The standard axiomatic foundation for mathematics — a list of axioms about sets (extensionality, pairing, union, power set, infinity, replacement, foundation) that avoid the Russell and Burali-Forti paradoxes and suffice to develop virtually all mathematics.
ZF (with Choice: ZFC) is the de facto standard foundation of modern mathematics; most mathematical questions are settled within ZFC; questions independent of ZFC (like the Continuum Hypothesis) require additional axioms or are genuinely undecidable.
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