INNER MODEL THEORETIC GEOLOGY. (21st July 2016)

• Record Type: Journal Article
• Authors: FUCHS, GUNTER; SCHINDLER, RALF
• Is Part Of: Journal of symbolic logic. Volume 81:Number 3(2016)
• Page Start: 972
• Page End: 996
• Publication Date: 2016-07-21
• Subjects: 03E35, -- 03E40, -- 03E45, -- 03E47, -- 03E55; inner models, -- mantle, -- forcing, -- large cardinals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2015.64
• Languages: English
• ISSNs: 0022-4812

Abstract

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.