Definition:Natural Numbers/Elements of Minimal Infinite Successor Set
< Definition:Natural Numbers(Redirected from Definition:Natural Numbers as Elements of Minimal Infinite Successor Set)
Definition
Let $\omega$ denote the minimal infinite successor 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 $\varnothing$ and successor sets, we thus define:
- $0 := \varnothing = \left\{{}\right\}$
- $1 := 0^+ = 0 \cup \left\{{0}\right\} = \left\{{0}\right\}$
- $2 := 1^+ = 1 \cup \left\{{1}\right\} = \left\{{0, 1}\right\}$
- $3 := 2^+ = 2 \cup \left\{{2}\right\} = \left\{{0, 1, 2}\right\}$
- $\vdots$
This can be expressed in detail as:
- $0 := \varnothing$
- $1 := \left\{{\varnothing}\right\}$
- $2 := \left\{{\varnothing, \left\{{\varnothing}\right\}}\right\}$
- $3 := \left\{{\varnothing, \left\{{\varnothing}\right\}, \left\{{\varnothing, \left\{{\varnothing}\right\}}\right\}}\right\}$
- $\vdots$
Also see
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
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.7$: 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