# Inner model theory

In

set theory ,**inner model theory**is the study of certain models ofZFC or some fragment or strengthening thereof. Ordinarily these models are transitivesubset s or subclasses of thevon Neumann universe "V", or sometimes of ageneric extension of "V". Inner model theory studies the relationships of these models todeterminacy ,large cardinal s, anddescriptive set theory . Despite the name, it is considered more a branch of set theory than ofmodel theory .**Examples**The first example of an inner model was the

constructible universe "L" developed byKurt Gödel . Every model "M" of ZFC has an inner model "L"^{M}satisfying theaxiom of constructibility , and this will be the smallest inner model of "M" containing all the ordinals of "M". Regardless of the properties of the original model, "L"^{"M"}will satisfy thegeneralized continuum hypothesis and combinatorial axioms such as ◊.**Consistency results**One important use of inner models is the proof of consistency results. If it can be shown that every model of an axiom "A" has an inner model satisfying axiom "B", then if "A" is

consistent , "B" must also be consistent. This analysis is most useful when "A" is an axiom independent of ZFC, for example alarge cardinal axiom ; it is one of the tools used to rank axioms byconsistency strength .**References*** Citation

last1=Jech

first1=Thomas

author1-link=Thomas Jech

title=Set Theory

publisher=Springer-Verlag

location=Berlin, New York

series=Springer Monographs in Mathematics

year=2003

* Citation

last1=Kanamori

first1=Akihiro

title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings

publisher=Springer-Verlag

location=Berlin, New York

edition=2nd

isbn=978-3-540-00384-7

year=2003

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