Axiom:Axiom of Infinity/Class Theory

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

• Equivalence of Formulations of Axiom of Infinity for Zermelo Universe

Historical Note

Mathematicians had unsuccessfully attempted to prove the existence of an infinite set using existing axioms.

Ernst Zermelo determined in $1908$ that it was necessary to assume the existence of such a set by creating an axiom specifically for that task.

Hence his introduction of the Axiom of Infinity.