I've got 2 questions concerning the axiom of infinity $AI$.
- Given the axiom of empty set $AE$, it's obviously not necessary that the set prescribed by $AI$ is unique. Can $AC$ or some another condition make this set to be unique?
- Does $ZF$ or $ZFC$ imply that $\mathbb{N}$ contains exactly $\emptyset,\{\emptyset\},\{\emptyset\,\{\emptyset\}\},...$, i.e., can $\mathbb{N}$ contain some element of another type? The another variant of this question. If $ZFC$ has a model, is it necessary that all elements of $\mathbb{N}$ are of the given type?
Addition 1. I'm using the following variant of $AI$: there is an inductive set $x$, i.e. $\emptyset\in x$ and $\forall(y\in x)\exists(z\in x):z=y\cup\{y\}$.
Addition 2. I also consider $\mathbb{N}$ to be the intersection of all inductive sets.
P.S. If the full answer was given before, then please can you send me the link? Here's something close, but it's not precisely my question: Could the natural numbers be unique after all?
The set isn't unique. The natural numbers are usually defined to be the intersection of all sets satisfying $\emptyset \in S \wedge \forall x \in S, \{x\}\cup x \in S$. The axiom says such a set exists. The natural numbers will be some subset of the set shown to exist by AI.