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- Title
STRUCTURAL PROPERTIES OF THE STABLE CORE.
- Authors
FRIEDMAN, SY-DAVID; GITMAN, VICTORIA; MÜLLER, SANDRA
- Abstract
The stable core, an inner model of the form $\langle L[S],\in , S\rangle $ for a simply definable predicate S , was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in V , then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots , C_n]$ for specially nested class clubs $C_1, \dots , C_n$ , like those arising in the stable core, generalizing recent results of Welch [29].
- Subjects
QUANTUM dots; CLUBS
- Publication
Journal of Symbolic Logic, 2023, Vol 88, Issue 3, p889
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2023.10