This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and selfevidence. Individual axioms are almost always part of a larger axiomatic system.
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Introduction to Higher Mathematics  Lecture 2: Introduction to Proofs

supremum infimum part I Real Analysis Mathematics

Verifying vector space axioms 1 to 4  Example of C^n and Tutorial Q6
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ZF (the Zermelo–Fraenkel axioms without the axiom of choice)
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology.
 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 schema of specification
See also Zermelo set theory.
Axiom of choice
With the Zermelo–Fraenkel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable.
Equivalents of AC
Stronger than AC
Weaker than AC
 Axiom of countable choice
 Axiom of dependent choice
 Boolean prime ideal theorem
 Axiom of uniformization
Alternates incompatible with AC
Other axioms of mathematical logic
 Von Neumann–Bernays–Gödel axioms
 Continuum hypothesis and its generalization
 Freiling's axiom of symmetry
 Axiom of determinacy
 Axiom of projective determinacy
 Martin's axiom
 Axiom of constructibility
 Rankintorank
 Kripke–Platek axioms
 Diamond principle
Geometry
 Parallel postulate
 Birkhoff's axioms (4 axioms)
 Hilbert's axioms (20 axioms)
 Tarski's axioms (10 axioms and 1 schema)
Other axioms
 Axiom of Archimedes (real number)
 Axiom of countability (topology)
 Dirac–von Neumann axioms
 Fundamental axiom of analysis (real analysis)
 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)
 Action axiom (praxeology)