Inner models with large cardinal features usually obtained by forcing

Abstract

Abstract. We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ + , another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that H V κ + ⊆ HOD W . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Equivalence Relations Which Are Borel Somewhere

“…Secondly, the statement that f enumerates [x] E is Π 1 1 and hence absolute between (the countable model) M [x] and V . One can ask the same question for Π 1 1 equivalence relations with all classes countable. However, the above proof can not be applied.…”

“…(ZFC) If I is a σ-ideal such that P I is proper and E is a thin Σ 1 1 equivalence relation, then there exists a I + ∆ 1 1 set C such that C is contained in a single E-class.…”

“…The basic question addressed here in its most naive form is: Question 1.1. If E is an equivalence relation on a Polish space X, is there a large and nice set C ⊆ X such that E C is a Δ 1 1 equivalence relation? Here, E C = E ∩ (C × C ).…”

“…One does not want to hide any complexity of E C inside the set C . Therefore C should be "nice" in the sense that it is a Δ 1 1 subset of X . Every equivalence relation restricted to a countable set is a Δ 1 1 equivalence relation.…”

Level by Level Inequivalence, Strong Compactness, and GCH

“…Models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds have been constructed in [3,Theorem 2], [5,Theorem 2], and [2,Theorem 1]. (See also [9]. Note that the dual notion of level by level equivalence between strong compactness and supercompactness was first studied by the author and Shelah in [10], to which we refer readers for additional details.)…”