Craig admits that the
concept of an actual infinity in mathematics is perfectly consistent. But he
fails to show that there is anything logically inconsistent about an actual
infinity existing in reality. Moreover, in some of his examples he even fails to
show that there is some nonlogical absurdity.
Consider his example
of the library with an infinite number of books. Craig maintains that since
each book had a number on its spine, no new book could be added to the library.
But this, he concludes, is absurd (presumably in some nonlogical sense), since
“entities that exist in reality can be numbered.” This argument is unsound,
however, for books can be added and numbered by simply renumbering the books
already in the library. The new books would then be given the numbers of old
books—the books that had already been assigned numbers—and the old books would
be assigned new numbers.
Although it is
condensed and enthymatic, Craig’s argument that one cannot construct an actual
infinity by successive addition can perhaps be reconstructed as follows:
(1) For any point, it
is impossible to begin at that point and construct an actual infinity by
successive addition.
(2) In order to
construct an actual infinity by successive addition, it is necessary to begin
at some point.
(3) Therefore, an
actual infinity cannot be constructed by successive addition.
It should be clear
that (2) begs the question, since there is an alternative—namely, that an
actual infinity can be constructed by successive addition if the successive
addition is beginningless. To suppose that an infinity cannot be constructed in
this way is to assume exactly that is at issue. (Michael Martin, Atheism: A
Philosophical Justification [Philadelphia: Temple University Press, 1990],
105)
