Level by Level Inequivalence, Strong Compactness, and GCH

Abstract

We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

Some Remarks on Tall Cardinals and Failures of GCH

“…On [8, page 35], it is stated that Woodin can obtain a model for the theory "ZFC + GCH fails everywhere" (in fact, for the theory "ZFC + 2 δ = δ ++ for every cardinal δ") starting from a ℘ 2 (κ) hypermeasurable cardinal κ (also known as a κ + 2 strong cardinal κ). 2 We conjecture that the consistency of the first two of the above theories can be established relative to the existence of a cardinal λ which is strong up to a ℘ 2 (κ) hypermeasurable cardinal κ and the consistency strength of the last theory can be established relative to a ℘ 2 (κ) hypermeasurable cardinal κ which is a limit of cardinals λ which are strong up to κ. However, it is unclear if these assumptions will provide equiconsistencies in each case.…”

Precisely controlling level by level behavior

“…We remark that the exact hypotheses used to prove Theorems and are together with the existence of cardinals such that κ 1 is supercompact and κ 2 is inaccessible. In particular, we first force to construct the ground model V mentioned in the statements of Theorems and (which will be the model witnessing the conclusions of [, Theorem 3]), and then force over this model to complete the proofs of these theorems. To avoid excessive technicalities, we have chosen to state these theorems as we did above.…”

“…These ideas were first introduced and studied in . Models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds may be found in [, Theorem 2], [, Theorem 2], [, Theorem 1], [, Theorems 1–3], [, Theorem 32(2)], and [, Theorem ].…”