# Math Help - Zermelo Universe from Notes on Set Theory

1. ## Zermelo Universe from Notes on Set Theory

If I have some universe W and I say that it is closed under the Axiom of Infinity, what exactly does that mean?

Does it mean that W contains the natural numbers and the set I that contains the empty set and the singleton of each of its members?

I just need a little clarification.

2. "Closed" under a property, means the result obtained after using the property, belongs to the considered universe W.

So, closed under the axiom of infinity means you can construct an inductive set, and this set is a member of W.

Now since the universal set W you are examining need not contain numbers, the inductive set you obtain need not be the set of natural numbers.

For example, say your set contains the ordinal $\omega_0$. Using the axiom, your universe contains all the ordinals up to $\omega_1$