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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Set Theory as a foundational system for mathematics. ZF, ZFC and ZF with atoms. Relative consistency of the Axiom of Choice, the Continuum Hypothesis, the reals as a countable union of countable sets,
Ce cours entend exposer les fondements de la géométrie à un triple titre :
1/ de technique mathématique essentielle au processus de conception du projet,
2/ d'objet privilégié des logiciels de concept
Delves into ancient geometric means, transformations, and projective geometry's birapports.
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons , α ∈ κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by . A cardinal κ is called α-Erdős if for every function f : κ< ω → {0, 1}, there is a set of order type α that is homogeneous for f . In the notation of the partition calculus, κ is α-Erdős if κ(α) → (α)< ω. The existence of zero sharp implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal".
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations of this axiom are explored in inner model theory.
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