Categoricity: Difference between revisions
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#REDIRECT [[Categorical theory]] |
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In [[mathematics]], a [[theory (model theory)|theory]] in the sense of [[model theory]] is '''categorical''' if it has one and only one [[model (model theory)|model]] up to [[isomorphism]]. A theory is κ-'''categorical''' if it has one and only one model of [[cardinal number|cardinality]] κ up to isomorphism. |
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==History and motivation== |
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The notion of categoricity was introduced by [[Oswald Veblen]] in 1904 to describe a theory with a structure strong enough to force all of its models to be isomorphic. It follows from the definition above and the [[Löwenheim-Skolem theorem]] that any [[first-order theory]] with a model of infinite [[cardinal number|cardinality]] cannot be categorical. One is then immediately led to the more subtle notion of κ-categoricity, which asks: for which cardinals κ is there exactly one model of cardinality κ of the given theory ''T'' up to isomorphism? This is a deep question and significant progress was only made in 1954 when [[Jerzy Łoś]] noticed that, at least for [[complete theory|complete theories]] ''T'' over countable [[formal language|languages]] with at least one infinite model, he could only find three ways for ''T'' to be κ-categorical at some &kappa: |
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*''T'' is '''totally categorical''', ''i.e.'' ''T'' is κ-categorical for all infinite [[cardinal number|cardinal]]s κ. |
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*''T'' is '''uncountably categorical''', ''i.e.'' ''T'' is κ-categorical if and only if κ is an [[countable|uncountable]] cardinal. |
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*''T'' is '''countably categorical''', ''i.e.'' ''T'' is κ-categorical if and only if κ is a countable cardinal. |
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In other words, he observed that, in all the cases he could think of, κ-categoricity at any uncountable cardinal implied κ-categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating in [[Michael Morley]]'s famous result that these are in fact the only possibilities. The theory was subsequently extended and refined by [[Saharon Shelah]] in the 1970s and beyond, leading to [[stability theory]] and Shelah's more general programme of [[classification theory]]. |
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==Reference== |
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*[http://plato.stanford.edu/archives/sum2005/entries/modeltheory-fo Hodges, Wilfrid, "First-order Model Theory", The Stanford Encyclopedia of Philosophy (Summer 2005 Edition), Edward N. Zalta (ed.).] |
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[[Category:Model theory]] |
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Latest revision as of 07:43, 7 December 2018
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