Definition:Natural Numbers/Von Neumann Construction
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Definition
Let $\omega$ denote the minimally inductive set.
The natural numbers can be defined as the elements of $\omega$.
Following Definition 2 of $\omega$, this amounts to defining the natural numbers as the finite ordinals.
In terms of the empty set $\O$ and successor sets, we thus define:
\(\ds 0\) | \(:=\) | \(\ds \O = \set {}\) | ||||||||||||
\(\ds 1\) | \(:=\) | \(\ds 0^+ = 0 \cup \set 0 = \set 0\) | ||||||||||||
\(\ds 2\) | \(:=\) | \(\ds 1^+ = 1 \cup \set 1 = \set {0, 1}\) | ||||||||||||
\(\ds 3\) | \(:=\) | \(\ds 2^+ = 2 \cup \set 2 = \set {0, 1, 2}\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
\(\ds n + 1\) | \(:=\) | \(\ds n^+ = n \cup \set n\) |
This can be expressed in detail as:
\(\ds 0\) | \(:=\) | \(\ds \O = \set {}\) | ||||||||||||
\(\ds 1\) | \(:=\) | \(\ds \set \O\) | ||||||||||||
\(\ds 2\) | \(:=\) | \(\ds \set {\O, \set \O}\) | ||||||||||||
\(\ds 3\) | \(:=\) | \(\ds \set {\O, \set \O, \set {\O, \set \O} }\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) |
Successor Mapping
The mapping $s: \N \to \N$ defined thus as:
- $\forall n \in \N: \map s n = n + 1$
is the successor mapping on $\N$.
Also see
Source of Name
This entry was named for John von Neumann.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 8$: Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 2$. Sets of sets: Exercise $3$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers
- 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