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In Mathematics , the constructible universe (or '''Gödel's constructible universe'''), denoted '''L''', is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1940 paper ''Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory''. In this, he proved that the constructible universe is an Inner Model of Set Theory , 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 Axiom s of set theory. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. WHAT IS L? L can be thought of as being built in "stages" resembling Von Neumann 's Universe . The stages are indexed by Ordinal s; unlike von Neumann's construction, where one takes at a successor stage +1 the full power set of the previous stage (i.e., the set of all subsets of the previous stage) in Gödel's construction one uses only the subsets of the previous stage definable by a formula with parameters in the language of set theory with the quantifiers interpreted to range over the sets of 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. | ||
| If Z Is An Element Of L<sub>&alpha</sub>, Then Z | {yyεL<sub>&alpha</sub> and yεz} ε Def (L<sub>&alpha</sub>) = L<sub>&alpha+1</sub> So L<sub>&alpha</sub> is a subset of L<sub>&alpha+1</sub> which is a subset of the power set of L<sub>&alpha</sub> Consequently, this is a tower of nested transitive sets But L itself is a proper class |
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| :{} | L<sub>0</sub> = {yyεL<sub>0</sub> and y=y} ε L<sub>1</sub> So {} ε L Since the element relation is the same and no new elements were added, this is the empty set of L |
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| :If XεL And YεL, Then There Is Some Ordinal &alpha Such That XεL<sub>&alpha</sub> And YεL<sub>&alpha</sub> Then {x,y} | {ssεL<sub>&alpha</sub> and (s=x or s=y)} ε L<sub>&alpha+1</sub> Thus {x,y} ε L and it has the same meaning for L as for V |
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| :If X ε L<sub>&alpha</sub>, Then Its Elements Are In L<sub>&alpha</sub> And Their Elements Are Also In L<sub>&alpha</sub> So Y Is A Subset Of L<sub>&alpha</sub> Y | {ssεL<sub>&alpha</sub> and there exists zεx such that sεz} ε L<sub>&alpha+1</sub> Thus y ε L |
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| :By Induction On Subformulas Of P, One Can Show That There Is An &alpha Such That L<sub>&alpha</sub> Contains S And Z<sub>1</sub>,,z<sub>n</sub> And (P Is True In L<sub>&alpha</sub> Iff P Is True In L (this Is Called The "Reflection Principle")) So {xxεS And P(x,z<sub>1</sub>,,z<sub>n</sub>) In L} | {xxεL<sub>&alpha</sub> and xεS and P(x,z<sub>1</sub>,,z<sub>n</sub>) in L<sub>&alpha</sub>} ε L<sub>&alpha+1</sub> Thus the subset is in L |
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| Axiom Of Replacement: Given Any Set S And Any Mapping (formally Defined As A Proposition P(x,y) Where P(x,y) And P(x,z) Implies Y | z), {y there exists xεS such that P(x,y)} is a set |
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| :Let Q(x,y) Be The Formula Which Relativizes P To L, Ie All Quantifiers In P Are Restricted To L Q Is A Much More Complex Formula Than P, But It Is Still A Finite Formula And We Can Apply Replacement In V To Q So {yyεL And There Exists XεS Such That P(x,y) In L} | {y there exists xεS such that Q(x,y)} is a set in V and a subclass of L Again using the axiom of replacement in V, we can show that there must be an &alpha such that this set is a subset of L<sub>&alpha</sub> ε L<sub>&alpha+1</sub> Then one can use the axiom of separation in L to finish showing that it is an element of L |
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| There Is A Formula Of Set Theory Which Expresses The Idea That X | L<sub>&alpha</sub> It has only free variables for X and &alpha Using this we can expand the definition of each constructible set If sεL<sub>&alpha+1</sub>, then s = {yyεL<sub>&alpha</sub> and Φ(y,z<sub>1</sub>,,z<sub>n</sub>) in (L<sub>&alpha</sub>,ε)} for some formula Φ and some z<sub>1</sub>,,z<sub>n</sub> in L<sub>&alpha</sub> This is equivalent to saying that: for all y, yεs iff exists X such that X=L<sub>&alpha</sub> and yεX and Ψ(X,y,z<sub>1</sub>,,z<sub>n</sub>) where Ψ(X,) is the result of restricting each quantifier in |
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