Definition:Natural Numbers/Von Neumann Construction/Successor Mapping
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Definition
Let $\omega$ denote the minimally inductive set.
Let the natural numbers $\N$ be modelled using the Von Neumann construction as the elements of $\omega$:
\(\ds 0\) | \(:=\) | \(\ds \O = \set {}\) | ||||||||||||
\(\ds 1\) | \(:=\) | \(\ds 0^+ = 0 \cup \set 0 = \set 0\) | ||||||||||||
\(\ds 2\) | \(:=\) | \(\ds 1^+ = 1 \cup \set 1 = \set {0, 1}\) | ||||||||||||
\(\ds 3\) | \(:=\) | \(\ds 2^+ = 2 \cup \set 2 = \set {0, 1, 2}\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
\(\ds n + 1\) | \(:=\) | \(\ds n^+ = n \cup \set n\) |
The mapping $s: \N \to \N$ defined thus as:
- $\forall n \in \N: \map s n = n + 1$
is the successor mapping on $\N$.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries