### Craig’s Case Against the Existence of Actual Infinites, Part 5: More Objects than Numbers (Revised)

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

Consider the following argument from Craig:

"Suppose...that each book in the library has a number printed on its spine so as to create a one-to-one correspondence with the natural numbers. Because the collection is actually infinite, this means that every possible natural number is printed on some book. Therefore, it would be impossible to add another book to this library. For what would be the number of the new book? . . .Every possible number already has a counterpart in realty, for corresponding to every natural number is an already existent book. Therefore, there would be no number for the new book. But this is absurd, since entities that exist in reality can be numbered."[1]

We can put the argument more carefully as follows:

1. If concrete actual infinites are possible, then a library L with an infinite set of books is possible.
2. If L is possible, then it’s possible to assign a unique natural number to each book in L.
3. If it’s possible to assign a unique natural number to each book in L, then it’s possible to assign all the natural numbers to books in L without remainder.
4. If it’s possible to assign all the natural numbers to books in L without remainder, then if it’s possible to add a new book B to L, then it’s impossible to assign a unique natural number to B.
5. It’s possible to add B to L.
6. It’s possible to assign a unique natural number to B.
Therefore,
7. Concrete actual infinites are impossible.

This argument is valid.[2] Furthermore, (1), (2), (3), and (5) look to be true. Unfortunately, (4) seems false. Thus, consider library L again. Now suppose we reassign the natural numbers to the books in L as follows: assign ‘2’ to the first book, ‘3’ to the second book, and so on all the way through the rest of the books in L. Then we can free up ‘1’ to be assigned to the new book. But if so, then it is possible to assign a unique natural number to B, in which case (4) is false.[3] What does seem true, though, is not (4) but rather:

(4’) If it’s possible to assign all the natural numbers to books in L without remainder, then if it’s possible to add a new book B to L, then it’s impossible to assign a unique natural number to B if we hold fixed the original assignment of numbers to books.

Let’s revise the argument accordingly. To preserve the argument’s validity, we’ll also need to revise (6) to account for the new qualification:

(6’) It’s possible to assign a unique natural number to B (even) if we hold fixed the original assignment of numbers to books.

Does the revised version of Craig’s argument fare any better?

No, it doesn’t. For while (4’) seems clearly true, (6’) seems clearly false. For if we hold fixed the assignment of natural numbers to the books in L prior to the addition of B, then of course no unique natural number remains that can be assigned to B. But the problem here lies not with actual infinites, but rather with the internal coherence of Craig’s assertion that it must be possible to assign a unique natural number to a new book, even under the stipulation that all the unique natural numbers have already been assigned to other books.

What went wrong in Craig's argument? Recall premise (6) in the initial version of the argument:

(6) It’s possible to assign a unique natural number to B.

As Craig indicates in the quoted passage above, he accepts (6) on the grounds that:

(a) Any entity that exists in reality can be uniquely numbered.

This seems to be true. But (a) is not what Craig needs to derive (6). What he needs instead is

(b) Any entity that exists in reality can be uniquely numbered via a natural number.

Unfortunately, (b) looks to be false. For consider a library L’ that contains a set of books that can be put in a 1-1 correspondence with the irrational numbers. Such a set of books would be non-denumerably infinite; that is, it’d be actually infinite, but it couldn’t be put into a 1-1 correspondence with the natural numbers. Therefore, while all such books in L’ can be uniquely numbered, they can’t all by uniquely numbered via the natural numbers.

This example illustrates two salient points: (i) some sets of entities can’t be uniquely numbered via the natural numbers, and (ii) such entities can yet be uniquely numbered via other numbers, as there are more numbers than just the naturals. But given (i) and (ii), the door is open for numbering the new book in Craig’s library with a unique non-natural number.

We can sum up the problem with Craig's argument as follows. Either we hold fixed the assignment of natural numbers to books in Craig's infinite library or we don't. If we don't, then it's possible to reassign the natural numbers so as to free up a unique natural number for the new book. On the other hand, if we do hold fixed the original assignment of numbers to books, then it is impossible to assign a unique natural number to the new book. But of course there are more numbers than the naturals, and the new book can be numbered with one of these. If Craig yet demands that the new book be numbered with a natural number, even after all the natural numbers have been assigned to other books, then the problem lies not with the possibility of his infinite library, but rather with the coherence of the task demanded for it. Either way, then, Craig's argument is unsuccessful.

-------------------------------------
[*] Thanks to D.D. for reminding me of this argument of Craig's.
[1] Craig, William Lane. The Kalam Cosmological Argument (London: MacMillan, 1979), p. 83.
[2] Proof:

Let:
P=concrete actual infinites are possible
Q= a library L with an infinite number of books is possible
R=It’s possible to assign a unique natural number to each book in L
S=It’s possible to assign all the natural numbers to books in L without remainder
T= It’s possible to add a new book B to L
U=It’s impossible to assign a unique natural number to B

Then we have:

1. P -> Q Premise
2. Q -> R Premise
3. R -> S Premise
4. S -> (T - > U) Premise
5. T Premise
6. ~U Premise
----------------------------
7. P Assumption
8. Q 1, 7 MP
9. R 2, 8 MP
10. S 3, 9 MP
11. T -> U 4, 10 MP
12. U 5, 11 MP
13. ~T 6, 11 MT
14. T 5 R
15. ~P 7-14 ~I
Q.E.D.
[3] Wes Morriston makes this criticism in "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. 149-150.

Dr. Rizz said...

Dear ex-,

I have a problem with your objection that is hard to put into words. Let's say that you go ahead and free up '1' for the new book by bumping the number of all of the other books up by one. Wouldn't there be a book in the series that can't get a new number since all of the numbers are taken? Of course we can't say that the last member of the series can't get a new number because there is no last number. But, if all of the numbers have been used up don't we know that some book or other can't get a new number? I think that this is Craig's intuition as well.

exapologist said...

Hi Dr. Rizz. ;)

It surely seems true that if you have have, say, three books and three numbers (say, {1,2,3}), and you assign each of these numbers to a book, then you can't free up a number for a new book without leaving one of the original three books without a number.

However, it seems to me that the conditions that generate this fact about finite sets of books and numbers don't apply to infinite sets of books and numbers. For what seems to generate the problem for the former is that when each book "gives up" their number to another book in the set, at least one book cannot receive a number from the next book down the line. And the reason for this is that there is no other book down the line from which it can receive a replacement number.

But these conditions aren't present in the case of infinite sets of books and numbers. For there is a book next to each and every book without remainder (at least in one direction) in such cases. And because of this, each and every book can receive a new number from the next book down the line when they "give up" their number.

In short, it seems to me that the conditions that give rise to the problem of a book without a number in cases of finite sets of books and numbers don't apply in cases of infinite sets of books and numbers. And because of this, I don't share Craig's intuition.

Dr. Rizz said...

Dear Ex-,

If you are correct then it appears that it must be incoherent to speak of all the numbers in an infinite set being used up. I feel your intuition: for every number n that is assigned to a book, there is another number n+1 to which that book can be reassigned since the series simply goes on forever. Yet, I also (intellectually) feel the intuition that you can't can't add a natural number to all the natural numbers, and hence if every single one has been assigned, you can't create more space by shifting every book assignment down the number line. Hmm....

exapologist said...

Hi Dr. Rizz,

I'm not sure I quite follow you. Perhaps you can spell out the incoherence or logical contradiction explicitly (i.e., P & ~P)?

Dr. Rizz said...

I will give it a shot....

(p1) If all of the natural numbers have been assigned to a book in the library, then there are no numbers left to assign to a newly arrived book.
(p2) For all books assigned to a number n, these book can be re-assigned to another number n+1.
(p3) If all books assigned to a number n can be re-assigned to another number n+1, then a newly arrived book can be assigned number 1 which would then be unassigned.
(c) From (p2) and (p3) it follows that......

Okay, I see your point. What follows is not the denial of (p1).

### Review of Draper and Schellenberg (eds.), <I>Renewing Philosophy of Religion: Exploratory Essays</I>

Adam Green reviews the book for NDPR.