Kalam, Actual Infinites, and Set Theory
In order to establish the second premise of the Kalam Cosmological Argument (“the universe began to exist”), William Lane Craig often argues against the idea of an “actual infinite” (i.e. an infinite amount of things/moments existing in reality). To do this, Craig shows that basic arithmetic cannot make sense of infinity, therefore supposing an “actual infinite” is absurd. Philosopher Colin Howson objects:
Lane Craig uses an argument that originates with Kant to ‘establish’ that time cannot be infinite in the past and still proceed into the future, on the ground that an actual infinite cannot exist because, among other reasons, if it did it would be impossible to add to it. But this claim is vitiated by the facts that (i) in contemporary set theory it is easy to show that there exists a sequence of infinite discrete ordered sets each with a greatest but no smallest member, each set extending its predecessor by an additional largest element; and (ii) the things in the domain of any consistent theory, as set theory is thought to be, are possible existents. Adducing similar observations, the distinguished philosopher of physics Michael Redhead concludes a review of Lane Craig’s argument with the remark that it, ‘seems a total muddle’. (Objecting to God, Pg. 92-93)
In other words, Howson argues that set theory makes sense of infinity in a way that arithmetic does not. And since set theory is consistent, then it is possible that “actual infinites” do exist.