VARSOVIAN MODELS I. (1st August 2018)

• Record Type: Journal Article
• Authors: SARGSYAN, GRIGOR; SCHINDLER, RALF
• Is Part Of: Journal of symbolic logic. Volume 83:Number 2(2018)
• Page Start: 496
• Page End: 528
• Publication Date: 2018-08-01
• Subjects: 03E15, -- 03E45, -- 03E60; mouse, -- inner model theory, -- descriptive set theory, -- hod mouse; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2018.5
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: Let M sw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], M sw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and M sw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of M sw via its iteration strategy Σ. We here show that M sw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence M sw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}}, {\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it. These results generalize to models of the form M s ( x ) for a cone of reals x, where M s ( x ) denotes the least iterable inner model with a strong cardinal containing x . In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K, {\rm{\bar{\Sigma }}}]$ .