Archive for Mathematical Logic最新文献

We provide a general criterion for Fraenkel–Mostowski models of ({textsf{ZFA}}) (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (({textsf{LW}})), and look at six models for ({textsf{ZFA}}) which satisfy this criterion (and thus ({textsf{LW}}) is true in these models) and “every Dedekind finite set is finite” (({textsf{DF}}={textsf{F}})) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (({textsf{MC}}_{aleph _{0}}^{aleph _{0}})) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of ({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}}) which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 ({textsf{AC}}_{textrm{fin}}^{{textsf{WO}}}) is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which (2{mathfrak {m}} = {mathfrak {m}}) for every infinite cardinal number ({mathfrak {m}}). We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic (textbf{IL}). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form (phi ^mathcal {M}le r), and the open diagram, which encapsulates strict inequalities of the form (phi ^mathcal {M}< r). We show that the closed and open (Sigma _N) diagrams are (Pi ^0_{N+1}) and (Sigma ^0_N) respectively, and that the closed and open (Pi _N) diagrams are (Pi ^0_N) and (Sigma ^0_{N + 1}) respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space ({mathcal {X}}), and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space (mathbb {N}times {mathcal {X}}).

This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in (mathrm RCA_{0}), showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in (mathrm RCA_{0}).

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class (Gamma ), then its (Gamma )-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a ( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)-function, then one can find its ( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.

Starting from a model with a Laver-indestructible supercompact cardinal (kappa ), we construct a model of (ZF+DC_{kappa }) where there are no (kappa )-mad families.

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).

It is shown that all the countable superatomic boolean algebras of finite rank have the small index property.