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Successor levels of the Jensen hierarchy

✍ Scribed by Gunter Fuchs

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
216 KB
Volume
55
Category
Article
ISSN
0044-3050

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X β€² be the closure of X βˆͺ {X } under rudimentary functions. Given a Ξ£~0~‐formula Ο† (x) and a code c for a rudimentary function f, T (Ο†, c, $ \vec x $) is a Ξ£~Ο‰~ ‐formula such that for any $ \vec a $ ∈ X, X β€² ⊧ Ο† [f ($ \vec a $)] iff X ⊧ T (Ο†, 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 Ξ£~Ο‰~ extender ultrapower of X with respect to some extender F that also is an extender on X β€², then the closure of Y βˆͺ {Y } under rudimentary functions is the Ξ£~0~ extender ultrapower of Xβ€² with respect to F, and the ultrapower embeddings agree on X. (Β© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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