# Von Neumann Construction of Natural Numbers is Minimally Inductive

## Theorem

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.

$\omega$ is a minimally inductive class under the successor mapping.

## Proof

Consider Peano's axioms:

**Peano's Axioms** are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$.

Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, the existence of $s$ and the existence of $0$ were considered the first two of **Peano's Axioms**:

\((\text P 1)\) | $:$ | \(\ds 0 \in P \) | $0$ is an element of $P$ | ||||||

\((\text P 2)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \in P \) | For all $n \in P$, its successor $\map s n$ is also in $P$ |

The other three are as follows:

\((\text P 3)\) | $:$ | \(\ds \forall m, n \in P:\) | \(\ds \map s m = \map s n \implies m = n \) | $s$ is injective | |||||

\((\text P 4)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \ne 0 \) | $0$ is not in the image of $s$ | |||||

\((\text P 5)\) | $:$ | \(\ds \forall A \subseteq P:\) | \(\ds \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \) | Principle of Mathematical Induction: | |||||

Any subset $A$ of $P$, containing $0$ and | |||||||||

closed under $s$, is equal to $P$ |

From Inductive Construction of Natural Numbers fulfils Peano's Axioms, $\omega$ fulfils Peano's axioms.

We note that from Peano's Axiom $\text P 1$: $0 \in P$:

- $\O \in \omega$

We acknowledge from Peano's Axiom $\text P 2$: $n \in P \implies \map s n \in P$:

- the successor mapping defines that $n^+ := n \cup \set n$

and from Peano's Axiom $\text P 5$: Principle of Mathematical Induction the result follows.

$\blacksquare$

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications