Axiom:Axiom of Infinity

Axiom

Set Theory

There exists a set containing:

$(1): \quad$ a set with no elements

$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$

Class Theory

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.

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.