CHARACTERIZATION OF □_κ IN CORE MODELS

Abstract

We present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

Large Cardinals, Inner Models, and Determinacy: An Introductory Overview

“…Jensen first established γ for all γ ∈ Card, in L. This was then established in the core models over the years as they were developed (in K DJ by the author, in K Steel by Schimmerling-Zeman [26], and with other results by other set theorists for intermediate models which we have not defined, see also [25] for an overview.) As intimated, principles such as γ can be seen as identifying very uniform singularising functions for singular ordinals.…”

Square principles in ℙmax extensions

“…The heart of the construction takes place in HOD and is a straightforward combination of standard constructions of square sequences as developed in Jensen [Jen72], Schimmerling-Zeman [SZ04], and Zeman [Zem10], adapted to the context of strategic extender models as developed in Sargsyan [Sara]. In order to stay close to the constructions in Schimmerling-Zeman [SZ04] and Zeman [Zem10], we use fine structure notation and terminology as in Zeman [Zem02]; the rest of the notation and terminology is consistent with that in Mitchell-Steel [MS94], Steel [Ste96], and Sargsyan [Sara].…”

More fine structural global square sequences

“…For successor cardinals κ = µ + where µ is not subcompact one uses ♦ κ (A) and (κ, A) for a suitable stationary A ⊆ κ; here the (κ, A)-sequence is obtained from a µ -sequence whose existence is guaranteed by [6]. If µ is subcompact then µ is inaccessible, so GCH in L[E] makes it possible to construct a Suslin κ-tree "naively" by using only a ♦ κ (S κ µ )-sequence 1 to seal off large antichains at limit stages of cofinality κ in the construction, and adding all possible branches at limit stages of cofinality smaller than κ.…”

“…The paper builds on previous work on fine structural square sequences in extender models, in particular on [5,6] and…”