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In [[mathematical logic]], suppose ''T'' is a [[Theory (mathematical logic)|theory]] in the language 


:<math>L = \langle \in \rangle</math>


of [[set theory]].
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If ''M'' is a [[model theory|model]] of <math>L</math> describing a [[set theory]] and ''N'' is a [[class (set theory)|class]] of ''M'' such that
 
:<math> \langle N, \in_M, \ldots \rangle</math>
 
is a model of ''T'' containing all [[ordinal number|ordinals]] of ''M'' then we say that  ''N'' is an '''inner model''' of ''T'' (in ''M'').<ref>{{cite book | last = Jech | first = Thomas | title = Set Theory | publisher = Springer-Verlag | location = Berlin | year = 2002 | isbn = 3-540-44085-2 }}</ref> Ordinarily these models are [[transitive set|transitive]] [[subset]]s or [[subclass (set theory)|subclass]]es of the [[von Neumann universe]] ''V'', or sometimes of a [[generic extension]] of ''V''
 
This term '''''inner model''''' is sometimes applied to models which are [[class (set theory)|proper classes]]; the term '''set model''' is used for models which are sets.
 
A model of set theory is called '''standard''' if the element relation of the model is the actual element relation restricted to the model. A model is called  '''transitive''' when it is standard and the base class is a [[transitive class]] of sets. A model of set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models are transitive, transitive models are standard, and standard models are [[well-founded set|well-founded]].
 
The assumption that there exists a standard model of [[ZFC]] (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model
called the '''[[Minimal model (set theory)|minimal model]]''' contained in all standard models. The minimal model contains no standard model (as it is minimal) but (assuming the [[consistency]] of ZFC) it contains
some model of ZFC by the [[Gödel completeness theorem]]. This model is necessarily not well founded otherwise its [[Mostowski collapse]] would be a standard model. (It is not well founded as a relation in the universe, though it
satisfies the [[axiom of foundation]] so is "internally" well founded. Being well founded is not an absolute property.<ref>{{cite book | last = Kunen | first = Kenneth | title = Set Theory | publisher = North-Holland Pub. Co | location = Amsterdam | year = 1980 | isbn = 0-444-86839-9 }}, Page 117</ref>)
In particular in the minimal model  there is a model of ZFC but there is no  standard model of ZFC.
 
==Use==
Usually when one talks about inner models of a theory, the theory one is discussing is [[ZFC]] or some extension of ZFC (like ZFC&nbsp;+&nbsp;<math>\exists</math> a [[measurable cardinal]]). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of [[subtheories]] of ZFC (like [[Zermelo–Frankel set theory|ZF]] or [[Kripke–Platek set theory|KP]]) as well.
 
==Related ideas==
It was proved by [[Gödel|Kurt Gödel]] that any model of ZF has a least inner model of ZF (which is also an inner model of ZFC&nbsp;+&nbsp;[[Generalized Continuum Hypothesis|GCH]]), called the [[constructible universe]], or&nbsp;''L''.  
 
There is a branch of set theory called [[inner model theory]] which studies ways of constructing least inner models of theories extending&nbsp;ZF. Inner model theory has led to the discovery of the exact [[consistency strength]] of many important set theoretical properties.
 
==References==
{{reflist}}
 
==See also==
*[[Forcing_(mathematics)#Countable_transitive_models_and_generic_filters|Countable transitive models and generic filters]]
 
{{DEFAULTSORT:Inner Model}}
[[Category:Inner model theory]]

Revision as of 14:39, 9 February 2014


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