Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Download
Faster access than browser!
 

Constructible universe

+ Save concept

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. [1]

66 relations: Arithmetical hierarchy, Axiom, Axiom of choice, Axiom of constructibility, Axiom of empty set, Axiom of extensionality, Axiom of global choice, Axiom of infinity, Axiom of pairing, Axiom of power set, Axiom of regularity, Axiom of union, Axiom schema of replacement, Axiom schema of specification, Bijection, Bounded quantifier, Church–Kleene ordinal, Class (set theory), Club set, Consistency, Continuum hypothesis, Descriptive set theory, Elementary equivalence, Equinumerosity, Formal language, Gödel numbering, Gödel operation, Hereditarily countable set, Hereditarily finite set, Hyperarithmetical theory, Inaccessible cardinal, Indiscernibles, Inner model, Kurt Gödel, L(R), Large cardinal, Löwenheim–Skolem theorem, Lexicographical order, Limit cardinal, Limit ordinal, List of large cardinal properties, Mahlo cardinal, Mathematics, Measurable cardinal, Minimal model (set theory), Ordinal definable set, Ordinal number, Parameter, Power set, Quantifier (logic), ..., Reflection principle, Regular cardinal, Ronald Jensen, Set (mathematics), Set theory, Statements true in L, Successor ordinal, Transfinite induction, Transitive set, Truth value, Von Neumann cardinal assignment, Von Neumann universe, Well-formed formula, Well-founded relation, Zermelo–Fraenkel set theory, Zero sharp. Expand index (16 more) »

Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them.

New!!: Constructible universe and Arithmetical hierarchy · See more »

Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

New!!: Constructible universe and Axiom · See more »

Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

New!!: Constructible universe and Axiom of choice · 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!!: Constructible universe and Axiom of constructibility · See more »

Axiom of empty set

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.

New!!: Constructible universe and Axiom of empty set · See more »

Axiom of extensionality

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.

New!!: Constructible universe and Axiom of extensionality · See more »

Axiom of global choice

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.

New!!: Constructible universe and Axiom of global choice · See more »

Axiom of infinity

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.

New!!: Constructible universe and Axiom of infinity · See more »

Axiom of pairing

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.

New!!: Constructible universe and Axiom of pairing · See more »

Axiom of power set

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.

New!!: Constructible universe and Axiom of power set · 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!!: Constructible universe and Axiom of regularity · See more »

Axiom of union

In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory.

New!!: Constructible universe and Axiom of union · 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!!: Constructible universe and Axiom schema of replacement · See more »

Axiom schema of specification

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.

New!!: Constructible universe and Axiom schema of specification · See more »

Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

New!!: Constructible universe and Bijection · See more »

Bounded quantifier

In the study of formal theories in mathematical logic, bounded quantifiers are often included in a formal language in addition to the standard quantifiers "∀" and "∃".

New!!: Constructible universe and Bounded quantifier · See more »

Christmas

Christmas is an annual festival commemorating the birth of Jesus Christ,Martindale, Cyril Charles.

New!!: Constructible universe and Christmas · See more »

Christmas and holiday season

The Christmas season, also called the festive season, or the holiday season (mainly in the U.S. and Canada; often simply called the holidays),, is an annually recurring period recognized in many Western and Western-influenced countries that is generally considered to run from late November to early January.

New!!: Constructible universe and Christmas and holiday season · See more »

Christmas Eve

Christmas Eve is the evening or entire day before Christmas Day, the festival commemorating the birth of Jesus.

New!!: Constructible universe and Christmas Eve · See more »

Christmas traditions

Christmas traditions vary from country to country.

New!!: Constructible universe and Christmas traditions · See more »

Church–Kleene ordinal

In mathematics, the Church–Kleene ordinal, \omega^_1, named after Alonzo Church and S. C. Kleene, is a large countable ordinal.

New!!: Constructible universe and Church–Kleene ordinal · 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!!: Constructible universe and Class (set theory) · See more »

Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded (see below) relative to the limit ordinal.

New!!: Constructible universe and Club set · See more »

Consistency

In classical deductive logic, a consistent theory is one that does not contain a contradiction.

New!!: Constructible universe and Consistency · See more »

Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.

New!!: Constructible universe and Continuum hypothesis · See more »

Descriptive set theory

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces.

New!!: Constructible universe and Descriptive set theory · See more »

Elementary equivalence

In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.

New!!: Constructible universe and Elementary equivalence · See more »

Equinumerosity

In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (a bijection) between them, i.e. if there exists a function from A to B such that for every element y of B there is exactly one element x of A with f(x).

New!!: Constructible universe and Equinumerosity · See more »

Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it.

New!!: Constructible universe and Formal language · See more »

Gödel numbering

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number.

New!!: Constructible universe and Gödel numbering · See more »

Gödel operation

In mathematical set theory, a set of Gödel operations is a finite collection of operations on sets that can be used to construct the constructible sets from ordinals.

New!!: Constructible universe and Gödel operation · See more »

Hereditarily countable set

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets.

New!!: Constructible universe and Hereditarily countable set · 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!!: Constructible universe and Hereditarily finite set · See more »

Hyperarithmetical theory

In recursion theory, hyperarithmetic theory is a generalization of Turing computability.

New!!: Constructible universe and Hyperarithmetical theory · 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!!: Constructible universe and Inaccessible cardinal · See more »

Indiscernibles

In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula.

New!!: Constructible universe and Indiscernibles · 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!!: Constructible universe and Inner model · See more »

Kurt Gödel

Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.

New!!: Constructible universe and Kurt Gödel · See more »

L(R)

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.

New!!: Constructible universe and L(R) · See more »

Large cardinal

In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.

New!!: Constructible universe and Large cardinal · See more »

Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.

New!!: Constructible universe and Löwenheim–Skolem theorem · See more »

Lexicographical order

In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the way words are alphabetically ordered based on the alphabetical order of their component letters.

New!!: Constructible universe and Lexicographical order · See more »

Limit cardinal

In mathematics, limit cardinals are certain cardinal numbers.

New!!: Constructible universe and Limit cardinal · See more »

Limit ordinal

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.

New!!: Constructible universe and Limit ordinal · See more »

List of large cardinal properties

This page includes a list of cardinals with large cardinal properties.

New!!: Constructible universe and List of large cardinal properties · See more »

Mahlo cardinal

In mathematics, a Mahlo cardinal is a certain kind of large cardinal number.

New!!: Constructible universe and Mahlo cardinal · See more »

Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

New!!: Constructible universe and Mathematics · See more »

Measurable cardinal

In mathematics, a measurable cardinal is a certain kind of large cardinal number.

New!!: Constructible universe and Measurable cardinal · See more »

Minimal model (set theory)

In set theory, the minimal model is the minimal standard model of ZFC.

New!!: Constructible universe and Minimal model (set theory) · See more »

New Year

New Year is the time or day at which a new calendar year begins and the calendar's year count increments by one.

New!!: Constructible universe and New Year · See more »

New Year's Day

New Year's Day, also called simply New Year's or New Year, is observed on January 1, the first day of the year on the modern Gregorian calendar as well as the Julian calendar.

New!!: Constructible universe and New Year's Day · See more »

New Year's Eve

In the Gregorian calendar, New Year's Eve (also known as Old Year's Day or Saint Sylvester's Day in many countries), the last day of the year, is on 31 December which is the seventh day of Christmastide.

New!!: Constructible universe and New Year's Eve · See more »

Ordinal definable set

In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first order formula.

New!!: Constructible universe and Ordinal definable set · 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!!: Constructible universe and Ordinal number · See more »

Parameter

A parameter (from the Ancient Greek παρά, para: "beside", "subsidiary"; and μέτρον, metron: "measure"), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

New!!: Constructible universe and Parameter · 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!!: Constructible universe and Power set · See more »

Quantifier (logic)

In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

New!!: Constructible universe and Quantifier (logic) · See more »

Reflection principle

In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets.

New!!: Constructible universe and Reflection principle · See more »

Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality.

New!!: Constructible universe and Regular cardinal · See more »

Ronald Jensen

Ronald Björn Jensen (born April 1, 1936) is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.

New!!: Constructible universe and Ronald Jensen · See more »

Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

New!!: Constructible universe and Set (mathematics) · See more »

Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

New!!: Constructible universe and Set theory · See more »

Statements true in L

Here is a list of propositions that hold in the constructible universe (denoted L).

New!!: Constructible universe and Statements true in L · See more »

Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.

New!!: Constructible universe and Successor ordinal · 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!!: Constructible universe and Transfinite induction · See more »

Transitive set

In set theory, a set A is called transitive if either of the following equivalent conditions hold.

New!!: Constructible universe and Transitive set · See more »

Truth value

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

New!!: Constructible universe and Truth value · See more »

Von Neumann cardinal assignment

The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers.

New!!: Constructible universe and Von Neumann cardinal assignment · 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!!: Constructible universe and Von Neumann universe · See more »

Well-formed formula

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

New!!: Constructible universe and Well-formed formula · See more »

Well-founded relation

In mathematics, a binary relation, R, is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

New!!: Constructible universe and Well-founded relation · 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!!: Constructible universe and Zermelo–Fraenkel set theory · See more »

Zero sharp

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe.

New!!: Constructible universe and Zero sharp · See more »

2018

2018 has been designated as the third International Year of the Reef by the International Coral Reef Initiative.

New!!: Constructible universe and 2018 · See more »

2019

2019 (MMXIX) will be a common year starting on Tuesday of the Gregorian calendar, the 2019th year of the Common Era (CE) and Anno Domini (AD) designations, the 19th year of the 3rd millennium, the 19th year of the 21st century, and the 10th and last year of the 2010s decade.

New!!: Constructible universe and 2019 · See more »

Redirects here:

Constructible hierarchy, Constructible subset, Godel constructibility, Godel constructible, Godel constructible universe, Godel's constructible universe, Godel's universe, Godels universe, Goedel constructibility, Goedel constructible, Goedel constructible universe, Goedel universe, Goedel's constructible universe, Goedel's universe, Goedels universe, Gödel constructibility, Gödel constructible, Gödel constructible universe, Gödel constructive set, Gödel's constructible universe, L (set theory), Set-theoretic constructibility.

References

[1] https://en.wikipedia.org/wiki/Constructible_universe

OutgoingIncoming
Hey! We are on Facebook now! »