Two‐cardinal diamond star

Our main results are: (A) It is consistent relative to a large cardinal that holds but fails. (B) If holds and are two infinite cardinals such that and λ carries a good scale, then holds. (C) If are two cardinals such that κ is λ‐Shelah and , then there is no good scale for λ.     

The first omitting cardinal for Magidority

An infinite cardinal λ is Magidor if and only if . It is known that if λ is Magidor then for some , and the first such α is denoted by . In this paper we try to understand some of the properties of . We prove that can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of being a successor of a singular cardinal with uncountable cofinality.     

The Hypothesis and a supercompact cardinal

In this paper, we prove that: if κ is supercompact and the Hypothesis holds, then there is a proper class of regular cardinals in which are measurable in . Woodin also proved this result independently 11 . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the Hypothesis and supercompact cardinals, large cardinals in are reflected to be large cardinals in in a local way, and reveals the huge difference between ‐supercompact cardinals and supercompact cardinals under the Hypothesis.     

Free sets for set‐mappings relative to a family of sets

Given a family of subsets of , we try to compute the least natural number n such that for every function there exists a bijection such that for all .     

The cofinality of the least Berkeley cardinal and the extent of dependent choice

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of . In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐ fails. We shall prove that this result is optimal for or . In particular, it will follow that the cofinality of the least Berkeley is independent of .     

Building prime models in fully good abstract elementary classes

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well‐behaved independence relation: Let be an almost fully good AEC that is categorical in and has the ‐existence property for domination triples. For any , the class of Galois saturated models of of size λ has prime models over every set of the form . This generalizes an argument of Shelah, who proved the result when λ is a successor cardinal.     

On Schauder equivalence relations

We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that and are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality .     

On the reducibility of isomorphism relations

We study the Borel reducibility of isomorphism relations in the generalized Baire space . In the main result we show for inaccessible κ, that if T is a classifiable theory and is stable with the orthogonal chain property (OCP), then the isomorphism of models of T is Borel reducible to the isomorphism of models of .     

On the (non) superstable part of the free group

In this short note we prove that a definable set X over is superstable only if .     

Lipschitz extensions of definable p‐adic functions

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function , where and , that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.     

Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for , a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any .     

The length of an intersection

A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length , of p . We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X , , then, letting , with , denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes the least initial ordinal greater than the ordinal . This inequality is optimal. This result answers questions of Forster.     

Decreasing sentences in Simple Type Theory

We present various results regarding the decidability of certain sets of sentences by Simple Type Theory (). First, we introduce the notion of decreasing sentence, and prove that the set of decreasing sentences is undecidable by Simple Type Theory with infinitely many zero‐type elements (); a result that follows directly from the fact that every sentence is equivalent to a decreasing sentence. We then establish two different positive decidability results for a weak subtheory of . Namely, the decidability of (a subset of Σ₁) and (the set of all sentences , where φ is strictly decreasing). Finally, we present some consequences for the set of existential‐universal sentences. All the above results have direct implications for Quine's theory of “New Foundations” () and its weak subtheory .     

On the strength of a weak variant of the axiom of counting

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of , and that proves the consistency of .     

Cluster expansion and the boxdot conjecture

The boxdot conjecture asserts that every normal modal logic that faithfully interprets  by the well‐known boxdot translation is in fact included in . We confirm that the conjecture is true. More generally, we present a simple semantic condition on modal logics L₀ which ensures that the largest logic where L₀ embeds faithfully by the boxdot translation is L₀ itself. In particular, this natural generalization of the boxdot conjecture holds for , , and in place of .     

Computational aspects of satisfiability in probability logic

We consider the complexity of satisfiability in ε‐logic, a probability logic. We show that for the relational fragment this problem is ‐complete for rational , answering a question by Terwijn. In contrast, we show that satisfiability in 0‐logic is decidable. The methods we employ to prove this fact also allow us to show that 0‐logic is compact, while it was previously shown that ε‐logic is not compact for .     

Indestructible strong compactness and level by level inequivalence

If are such that δ is indestructibly supercompact and γ is measurable, then it must be the case that level by level inequivalence between strong compactness and supercompactness fails. We prove a theorem which points to this result being best possible. Specifically, we show that relative to the existence of cardinals such that κ₁ is λ‐supercompact and λ is inaccessible, there is a model for level by level inequivalence between strong compactness and supercompactness containing a supercompact cardinal in which κ’s strong compactness, but not supercompactness, is indestructible under κ‐directed closed forcing. In this model, κ is the least strongly compact cardinal, and no cardinal is supercompact up to an inaccessible cardinal.     

Paraconsistent double negation as a modal operator

A paraconsistent modal‐like logic, , is defined as a Gentzen‐type sequent calculus. The modal operator in the modal logic can be simulated by the paraconsistent double negation in . Some theorems for embedding into a Gentzen‐type sequent calculus for and vice versa are proved. The cut‐elimination and completeness theorems for are also proved.     

Completeness of intermediate logics with doubly negated axioms

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic . By , we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus . We shall show that if is strongly complete for a class of Kripke models , then is strongly complete for the class of Kripke models that are ultimately in .     

Precisely controlling level by level behavior

We construct four models containing one supercompact cardinal in which level by level equivalence between strong compactness and supercompactness and level by level inequivalence between strong compactness and supercompactness are precisely controlled at each non‐supercompact measurable cardinal. In these models, no cardinal κ is ‐supercompact, where is the least inaccessible cardinal greater than κ.