Infinity axiom

The axiom of infinity is an axiom of set theory that postulates the existence of an inductive set . It is called the axiom of infinity, since inductive sets are also infinite sets at the same time . Ernst Zermelo published the first axiom of infinity in 1908 in the Zermelo set theory . It influenced all later set theories, especially the Zermelo-Fraenkel set theory (ZF), the most widespread set theory, which Zermelo's axiom of infinity adopted in a slightly modified form.

formulation

There is a set that contains the empty set and with each element also the set . ${\displaystyle A}$ ${\displaystyle \emptyset }$ ${\displaystyle x\in A}$ ${\displaystyle x\cup \{x\}}$

${\displaystyle \exists A\colon (\emptyset \in A\land \forall x\colon (x\in A\Rightarrow x\cup \{x\}\in A))}$

The axiom of infinity not only postulates the existence of an infinite set, but also specifies the structure of this infinite set.

Meaning for math

Natural numbers

The existence of at least one inductive set , together with the axiom of exclusion, also ensures the existence of natural numbers as a set: ${\displaystyle I}$

${\displaystyle \mathbb {N} :=\{x\in I\mid \forall z(z\,\,{\text{induktiv}}\implies x\in z)\}}$

The natural numbers are defined as the intersection of all inductive sets, as the smallest inductive set.

Infinite quantities

Without the axiom of infinity, ZF would only ensure that finite sets exist. No statements could be made about the existence of infinite sets. The infinity axiom, together with the power set axiom, ensures that uncountable sets such as B. gives the real numbers.

Individual evidence

1. ↑ Zermelo: Investigations on the fundamentals of set theory , 1907, in: Mathematische Annalen 65 (1908), 261–281; Axiom of the Infinite p. 266f.