Partial near supercompactness

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Abstract

A cardinal κ is nearly θ-supercompact if for every Aθ, there exists a transitive MZFC closed under <κ sequences with A,κ,θM, a transitive N, and an elementary embedding j:MN with critical point κ such that j(κ)>θ and jθN.2 This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly 2θ<κ-supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that θ<κ=θ, we can move to a forcing extension preserving all cardinals below θ++ where κ remains θ-supercompact but is not nearly θ+-supercompact. We will also show that if κ is nearly θ-supercompact for some θ2κ such that θ<θ=θ, then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result. A forcing poset is <κ-directed closed if it is γ-directed closed for all γ<κ in the sense of Jech (2003) [13, Def. 21.6]. We will prove that if κ is nearly θ-supercompact for some θκ such that θ<θ=θ, then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, we will show that if κ is nearly θ-supercompact for some θκ+ for which θ<θ=θ, then AD holds in L(R). In particular, if κ is nearly κ+-supercompact and 2κ=κ+, then AD holds in L(R).

MSC

03E35
03E55

Keywords

Nearly supercompact
Laver diamond
Indestructibility
Determinacy
Failure of the GCH

Cited by (0)

1

This article is adapted from the authorʼs dissertation.

2

Here, we use ZFC to mean the theory of ZFC without the Powerset axiom, but where ZFC is understood to be axiomatized with Collection instead of Replacement.