Von Neumann integers definition

Question

I am learning the ZF (C) set theory considering all the 6 standard axioms, here is how I defined the set of natural integers $\mathbb{N}$. Is it correct and how do I prove the Lemma 2 ?

Axiom of Infinity : There exist a set ($A$) s.t

($\emptyset \in A ) \wedge (\forall X \in A, X\cup \left\lbrace X\right\rbrace \in A)$

Lemma there exist a (unique) set $\mathbb{N}$ that satisfies the axiom and such that $\mathbb{N} \subset A$

Proof :

Let $\mathcal{B}$ the power set of all subsets B of $A$ which have the above property (exists, power set axiom), $\mathbb{N}=\bigcap_{B \in \mathcal B} B,$ then $\forall P \in B, P \, \cap A $ also has this property (ie. $P \, \cap A \in \mathcal{B}$). Thus, by definition $\mathbb{N} \subset P \, \cap A$. Whence $ \mathbb{N} \subset P$. (this also proves the uniqueness)

Lemma 2 : $\forall n \in \mathbb{N}^* \exists m \in \mathbb{N}^*, n=m\cup \left\lbrace m\right\rbrace$

This one I can't find a proof by myself

Can you help me ?

T.D

Answer 1

You're definitely thinking along the right lines, but I'm a bit puzzled by your 'definition' of $\mathbb{N}$, since $\mathbb{N}$ appears in the condition defining itself. (In fact, all sets $A$ satisfying the axiom of infinity should satisfy $\mathbb{N} \subseteq A$.) It's also not clear what kind of uniqueness you're trying to prove—$\mathbb{N}$ is the unique set such that... what?

Here's what I'd do.

• We start off doing what you did: fix a set $A$ satisfying the axiom of infinity, let $\mathcal{B}$ be the set of subsets of $A$ which satisfy the axiom of infinity, and define $\mathbb{N} = \bigcap \mathcal{B}$.
• Next, prove that $\mathbb{N}$ satisfies the axiom of infinity. To see that $\varnothing \in \mathbb{N}$, note that $\varnothing \in B$ for each $B \in \mathcal{B}$; and given $X \in \mathbb{N}$, you have for each $B \in \mathcal{B}$ that $X \in B$, and hence $X \cup \{ X \} \in B$. Piecing this together tells you that $\mathbb{N}$ satisfies the axiom of infinity.
• You might now want to prove that $\mathbb{N}$ is the unique set such that $\mathbb{N} \subseteq C$ for all sets $C$ satisfying the axiom of infinity. (You know this when $C \subseteq A$.) Proving that $\mathbb{N} \subseteq C$ for each $C$ satisfying the axiom of infinity goes just like the previous step; proving that $\mathbb{N}$ is the unique such is easy (let $\mathbb{N}'$ be another such set, and note that $\mathbb{N} \subseteq \mathbb{N}'$ and $\mathbb{N}' \subseteq \mathbb{N}$ by the condition we just proved).

Something is also awry with your 'Lemma 2'. I don't know what $\mathbb{N}^*$ means, and in any case the expression $\exists m \in \mathbb{N},~ n = m \cup \{ m \}$ is only true when $n \ne \varnothing$. So instead let's prove $$\forall n \in \mathbb{N} \setminus \{ \varnothing \},~ \exists m \in \mathbb{N}, ~ n = m \cup \{ m \}$$ Proceed by contradiction, and suppose there is some $\varnothing \ne n \in \mathbb{N}$ which is not of the form $m \cup \{ m \}$. Now define $\mathbb{N}' = \mathbb{N} \setminus \{ n \}$. You will see that $\mathbb{N}'$ satisfies the axiom of infinity. You can deduce a contradiction from this, using the facts we already proved earlier.