Measurable: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "A measurable cardinal is a certain type of large cardinal which possesses strong properties. It was one of the first large cardinal axioms to be developed, after inaccessible, Mahlo and weakly compact cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is \(\Pi^2_1\)-indescribable, but not \(\...") |
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A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\
Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\).
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