Faster access than browser!
78 relations: Actual infinity, Alfred Tarski, Axiom of constructibility, Axiom of regularity, Axiom schema of replacement, Bertrand Russell, Binary relation, Boolean algebra (structure), Cardinality, Cartesian product, Category of sets, Category theory, Class (set theory), Complement (set theory), Complemented lattice, Constructible universe, Construction of the real numbers, Countable set, De Morgan's laws, Dedekind cut, Dependent type, Domain of discourse, Element (mathematics), Empty set, Ernst Zermelo, Finite set, Finitism, Foundations of mathematics, Free object, Function (mathematics), Georg Cantor, Greatest and least elements, Grothendieck universe, Herbrand structure, Hereditarily finite set, Inaccessible cardinal, Infinity, Inner model, Intersection (set theory), Join and meet, Kurt Gödel, Leopold Kronecker, Mathematical proof, Mathematics, Metamathematics, Model theory, Morse–Kelley set theory, Naive set theory, Natural number, New Foundations, ..., NLab, Non-standard analysis, Non-standard model, Null set, Ordered pair, Ordinal number, Per Martin-Löf, Philosophy, Power set, Real analysis, Real line, Real number, Russell's paradox, Set theory, Structural induction, Subset, Tarski–Grothendieck set theory, Term (logic), Theorem, Topological space, Transfinite induction, Type theory, Union (set theory), Venn diagram, Von Neumann universe, Zermelo set theory, Zermelo–Fraenkel set theory, 0. Expand index (28 more) »
Actual infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects.
New!!: Universe (mathematics) and Actual infinity · See more »
Alfred Tarski
Alfred Tarski (January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.
New!!: Universe (mathematics) and Alfred Tarski · See more »
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.
New!!: Universe (mathematics) and Axiom of constructibility · See more »
Axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.
New!!: Universe (mathematics) and Axiom of regularity · See more »
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set.
New!!: Universe (mathematics) and Axiom schema of replacement · See more »
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.
New!!: Universe (mathematics) and Bertrand Russell · See more »
Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
New!!: Universe (mathematics) and Binary relation · See more »
Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
New!!: Universe (mathematics) and Boolean algebra (structure) · See more »
Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
New!!: Universe (mathematics) and Cardinality · See more »
Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
New!!: Universe (mathematics) and Cartesian product · See more »
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.
New!!: Universe (mathematics) and Category of sets · See more »
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
New!!: Universe (mathematics) and Category theory · See more »
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
New!!: Universe (mathematics) and Class (set theory) · See more »
Complement (set theory)
In set theory, the complement of a set refers to elements not in.
New!!: Universe (mathematics) and Complement (set theory) · See more »
Complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b.
New!!: Universe (mathematics) and Complemented lattice · See more »
Constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.
New!!: Universe (mathematics) and Constructible universe · See more »
Construction of the real numbers
In mathematics, there are several ways of defining the real number system as an ordered field.
New!!: Universe (mathematics) and Construction of the real numbers · See more »
Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
New!!: Universe (mathematics) and Countable set · See more »
De Morgan's laws
In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.
New!!: Universe (mathematics) and De Morgan's laws · See more »
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers.
New!!: Universe (mathematics) and Dedekind cut · See more »
Dependent type
In computer science and logic, a dependent type is a type whose definition depends on a value.
New!!: Universe (mathematics) and Dependent type · See more »
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
New!!: Universe (mathematics) and Domain of discourse · See more »
Element (mathematics)
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
New!!: Universe (mathematics) and Element (mathematics) · See more »
Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
New!!: Universe (mathematics) and Empty set · See more »
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics.
New!!: Universe (mathematics) and Ernst Zermelo · See more »
Finite set
In mathematics, a finite set is a set that has a finite number of elements.
New!!: Universe (mathematics) and Finite set · See more »
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.
New!!: Universe (mathematics) and Finitism · See more »
Foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.
New!!: Universe (mathematics) and Foundations of mathematics · See more »
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.
New!!: Universe (mathematics) and Free object · See more »
Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
New!!: Universe (mathematics) and Function (mathematics) · See more »
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
New!!: Universe (mathematics) and Georg Cantor · See more »
Greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
New!!: Universe (mathematics) and Greatest and least elements · See more »
Grothendieck universe
In mathematics, a Grothendieck universe is a set U with the following properties.
New!!: Universe (mathematics) and Grothendieck universe · See more »
Herbrand structure
In first-order logic, a Herbrand structure S is a structure over a vocabulary σ, that is defined solely by the syntactical properties of σ.
New!!: Universe (mathematics) and Herbrand structure · See more »
Hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.
New!!: Universe (mathematics) and Hereditarily finite set · See more »
Inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic.
New!!: Universe (mathematics) and Inaccessible cardinal · See more »
Infinity
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
New!!: Universe (mathematics) and Infinity · See more »
Inner model
In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.
New!!: Universe (mathematics) and Inner model · See more »
Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
New!!: Universe (mathematics) and Intersection (set theory) · See more »
Join and meet
In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S.
New!!: Universe (mathematics) and Join and meet · See more »
Kurt Gödel
Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.
New!!: Universe (mathematics) and Kurt Gödel · See more »
Leopold Kronecker
Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.
New!!: Universe (mathematics) and Leopold Kronecker · See more »
Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement.
New!!: Universe (mathematics) and Mathematical proof · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Universe (mathematics) and Mathematics · See more »
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods.
New!!: Universe (mathematics) and Metamathematics · See more »
Model theory
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.
New!!: Universe (mathematics) and Model theory · See more »
Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), 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).
New!!: Universe (mathematics) and Morse–Kelley set theory · See more »
Naive set theory
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
New!!: Universe (mathematics) and Naive set theory · See more »
Natural number
In mathematics, the natural 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!!: Universe (mathematics) and Natural number · See more »
New Foundations
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.
New!!: Universe (mathematics) and New Foundations · See more »
NLab
The nLab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from category theory and homotopy theory.
New!!: Universe (mathematics) and NLab · See more »
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.
New!!: Universe (mathematics) and Non-standard analysis · See more »
Non-standard model
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).
New!!: Universe (mathematics) and Non-standard model · See more »
Null set
In set theory, a null set N \subset \mathbb is a set that can be covered by a countable union of intervals of arbitrarily small total length.
New!!: Universe (mathematics) and Null set · See more »
Ordered pair
In mathematics, an ordered pair (a, b) is a pair of objects.
New!!: Universe (mathematics) and Ordered pair · See more »
Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
New!!: Universe (mathematics) and Ordinal number · See more »
Per Martin-Löf
Per Erik Rutger Martin-Löf (born May 8, 1942) is a Swedish logician, philosopher, and mathematical statistician.
New!!: Universe (mathematics) and Per Martin-Löf · See more »
Philosophy
Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.
New!!: Universe (mathematics) and Philosophy · See more »
Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
New!!: Universe (mathematics) and Power set · See more »
Real analysis
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.
New!!: Universe (mathematics) and Real analysis · See more »
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers.
New!!: Universe (mathematics) and Real line · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Universe (mathematics) and Real number · See more »
Russell's paradox
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.
New!!: Universe (mathematics) and Russell's paradox · See more »
Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
New!!: Universe (mathematics) and Set theory · See more »
Structural induction
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields.
New!!: Universe (mathematics) and Structural induction · See more »
Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
New!!: Universe (mathematics) and Subset · See more »
Tarski–Grothendieck set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.
New!!: Universe (mathematics) and Tarski–Grothendieck set theory · See more »
Term (logic)
In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.
New!!: Universe (mathematics) and Term (logic) · See more »
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.
New!!: Universe (mathematics) and Theorem · See more »
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
New!!: Universe (mathematics) and Topological space · See more »
Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
New!!: Universe (mathematics) and Transfinite induction · See more »
Type theory
In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.
New!!: Universe (mathematics) and Type theory · See more »
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
New!!: Universe (mathematics) and Union (set theory) · See more »
Venn diagram
A Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets.
New!!: Universe (mathematics) and Venn diagram · See more »
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets.
New!!: Universe (mathematics) and Von Neumann universe · See more »
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.
New!!: Universe (mathematics) and Zermelo set theory · See more »
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
New!!: Universe (mathematics) and Zermelo–Fraenkel set theory · See more »
0
0 (zero) is both a number and the numerical digit used to represent that number in numerals.
New!!: Universe (mathematics) and 0 · See more »
Redirects here:
Russell-style universe, Russell-style universes, Superstructure (mathematics), Tarski-style universe, Tarski-style universes, U-small, Universe (math), Universe (noospherics), Universe (set theory), Universe (set), Universe (sets), Universe set, Universes (mathematics).
References
[1] https://en.wikipedia.org/wiki/Universe_(mathematics)
