Inductive Construction of Natural Numbers fulfils Peano's Axioms

Theorem
Let $P$ denote the set of natural numbers by definition as an inductive set.

Then $P$ fulfils Peano's axioms.

Proof
By definition of inductive set:
 * $\O \in P$

By definition of the natural numbers, $\O$ is identified with $0$ (zero).

Thus holds.

Let $x$ be a natural number.

By definition, $x$ is an element of every inductive set.

Thus if $x \in P$ it follows that $x^+ \in P$.

Thus holds.

By Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity, holds.

For all $n$, $n$ is an element of $n^+$.

But $0$ is identified with the empty set $\O$.

By definition, $\O$ has no elements.

Therefore it is not possible for $\O$ to equal $n^+$ for any $n$.

Thus holds.

holds immediately by definition of inductive set.