Archive for Mathematical Logic最新文献

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano (Around the set-theoretical consistency of d-tameness of metric abstract elementary classes, arXiv:1508.05529, 2015) on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016).

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zheng’s work for the profinite graph to the setting of Fraïssé classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fraïssé class, its universal inverse limit structure has finite big Ramsey degrees under finite Baire-measurable colorings. For such Fraïssé classes satisfying free amalgamation as well as finite ordered tournaments and finite partial orders with a linear extension, we characterize the exact big Ramsey degrees.

This paper concerns product constructions within the continuous-logic framework of Ben Yaacov, Berenstein, Henson, and Usvyatsov. Continuous-logic analogues are presented for the direct product, direct sum, and almost everywhere direct product analyzed in the work of Feferman and Vaught. These constructions are shown to possess a number of preservation properties analogous to those enjoyed by their classical counterparts in ordinary first-order logic: for example, each product preserves elementary equivalence in an appropriate sense; and if for (iin mathbb {N}) (mathcal {M}_i) is a metric structure and the sentence (theta ) is true in (prod _{i=0}^kmathcal {M}_i) for every (kin mathbb {N}), then (theta ) is true in (prod _{iin mathbb {N}}mathcal {M}_i).

We show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs (C subset omega _1), (Supharpoonright C) is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with (diamondsuit ) that for every Souslin tree T there is a dense (X subseteq T) which does not contain a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982).

We show that the sQ-degree of a hypersimple set includes an infinite collection of (sQ_1)-degrees linearly ordered under (le _{sQ_1}) with order type of the integers and each c.e. set in these sQ-degrees is a hypersimple set. Also, we prove that there exist two c.e. sets having no least upper bound on the (sQ_1)-reducibility ordering. We show that the c.e. (sQ_1)-degrees are not dense and if a is a c.e. (sQ_1)-degree such that (o_{sQ_1}<_{sQ_1}a<_{sQ_1}o'_{sQ_1}), then there exist infinitely many pairwise sQ-incomputable c.e. sQ-degrees ({c_i}_{iin omega }) such that ((forall ,i);(a<_{sQ_1}c_i<_{sQ_1}o'_{sQ_1})).

In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of (textsf {ZFC}), then (textsf {DC}_{<kappa }) can be preserved in the symmetric extension of V in terms of symmetric system (langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle ), if ({mathbb {P}}) is (kappa )-distributive and ({mathcal {F}}) is (kappa )-complete. Further we observe that if (delta <kappa ) and V is a model of (textsf {ZF}+textsf {DC}_{delta }), then (textsf {DC}_{delta }) can be preserved in the symmetric extension of V in terms of symmetric system (langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle ), if ({mathbb {P}}) is ((delta +1))-strategically closed and ({mathcal {F}}) is (kappa )-complete.