List of axioms


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 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.

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
*Axiom of uniformization

Alternates incompatible with AC

*Axiom of real determinacy

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
*Rank-into-rank
*Kripke-Platek axioms

Geometry

*Parallel postulate
*Birkhoff's axioms
*Hilbert's axioms
*Tarski's axioms

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)


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