### Craig's Critique of the Existence of Actual Infinites in His Blackwell Companion Article, Part I

Craig's recent defense of his philosophical arguments against the existence of actual infinites is unsuccessful. Start with his interaction with David Yandell's reply. Yandell correctly points out that no contradiction arises when one subtracts different infinite subsets from an actually infinite set. Thus: subtracting all the even numbers from all the natural numbers = all the odd numbers; subtracting all the natural numbers from all the natural numbers = 0. Now of course, if subtracting the very same subset yielded different answers on different occasions, then that would be a contradiction. But of course that doesn't happen when one subtracts from actually infinite sets.

In reply, Craig admits that no contradiction results from the infinite subtractions in Yandell's illustrations. Nevertheless, he asserts that a contradiction is yet to be found with another aspect of such subtractions. In particular, it's to be found in the fact that "an identical number of members have been subtracted from the identical number and yet did not arrive at an identical result." (p. 112)

As those who follow the literature on the kalam argument know, however, Craig here fails to address a well-known, mainstream criticism from Wes Morriston[1] and Paul Draper[2] -- one that's been around for at least a decade. The criticism is that Craig's reply turns on an ambiguity in the notion of "an identical number" of members of a set. According to one construal, two sets fail to have an identical number of members just in case they can't be put into a one-to-one correspondence with one another. According to another construal, two sets fail to have an identical number of members just in case one of them has all the members of the other, and others besides. The problem is that Craig is relying on the former construal, while the latter construal entails that the two cases do not involve an identical number of members that are subtracted. Referring to the coarse-grained similarity of the two subtracted subsets (viz., correspondence) conceals this crucial difference. But once one shifts one's focus from sameness of cardinality to the finer-grained difference in the particular members of the two subtracted subsets (the set of all natural numbers has all the members of the set of all even numbers, plus all the odd numbers besides), the intuitiveness of Craig's assertion disappears. For then one would of course expect different results when the sets removed in the two scenarios are different in this crucial way. Craig has thus failed to show a contradiction or absurdity in the notion of an instantiated actual infinite. Here, as elsewhere, Craig is craigging.

(We've covered this point in more detail on other occasions (here, for example). An index of criticisms of virtually all of his other key a priori arguments against the existence and traversability of actual infinites can be found here. A critique of his reply to Mackie's criticisms can be found here.)
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[1] See, for example, Morriston, Wes."Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 153-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 299-300
[2] Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008).

Angra Mainyu said...

Hi, EA.

Decisive points. I fully agree of course.

I wonder whether Craig will ever address this defeater to this particular argument of his, and whether he will ever drop the related claim that mathematicians "recognize" that actual infinities result in "self-contradictions" - a claim that he repeats many times in his debates and on his website, as a quick Google search shows.

I would just add that according to the first construal (i.e., the cardinality sense of "identical number"), there is no contradiction either, so as you say, Craig would need to equivocate in order to get the contradiction.

For instance, in the first sense of "identical number", if we subtract the set E of even numbers from the set of natural numbers N, the result is the (denumerably) infinite set of odd numbers O. If we subtract the set A of numbers greater than 1 from the set of natural numbers, the result is the set {1}.
We're subtracting an identical number in the sense of cardinality in both cases, and we do not get an identical result in terms of cardinality. But that's not a contradiction at all, since in that sense of "identical number", to say that we subtract an identical number in both cases and we do not get an identical result just means that there is a bijection between the set we subtract in one case and the set we subtract in the other case, but there is no bijection between the set that remains in one case, and the set that remains in the other. But that's not contradictory. In fact, it's clearly true in this case that there is a bijection between E and A, but not between O and {1}.

exapologist said...

Nice point, Angra.

Dima said...

Thank you for an interesting post, Exapologist!

I would be interested in your feedback on the following questions/thoughts of mine re what you have written:

1. "The criticism is that Craig's reply turns on an equivocation on the notion of "an identical number" of members of a set."
I don't think that charging Craig with equivocation is justified in this case. For him to commit this fallacy, he would need to use the same term in more than one sense in the course of his argument, not merely employing a term which could legitimately be used in more than one sense depending on a context. On the contrary, it seems to me that throughout his argument, Craig is using the term "identical number" consistently, according to your first construal ("two sets fail to have an identical number of members just in case they can't be put into a one-to-one correspondence with one another"). Perhaps you could point me to a step in his argument where he is equivocating.

2. "According to another construal, two sets fail to have an identical number of members just in case one of them has all the members of the other, and others besides".
Does this construal deny what appears to be an essential element of Cantor's set theory, namely, that an infinite set A is different from a finite set B just in case it is possible for A to contain a subset A1 which could be put into a one-to-one correspondence with A? If it does deny this, then could you please refer me to mathematicians who reject this claim of Cantor's theory? If it does not deny this, then it seems to me the following follows:
(1) If the members of A1 and A can be put into a one-to-one correspondence with one another, then A1 and A have an identical (equivalent, same)number of members.
(2) If A2 is also a subset of A and their members can also be put in a one-to-one correspondence with one another, then A2 and A also have an identical number of members.
(3) If the number of members (elements) in A = the number of members in A1 AND the number of members in A = the number of members in A2, then the number of members in A1 = the number of members in A2.

If (1) - (3) are true, then Craig's point about the absurdity of subtraction stands, or so at least it seems to me. Please share your thoughts on this and point me to any resources (preferribly by professional mathematicians who work in the field) which would correct my reasoning above.

Dima

Dima said...

Angra Mainyu,

"But that's not a contradiction at all, since in that sense of "identical number", to say that we subtract an identical number in both cases and we do not get an identical result just means that there is a bijection between the set we subtract in one case and the set we subtract in the other case, but there is no bijection between the set that remains in one case, and the set that remains in the other. But that's not contradictory."

The relevant point you seem to be missing here is that in addition to there being a bijection between E and A, there is also a bijection between N and E as well as N and A. Now that does seem to lead to a contradiction.

Dima

Dima said...

Angra Mainyu,

"But that's not a contradiction at all, since in that sense of "identical number", to say that we subtract an identical number in both cases and we do not get an identical result just means that there is a bijection between the set we subtract in one case and the set we subtract in the other case, but there is no bijection between the set that remains in one case, and the set that remains in the other. But that's not contradictory."

The relevant point you seem to be missing here is that in addition to there being a bijection between E and A, there is also a bijection between N and E as well as N and A. Now that does seem to lead to a contradiction.

Dima

Angra Mainyu said...

Dima,

N is the set of natural numbers.
E is the set of even natural numbers.
O is the set of odd natural numbers, so O = N\E (i.e., N minus E).
A is the set of natural numbers greater than 1.
So, A = N\{1}
E = N\O
O = N\E.
{1} = N\A.
{} = N\N.

Let's consider the following statements:

S1: There is a bijection between N and E.
S2: There is a bijection between N and O.
S3: There is a bijection between N and A.
S4: There is a bijection between A and E.
S5: There is a bijection between A and O.
S4: There is a bijection between N\E and A.
S5: There is a bijection between N\O and A.
S6: There is a no bijection between N\O and N\A.
S7: There is no bijection between N\E and N\A.
S8: There is no bijection between N\N and N\A.
S9: There is no bijection between N\N and N\E.
S10: There is no bijection between N\N and N\O.

All statements S1-S10 are true, so their conjunction entails no contradiction.
Now, let's define for any two sets X and Y, 'X and Y have an identical number of elements' to mean 'There is a bijection between X and Y'.
Then, you may re-write S1-S10 in terms of having or not having identical numbers, and that still won't result in any contradiction, since that's only a change in notation.

Angra,

exapologist said...

Hi, Dima.

Best,
EA

GGDFan777 said...

Here is another and I think stronger argument against the existence of an actual infinity. The article by philosopher Casper Storm Hansen of the University of Aberdeen called "New Zeno and Actual Infinity" can be found here: