Axiom of Infinity

The axiom of infinity asserts the existence of a Successor Set

We can then show that there’s a smallest successor set, i.e. we can prove:

To prove it, In a model, pick a successor set using (Inf). Form the set

Since we can form . It’s easy to check that is the smallest successor set, which we denote by .

Every successor set contained in is i.e.

where is a shorthand for for any formula

So inside we have true induction! We call this -induction But still, from the outside there may be subsets of that are not sets in