 Since conclusion 8 follows validly, if premises 6 and 7 are true the argument is sound. In defense of premise 6, he defines an actual infinite as a determinate totality that occurs when a part of a system can be put into a one-to-one correspondence with the entire system (Craig and Sinclair 2009: 104). Craig argues that if actual infinites that neither increase nor decrease in the number of members they contain were to really exist, rather absurd consequences would result. For example, imagine a library with an actually infinite number of books. Suppose that the library also contains an infinite number of red and an infinite number of black books, so that for every red book there is a black book, and vice versa. It follows that the library contains as many red books as the total books in its collection and as many red books as red and black books combined. However, this is absurd; in reality the subset cannot be equivalent to the entire set. Likewise, in a real library by removing a certain number of books we reduce the overall collection. However, if infinites are actual, a library with an infinite number of books would not be reduced in size at all by removal of a specific number of books (short of all of them or all but a specific number), for example, all the red books or those with even catalogue numbers (Craig and Smith 1993: 11–16). The absurdities resulting from attempting to apply basic arithmetical operations, functional in the real world, to infinities suggest that although actual infinites can have an ideal existence, they cannot really exist.