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Concept
Constructible universe
Summary
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
can be thought of as being built in "stages" resembling the construction of von Neumann universe, . The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes to be the set of all subsets of the previous stage, . By contrast, in Gödel's constructible universe , one uses only those subsets of the previous stage that are:
definable by a formula in the formal language of set theory,
with parameters from the previous stage and,
with the quantifiers interpreted to range over the previous stage.
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.
Define the Def operator:
is defined by transfinite recursion as follows:
If is a limit ordinal, then Here means precedes .
Here Ord denotes the class of all ordinals.
If is an element of ,
then . So is a subset of , which is a subset of the power set of . Consequently, this is a tower of nested transitive sets. But itself is a proper class.
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Ontological neighbourhood
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