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# Inner model theory

In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe "V", or sometimes of a generic extension of "V". Inner model theory studies the relationships of these models to determinacy, large cardinals, and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory.

Examples

The first example of an inner model was the constructible universe "L" developed by Kurt Gödel. Every model "M" of ZFC has an inner model "L"M satisfying the axiom 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 the generalized 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 a large cardinal axiom; it is one of the tools used to rank axioms by consistency strength.

References

* Citation
last1=Jech
first1=Thomas
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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