# Axioms This is a description of all distinct axioms reoccuring in mathematics. Some axioms are tied only to a single particular subject or situation, and some are more general. Out of the more general ones, the set of *the most common axioms* consists of the very familiar axioms that occur in many different mathematical areas. ## The Most Common Axioms (MCA) 1. Identity The first place goes to identity. Almost everybody's got one, except semis. 1. Associativity The second place goes to associativity. Hardly a surprise, since this guy has appeared in blockbusters such as "pretty much everyhing that has got to do with numbers", "operators, actually", "operation: binary", to name just a few. ## Axiom list @wolfram.com absorption law absorption identity algebra of random variables zermelo-fraenkel axioms axiom of choice axiom of extensionality axiom of foundation axiom of infinity axiom of replacement axiom of subsets axiom of the empty set axiom of the power set axiom of the sum set axiom of the unordered pair axioms of subsets congruence axioms continuity axioms de morgan's laws eilenberg-steenrod axioms long exact sequence of a pair axiom homotopy axiom excision axiom dimension axiom euclid's postulates equidistance postulate field axioms hausdorff axioms hilbert's axioms incidence axioms induction axiom kolmogorov's axioms ordering axioms parallel postulate peano's axioms peano arithmetic playfair's axiom presburger arithmetic probability axioms proclus' axiom ## THE LIST Identity Totality/Closure Associativity Commutativity Distributivity Invertability Idempotency Absorption Annihilation Cancellation Domination Complement Connex Euclidean Involution Linearity Monotonicity Reflexivity Symmetry Transitivity Trichotomy Functional (left-total) Serial (right-unique) Well-ordering Well-formedness Well-foundedness Well-definedness ## PAGES [absorption](/math-debrief/900-index/enums/enum-axioms/enum-axioms/absorption.md) [annihilation](/math-debrief/900-index/enums/enum-axioms/enum-axioms/annihilation.md) [associativity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/associativity.md) [cancellation](/math-debrief/900-index/enums/enum-axioms/enum-axioms/cancellation.md) [closure](/math-debrief/900-index/enums/enum-axioms/enum-axioms/closure.md) [commutativity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/commutativity.md) [complement](/math-debrief/900-index/enums/enum-axioms/enum-axioms/complement.md) [connex](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/connex.md) [distributivity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/distributivity.md) [domination](/math-debrief/900-index/enums/enum-axioms/enum-axioms/domination.md) [euclidean](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/euclidean.md) [idempotency](/math-debrief/900-index/enums/enum-axioms/enum-axioms/idempotency.md) [identity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/identity.md) [invertibility](/math-debrief/900-index/enums/enum-axioms/enum-axioms/invertibility.md) [involution](/math-debrief/900-index/enums/enum-axioms/enum-axioms/involution.md) [linearity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/linearity.md) [monotonicity](/math-debrief/900-index/enums/enum-axioms/enum-axioms/monotonicity.md) [reflexivity](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/reflexivity.md) [symmetry](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/symmetry.md) [totality](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/totality.md) [transitivity](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/transitivity.md) [trichotomy](https://github.com/mandober/debrief.math/tree/22d3b35bdafab8f414b8c4da51d1fb13b55aa733/101-fundamentals/enums/enum-axioms/enum-axioms/trichotomy.md) [well-definedness](/math-debrief/900-index/enums/enum-axioms/enum-axioms/well-definedness.md) [well-formedness](/math-debrief/900-index/enums/enum-axioms/enum-axioms/well-formedness.md) [well-foundedness](/math-debrief/220-relation-theory/relation-types/well-founded-relation.md) [well-ordering](/math-debrief/900-index/enums/enum-axioms/enum-axioms/well-ordering.md) --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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