Natural Numbers are Elements of the Minimal Infinite Successor Set

Theorem

The natural numbers are elements of the minimal infinite successor set $\omega$.

Thus it follows that $\N = \omega$.

Proof

From Natural Numbers as Successor Sets, we have that for all $n \in \N$:

$0 \in n$

$n^+ = n \cup \left\{{n}\right\}$

which characterises elements of any infinite successor set.

From the definition of the minimal infinite successor set, $\omega$ is itself an infinite successor set.

Hence:

$\forall n \in \N: n \in \omega$

By definition of subset, it follows that:

$\N \subseteq \omega$

But from the Axiom of Infinity together with Natural Numbers as Successor Sets, it follows that, by its method of construction, $\N$ is itself an infinite successor set.

From the definition of the minimal infinite successor set it follows that:

$\omega \subseteq \N$

By Equality of Sets it follows that $\N = \omega$.

$\blacksquare$

Notation

Rather than expressing the set $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ as being equal to $\omega$, Paul R. Halmos: Naive Set Theory (1960) goes a further step and suggests using $\omega$ for the set of all natural numbers.

This use of $\omega$ is usually seen for the order type of the natural numbers, that is, $\left({\N, \le}\right)$ where $\le$ is the usual ordering on the natural numbers.

The two definitions of $\omega$ are seen to coincide via the definition and analysis of the concept of the ordinal, itself an interpretation from a slightly different direction of the concept of a successor set.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers