> you can't use anything infinite, including natural numbers. This is false, you don't need the axiom of infinity to prove theorems about all natural numbers.

> you can't use anything infinite, including natural numbers. This is false, you don't need the axiom of infinity to prove theorems about all natural numbers. https://math.stackexchange.com/questions/2850064/does-countable-induction-over-mathbb-n-require-the-axiom-of-infinity https://math.stackexchange.com/questions/2850064/does-counta... Particularly: > Notice here, we are just treating the natural numbers as a class, not necessarily a set. In fact we can prove from ZF minus infinity that any of the (translations of the) axioms of PA, including all the induction axioms, hold in this class.