The first premise of the Kalam cosmological argument has come under fire in the last few years. The premise states that the universe had a beginning, and one of two prominent arguments for it turns on the claim that an actual infinite collection of entities cannot exist. After stating the Kalam cosmological argument and the two approaches to defending its first premise, I respond to two objections against the notion that an actual infinite collection is impossible: a Platonistic objection from abstract objects and a set-theoretic objection from an ambiguity in the definition of ‘=’ and ‘<’ as applied to sets. The thought-experiment involving Hilbert's Hotel is central to the dialectic, and the discussion clarifies its use in supporting the Kalam cosmological argument.