# Proving equivalence of 3 facts without axiom of infinity

I am reading Jech's book on set theory. In chapter 2, ex 2.4 says:

(Without axiom of infinity). Let $\omega =$ least limit $\alpha \neq 0$ if it exists, $\omega = Ord$ otherwise. Prove that the following statements are equivalent:

1. There is an inductive set

2. There exists an infinite set.

3. $\omega$ is a set.

For $(2) \Rightarrow (3)$, apply replacement to the set of all finite subsets of $X$.

My questions are:

1. Isn't (2) exactly axiom of infinity. If so, when i prove $(2) \Rightarrow (3)$ can I assume I have axiom of infinity?

2. I guess I can use the following facts?: All other axioms, $\mathbb N$ is the intersection of all inductive sets.

3. Can I use anything else?

4. Can I assume the "names" of elemtnts in a set are not important? For example, Is $\{0,1,2,3,4,...\}$ identical to $\{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \},... \}$ which is identical to $\{ a, \{ a \}, \{ a, \{ a \} \}... \}$?

Another question: How do I make new lines here in the stackexchange editor window? is there a special command for that?

Thank you

• Concerning 1.: No. The precise formulation of the axiom of infinity is given on page 13. In particular, the axiom of infinity is (1), i.e. that there exists an inductive set. – William Sep 15 '14 at 7:13
• Note that in (2), I believe being infinite here means not in bijection with a finite cardinal. – William Sep 15 '14 at 7:15

Of the three implications, $1\implies 2$ is trivial, by noting that no finite set is inductive; and $3\implies 1$ is trivial by noting that $\omega$ is an inductive set. So the only part which left is $2\implies 3$.
You have to show now, why given that there exists an infinite set, we can deduce the existence of $\omega$. If you already know what a transitive closure is, and what is the rank of a set, then this should be sufficient for a quick proof. If you don't you might have to do this slightly more "by hand".
Here's a small hint: Let $X$ be an infinite set. Use the power set and separation axioms to conclude that the set $\{A\subseteq X\mid A\text{ is finite}\}$ exists. Now use replacement to prove that $\omega$ exists.