"But mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions. For example, what is infinity minus infinity? Mathematically, you get self-contradictory answers."

It's hard to know what Craig really means here, because it is so confused. Mathematicians

*routinely*study "an actually infinite number of things", such as the natural numbers, the real numbers, and the complex numbers. No contradictions are involved.

But maybe Craig is talking about an actual infinity of things in

*nature*. Then he shouldn't be talking about mathematicians, but physicists. Even here, physicists

*do*discuss an actual physical infinity - without contradictions - such as Malament-Hogarth spacetime. Examples like Hilbert's hotel, that are often proffered as insoluble paradoxes, only show that infinity needs to be treated with care and may result in scenarios that seem counter-intuitive. But so does relativity.

Infinity minus infinity is not "self-contradictory", any more than 1/0 is "self-contradictory". Lane seems not to understand that not all functions are everywhere defined. The subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞). What's so hard to understand about that?

*Addendum*As I listen to more of the debate, it seems Craig retreats a bit from his claim about mathematics. He still seems to think that an actual infinite number of objects in the universe creates "contradictions", but he doesn't say explicitly what those contradictions are.