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# 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. [1]

63 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, Bounded quantifier, Church–Kleene ordinal, Class (set theory), Club set, Consistency, Continuum hypothesis, Descriptive set theory, Elementary equivalence, Equinumerosity, Formal language, Gödel numbering, 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, ... Expand index (13 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.

## Axiom

An axiom or postulate is a premise or starting point of reasoning.

## 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.

## Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.

## 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.

## 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.

## 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.

## 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.

## 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.

## Axiom of power set

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

## 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.

## Axiom of union

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 of replacement

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.

## 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.

## Bounded quantifier

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

## Church–Kleene ordinal

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

## 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.

## 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.

## Consistency

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

## Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets.

## Descriptive set theory

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

## 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.

## Equinumerosity

In mathematics, two sets 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).

## Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols that may be constrained by rules that are specific to it.

## 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.

## Hereditarily countable set

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

## Hereditarily finite set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

## Hyperarithmetical theory

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

## Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal.

## Indiscernibles

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

## Inner model

In mathematical logic, suppose T is a theory in the language of set theory.

## Kurt Gödel

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

## 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.

## Large cardinal

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

## 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.

## 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 the alphabetical order of words is based on the alphabetical order of their component letters.

## Limit cardinal

In mathematics, limit cardinals are certain cardinal numbers.

## Limit ordinal

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

## Mahlo cardinal

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

## Mathematics

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

## Measurable cardinal

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

## Minimal model (set theory)

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

## 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.

## Ordinal number

In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set.

## Parameter

A parameter (from the Ancient Greek παρά, "para", meaning "beside, subsidiary" and μέτρον, "metron", meaning "measure"), in its common meaning, is a characteristic, feature, or measurable factor that can help in defining a particular system.

## Power set

In mathematics, the power set (or powerset) of any set, written, ℘(),, or 2''S'', is the set of all subsets of, including the empty set and itself.

## Quantifier (logic)

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

## 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.

## Regular cardinal

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

## Set (mathematics)

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

## Set theory

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

## Statements true in L

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

## Successor ordinal

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

## Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

## Transitive set

In set theory, a set A is transitive, if and only if.

## 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.

## Von Neumann cardinal assignment

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

## 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.

## Well-formed formula

In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word (i.e. a finite sequence of symbols from a given alphabet) that is part of a formal language.

## Well-founded relation

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S⊆X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than") for the rest of the s ∈ S. (Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.) Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x0, x1, x2,...

## Zermelo–Fraenkel set theory

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.

## 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.

## References

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