# List of axioms

Appearance

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

## 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.

*With the Zermelo–Fraenkel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable.*

### Equivalents of AC

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

### 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
- Rank-into-rank
- Kripke–Platek axioms
- Diamond principle

- 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)
- sheaf theory)
- )
- )
- Kuratowski closure axioms (topology)
- natural numbers)
- Probability axioms
- Separation axiom (topology)
- Wightman axioms (quantum field theory)
- Action axiom (praxeology)