Mathematics

The minimum classical extension S +g of a classically sound theory S based on intuitionistic logic, defined by adding to S the Gentzen negative interpretations of its mathematical axioms, contains a faithful translation S g of the classical version S + (--A -> A) of S. S g may be thought of as the classical content of S. First and second order intuitionistic arithmetic contain their classical contents, but intuitionistic recursive analysis cannot prove the negative interpretation of its quantifier-free countable choice axiom. Variants of Kuroda's double negation shift principle (including the Gödel-Dyson-Kreisel axiom equivalent to the weak completeness of intuitionistic predicate logic), and doubly negated characteristic function principles, provide neat characterizations of the minimum classical extensions of classically sound subsystems of Kleene's intuitionistic analysis I which are of interest to constructive mathematicians. Markov's recursive analysis is its own minimum classical extension, and Bishop's constructive analysis has the same classical content as the neutral subsystem B of Kleene's I.

In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. We will show that our large cardinal assumption is optimal, i.e. if every Kurepa tree has an Aronszajn subtree then ω 2 is inaccessible in the constructible universe \textsc{L}. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree T such that if U⊂T is a Kurepa tree with the inherited order from T , then U has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume MA ω 2 holds and ω 2 is not a Mahlo cardinal in $\textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's ρ function which might be useful in other contexts.

A counterexample to Naimark's problem can be constructed without using Jensen's diamond principle. We also construct, using our weakening of diamond, a separably represented, simple \textrm{C}^* -algebra with exactly m inequivalent irreducible representations for all m\geq 2 . Our principal technical contribution is the introduction of a forcing notion that generically adds an automorphism of a given \textrm{C}^* -algebra with a prescribed action on its space of pure states.

Constraint satisfaction problems for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k-types, for a sufficiently large k. We give sufficient conditions when this method can be applied and illustrate how to use the general results to prove a new complexity dichotomy for first-order expansions of the basic relations of the spatial reasoning formalism RCC5.

For an arbitrary forcing class Γ , the Γ -fragment of Todorcevic's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for Γ implies the Γ -fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.

The system of intuitionistic modal logic IEL − was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic \cite{Artemov}. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL − as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study compelete Kripke-Joyal-style semantics for predicate extensions of IEL − and related logics using Dedekind-MacNeille completions and modal cover systems introduced by Goldblatt \cite{goldblatt2011cover}. The paper extends the conference paper published in the LFCS'20 volume \cite{rogozin2020modal}.

Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory ZFC 2 , we investigate when those models are fully categorical, characterized by the addition to ZFC 2 either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. The heights of these models, we define, are the categorical large cardinals. We subsequently consider various philosophical aspects of categoricity for structuralism and realism, including the tension between categoricity and set-theoretic reflection, and we present (and criticize) a categorical characterization of the set-theoretic universe ⟨V,∈⟩ in second-order logic.

By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of countable Heyting algebras. We show that this characterization no longer holds in the uncountable case. Our results have analogues for co-Heyting algebras and bi-Heyting algebras, and they easily generalize to the setting of p-algebras.

We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.

Assume ZFC. Let κ be a cardinal. A <κ -ground is a transitive proper class W modelling ZFC and such that V is a generic extension of W via a forcing P∈W of cardinality <κ . The κ -mantle is the intersection of all <κ -grounds. We prove that certain partial choice principles in the κ -mantle are the consequence of κ being inaccessible/weakly compact, and some other related facts.