Mathematical Logic Quarterly最新文献

In [3], Shi proved that there exist incomparable Zermelo degrees at γ if there exists an ω-sequence of measurable cardinals, whose limit is γ. He asked whether there is a size $�{\gamma }^{\omega }$ antichain of Zermelo degrees. We consider this question for the $�V_{\gamma }$-degree structure. We use a kind of Prikry-type forcing to show that if there is an ω-sequence of measurable cardinals, then there are $�{\gamma }^{\omega }$-many pairwise incomparable $�V_{\gamma }$-degrees, where γ is the limit of the ω-sequence of measurable cardinals.

Assuming $�\mathrm{GCH}$, we show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under a one step iteration of the Hechler forcing for adding a dominating κ-real. Moreover, we show that if κ is strongly unfoldable, $��2^{\kappa }�=�{\kappa }^{+}�$ and λ is a regular cardinal such that , then there is a set generic extension in which .

Let G be a definable group in a p-adically closed field M. We show that G has finitely satisfiable generics ($�\text{fsg}$) if and only if G is definably compact. The case $��M�=�Q_p�$ was previously proved by Onshuus and Pillay.

We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if $��\lambda �\ge �2^{\kappa }�$, then (a) $�I_{�\kappa �,�\lambda �}$ is not (λ, 2)-distributive, and (b) $��I_{�\kappa �,�\lambda �}^{+}�\to �{�\left(�I_{�\kappa �,�\lambda �}^{+}�\right)�}_{\omega }^2�$ does not hold.

We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and an intuitionistic tableaux method.

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.