List of axioms

This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.

Zermelo-Frankel axioms

These are the de facto standard axioms for contemporary mathematics

• Axiom of extensionality
• Axiom of empty set
• Axiom of pairing
• Axiom of union
• Axiom of infinity
• Axiom schema of replacement
• Axiom of power set
• Axiom of regularity
• Axiom of separation
• Axiom schema of specification

See also Zermelo set theory.

Axiom of choice

With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable

Equivalents of AC

• Hausdorff maximality theorem
• Well-ordering principle
• Zorn's lemma

Weaker than AC

• Axiom of countable choice
• Axiom of dependent choice
• Boolean prime ideal theorem

Alternates incompatible with AC

• Axiom of real determinacy

Other axioms of mathematical logic

• Von Neumann-Bernays-Gödel axioms
• Continuum hypothesis
• Generalized continuum hypothesis
• Freiling's axiom of symmetry
• Axiom of determinacy
• Axiom of projective determinacy
• Martin's axiom
• Axiom of constructibility
• Rank-into-rank
• Kripke-Platek axioms

Geometry

• Parallel postulate
• Birkhoff's axioms
• Hilbert's axioms

Other axioms

• Axiom of countability (topology)
• Gluing axiom (sheaf theory)
• Haag-Kastler axioms (quantum field theory)
• Huzita's axioms (origami)
• Kuratowski closure axioms (topology)
• Peano's axioms (natural numbers)
• Probability axioms
• Separation axiom (topology)
• Wightman axioms (quantum field theory)