ZFCMengelehre

ZFC Mengenlehre, short for Zermelo-Fraenkel-Mengenlehre mit Auswahl (ZFC), is the standard axiomatic foundation for contemporary mathematics. It formalizes set theory in the language of first-order logic with the membership relation and provides a rigorous framework in which most mathematical objects can be defined and manipulated. The theory aims to avoid paradoxes by restricting how sets are constructed.

The core of ZFC consists of several axioms, many of which are schemas. Key axioms include Extensionality (sets with the same elements are identical), Foundation (no infinite descending membership chains), Pairing (for any two sets there is a set containing exactly them), Union (a set of sets can be flattened into a single set of their elements), Power Set (every set has a set of all its subsets), Infinity (there exists an infinite set, typically modeling the natural numbers), and the axioms schemes Separation and Replacement. Separation allows forming subsets by definable properties, while Replacement ensures image sets exist under definable functions. The Axiom of Choice completes the standard ZF framework, asserting the existence of choice functions for suitable collections.

ZFC supports the cumulative hierarchy of sets and serves as the backbone for most of mainstream mathematics. It provides a common language for defining numbers, functions, sets of real numbers, and beyond. While extensively successful, ZFC is known to be incomplete: there are statements independent of ZFC, meaning neither provable nor disprovable from the axioms. Gödel showed that AC cannot be disproved from ZF, and Cohen demonstrated that AC cannot be proven from ZF. Consequently, ZFC’s consistency cannot be established within ZFC itself, though models in which AC holds (such as the constructible universe L) and models where it fails (under certain assumptions) are both possible.