We study Structural Reflection beyond Vopěnka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some form of what we call Exact Structural Reflection (ESR). Namely, given cardinals κ<λ and a class C of structures of the same type, the corresponding instance of ESR asserts that for every structure A in C of rank λ, there is a structure B in C of rank κ and an elementary embedding of B into A. Inspired by the statement of Chang's Conjecture, we also introduce and study sequential forms of ESR, which, in the case of sequences of length ω, turn out to be very strong. Indeed, when restricted to Π1-definable classes of structures they follow from the existence of I1-embeddings, while for more complicated classes of structures, e.g., Σ2, they are not known to be consistent. Thus, these principles unveil a new class of large cardinals that go beyond I1-embeddings, yet they may not fall into Kunen's Inconsistency.

Abstract We study closure properties of measurable ultrapowers with respect to Hamkin’s notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.

Abstract With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$ , we show that for all infinite, regular cardinals $\lambda < \kappa$ , the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$ -Aronszajn trees exist and all such trees contain ${\aleph _0}$ -ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$ .

Abstract We study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ . We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$ , assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

Abstract The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

Abstract This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

Abstract Let κ be an infinite cardinal. A subset of (κκ)n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κκ)n+1 of height κ. We define and -subsets of (κκ)n as usual. Given an uncountable regular cardinal κ with κ = κ<κ and an arbitrary subset A of κκ, we show that there is a <κ-closed forcing ℙ that satisfies the κ+-chain condition and forces A to be a -subset of κκ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof. (i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L . (ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a -subset of κκ × κκ. (iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T. (iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κκ) under <κ-closed forcings that satisfy the κ+-chain condition is inconsistent. In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.