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) =ℵ_{0}

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.

Hi Chad,

I’m not sure this is much of an objection:

But that is a problem for one who believes in the existence of actually infinite sets of things, not a problem for Craig. That is, were we to grant the notion that such sets could exist, then it would be possible to perform such an operation in the real world. But, then it would be possible to achieve absurd results.

Comment by Reidish — July 25, 2010 @ 1:04 pm |

Thanks for your comment, Reidish.

I’m not convinced my “objection” is a good one, but I’m not sure your reply quite does it justice.

My point is that if we grant for the sake of a reductio that such sets could exist, then it would

not necessarily be possibleto perform such operations in the real world, as Craig assumes.As I see it, Craig identifies two absurdities that would result should an actual infinite exist in reality. The first absurdity is that of adding or subtracting any number to or from an infinite set and ending up with no more or less members than what the set originally had. This corresponds to operation (i) in the post. Call this the “same results” absurdity. The second absurdity is that of subtracting identical quantities from identical quantities and ending up with contradictory results. The contradiction is between operations (i) and (ii) in the post: same quantities, different results. Call this the “contradiction” absurdity. Craig emphasizes the contradiction absurdity as the real problem, as is evident in the following remarks:

But my point is that one only gets the contradiction absurdity by assuming one actually can subtract an infinite number in the real world (i.e., that one can actually remove an infinite number of books). But why think someone can actually do that? Well, suppose we modify the example to include an infinite number of people, each of whom check out one book starting with 4 and above. But now the problem is that if we had an infinite number of people, there should have been a one-to-one correspondence between persons and books so that no remainder would be possible upon removal; hence, no contraction.

What I think this illustrates is that there is something even deeper awry with instantiated infinite sets than possibly to perform inverse operations with them.

Comment by Chad McIntosh — July 28, 2010 @ 2:16 pm |

I take it back! You would get a contraction in the modified case! Supposing we had an infinite number of people each of whom correspond to a book in an infinite library, we could have one case where every

odd-numbered personchecked out their corresponding book and another case whereevery personchecked out their corresponding book. Again, wherem= the number of books in an infinite library,n= the number of odd-numbered books, the contraction absurdity follows from these two cases:because

nandmare the same number.Comment by Chad McIntosh — July 28, 2010 @ 2:43 pm |