Axiom of infinity

An axiom of set theory asserting that an inductive set exists. It guarantees the existence of an infinite set and supports the construction of the natural numbers.

Inductive set: A set containing the empty set and containing the successor of each of its elements. The axiom asserts that at least one such set exists.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics whose objects are sets and whose axioms govern their existence and construction. The axiom of infinity is one of its standard existence axioms.
Ernst Zermelo: A German mathematician who formulated an early axiomatic set theory in 1908. His system included an axiom guaranteeing an infinite set.
Axiom of choice: An axiom asserting that a choice function exists for every family of nonempty sets. It is another prominent set-existence principle, but is independent of infinity in standard set theory.
Empty set: The unique set containing no elements. It serves as the initial element required by the axiom's inductive-set condition.
Axiom of pairing: An axiom asserting that any two sets belong together to some set. It helps construct successor sets from existing sets.
Abraham Fraenkel: A mathematician whose work helped develop the axiomatic theory now called Zermelo–Fraenkel set theory. The modern framework retaining this axiom bears his name alongside Zermelo's.
Finite set: A set that has the same size as some natural number. The axiom rules out a universe containing only finite sets.
Successor ordinal: An ordinal formed by adjoining an ordinal as a new largest element. Successor formation supplies the closure operation used to define an inductive set.
Axiom of union: An axiom asserting that the union of a set of sets is itself a set. It combines successor constructions when building the natural numbers.

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Large cardinalRelated: It guarantees infinite sets, but does not establish the stronger cardinal properties at issue here.