Joan Bagaria

Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenkas principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenkas principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.

An Ω-logic Primer

In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ω-logic and the Ω-conjecture.

Bounded forcing axioms and the continuum

Abstract We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (ω2,ω2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensens combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM ℵ 3 implies ℵ 2 ℵ 1 =ℵ 2 , as well as a stationary reflection principle which has many of the consequences of Martins Maximum for objects of size ℵ 2 . Finally, we give an example of a so-called boldface bounded forcing axiom implying 2 ℵ 0 =ℵ 2 .

Group radicals and strongly compact cardinals

We answer some natural questions about group radicals and torsion classes, which involve the existence of measurable cardinals, by constructing, relative to the existence of a supercompact cardinal, a model of ZFC in which the first ω1-strongly compact cardinal is singular.

Determinacy and weakly Ramsey sets in Banach spaces

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of c 0 we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper Weakly Ramsey sets in Banach spaces.

Proper forcing extensions and Solovay models

Abstract.We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength of the absoluteness of under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.

On coding uncountable sets by reals

If A ⊆ ω1, then there exists a cardinal preserving generic extension [A ][x ] of [A ] by a real x such that 1) A ∈ [x ] and A is Δ1HC (x) in [x ]; 2) x is minimal over [A ], that is, if a set Y belongs to [x ], then either x ∈ [A, Y ] or Y ∈ [A ]. The forcing we use implicitly provides reshaping of the given set A (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generic Vopĕnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures

ON

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