Supercompactness and partial level by level equivalence between strong compactness and strongness

Abstract

Abstract.We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if δ +α is regular, δ is δ +α strongly compact iff δ is δ +α+1 strong, except possibly if δ is a limit of cardinals γ which are δ +α strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible. Introduction and preliminaries.In [1], the following theorem was proven.Theorem 1. Let V "ZFC + κ is supercompact + There is no pair of cardinals δ < λ such that δ is λ supercompact and λ is measurable". There is then a partial ordering P ⊆ V such that V P "ZFC + GCH + There is no pair of cardinals δ < λ such that δ is λ supercompact and λ is measurable + κ is both the least strongly compact and least strong cardinal (so κ is notThis theorem provides a counterpoint to the main result of [5], which is as follows.There is then a partial ordering P ⊆ V such that V P "ZFC + GCH + K is the class of supercompact cardinals + For every pair of regular cardinals κ < λ, κ is λ strongly compact iff κ is λ supercompact, except possibly if κ is a limit of cardinals δ which are λ supercompact".2000 Mathematics Subject Classification: 03E35, 03E55. Key words and phrases: supercompact cardinal, strongly compact cardinal, strong cardinal, measurable cardinal, non-reflecting stationary set of ordinals, level by level equivalence between strong compactness and supercompactness, level by level equivalence between strong compactness and strongness.[123]