Constructible universe

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the constructible universe (or Gödel's constructible universe), denoted by 
  
    
      
        L
        ,
      
    
    {\displaystyle L,}
  
 is a particular <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> that can be described entirely in terms of simpler sets. 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is the union of the constructible hierarchy 
  
    
      
        
          L
          
            α
          
        
      
    
    {\displaystyle L_{\alpha }}
  
. It was introduced by <a href="/facts/Kurt_G%25C3%25B6del/P14DqKqK">Kurt Gödel</a> 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 <a href="/facts/Inner_model/9WKoWvCe">inner model</a> of ZF set theory (that is, of <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a> with the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> excluded), and also that the axiom of choice and the <a href="/facts/Continuum_hypothesis/R0yGqHND">generalized continuum hypothesis</a> are true in the constructible universe. This shows that both propositions are <a href="/facts/Consistent/IHkdnA9M">consistent</a> with the basic <a href="/facts/Axiom/mLjP14Mc">axioms</a> 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.
</p>

Constructible universe open-in-new

Constructible universe