# Gödel constructive set

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Gödel constructible set, constructible set

A set arising in the process of constructing sets described below. Let be a set and let . Consider the first-order language containing one -place predicate symbol denoting the relation , and individual constants, denoting the elements of the set (for each its constant is ). The statement "the formula f of the language LR, X is valid in the model M=X, R" is written: A set is called definable in the model (or -definable) if there exists a formula of with one free variable such that Let denote the set of all -definable sets. To each ordinal number is associated the set defined recursively by the relation where is the membership relation restricted to the set . Hence it follows that  A set is called constructible if there exists an ordinal number such that . The class of all constructible sets is denoted by . In 1938, K. Gödel defined and introduced the following axiom of constructibility: Every set is constructible. On the basis of the axioms of , he proved that in all axioms of hold and also the axiom of constructibility, and that the axiom of choice and the generalized continuum hypothesis ( "for every ordinal number a one has 2a=a+ 1" ) follow in from the axiom of constructibility.

The class can also be characterized as the smallest class that is a model of and contains all the ordinal numbers; there are other ways of defining (see ). The relation can be expressed by a formula in the language , which is moreover of a simple syntactic structure (a so-called -formula, cf. ).

Some results relating to constructible sets. The set of constructible real numbers (cf. Constructive real number), that is, the set where is the set of all real numbers, that is, sequences of zeros and ones, is a -set (see ). It has been shown that the axiom of constructibility implies the existence of a Lebesgue non-measurable set of real numbers of type (see ), the negation of the Suslin hypothesis and the non-existence of measurable cardinal numbers (see ).

How to Cite This Entry:
Gödel constructive set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%B6del_constructive_set&oldid=14011
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article