Mathematical Logic Quarterly最新文献

The purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model construction technique. In particular, we will show that Łoś's theorem can be construed as a specific case of Cohen's truth lemma, and we isolate the weakest conditions a filter must satisfy in order for the truth lemma to work.

We study logics determined by matrices consisting of a De Morgan lattice with an upward closed set of designated values, such as the logic of non-falsity preservation in a given finite Boolean algebra and Shramko's logic of non-falsity preservation in the four-element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice-theoretic notion of an n-filter. We study the logics of all (complete, consistent, and classical) n-filters on De Morgan lattices, which are non-adjunctive generalizations of the four-valued logic of Belnap and Dunn (of the three-valued logics of Priest and Kleene, and of classical logic). We then show how to find a finite Hilbert-style axiomatization of any logic determined by a finite family of prime upsets of finite De Morgan lattices and a finite Gentzen-style axiomatization of any logic determined by a finite family of filters on finite De Morgan lattices. As an application, we axiomatize Shramko's logic of anything but falsehood.

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0-h^mix-minimality (which, in equi-characteristic 0, amounts to 0-h-minimality).

In the context of constructive reverse mathematics, we show that weak Kőnig's lemma ($�\mathrm{WKL}$) implies that every pointwise continuous function $��f�:�[�0�,�1�]�\to �R�$ is induced by a code in the sense of reverse mathematics. This, combined with the fact that $�\mathrm{WKL}$ implies the Fan theorem, shows that $�\mathrm{WKL}$ implies the uniform continuity theorem: every pointwise continuous function $��f�:�[�0�,�1�]�\to �R�$ has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier-free axiom of choice.

A Hausdorff space $��(�X�,�T�)�$ is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every $��x�,�y�\in �X�$ with $��x�\ne �y�$, $��F�(�x�,�y�)�=�(�U�,�V�)�$, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in $�\mathrm{ZF}$, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice ($�\mathrm{AC}$):

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak p-morphisms. It is well-known that orthomodular lattices provide an algebraic semantics for the quantum logic $�Q$. Hence, as an application of our duality, we develop a topological semantics for $�Q$ using orthomodular spaces and prove soundness and completeness.

We show that the countable universal ω-categorical bowtie-free graph admits generic automorphisms. Moreover, we show that this graph is not finitely homogenisable.

While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik-reducible to J itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.

This paper introduces the notion of logical quantale module. It proves that there is a dual equivalence between the category of logical quantale modules and the category of quantum B-modules, in the way that every quantum B-module admits a natural embedding into a logical quantale module, the enveloping quantale module.

We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle-third set (Theorem 2.9). If it has burden 2 and both an infinite discrete set D and a dense-codense set X are definable, then translates of X must witness the Independence Property (Theorem 2.26). In the last section, an explicit example of an ordered Abelian group of burden 2 is given in which both an infinite discrete set and a dense-codense set are definable.