Axiom:Axiom of Infinity/Class Theory
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Axiom
Formulation 1
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.
Formulation 2
There exists an inductive set.
Formulation 3
Not every set is a natural number.