List of axioms
Encyclopedia
This is a list of axiom
s 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
.
standard axioms for contemporary mathematics
or set theory
. They can be easily adapted to analogous theories, such as mereology
.
See also Zermelo set theory
.
Other axioms of mathematical logic
Geometry
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s as that term is understood in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
and self-evidence
Self-evidence
In epistemology , a self-evident proposition is one that is known to be true by understanding its meaning without proof....
. Individual axioms are almost always part of a larger axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
.
Zermelo–Fraenkel axioms
These are the de factoDe facto
De facto is a Latin expression that means "concerning fact." In law, it often means "in practice but not necessarily ordained by law" or "in practice or actuality, but not officially established." It is commonly used in contrast to de jure when referring to matters of law, governance, or...
standard axioms for contemporary mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
or set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. They can be easily adapted to analogous theories, such as mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
.
- Axiom of extensionalityAxiom of extensionalityIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
- Axiom of empty setAxiom of empty setIn axiomatic set theory, the axiom of empty set is an axiom of Zermelo–Fraenkel set theory, the fragment thereof Burgess calls ST, and Kripke–Platek set theory.- Formal statement :...
- Axiom of pairingAxiom of pairingIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory.- Formal statement :...
- Axiom of unionAxiom of unionIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x...
- Axiom of infinityAxiom of infinityIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
- Axiom schema of replacementAxiom schema of replacementIn set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
- Axiom of power setAxiom of power setIn mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...
- Axiom of regularityAxiom of regularityIn mathematics, the axiom of regularity is one of the axioms of Zermelo-Fraenkel set theory and was introduced by...
- Axiom schema of specificationAxiom schema of specificationIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory...
See also Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
.
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
- Hausdorff maximality theorem
- Well-ordering principleWell-ordering principleThe "well-ordering principle" is a property of ordered sets. A set X is well ordered if every subset of X has a least element. An example of a well ordered set is the set of natural numbers. An example of set that is not well ordered is the set of integers...
- Zorn's lemmaZorn's lemmaZorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
Weaker than AC
- Axiom of countable choiceAxiom of countable choiceThe axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory, similar to the axiom of choice. It states that any countable collection of non-empty sets must have a choice function...
- Axiom of dependent choiceAxiom of dependent choiceIn mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis...
- Boolean prime ideal theoremBoolean prime ideal theoremIn mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...
- Axiom of uniformization
Other axioms of mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
- Von Neumann-Bernays-Gödel axioms
- Continuum hypothesisContinuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
and its generalization - Freiling's axiom of symmetryFreiling's axiom of symmetryFreiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidsonbut the mathematics behind it goes back to Wacław Sierpiński....
- Axiom of determinacyAxiom of determinacyThe axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
- Axiom of projective determinacyAxiom of projective determinacyIn mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect information of length ω in which the players play natural numbers, if...
- Martin's axiomMartin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
- Axiom of constructibilityAxiom of constructibilityThe axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
- Rank-into-rankRank-into-rankIn set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms :...
- Kripke-Platek axioms
GeometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
- Parallel postulateParallel postulateIn geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
- Birkhoff's axiomsBirkhoff's axiomsIn 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the...
- Hilbert's axiomsHilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
- Tarski's axiomsTarski's axiomsTarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory . Other modern axiomizations of Euclidean geometry are those by Hilbert and George...
Other axioms
- Axiom of Archimedes (real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
) - Axiom of countabilityAxiom of countabilityIn mathematics, an axiom of countability is a property of certain mathematical objects that requires the existence of a countable set with certain properties, while without it such sets might not exist....
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Fundamental axiom of analysis (real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
) - Gluing axiomGluing axiomIn mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor...
(sheaf theory) - Haag-Kastler axioms (quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) - Huzita's axioms (origamiOrigamiis the traditional Japanese art of paper folding, which started in the 17th century AD at the latest and was popularized outside Japan in the mid-1900s. It has since then evolved into a modern art form...
) - Kuratowski closure axiomsKuratowski closure axiomsIn topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Peano's axioms (natural numbers)
- Probability axiomsProbability axiomsIn probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....
- Separation axiomSeparation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Wightman axiomsWightman axiomsIn physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...
(quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
)