Axiom:Axiom of Infinity/Class Theory/Formulation 1
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Axiom
Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction:
\(\ds 0\) | \(:=\) | \(\ds \O\) | ||||||||||||
\(\ds 1\) | \(:=\) | \(\ds 0 \cup \set 0\) | ||||||||||||
\(\ds 2\) | \(:=\) | \(\ds 1 \cup \set 1\) | ||||||||||||
\(\ds 3\) | \(:=\) | \(\ds 2 \cup \set 2\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
\(\ds n + 1\) | \(:=\) | \(\ds n \cup \set n\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) |
Then $\omega$ is a set.
Also see
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 2$ Definition of the Natural Numbers: $A_7$