Second Version: Adding Without Making Larger

(The rest of the posts in this series can be found here.)

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books is possible.

2. If a library with an actually infinite number of books is possible, then adding any finite or denumerably infinite amount of books to the library would result in it having just as many books as before.

3. Adding any finite or denumerably infinite amount of books to the library couldn't result in it having just as many books as before.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? As with the argument in the previous post, the notion of there being “just as many” books is ambiguous in (2) and (3).[1] On the one hand, one might mean that there are "just as many" books before the additions as after in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites.

Now as noted in the previous post, Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. Again, we'll consider those in later posts in this series. For now, though, we’ll just note again that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are "just as many" books before the additions as after in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the resultant collection of books after each addition is one that has other books besides the original set, viz., the ones added.

As with the argument discussed in the previous post, the upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

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[1] As noted in the previous post, at least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; 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). (Notes on Draper's paper can be found here and here.)

(The rest of the posts in this series can be found here.)

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books is possible.

2. If a library with an actually infinite number of books is possible, then adding any finite or denumerably infinite amount of books to the library would result in it having just as many books as before.

3. Adding any finite or denumerably infinite amount of books to the library couldn't result in it having just as many books as before.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? As with the argument in the previous post, the notion of there being “just as many” books is ambiguous in (2) and (3).[1] On the one hand, one might mean that there are "just as many" books before the additions as after in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites.

Now as noted in the previous post, Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. Again, we'll consider those in later posts in this series. For now, though, we’ll just note again that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are "just as many" books before the additions as after in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the resultant collection of books after each addition is one that has other books besides the original set, viz., the ones added.

As with the argument discussed in the previous post, the upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

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[1] As noted in the previous post, at least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; 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). (Notes on Draper's paper can be found here and here.)

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