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Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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 Vopěnka’s 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 Lévy 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...

Easton functions and supercompactness

Brent Cody, Sy-David Friedman, Radek Honzik (2014)

Fundamenta Mathematicae

Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving...

Extending real-valued functions in βκ

Alan Dow (1997)

Fundamenta Mathematicae

An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality c and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.

Extensions with the approximation and cover properties have no new large cardinals

Joel David Hamkins (2003)

Fundamenta Mathematicae

If an extension V ⊆ V̅ satisfies the δ approximation and cover properties for classes and V is a class in V̅, then every suitably closed embedding j: V̅ → N̅ in V̅ with critical point above δ restricts to an embedding j ↾ V amenable to the ground model V. In such extensions, therefore, there are no new large cardinals above δ. This result extends work in [Ham01].

Forcing with ideals generated by closed sets

Jindřich Zapletal (2002)

Commentationes Mathematicae Universitatis Carolinae

Consider the poset P I = Borel ( ℝ ) ∖ I where I is an arbitrary σ -ideal σ -generated by a projective collection of closed sets. Then the P I extension is given by a single real r of an almost minimal degree: every real s ∈ V [ r ] is Cohen-generic over V or V [ s ] = V [ r ] .

Fragments of strong compactness, families of partitions and ideal extensions

Laura Fontanella, Pierre Matet (2016)

Fundamenta Mathematicae

We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for κ ( λ ) of the canonical Ramsey theorem for pairs.

Generalized E-algebras via λ-calculus I

Rüdiger Göbel, Saharon Shelah (2006)

Fundamenta Mathematicae

An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra E n d R A of the R-module R A , taking any a ∈ A to the right multiplication a r ∈ E n d R A by a, is an isomorphism of algebras. In this case R A is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite...

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