ZermeloFraenkelaxiomas

The Zermelo-Fraenkel axioms, often abbreviated as ZF, form the standard foundation for much of modern set theory and, by extension, for much of mathematics. They were developed to provide a rigorous basis for constructing sets while avoiding paradoxes that plagued naive set theory in the early 20th century. The theory is usually formulated in first-order logic with a single primitive relation, membership, denoted ∈, and expresses how sets can be built from and related to one another.

A typical ZF system includes several core principles. Extensionality says that two sets are equal if they have the same elements. Foundation (Regularity) forbids certain kinds of circular membership, ensuring a well-founded universe of sets. Basic constructors are provided by Pairing, Union, and Power Set, which allow the creation of new sets from existing ones. Infinity asserts the existence of an infinite set, ensuring the presence of natural numbers. In addition, ZF contains two schemas: Separation (restricted comprehension) and Replacement, which govern the formation of subsets and the images of sets under definable functions, respectively. Together, these axioms create a robust framework in which most familiar mathematical objects can be defined and manipulated.

The axiom of choice (AC) is independent of ZF; adding AC yields ZFC. AC states that for any collection of nonempty sets, a choice function selections an element from each set. Gödel showed that ZF cannot prove the negation of AC if ZF is consistent, and Cohen showed that AC cannot be proved from ZF. ZFC is the most commonly used foundation for mainstream mathematics, and the ZF hierarchy also underlies the idea of the cumulative hierarchy of sets, built by iterating the powerset operation along the ordinals.