Successor Mapping on Natural Numbers is Progressing
Jump to navigation Jump to search
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.
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.
By definition, the successor mapping on $\omega$ is indeed an example of a successor mapping.
The result follows from Successor Mapping is Progressing.
- 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