Here is an objection to Craig-style arguments against the existence of an actual infinite “in the real world”. It is often said that inverse operations on transfinite numbers are prohibited. But this is not entirely true. As Nowacki explains, “inverse operations are prohibited for transfinite numbers in cases where performing the operation would yield a result that is not actually infinite.” (Kalam Cosmological Argument for God, p. 281, n.12. My emphasis). So where m = the number of books in an infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher, the following is permitted:
(i) (m – n) = ℵ0
while the following is not:
(ii) (m – o) = 4
because (ii) yields a finite number. If inverse operations yielding finite numbers were permitted, then it would follow that we get inconsistent results by subtracting the same number from m since n = o (both n and o are infinite), and this of course is why operations like (ii) are prohibited. But this leads me to think the following.
Craig’s point is that while inverse operations yielding finite numbers may be prohibited on paper, nothing would prevent us from performing them in the real world (say, by checking out a book from the library). But is this right? What would performing such an operation “in the real world” really amount to? What (ii) makes clear is that one would have to check out an infinite number of books to get a finite difference, not just any finite number of books. But then does Craig’s point follow that “nothing would prevent us from performing them in the real world”? Here’s something that might prevent us: having to perform an infinite task (check out an infinite number of books).
I’m not convinced that this objection works, but I don’t know the answer to it.