On level by level equivalence and inequivalence between strong compactness and supercompactness

Abstract

Abstract. We prove two theorems, one concerning level by level inequivalence between strong compactness and supercompactness, and one concerning level by level equivalence between strong compactness and supercompactness. We first show that in a universe containing a supercompact cardinal but of restricted size, it is possible to control precisely the difference between the degree of strong compactness and supercompactness that any measurable cardinal exhibits. We then show that in an unrestricted size universe containing many supercompact cardinals, it is possible to have significant failures of GCH along with level by level equivalence between strong compactness and supercompactness, except possibly at inaccessible levels.

Failures of GCH and the level by level equivalence between strong compactness and supercompactness

“…Note, however, that here, there is only one supercompact cardinal in the universe, and no cardinal λ > κ is measurable. There is also [1,Theorem 3] in which there can be an arbitrary number of supercompact cardinals in the universe, 2 δ = δ + whenever δ isn't inaccessible, 2 δ = δ ++ whenever δ is inaccessible, and δ is λ strongly compact iff δ is λ supercompact, except possibly if λ is inaccessible, or for λ > δ a regular cardinal, δ is a witness to the Menas exception at λ. In this model, though, there is a question whether there is precise level by level equivalence between strong compactness and supercompactness.…”

“…Intuitively, if G is V -generic over A, then G first selects an element of A (or as Hamkins says in [13], "holds a lottery among the posets in A") and then forces with it. 1) We mention that we are assuming familiarity with the large cardinal notions of measurability, strongness, strong compactness, and supercompactness. Interested readers may consult [17] for further details.…”

Indestructibility, instances of strong compactness, and level by level inequivalence

“…4 An outline of this argument is as follows. will not provide as many details as in [5], although we will give a reasonably complete proof. Let …”

Tallness and level by level equivalence and inequivalence

“…1 Specifically, we prove the following two theorems. 1 Hamkins has shown in [9, Lemma 2.1] that if δ is λ tall, then δ is (λ) δ tall. He has further shown in [9, Theorem 2.11] that if δ is λ strongly compact, then δ is λ + tall.…”

“…Models containing supercompact cardinals in which level by level equivalence holds were first constructed in [4]. A model with exactly one supercompact cardinal in which level by level inequivalence holds was constructed in [1]. A theorem of Magidor (see [4,Lemma 7]) shows that if κ is supercompact, then there are always cardinals δ < λ < κ such that λ is singular of cofinality greater than or equal to δ, δ is λ strongly compact, but δ is not λ supercompact.…”