Annals of Pure and Applied Logic最新文献

This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.

We study subsets of countable recursively saturated models of $\mathrm{PA}$ which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets X such that there is a satisfaction class S where S behaves correctly on an idempotent disjunction of length c if and only if $c\in X$. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.

It is known that an algebra is geometrically equivalent to any of its filterpowers if it is $q_{\omega }$-compact. We present an explicit description for the radicals of systems of equation over an algebra A and then we prove the above assertion by an elementary new argument. Then we define $q_{\kappa }$-compact algebras and κ-filterpowers for any infinite cardinal κ. We show that any $q_{\kappa }$-compact algebra is geometric equivalent to its κ-filterpowers. As there is no algebraic description of the κ-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.

Social welfare relations satisfying Pareto and equity principles on infinite utility streams have revealed a non-constructive nature, specifically by showing that in general they imply the existence of non-Ramsey sets and non-Lebesgue measurable sets. In [4, Problem 11.14], the authors ask whether such a connection holds with non-Baire sets as well. In this paper we answer such a question showing that several versions of Pareto principles acting on different utility domains imply the existence of non-Baire sets. Furthermore, we analyze in more details the needed fragments of AC and we start a systematic investigation of a social welfare diagram in a similar fashion done in the past decades concerning cardinal invariants and regularity properties of the reals. In doing that we use tools from forcing theory, such as specific tree-forcings (in particular variants of Silver and Mathias forcings) and Shelah's amalgamation.

For every number $n\ge 2$, let ${\Gamma }_n$ be the hypergraph on $R^n$ of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of ${\Gamma }_n$ is countable while that of ${\Gamma }_{n+1}$ is not.

We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey degrees for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in ${\mathrm{ACA}}_0^{+}$.

We present a detailed formalization of Lipschitz and Wadge games in the context of second order arithmetic and we investigate the logical strength of Lipschitz and Wadge determinacy, and the tightly related Semi-Linear Ordering principle, for the first levels of the Hausdorff difference hierarchy in the Cantor space. As a result, we obtain characterizations of ${\mathrm{WKL}}_0$ and ${\mathrm{ACA}}_0$ in terms of these determinacy principles.

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over $N$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\exists y^{\sigma }\phi \left(y\right)$, where the variable y ranges over objects of finite type σ. This applies in particular to $\mathrm{CZF}$ (Constructive Zermelo-Fraenkel set theory) and $\mathrm{IZF}$ (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

In this paper, it is shown that the fact that the BPI holds in the Cohen's symmetric model can be used as an equal substitute for the Halpern-Läuchli theorem. Also, some alternatives to the Halpern-Läuchli theorem in the form of absoluteness theorems for a certain class of statements are consequences¹ of such consistency results. The article also contains a new proof of the Halpern-Läuchli theorem.

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.