List of axioms
From Wikipedia, the free encyclopedia
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.
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[edit] Zermelo-Frankel axioms
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 of separation
- Axiom schema of specification
See also Zermelo set theory.
[edit] Axiom of choice
With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable.
[edit] Equivalents of AC
[edit] Weaker than AC
- Axiom of countable choice
- Axiom of dependent choice
- Boolean prime ideal theorem
- Axiom of uniformization
[edit] Alternates incompatible with AC
[edit] 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
[edit] Geometry
[edit] Other axioms
- Axiom of Archimedes (real number)
- Axiom of countability (topology)
- 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)
