ZFCssä

ZFC is the standard axiomatic system for set theory. ZFC is an abbreviation for Zermelo-Fraenkel set theory with the Axiom of Choice. It is the most commonly used foundation for mathematics. The system was developed by Ernst Zermelo and Abraham Fraenkel in the early 20th century. ZFC is a first-order theory with a specific signature, consisting of a single binary relation symbol, often denoted by '$\in$', which represents set membership.

The axioms of ZFC are a set of statements that are assumed to be true and from which all other theorems of set theory can be derived. These axioms define the properties of sets and how they can be formed and manipulated. Key axioms include the Axiom of Extensionality, which states that two sets are equal if and only if they have the same elements, and the Axiom of Pairing, which asserts that for any two sets, there exists a set containing precisely those two sets. Other important axioms include the Axiom of Union, the Axiom of Separation, the Axiom of Replacement, and the Axiom of Infinity.

The Axiom of Choice (AC) is a particularly significant axiom within ZFC. It states that for any collection of non-empty sets, there exists a function that chooses exactly one element from each set. While seemingly intuitive, the Axiom of Choice has non-constructive implications and led to considerable debate among mathematicians. Despite this, it is widely accepted and used in modern mathematics because it simplifies many proofs and allows for the establishment of fundamental results in various fields.

ZFC provides a robust framework for understanding mathematical objects, as virtually all mathematical concepts can be defined in terms of sets. It forms the bedrock upon which much of modern mathematics is built.