Successor Mapping on Natural Numbers is Progressing

Theorem

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

Let $s: \omega \to \omega$ denote the successor mapping on $\omega$.

Then $s$ is a progressing mapping.

Proof 1

By definition of the von Neumann construction:

$n^+ = n \cup \set n$

from which it follows that:

$n \subseteq n^+$

Hence the result by definition of progressing mapping.

$\blacksquare$

Proof 2

By definition, the successor mapping on $\omega$ is indeed an example of a successor mapping.

The result follows from Successor Mapping is Progressing.

$\blacksquare$

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