Axiom of infinity

The axiom of infinity is an axiom of set theory, which postulates the existence of an inductive set. It's called axiom of infinity, since inductive quantities are also both infinite sets. The first axiom of infinity Ernst Zermelo published in 1908 in the Zermelo set theory. It has influenced all subsequent lot of theories, in particular the Zermelo -Fraenkel set theory (ZF ), the most widely used set theory, which took over Zermelo's axiom of infinity in slightly modified form.

Formulation

There is a set containing the empty set and with each item and the quantity.

Thus, the axiom of infinity postulates not only the existence of an infinite set, but is also the structure of this infinite set before.

Importance for mathematics

Natural Numbers

The existence of at least one inductive amount the existence of the natural numbers is ensured as a set together with the axiom:

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

Infinite sets

Without axiom of infinity would only be ensured in ZF that finite sets exist. About the existence of infinite sets could not make any statements. The axiom of infinity ensures together with the power set axiom, that such as are also uncountable sets the real numbers.

Credentials

Axiom schemes: axiom | replacement axiom

Set theory

Inductive set (axiom of infinity) Axiom schema of specification Zermelo–Fraenkel set theory Axiom of regularity Unordered pair Axiom schema of replacement

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