Зермело-Френкелдің жиым теориясының шексіздік аксиомасы

Axiom of Zermelo Fraenkel set theory

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.

The axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent. (To see why, note that ZFC Con(ZFC − Infinity) and use Gödel's Second incompleteness theorem.) The negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent. (This is tantamount to saying that ZFC is consistent, if the other axioms are consistent.) We believe this, but cannot prove it (if it is true). Indeed, using the von Neumann universe, we can build a model of ZFC − Infinity + (¬Infinity). It is , the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of the empty set is not taken as a part of this system (since it can be derived from ZF + Infinity), then the empty domain also satisfies ZFC − Infinity + ¬Infinity, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists. The cardinality of the set of natural numbers, aleph null , has many of the properties of a large cardinal. Thus the axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called stronger axioms of infinity.