Mathematical Logic Quarterly最新文献

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies . Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property, and for each there exists such that every tree of height μ with λ nodes has less than $�{\kappa }^{\ast }$ branches.

We consider so-called extremal numberings that form the greatest or minimal degrees under the reducibility of all A-computable numberings of a given family of subsets of $�N$, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A-computable numberings, respectively. The main question of this paper is when a universal or a minimal A-computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing degree of a set A is hyperimmune if and only if every universal A-computable numbering satisfies the Recursion Theorem. Next we prove that any universal A-computable numbering satisfies the Recursion Theorem with parameters if A computes a non-computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is precomplete if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given A-computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if A is high, then any infinite A-computable family has a minimal A-computable numbering that satisfies the Recursion Theorem.

Läuchli and Pincus showed that existence of algebraic completions of all fields cannot be proved from Zermelo-Fraenkel set theory alone. On the other hand, important special cases do follow. In particular, I show that an algebraic completion of $�Q_p$ can be constructed in Zermelo-Fraenkel set theory.

The notion of a critical successor [5] in relational semantics has been central to most classic modal completeness proofs in interpretability logics. In this paper we shall work with a more general notion, that of an assuring successor. This will enable more concisely formulated completeness proofs, both with respect to ordinary and generalised Veltman semantics. Due to their interesting theoretical properties, we will devote some space to the study of a particular kind of assuring labels, the so-called full labels and maximal labels. After a general treatment of assuringness, we shall apply it to obtain a completeness result for the modal logic $�\mathrm{ILP}$ w.r.t. generalised semantics for a restricted class of frames.

We define a potentialist system of $�\mathrm{ZF}$-structures, i.e., a collection of possible worlds in the language of $�\mathrm{ZF}$ connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just $�\mathrm{ZF}$. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory $��S�4�.�2�$. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory $��S�5�$, both for assertions in the language of $�\mathrm{ZF}$ and for assertions in the full potentialist language.

We study the propositional logic $�S_{\mathrm{DN}}$ associated with the variety of distributive nearlattices $�\mathrm{DN}$. We prove that the logic $�S_{\mathrm{DN}}$ coincides with the assertional logic associated with the variety $�\mathrm{DN}$ and with the order-based logic associated with $�\mathrm{DN}$. We obtain a characterization of the reduced matrix models of logic $�S_{\mathrm{DN}}$. We develop a connection between the logic $�S_{\mathrm{DN}}$ and the $��\{�\wedge �,�\vee �,�\top �\}�$-fragment of classical logic. Finally, we present two Hilbert-style axiomatizations for the logic $�S_{\mathrm{DN}}$.

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal κ, there are no $��{\Sigma }_1^1��\left(�\kappa �\right)��$ κ-mad families.

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.