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.


Also see