Natural Numbers as Successor Sets

Theorem

The natural numbers $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ can be defined by means of successor sets:

$0 := \varnothing = \left\{{}\right\}$

$1 := 0^+ = 0 \cup \left\{{0}\right\} = \left\{{0}\right\}$

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

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

$\vdots$

Proof

From the definition of a successor set:

The successor (set) of $S$ is defined and denoted:

$S^+ := S \cup \left\{{S}\right\}$

This can be seen to be the structure of the elements of $\N$ as defined above.

Thus every element of $\N$ but $0$ can be seen to be the successor of some other element of $\N$.

$\blacksquare$

Cardinality

When the natural numbers are defined as here, the cardinality function can be viewed as the identity mapping on $\N$.

That is:

$\forall n \in N: \left|{n}\right| := n$

Also see

• Natural Numbers are Elements of the Minimal Infinite Successor Set

• Minimal Infinite Successor Set Fulfils Peano Axioms

• Natural Numbers from Nesting of Subsets for an alternative (arguably less useful) method of axiomatization of $\N$

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers
• AN INTRODUCTION TO SET THEORY. 2008 Professor William A. R. Weiss.
• Irving M. Copi: Symbolic Logic: 5th Edition (1979) p 205