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Successor levels of the Jensen hierarchy
Author(s) -
Fuchs Gunter
Publication year - 2009
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710077
Subject(s) - ultraproduct , successor cardinal , mathematics , closure (psychology) , extender , function (biology) , hierarchy , combinatorics , transitive closure , discrete mathematics , mathematical analysis , chemistry , organic chemistry , evolutionary biology , economics , market economy , polyurethane , biology
I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X ′ be the closure of X ∪ { X } under rudimentary functions. Given a Σ 0 ‐formula φ ( x ) and a code c for a rudimentary function f , T ( φ , c , $ \vec x $ ) is a Σ ω ‐formula such that for any $ \vec a $ ∈ X , X ′ ⊧ φ [ f ( $ \vec a $ )] iff X ⊧ T ( φ , c , $ \vec x $ )[ $ \vec a $ ]. I make this precise and show relativized versions of this. As an application, I prove that under certain conditions, if Y is the Σ ω extender ultrapower of X with respect to some extender F that also is an extender on X ′, then the closure of Y ∪ { Y } under rudimentary functions is the Σ 0 extender ultrapower of X′ with respect to F , and the ultrapower embeddings agree on X . (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)