Editorial Math. Logic Q. (IF 0.327) Pub Date : 2020-04-02
Dear readers of Mathematical Logic Quarterly: Dr Hugo Nobrega has been the Editorial Assistant of our journal for four years, from 2016 to 2019, and has now handed over to his successor, Dr Thomas Piecha from the Eberhard Karls Universität Tübingen, who started to work for the journal on New Year's Day 2020. We thank Hugo for four years of excellent service for the journal and welcome Thomas as our
The torsion‐free part of the Ziegler spectrum of orders over Dedekind domains Math. Logic Q. (IF 0.327) Pub Date : 2020-03-24 Lorna Gregory; Sonia L'Innocente; Carlo Toffalori
We study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.
Computability of graphs Math. Logic Q. (IF 0.327) Pub Date : 2020-03-19 Zvonko Iljazović
We consider topological pairs ( A , B ) , B ⊆ A , which have computable type, which means that they have the following property: if X is a computable topological space and f : A → X a topological imbedding such that f ( A ) and f ( B ) are semicomputable sets in X, then f ( A ) is a computable set in X. It is known, e.g., that ( M , ∂ M ) has computable type if M is a compact manifold with boundary
On the Herbrand functional interpretation Math. Logic Q. (IF 0.327) Pub Date : 2020-03-16 Paulo Oliva; Chuangjie Xu
We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation
Local weak presaturation of the strongly non‐stationary ideal Math. Logic Q. (IF 0.327) Pub Date : 2020-03-16 Masahiro Shioya; Naoki Yamaura
We give a model of set theory in which the strongly non‐stationary ideal over ℘ μ μ is weakly presaturated below some canonical set. Here μ is a regular uncountable cardinal. The model is the forcing extension with the Lévy collapse of a Woodin cardinal to the successor of μ. This improves on results of Goldring and of the first author.
Lowness for isomorphism, countable ideals, and computable traceability Math. Logic Q. (IF 0.327) Pub Date : 2020-03-16 Johanna N. Y. Franklin; Reed Solomon
We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune free degrees, lowness for isomorphism is entirely independent of computable traceability.
On weak square, approachability, the tree property, and failures of SCH in a choiceless context Math. Logic Q. (IF 0.327) Pub Date : 2020-03-16 Arthur W. Apter
We show that the consistency of the theories ZF + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “both □ ℵ ω ∗ and AP ℵ ω fail” and ZF + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “ ℵ ω + 1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context
Expansions of the p‐adic numbers that interpret the ring of integers Math. Logic Q. (IF 0.327) Pub Date : 2020-03-11 Nathanaël Mariaule
Let Q p ∼ be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of Q p ∼ expanded by two predicates interpreted by multiplicative subgroups α Z and β Z where α , β ∈ N are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that
Cohen forcing and inner models Math. Logic Q. (IF 0.327) Pub Date : 2020-03-09 Jonas Reitz
Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add ( κ , 1 ) W . The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality
On p‐adic semi‐algebraic continuous selections Math. Logic Q. (IF 0.327) Pub Date : 2020-02-25 Athipat Thamrongthanyalak
Let E ⊆ Q p n and T be a set‐valued map from E to Q p m . We prove that if T is p‐adic semi‐algebraic, lower semi‐continuous and T ( x ) is closed for every x ∈ E , then T has a p‐adic semi‐algebraic continuous selection. In addition, we include three applications of this result. The first one is related to Fefferman's and Kollár's question on existence of p‐adic semi‐algebraic continuous solution
A two‐dimensional metric temporal logic Math. Logic Q. (IF 0.327) Pub Date : 2019-12-26 Stefano Baratella; Andrea Masini
We introduce a two‐dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut elimination theorem down to formulas of a certain type.
New substitution bases for complexity classes Math. Logic Q. (IF 0.327) Pub Date : 2019-12-23 Stefano Mazzanti
The set AC 0 ( F ) , the AC 0 closure of F, is the closure with respect to substitution and concatenation recursion on notation of a set of basic functions comprehending the set F. By improving earlier work, we show that AC 0 ( F ) is the substitution closure of a simple function set and characterize well‐known function complexity classes as the substitution closure of finite sets of simple functions