Difference between revisions of "Axiom of constructibility"

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{{DISPLAYTITLE: V = L}}
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{{DISPLAYTITLE: The axiom of constructibility, $V = L$}}
  
$V = L$, also known as the axiom of constructibility, is the assertion that the universe of all sets is exactly the universe of all constructible sets.  It is minimalistic in the sense that any inner model $M$ of ZF must contain all sets from Gödel's constructible universe $L$.  The axiom is compatible with some of the smaller large cardinal notions such as weak compactness but is ''not'' compatible with any large cardinal notion implying the existence of [[Zero_sharp|$0^{\sharp}$]] such as measurability.
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The axiom of constructibility, written $V=L$, is the assertion that the universe of all sets is exactly the universe of all constructible sets.  It is minimalistic in the sense that any inner model $M$ of ZF must contain all sets from Gödel's constructible universe $L$.  The axiom is compatible with some of the smaller large cardinal notions such as weak compactness but is ''not'' compatible with any large cardinal notion implying the existence of [[Zero_sharp|$0^{\sharp}$]] such as measurability.
  
 
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Revision as of 04:55, 3 January 2012


The axiom of constructibility, written $V=L$, is the assertion that the universe of all sets is exactly the universe of all constructible sets. It is minimalistic in the sense that any inner model $M$ of ZF must contain all sets from Gödel's constructible universe $L$. The axiom is compatible with some of the smaller large cardinal notions such as weak compactness but is not compatible with any large cardinal notion implying the existence of $0^{\sharp}$ such as measurability.


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