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VARSOVIAN MODELS I

Published online by Cambridge University Press:  01 August 2018


GRIGOR SARGSYAN
Affiliation:
DEPARTMENT OF MATHEMATICS RUTGERS UNIVERSITY NEW BRUNSWICK, NJ 08901, USAE-mail:gs481@math.rutgers.edu
RALF SCHINDLER
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG WWU MÜNSTER MÜNSTER, GERMANYE-mail:rds@wwu.edu
Corresponding

Abstract

Let Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$ , and Msw 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 Msw via its iteration strategy Σ.

We here show that Msw 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 Msw 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 Ms (x) for a cone of reals x, where Ms (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 $L[K,{\rm{\bar{\Sigma }}}]$ .


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Copyright
Copyright © The Association for Symbolic Logic 2018 

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