Query: standard axioms of set theory

Relevant passages:
AD+ <=SEP=> In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DCR (the axiom of dependent choice for real numbers), states two things: Every set of reals is ∞-Borel. For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A] is determined.

Ackermann set theory <=SEP=> Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.

Aczel's anti-foundation axiom <=SEP=> In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel (1988), as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to a unique set.  In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom.

Alternative set theory <=SEP=> Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory. Some of the alternative set theories are:the theory of semisetsthe set theory New FoundationsPositive set theoryInternal set theorySpecifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students.

Axiom <=SEP=> An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning.

Axiom of constructibility <=SEP=> The 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. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties).

Axiom of countable choice <=SEP=> The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that any countable collection of non-empty sets must have a choice function.

Axiom of determinacy <=SEP=> In mathematics, the axiom of determinacy (abbreviated as AD) 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 empty set <=SEP=> In axiomatic set theory, the axiom of empty set is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo set theory and Zermelo–Fraenkel set theory, with or without the axiom of choice.

Axiom of extensionality <=SEP=> In 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.

Axiom of global choice <=SEP=> In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set.

Axiom of infinity <=SEP=> In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory.  It guarantees the existence of at least one infinite set, namely a set containing the natural numbers.

Axiom of pairing <=SEP=> In 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.

Axiom of projective determinacy <=SEP=> In 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 the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a winning strategy.The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD), which contradicts the axiom of choice, it is not known to be inconsistent with ZFC.

Axiom of real determinacy <=SEP=> In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following:Consider infinite two-person games with perfect information.

Axiom of union <=SEP=> In 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. Together with the axiom of pairing this implies that for any two sets, there is a set that contains exactly the elements of both.

Axiom schema <=SEP=> In mathematical logic, an axiom schema (plural: axiom schemata) generalizes the notion of axiom.

Axiom schema of replacement <=SEP=> In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZFC.The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements.

Axiom schema of specification <=SEP=> In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema.

Axiomatic system <=SEP=> 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. An axiomatic system that is completely described is a special kind of formal system; usually though, the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans.

Boolean prime ideal theorem <=SEP=> In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given 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 sets is known as the ultrafilter lemma.

Borel set <=SEP=> In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.  Borel sets are named after Émile Borel.For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra.

Choice function <=SEP=> A choice function (selector, selection)  is a mathematical function f  that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S.  In other words, f is a choice function for X if and only if it belongs to the direct product of X.

Constructive set theory <=SEP=> Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first-order language of classical set theory, and although of course the logic is constructive, there is no explicit use of constructive types. Rather, there are just sets, thus it can look very much like classical mathematics done on the most common foundations, namely the Zermelo–Fraenkel axioms (ZFC).

Countable set <=SEP=> In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number.Some authors use countable set to mean countably infinite alone.

Frege's propositional calculus <=SEP=> In mathematical logic Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus,  in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege).

General set theory <=SEP=> General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.

Glossary of set theory <=SEP=> This is a glossary of set theory.

Gluing axiom <=SEP=> In 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 functorF: O(X) → Cto a category C which initially one takes to be the category of sets.

Ground axiom <=SEP=> In set theory, the ground axiom was introduced by Hamkins (2005) and Reitz (2007). It states that the universe is not a nontrivial set forcing extension of an inner model.

Group (mathematics) <=SEP=> In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.  The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer.

Group structure and the axiom of choice <=SEP=> In mathematics a group is a set together with a binary operation on the set called multiplication that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form states that every set can be wellordered.In ZF set theory, i.e. ZFC without the axiom of choice, the following statements are equivalent: For every nonempty set X there exists a binary operation • such that (X, •) is a group. The axiom of choice is true.↑

Internal set theory <=SEP=> Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment.  Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional  axioms for sets.

Kripke–Platek set theory <=SEP=> The Kripke–Platek axioms of set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, are a system of axiomatic set theory based on the ideas of Saul Kripke (1964) and Richard Platek (1966).KP is considerably weaker than Zermelo–Fraenkel set theory (ZFC), and can be thought of as roughly the predicative part of ZFC.  The consistency strength of KP with an axiom of infinity is given by the Bachmann–Howard ordinal.

Lebesgue measure <=SEP=> In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration.

List of axioms <=SEP=> 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.

List of first-order theories <=SEP=> In mathematical logic, a first-order theory is given by a set of axioms in somelanguage. This entry lists some of the more common examples used in model theory and some of their properties.

List of mathematical logic topics <=SEP=> This is a list of mathematical logic topics, by Wikipedia page.For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms.

List of mathematics categories <=SEP=> This list contains categories related to mathematics.

Mathematical logic <=SEP=> Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.  It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.  The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.

Model theory <=SEP=> In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. We call a set of sentences in a formal language a theory;  a model of a theory is a structure (e.g.

Morse–Kelley set theory <=SEP=> In the foundation of mathematics, Morse–Kelley set theory (MK) or Kelley–Morse set theory (KM) or Quine–Morse set theory (QM) or the system of Quine and Morse is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG).

Naive set theory <=SEP=> Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language.

Natural number <=SEP=> In mathematics, the natural numbers (sometimes called the whole numbers) are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

New Foundations <=SEP=> In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998).

Non-measurable set <=SEP=> This page gives a general overview of the concept of non-measurable sets. For a precise definition of measure, see Measure (mathematics). For various constructions of non-measurable sets, see Vitali set, Hausdorff paradox, and Banach–Tarski paradox.In mathematics, a non-measurable set is a set which cannot be assigned a meaningful \size\"."

Non-well-founded set theory <=SEP=> Non-well-founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of well-foundedness.

Peano axioms <=SEP=> In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

Power set <=SEP=> In mathematics, the power set (or powerset) of any set S, written P(S), ℘(S), P(S), ℙ(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.Any subset of P(S) is called a family of sets over S.

Proper forcing axiom <=SEP=> In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.

Robinson arithmetic <=SEP=> In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of Peano arithmetic (PA), first set out in R. M. Robinson (1950). Q is essentially PA without the axiom schema of induction. Since Q is weaker than PA but it has the same language, it is incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.

Semiset <=SEP=> In set theory, a semiset is a proper class that is contained in a set.The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972). It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is precluded by the axiom of separation. The concept of semisets opens the way for a formulation of an alternative set theory.Semisets can be used to represent sets with imprecise boundaries.

Set-theoretic definition of natural numbers <=SEP=> Several ways have been proposed to define the natural numbers using set theory.

Set Theory: An Introduction to Independence Proofs <=SEP=> Set Theory: An Introduction to Independence Proofs is a textbook and reference work in set theory by Kenneth Kunen.  It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees, Suslin's problem, ◊, and Martin's axiom. It develops some basic model theory (rather specifically aimed at models of set theory) and the theory of Gödel's constructible universe L.The book then proceeds to describe the method of forcing.

Set theory <=SEP=> Set theory is the branch of mathematical logic that studies sets, which informally 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. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s.

Tarski–Grothendieck set theory <=SEP=> Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.  It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom which states that for each set there is a Grothendieck universe it belongs to (see below).

Timeline of category theory and related mathematics <=SEP=> This is a timeline of category theory and related mathematics.

Universal set <=SEP=> In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set.It is often symbolized by the Greek letter xi.

Universe (mathematics) <=SEP=> In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains (as elements) all the entities one wishes to consider in a given situation. There are several versions of this general idea, described in the following sections.

Von Neumann–Bernays–Gödel set theory <=SEP=> In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC). A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes proper classes, objects having members but that cannot be members of other entities.

Wightman axioms <=SEP=> In physics the Wightman axioms (also called  Gårding–Wightman axioms) are an attempt at a mathematically rigorous formulation of quantum field theory.

Zermelo set theory <=SEP=> 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.  This article sets out the original axioms, with the original text (translated into English) and original numbering.

Zermelo–Fraenkel set theory <=SEP=> In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell's paradox. Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice (Ciesielski 1997).