Zermelo set theory

Zermelo set theory (Z) is the axiomatic foundation of mathematics formulated by Ernst Zermelo in 1908, comprising the axioms of Extensionality, Pairing, Union, Power Set, Infinity, and Separation. It is strictly weaker than ZFC: it lacks both the Axiom of Replacement and the Axiom of Foundation, which means it cannot prove the existence of ordinals beyond ω or guarantee the well-foundedness of the set-theoretic universe.

Zermelo set theory is sufficient for most of classical mathematics — arithmetic, analysis, geometry — but it stalls at the threshold of the transfinite. The inability to construct V_ω+ω or higher stages of the von Neumann universe makes it a foundation for the finite and the countably infinite, but not for the full architecture of modern set theory. It is the minimal system that captures ordinary mathematical practice, and its limitations reveal precisely where stronger axioms become necessary.