Definition:Natural Numbers/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 $\O$ and successor sets, we thus define:

$0 := \O = \set {}$

$1 := 0^+ = 0 \cup \set 0 = \set 0$

$2 := 1^+ = 1 \cup \set 1 = \set {0, 1}$

$3 := 2^+ = 2 \cup \set 2 = \set {0, 1, 2}$

$\vdots$

This can be expressed in detail as:

$0 := \O$

$1 := \set \O$

$2 := \set {\O, \set \O}$

$3 := \set {\O, \set \O, \set {\O, \set \O} }$

$\vdots$

Also see

• Definition:Minimal Infinite Successor Set

• Minimal Infinite Successor Set forms Peano Structure

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