What is axiom of Infinity?

An axiom of a formal theory or of a theory with an interpretation (thematic theory) which ensures the presence of infinite objects in the theory. Thus, the axiom of infinity in some system of axiomatic set theory ensures the existence of an infinite set.

What is Zermelo-Fraenkel 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.

What axioms are used in set theory?

In set theory the so-called axioms of higher infinity, which postulate the existence of sets of high cardinality, are employed as well: The axiom on the existence of an inaccessible cardinal, the axiom on the existence of a measurable cardinal, etc.

Is Dedekind's axiom of Infinity equivalence?

With the aid of the axiom of choice it is easily shown that Dedekind's axiom of infinity is equivalent to the other above-mentioned forms of the axiom of infinity. It is known, however, that this equivalence cannot be proved by usual set-theoretic means without the use of the axiom of choice.