Axiom of infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.

Formal statement

Using first-order logic primitive symbols, the axiom can be expressed as follows:

If the notations of both set-builder and empty set are allowed:

Some mathematicians may call a set built this way an inductive set.

A natural language formulation of this axiom may read as: "There exists a set <big>𝐈</big> such that the empty set is an element of it, and for every element of <big>𝐈</big>, there exists an element of <big>𝐈</big> such the elements of consist of itself and the elements of ."

Interpretation and consequences

This axiom is closely related to the von Neumann construction of the natural numbers in set theory, in which the successor of x is defined as x ∪ {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set:

The number 1 is the successor of 0:

Likewise, 2 is the successor of 1:

and so on: