Zermelo–Fraenkel set theory

Summary

Zermelo–Fraenkel set theory, ZF, is an axiomatic theory of sets in first-order logic with one binary relation $\in$. Its axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas. ZF with the axiom of choice is ZFC, the standard foundation of most of mathematics.

Remarks

The axiom of extensionality says that a set is determined by its elements. Separation forms only subsets $\{x \in A \mid P(x)\}$ of a given set $A$. This avoids Russell's paradox. The continuum hypothesis is independent of ZFC, if ZFC is consistent.

Explanations

L1 human

ZF is a short list of rules that says which sets exist. For example, for two sets there is a set that contains both, and for each set there is the set of its subsets. From these rules, one can build the numbers, functions and most other objects of mathematics.