Definition:Natural Numbers/Von Neumann Construction

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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:

\(\displaystyle 0\) \(:=\) \(\displaystyle \O = \set {}\)
\(\displaystyle 1\) \(:=\) \(\displaystyle 0^+ = 0 \cup \set 0 = \set 0\)
\(\displaystyle 2\) \(:=\) \(\displaystyle 1^+ = 1 \cup \set 1 = \set {0, 1}\)
\(\displaystyle 3\) \(:=\) \(\displaystyle 2^+ = 2 \cup \set 2 = \set {0, 1, 2}\)
\(\displaystyle \) \(\vdots\) \(\displaystyle \)
\(\displaystyle n + 1\) \(:=\) \(\displaystyle n^+ = n \cup \set n\)


This can be expressed in detail as:

\(\displaystyle 0\) \(:=\) \(\displaystyle \O = \set {}\)
\(\displaystyle 1\) \(:=\) \(\displaystyle \set \O\)
\(\displaystyle 2\) \(:=\) \(\displaystyle \set {\O, \set \O}\)
\(\displaystyle 3\) \(:=\) \(\displaystyle \set {\O, \set \O, \set {\O, \set \O} }\)
\(\displaystyle \) \(\vdots\) \(\displaystyle \)


Also see


Source of Name

This entry was named for John von Neumann.


Sources