Archive for Mathematical Logic最新文献

An appropriate framework is put forward for the construction of (lambda )-models with (infty )-groupoid structure, which we call homotopic (lambda )-models, through the use of an (infty )-category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and (lambda )-calculus, in the sense that the concept of proof (path) of equality of (lambda )-terms is raised to higher proof (homotopy).

It is generally accepted that H. Friedman’s gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we call ‘Bachmann–Howard derivative’. In the present paper, we focus on the unary case, i.e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences.

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})).