Suppose A is a finite set.  Suppose ⊆ A and ⊆ A, and that |B| = |C|.  Then |B| = |C|.

Theorem 2.  Suppose A and B are finite sets and that |A| = |B|.  Suppose ⊆ A and  B, and that |C| = |D|.  Then |C| = |D|.

These might look complicated (mathematical notation can sometimes have that effect).  But all the theorems really say is that if you have two collections, each with the same finite number of objects, and you then remove the same number of objects from each collection, then you will end up with the same number of objects in each case.  Theorem 1 is the reason why you get the same number of apples when you remove 3 apples from 5, no matter which 3 you remove.  And Theorem 2 is the reason why you get the same answer if you used bananas instead of apples.

Because of these theorems, we are able to define subtraction in a more direct way, without having to refer to addition at all.  For example, we can define 9 – 5 to be “the number of objects left when you take 5 objects away from any collection of 9 objects”.  Because of the theorems, we know we are going to get the same answer no matter which collection of 9 objects we start with, and no matter which 5 objects we take away.

But here is where it gets interesting.  You’ll notice that Theorems 1 and 2 specifically require the sets A and B to be finite.  The theorems cannot be proved for infinite sets.  In fact, the theorems are false in the context of infinite sets.  Instead, we have interesting situations like the one Craig referred to with the infinite coin collection.  Here is another theorem from set theory:

Theorem 3.  Suppose A is an infinite set.  Then there exist subsets ⊆ A and ⊆ A such that |B| = |C| but |B| ≠ |C|.

This theorem says that given any infinite collection, it is possible to remove an equal number of objects from the collection in two different ways, resulting in a different number of objects left over.  Craig has kindly supplied us with an example of this theorem in action.  His analogy with coins can be simplified with mathematical notation.  So, let

  • A be the set of all positive integers: 1, 2, 3, 4, 5, 6, …
  • be the set of all odd positive integers: 1, 3, 5, 7, 9, 11, …
  • C be the set of all integers greater than three: 4, 5, 6, 7, 8, 9, …

Then we see that ⊆ A and ⊆ A.  Also, we clearly have |B| = |C|, since both and are infinite.  However:

  • B is the set of all even positive integers: 2, 4, 6, 8, 10, …
  • C is the set of the first three positive integers: 1, 2, 3.

So here we have |B| = ∞, while |C| = 3.  In other words, |B| ≠ |C| even though |B| = |C|.

Examples like these show that we would not be able to define ∞ – ∞ as “the number of objects left when you take infinitely many objects away from any infinite collection of objects”, since you could get different answers depending on which objects you chose to take away.  But this is the only problem, if it is indeed deemed a problem (remember, we can’t define 0 ÷ 0 either, and this is not regarded as problematic by any mathematician in the world, as far as I know).  It is worth quoting Craig again on this point, to remember exactly what he said about this:

You can slap the hand of the mathematician who tries to subtract infinity from infinity, but you can’t stop someone from taking away a certain number of coins.  And the contradiction is that you have identical quantities, you subtract identical quantities, and you come up with non-identical results… and I submit in reality that’s absurd.

I agree that a mathematician who thought it was possible to define ∞ – ∞ should have his or her hand slapped (metaphorically, of course).  And I agree that you can’t stop someone from taking away a certain number of coins, no matter how big their coin collection is.  But there is no contradiction here at all.  To think otherwise is to grossly misunderstand what is going on.  The fact that ∞ – ∞ has no unambiguous meaning does not prohibit someone with an infinite coin collection from giving away infinitely many of their coins.  All it means is that the number of coins they have left after doing so will depend on which coins they gave away.  If Bob has an infinite collection of coins, then he may give infinitely many of them away if he likes.  However, if he tells Wendy that he has done so, she will not be able to determine how many coins are left – at least not without knowing which coins were given away.  But there is nothing to prohibit Bob from giving away as many coins as he wants, even though Craig seems to be saying there is.

The above discussion makes it clear that Craig is (deliberately or accidentally) equivocating on the phrase “take away”.  For example, the sentence “you can’t take away infinity from infinity” could be interpreted in (at least) two different ways:

  1. you can’t define ∞ – ∞, or
  2. you can’t take infinitely many objects away from an infinite collection.

As we have seen, Statement 1 is true, but Statement 2 is false.  More importantly though, Statement 1 has no bearing on Statement 2, even though Craig claims that it does.

5.  Conclusion

Craig’s problems seem to arise from the view that Theorems 1 and 2 should apply to infinite collections (if they exist), and not just finite collections.  But they don’t, as the example of the infinite coin collection clearly illustrates.  But far from demonstrating that it is therefore impossible for a person with an infinite coin collection to give away infinitely many coins, all it shows is that it would not be possible to determine how many coins were left after such a generous act, unless we knew precisely which coins had been given away.  In fact, all these considerations show that life would be very interesting for someone with an infinite coin collection, even if most economic models couldn’t cope with an infinitely wealthy tycoon.

But, more importantly, the impossibility of unambiguously defining ∞ – ∞ has no bearing whatsoever on the possibility or impossibility of the “real existence of an actually infinite number of things”.  Craig’s argument against the existence of actual infinite collections therefore fails.  Maybe such collections are impossible.  Maybe time couldn’t be infinite in the past.  But one thing is abundantly clear: Craig’s arguments go nowhere towards demonstrating it.

  • im-skeptical

    I read in one of Craig’s papers (I can’t recall the title now) that he believes there is something called ontological time. This is supposedly God’s time, and it has an infinite past. He claims it is synchronized with physical time (if that’s possible) during the time of the existence of the universe. What is really striking about this is that he could apply the same argument to “prove” that an infinite timespan is metaphysically impossible for ontological time, as well as for physical time, but for some reason he has failed to do so.

    • D Rizdek

      When Craig does this…tries to apply SOME FORM of time to his god, I believe the saying…”When you are in a hole, stop digging” applies. Dr Craig is, by many measures very successful at what he does. He presents intuitive lines of reasoning to justify a preconceived notion of god and provides a shelter for the flock against the skepticism of the world. IOW, he makes believers feel good about their beliefs. But I THINK he could do this without making up some sort of “ontological time” which no one understands and creates more problems that it solves…even for the theist. What does that move get him?

      Maybe the wear and tear of setting himself up as the bastion, the frontier stronghold so-to-speak, against the atheist hordes is finally taking its toll.

      • Reasonably Faithless

        Some good points. But I think that, ultimately, Craig *must* invent something like “ontological time”. Craig claims a lot of (at least seemingly) contradictory things – eg, there are no infinite collections, yet God’s power, knowledge, goodness, etc are infinite. As such, he simply cannot explain these in simple terms. It is therefore necessary to introduce some kind of mystical idea that somehow allows these “tensions” to seem amazing and mysterious, rather than flat out contradictory. But, as we know very well, you can’t just appeal to another mystery to explain the first mystery.

        • Mike D

          He does that quite a bit. Another example is a non-physical analog of causality to “explain” how causality could apply to the universe; “ontological time”, or “non-physical time”, in a concept he invented to explain how the universe could begin to exist if no time existed.

          My problem with such speculations isn’t that they’re not possible (it’s anyone’s guess what’s possible “beyond” the physical universe), but that literally the only reason to introduce such speculative phenomena is because without the assumption that they exist, his clever-sounding syllogisms fall apart. But he’s clearly just begging the question.

    • Greg G.

      Craig argues that there cannot be an infinite number of events going back into the past. If so, and since thoughts are events, what was God’s first thought? Dan Barker stumped Hamza Tzortzis, the Muslim WLC, with that question.

      • Reasonably Faithless

        Yes, it’s hard to see how God didn’t “begin to exist” if there was no time “before” the universe, and no time at which God existed but the universe did not. I suppose Craig would say that “before” the universe, with no time around, God’s thoughts had no limits – but that once the universe was created, God started having thoughts – it would be interesting to hear what Craig thinks God’s first thought might have been, but I suspect he would say he didn’t know, and that it was irrelevant.

        • Greg G.

          Yes, the specific thought is irrelevant but the question of there being a first thought, and when, is relevant.
          If there was no time, there couldn’t be a prior decision to create. There would be no distinction between a decision and a post hoc excuse for having created the universe without intending to do so ahead of “time”.
          Causing creation presents another challenge. A cause acting on nothing produces no effect so the extrapolation from things that begin to exist having a cause is talking apples and oranges. Things that are caused to exist are changes caused to other things that already exist. The only way around that is if the caused acted on the only thing that already existed but that leads to pantheism and they don’t want to go there.

          • Reasonably Faithless

            Good points, and yes – they must go in a lot of directions they don’t want to go!

      • Eric Breaux
        • Greg G.

          Thanks for the link. I have always thought of Calvinism as the logical conclusion of Christianity and should serve as an adequate reductio ad absurdum refutation of it.

          The description of God as outside of time and therefore unchanging is much like the idea of the description of the multiverse as a space-time continuum. Adding knowledge or consciousness to the description is presumptuous, like wishful thinking. That Calvinistic description then reduces to a grand form of pantheism.

  • DRC

    About the definition of 0/0…
    I used to think that if any variable was found to be 0/0, then this was a road block; a failure to find an answer. I’ve since come to the conclusion that 0/0 actually provides very important information about the system you’re looking at. A simple example:

    x – x = 0
    Now we solve for x:

    The answer is not saying that we’re unable to find x. Rather, it’s saying that *every* value of x is true (you can check this by putting any value into the original equation). 0/0 carries very important information which is just as useful as finding out that x=5 or 3+2i. We shouldn’t think of 0/0 as being a problem of any sort. It’s a legitimate answer to many problems, and it provides very useful information.

    • Reasonably Faithless

      Those deductions are only valid until the last. Once we hit x*0=0, we know that any value of x is a solution. If 0/0 was meant to be a meaningful expression, then we could do something like this:


      So this would imply that 0/0=1. But of course we could have replaced 1 with anything we like.

      • DRC

        Good point. Your example shows that allowing 0/0 can result in a *loss* of information about x. That’s no good!

        • Reasonably Faithless

          True, although it was really multiplying by 0 that resulted in the loss of information. All of your steps are reversible. But, since the equation you started with was essentially 0=0, it’s no surprise that any value of x (or y,z,a,b,c,…) “satisfies” it ;)

  • josh

    Zeno’s paradox (of Achilles and the Tortoise) must give Craig fits. He must not believe there are an actual, real, present infinite number of points in a finite space. And I guess he thinks that past time we have witnessed is really a finite number of discrete moments.

    I think part of the confusion people (including Craig, but most have the sense not to argue metaphysics based on their ignorance) have is that inifinity is not a number in the sense that 42 is a number. Infinity is a concept which means ‘without an end’, ‘no stopping point’,etc. Infinite sets do not have a numerical size. They are without end. Nonetheless, we can subtract or add one well defined set with another, the result is a new set which may or may not be infinite in size. We can also sometimes have a well defined map for every element in one infinite set to another, and sometimes not, which is what we mean when we speak of different ‘sizes’ of infinity. But they aren’t numerical sizes.

    All of this is said above with a little more formalism and rigor, but I do think the accepted terminology of infinite numbers and and bigger and smaller infinities is a bit off-putting.

    In terms of ‘real’ infinities, you have to ask ‘What are the conditions under which I conclude that something is ‘really’ infinite or ‘really’ finite?’ To argue that a real infinity is impossible you have to say under what conditions one COULD conclude that a potential infinite is real. If there aren’t any then you’ve just defined ‘real infinite’ out of existence and ‘finite’ as ‘not proven infinite’.

    For example, space looks infinite and might well be, I think. But I don’t know how to prove that space is infinite except to say that there is no logical contradiction in it having no boundaries and I haven’t found any. But if I say that it is impossible for space to be infinite then I have to prove that logically it must have discoverable boundaries. How the hell could I do that?

    Which of course has no logical force for the sort of arguments Craig wants to make. But that’s too empirical and skeptical for guys like Craig, who would rather cram reality into their preformed notions.

    • Reasonably Faithless

      Very interesting thoughts – thanks for sharing them. And yes, Craig seems to be obsessed with the simplest of “problems” with infinities. There are things that should keep him up much later at night than infinite coin collections. Singular cardinals, large cardinal axioms, the Ross Urn paradox, just to name a few – and Zeno ones are great too. I think it is an interesting question about whether space and time are discrete or continuous. Because if time had the structure of the real numbers (or even the rationals), then there really would be infinitely many past events even if time only went back a finite distance.

  • Gabe Czobel

    Although it is edifying to analyse, in sober detail, Craig’s various simplistic forays into such esoteric areas as the mathematics of infinity, or Big Bang cosmology, or the syllogistic logic of the Kalam argument, such dissection of his arguments on mathematical and scientific esoterica actually falls into the cleverly set trap of the very smoke and mirrors which constitute the bulk of his method in his writings and debates. He puts on a great show of confident and
    measured erudition on such weighty matters, full of abstruse terminology and endless oversimplified syllogisms (I call them “sillygisms” as he uses them), to impress the heck out of the gullible, and often flummox even learned debate opponents (Lawrence Kraus is an example) because Craig’s errors and lack of depth in such matters cannot be clarified to a general audience in a few simple sentences and retorts. And if his opponent even attempts such clarification, which most cannot resist, Craig has already scored his point by tying up his opponent in a side issue while diverting attention away from the fact that none of his arguments actually clearly point to the fairly specific notion of the god that is supposed to be at the center of contention.

    For instance, even if you grant Craig that real infinities do not exist, the Kalam argument simply concludes that the universe began. This is where the real work should start where Craig would need to convincingly show that this beginning of the universe could only be caused by his particular deity to whom he would attribute a set of specific traits such as infinite power, wisdom, love (oops! – didn’t we just agree infinities do not exist?), is really three-in-one, rests on Sundays and expects his followers to do the same, gets angry at a whole list of forbidden
    activities, and so on in endless tedium. None of these large number of traits could be read into the idea that the universe began to exist, except for the one regarding Genesis, and even that only by a liberal interpretation, since Genesis does not depict anything like the Big Bang. Or, to put it another way, how can we distinguish the claim that Craig’s deity was the cause of the universe’s beginning, from an opposing claim that it was the work of some different deity who only had enough power to create a universe, and no more, wants us to rest on Tuesdays, is really ten-in-one, created us for its (their) wanton amusement, etc? Of course, any such alternate claims are limited only by our imagination. By odds, Craig’s god, or any specific complex deity, is very unlikely, all things being roughly equal.

    • Reasonably Faithless

      Thanks for reading, and replying! You made some very good points there. My plan with this blog is to do something I could never do in a debate, and that is to one-by-one refute each of Craig’s points – even the tangential ones. You’re also right that one could grant quite a lot to Craig, and he would still have his work cut out – and you’re absolutely right that none of his arguments prove anything like what he defines God to be (maximally great, etc). I made this point in this post about the modal ontological argument:

  • John W. Loftus

    I’m not a mathematician by any stretch but this was very readable to me. You made some great points I haven’t seen anywhere else before. I sure hope you get this series of yours published in the journals as separate articles on specific problems, or as a whole. Wow!

    • Reasonably Faithless

      Thanks, John – that means a lot!

  • GearHedEd

    Doesn’t Craig really just give his own game away when he says stuff like this:

    And the contradiction is that you have identical quantities, you subtract identical quantities, and you come up with non-identical results…

    Since when are two different infinities “identical”?

    • Reasonably Faithless

      By “identical quantities”, I think he means “collections of the same size”. In the case of the coin example, although the sets {1,3,5,7,…} and {4,5,6,7,…} are different, they still have the same size – ie, they’re not “different infinities”. But the whole point is that with infinite sets, you *can* remove subsets of the same size, and end up with a different number of elements remaining. For Craig to assert that “removing identical quantities from identical quantities leaves identical quantities”, he is already begging the question that all quantities are finite.

  • John Grove


  • kuartus

    Hello. I was having some thoughts on this subject and wanted to get some feedback. You say in your post,
    ” The fact that ∞ – ∞ has no unambiguous meaning does not prohibit someone with an infinite coin collection from giving away infinitely many of their coins. All it means is that the number of coins they have left after doing so will depend on which coins they gave away. If Bob has an infinite collection of coins, then he may give infinitely many of them away if he likes. However, if he tells Wendy that he has done so, she will not be able to determine how many coins are left – at least not without knowing which coins were given away. ”

    You say that the number of coins left depends on which coins were taken away. But suppose the coin collection consists of identical coins. There is no difference between any of the coins. In this case the only distinguishing factor between coins is how each coin has been numbered. Suppose you number the coins from 1 to infinity. You then take away the coins numbered with the even positive integers. The number of coins left accordingly would be infinite. But since we are dealing with real material coins, we could always renumber the coins in a different order. Since the number of even positive integers is equal to the number of positive integers greater than 3, we could renumber the coins we took away with the set of positive integers greater than 3. But now that we have renumbered the SAME coins we took away earlier, we go from having an infinite amount of coins left to just having 3 coins left. Just from arbitrarily renumbering the coins. This seems absurd. But if your argument is correct then it would seem to be possible. After all, the order in which we number the coins is completely arbitrary. We are dealing with real coins, not numbers. What do you think? I’m not saying my argument is correct. I probably dont know what I talking about. But some feedback would be appreciated.


    • Reasonably Faithless

      Hello kuartus, and thanks for your very interesting question!

      The point is that, even though you might have infinitely many coins that appear to be completely identical, that is all they are – identical in appearence. Even though one infinite subcollection of coins would look absolutely identical to another infinite subcollection, it doesn’t mean that when these two different subcollections are removed from the entire collection, there will be the same number of coins left over.

      For example, imagine I wrote down infinitely many o’s (meant to represent coins):


      And now, suppose I removed the first three of these:


      This time, imagine that instead I removed every second one:


      Although those three lists would look the same to anyone, they are actually made up of different coins. Removing the second collection from the first would leave:


      That’s it – only 3. But removing the third collection from the first would leave:


      Let me know if this doesn’t make sense!

      (Also, regarding your example of renumbering the coins, just because you have changed the labels of the coins previously labelled 2,4,6,… to 4,5,6,…, this doesn’t mean that the number of coins left will be 3. Previously you were left with coins labelled 1,3,5,…, but now if those coins are to have distinct labels on them, they would need to have labels that were not used before. Again, let me know if this isn’t clear.)

      • kuartus

        You wrote, ” Previously you were left with coins labelled 1,3,5,…, but now if those coins are to have distinct labels on them, they would need to have labels that were not used before. ”

        But supposing the coins we took away have been relabeled 4,5,6….. then the only labels left would be 1,2,3. With what would we label the rest of them with. We haven’t added any new coins to the collection, so the set of positive integers we used the first time should be sufficient to label them all the second time. But there doesn’t seem to be any left besides 1,2,3.

        • Reasonably Faithless

          No, this is begging the question. You’re saying that if we split an infinite collection of coins into two piles, one of which is infinite, then if we split an infinite set of labels into two subsets, one of which is infinite, we should be able to label the infinite pile of coins with the infinite subset, and use the labels from the other set to label the other pile. This would only be the case if the cardinality of the second subset was the same as the number of coins in the second pile.

          Think of it like this. You take an infinite collection of coins that all look the same, which we can represent by an infinite row of o’s:


          Now you label them (even if it just conceptually) by the natural numbers 1,2,3,4,5,… and you then break up the coin collection into two infinite piles – perhaps according to whether the label is even or odd. This leaves two infinite coin collections:


          It seems that you are claiming that since we were able to label the coins with the numbers 1,2,3,4,5,… before (by starting at the left-most coin, and running through the numbers 1,2,3,4,5,…), then if we begin labelling the coins in a different way, we should be able to complete the labelling. So, for example, suppose you labelled the coins in the first row 4,5,6,7,8,… from left to right. It seems you are claiming that you ought to be able to complete this partial labelling into a total labelling of the entire collection using just the labels from {1,2,3,4,5,…} that haven’t been used yet. But this is simply false.

          Essentially, this comes down to the (wrong) assumption that Theorem 2 (from the blog post) must apply to infinite sets. More specifically, you’re saying something like this:

          Suppose A and X are sets of the same size. Suppose B is a subset of A, and Y is a subset of X, and that f:B->Y is a bijection. Then there is a bijection g:A->X such that g(b)=f(b) for all b in B.

          But, again, this is simply false in the case of infinite sets (as the example you mention illustrates). Actually, it relates to my area of mathematics – semigroup theory. For any set X, one may consider the semigroup of all bijections between subsets of X – this is usually denoted I(X). You can also consider the set of all restrictions of all permutations of X to subsets of X – this is usually called F(X). It is a theorem that I(X)=F(X) if and only if X is finite. When X is infinite, F(X) is a proper subset of I(X).

          For example, consider X={1,2,3,4,5,…}. Let A={4,5,6,7,8,…} and B=X={1,2,3,4,5,…}, and consider the bijection f:A->B defined by f(a)=a-1 for all a in A. Then there is no permutation of X that has f as one of its restrictions – that is, f belongs to I(X) but not to F(X). This deals with Craig’s objections.

          For your more interesting example, it works more like this. Let L={4,5,6,7,8,…}, E={2,4,6,8,…} and O={1,3,5,7,…}. So X is the union of E and O. Define g:L->E by g(a)=2(a-3). (Essentially, g just maps the elements of L onto the elements of E in the order they are listed above.) Then, again, it is impossible to find a permutation of X that has g as one of its restrictions. In other words, thinking of the decomposition of X into the disjoint union of E and O as representing the splitting of the set of coins into two infinite subsets, and thinking of the function g:L->E as a way to label the coins from one collection with the elements of L, to say that g belongs to I(X) but not to F(X) is to say that it is not possible to start labelling the coins in the way you suggest and expect to be able to complete the labelling.

          I know that’s all a little bit technical, so please let me know if something needs to be clarified. The short answer is this, though. Just because you know a certain task can be performed in one way, it doesn’t mean that you can perform the task in another way. An analogy might be helpful. You know you can paint the floor of a room without getting footprints on the paint by starting in a corner and finishing near the door. But this doesn’t mean that you can start by painting around any doors and windows, and then expect to be able to complete the same task. Do you know what I’m trying to say?

          • kuartus

            I think I see where I went wrong in my assumption that the set of positive integers should be enough to complete the relabeling of the two new subsets. But let me present a simpler scenario. Supposed we have an infinite collection of coins which we initially label with the set of positive integers 1,2,3,4,5,6……..
            We then take away all the coins labeled with the set of integers greater than 5. We now have two subsets. One subset is infinite and the other subset is finite with just 5 members. Now we proceed to relabel the infinite subset with the set of positive integers 1,2,3,4……..
            Obviously there are no positive integers left to relabel the remaining finite subset. So my question is, what numbers should we use to label the remaining five coins? At this point I’m just curious.

          • Reasonably Faithless

            If all you wanted to do was put labels on the coins so that you could distinguish them from the others, you could label the as-yet-unlabelled coins with negative numbers: -1,-2,-3,-4,-5. Or you could use letters: a,b,c,d,e (there are many other possibilities). On the other hand, if you only wish to use natural numbers as labels, then by leaving out 5 coins and labeling the others 1,2,3,4,5,… and using up all the natural numbers, you’d have done the equivalent of painting yourself into a corner (note to self – that seems like a handy analogy).

      • kuartus

        BTW, what’s your take on Craig’s example of an immortal man counting down all the negative numbers from infinity past?

        • Reasonably Faithless

          I’ll probably do a separate post on this at some point in the future. But the short answer is that I don’t really see any logical problem with this scenario. Off the top of my head, I don’t remember all the things Craig has to say about it (do you have a link?). But I think he says things like “If he has counted infinitely many numbers to get to 0 today, and if he had also counted infinitely numbers by yesterday, then why didn’t he get all the way to 0 yesterday instead of today?”. But this is just silly. I’ll be happy to say more if you have a link.

          I said something about the possibility of God counting to infinity in this post:

          The arguments there can be slightly modified to show that God (conceived as the Greatest Conceivable Being) should be able to count down from infinity – in fact, he should be able to do so in a minute (no need for an infinite past).

  • strangerlook

    As a grad student in philosophy who’s not entirely familiar with this kind of debate, I wonder why Craig was motivated to reject the concept of actual infinity altogether (if he did). After all, what he needs to reject is just infinite past, and infinite past could be seen as implausible even when actually infinite collections are possible.

    Given A-thoery of time, in which Craig seems to believe, any point of time in the future should be accessible from *now* and *now* should be accessible from any point of time in the past, where t2 is accessible from t1 if and only if there could be a continuous time line between t1 and t2 and t2 is later than t1.

    A consequence of this would be that the only way for time to be infinite is to be potentially infinite. But the past cannot be potentially infinite. So it should be finite.

    There would be other ways to explain the incompatibility of infinite past and A-theory, but what is important is that accepting actually infinite collections of concrete objects is one thing and accepting infinite past is another.

    Anyway, this seems to be an independently interesting question regardless of the motivations of those who are debating on it.

    Another way out for Craig would be to be a deflationist about modality (as some notable philosophers like Siders do). Then it doesn’t really matter whether or not infinite past is *possible*; what matters is whether the past is infinite in this very world in which we inhabit. Then the debate turns into a purely empirical one, and empirical evidence from science would appear favourable to finite past.

    • Reasonably Faithless

      Thanks for your very interesting comments, strangerlook.

      I completely agree that actual infinites need not be impossible for an infinite past to be impossible. And I also agree that all that is required (for Craig) is that the past is in fact not infinite in our world, rather than completely impossible in any world. It seems that claiming there are no actual infinities whatsoever is the easy way for Craig to get what he wants (in much the same way a non-believer might like it to be the case that Jesus never existed in order to have no “problem” with Christianity at all).

      But I wonder why you say that the past cannot be potentially infinite? Wes Morriston has argued (very well, I think) that (i) if an infinite past is an actual infinite, then an infinite future would be an actual infinite; and (ii) if an infinite future is a potential infinite, then an infinite past would be a potential infinite. You can find some of Wes’s papers at his website here –

      Hope to hear from you again!

      • strangerlook

        Hi, thanks for your reply. I’m afraid I haven’t read Morriston yet (I will read soon), and for now I’ll just leave my current opinions about the two statements you put forward citing Morriston.

        First, I’m not sure what (i) does in this debate. The antecedent of (i) —that an infinite past is an actual infinite– is either necessarily false or not necessarily false:

        If it is necessarily false, then the whole conditional (i) is trivially true, lacking any substantial content at all.

        On the other hand, if Morriston could show that the antecedent was not necessarily false, then… there’d be nothing more Morriston has to show! If he successfully demonstrates that the antecedent of (i) is conceptually possible, then we don’t need to debate on this any further; if he succeeds, (i) is redundant.

        Thus, in either case, (i) seems pointless.

        Second, (ii) appears interesting, and, actually, very tricky. This is because the term ‘potentially infinite’ can be ambiguous when applied to time.
        Let us consider the consequent of the conditional (ii).

        On one reading, it means:

        (P1) An infinite past is a potential infinite [at a certain point of time t].

        On another reading, it would mean:

        (P2) An infinite past is a potential infinite [simpliciter].

        The A-theorist has no reason to reject P2. In fact, it is the A-thoerist who believes more than anyone that a past (if it exists) *grows* as time passes, in which case the past could be said to be potentially infinite unless its future has an end. So Craig would certainly agree to P2. What he should reject is P1. Then it is not entirely clear whether (ii), as it is now, is actually against what Craig has claimed, let alone whether it could be successfully justified. I hope (and believe) Morriston would make this point clearer, but at this point I see no genuine problem concerning (ii) as well as (i).

        • Reasonably Faithless

          My statements were far too sloppy, sorry. What I meant is that if one defines “actual infinite” in such a way that a beginningless past can properly be regarded as an actual infinite, then this would have the consequence of forcing an endless future to be an actual infinite too. Similarly for the other statement.

          Maybe before I say any more, I should check if that makes more sense.

          I think there are some interesting subtleties around all of this, given that (as you’ve hinted) A-theorists tend to think of the past as non-existent in some sense, while Craig defines an actual infinite in terms of existent things.

          • strangerlook

            An interesting point indeed. But it should first be noted that not all A-theorists are presentists, for any theory of time that accepts ‘being past’, ‘being present’, and ‘being future’ as genuine properties of the world is an A-theory, regardless of whether it ontologically commits itself to the events that instantiate those properties.
            If Craig is a presentist, however, he will have to reformulate his argument as you said. Yet this would not take much effort, as presentists have developed various ways to talk about past events without ontologically accepting them.

          • Reasonably Faithless

            That’s true, actually. I really wish I knew more about the various competing theories. Craig goes into a little detail in this interview (part of a longer video featuring others, including Quentin Smith) – – he seems to be advocating a presentist view.

  • Kris Riemens

    See this paper for another argument against the possibility of an actual infinity that I haven’t seen refuted by anyone yet:

    Looks interesting

    • Reasonably Faithless

      I love the New Zeno paradoxes! I have a few thoughts, and will certainly post about them in the future.

  • John Jagger

    Goodness gracious. Lets get REAL. Are you actually proposing that because various tricks on paper, sets, or otherwise defeat the fact that there cannot be an infinite past???

    Of course you’re not. So what are you actually doing but what the fearful atheist always does….trying desperately to patch together some ad hoc way to defeat the argument, all the while knowing its just a trick held together with tape and glue that has no business in reality.

    I have heard this pedantic tripe before as atheists trip over themselves trying to paint Craig as a nut and all it does is identify you as pathologically biased. No well known professional philosopher would ever claim such nonsense about Craig. They recognize him as one of the finest minds and most friendly colleague. Its just typical wannabe thinkers who use the internet to pat themselves on the back. Clap clap…your argument is embarrassing. It might be food for the daily roll call the bar room atheists must perform so they can sleep that night but in the end it amounts to nothing. Pretending Craig is crazy is just about all you got…otherwise you’d actually put up something that isnt a mind blowing display of Ockhams worst nightmare. Next time, reduce the common denominator. Reality is not taped together with the scribblings of the manic.

    • Reasonably Faithless

      Hi John, thanks for your message. My “embarrassing argument” has just been published in Faith and Philosophy, one of the top journals on philosophy of religion – you’ll find a link to the article here –

      I also note that you haven’t posted a criticism of any part of my argument, but merely told me how stupid it is. Do you have any actual critique? I have also never pretended Craig is crazy – I think he’s a nice guy (I shook hands with him last year), but I think that his arguments are unsound, as do plenty of professional philosophers. This post covers the reasons why I think one of those arguments is unsound. You’d be more than welcome to tell me where you think I’ve gone wrong. Cheers, James.

  • JacobBe5

    Thank you for pointing in such clear terms how Craig has been abusing (intentionally or not) mathematics.

    Couldn’t resist a little fun though abusing infinity:

    Let {A} be the integers > 0
    Let {B} be the integers < 0
    Both {A} and {B} are infinite in size
    {A}{B} = {0}
    I – A is infinite
    I – B is infinite
    I – A – B = 0

    Therefore I – 2∞ = 0 or ∞ – 2∞ = 0

  • Eric Breaux

    This is such a simple concept to understand that children instinctively comprehend it, but as the saying goes “ignorance is bliss”

    • Reasonably Faithless

      You do realise that the current blog (and the subsequent article in “faith and philosophy” that expands on it) refutes are large part of that article you linked to, right?

  • Benard Holborn

    Interesting article.

    I know that WLC uses coins as a way of illustrating why an actual infinite is impossible, but I don’t think it is always analogous to the point he is trying to make, regardless of whether his coin illustration is successful or not.

    A better way of looking at coins which is more in line with his point (i.e. that the universe cannot be infinite in the past) is if you were to subtract the coins one at a time. If you subtract them this way, you will never be able to subtract them all. In the same way, if the universe was infinite in the past, we would never have “now”.

    I know that, as you note, this is under the A-theory of time, but that is where the argument lives. In other words, it is valid under the A-theory of time. I grant you that it could be challenged if the B-theory of time is a better model, but that would be a different argument altogether.

    About your take on an apparent contradiction in WLC’s take on time you said this:

    “The reader will of course have noticed that Craig says here that “[t]he past is gone; it no longer exists”, but says above that “past events … are real” when arguing against an infinite past. ”

    This sounds like a contradiction, but I’m not so sure it is. It’s like this

    1) Dinosaurs are real.

    2) Dinosaurs do not exist.

    Both of these are true. Dinosaurs are real in the sense that they are conceptually real. They are not fantasy. Yet they no longer exist.

    You continue:

    “Since Craig’s problem is with infinite collections of real things, not with infinite collections of non-existent things, it seems that an infinite past should be no problem for him, even if it was granted that actual infinite collections were impossible. If he has a problem with an infinite past, then
    he may as well have a problem with an infinite future since, on his view, the
    number of future events is also infinite.
    (Craig believes in a never-ending afterlife and therefore, presumably,
    that the number of future events is greater than any finite number. Anyone who says that the number of future
    events is zero will be quickly proved wrong when the second hand creeps forward
    another tick.)”

    You seem to be missing something about what Craig believes. Yes, he doesn’t believe in actual infinities, but he does believe in potential infinities. Under this view, it is not problematic to have an infinite amount of future events. These events will go on for infinity, but will never at any given time be infinite. Just heading in that direction forever. His view of the afterlife then, is not contradictory.

    The rest of what you said in the section I quoted is likewise only problematic due to a lack of distinction between an actual infinite in the past and a potential infinite in the future.

    Perhaps you addressed these concerns already in the comments.

    • Reasonably Faithless

      Hi Benard,

      Thanks for your response.

      About your coins idea, I think this is exactly as disanalogous as Craig’s formulation. In particular, I agree that if you took away coins from an infinite collection, *starting with one coin* and then moving on to the second, then third, etc, you would never finish. But this is not at all like the situation with an infinite past (as far as I’m aware, nobody is arguing that the past began on some fixed day and there have been infinitely many days between then and now). Here, the analogous idea would be a situation in which we had always been removing coins from the collection. In this case, it would be perfectly possible to some day finish the task (or not – as always, it depends on the particular order in which the coins are removed). Or do you have something else in mind? If you elaborate further, I’d be happy to comment.

      About the idea that Craig’s “apparent contradiction” can be resolved by understanding “real” and “exist” in a certain way, I certainly take this point, and the dinosaur example is a good way to make it. However, I think this moves the problem elsewhere. If such a view might create a distinction between past and future events so that a “no infinities” premise might count against an infinite past but not an infinite future, then I think the “no infinities” premise becomes much more difficult to establish (I suspect the original one is impossible to establish anyway). In particular, are you saying that there could not possibly be an actual infinite number of “real” (but perhaps “nonexistent”) things? I’d be interested to hear this position argued for. In particular, it won’t do to argue that infinite coin/book collections are impossible, as these are about (alleged) problems with *existent* things. Do you think numbers are “real” even though they might not “exist”?

      Hope to hear back from you!