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INNER MODEL THEORETIC GEOLOGY

Published online by Cambridge University Press:  21 July 2016

GUNTER FUCHS  and
RALF SCHINDLER
GUNTER FUCHS
Affiliation:
COLLEGE OF STATEN ISLAND (CUNY) 2800 VICTORY BOULEVARD STATEN ISLAND NY 10314, USA THE GRADUATE CENTER OF CUNY 365 5TH AVENUE NEW YORK NY 10016, USA
RALF SCHINDLER
Affiliation:
INSTITUT FUER MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG UNIVERSITAET MUENSTER EINSTEINSTRASSE 62 48149 MUENSTER, GERMANY
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Abstract

One of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

Keywords

03E3503E4003E4503E4703E55inner modelsmantleforcinglarge cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 972 - 996
Copyright
Copyright © The Association for Symbolic Logic 2016 

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