Authors:Yurii Khomskii; Giorgio Laguzzi Pages: 1491 - 1506 Abstract: Publication date: August 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 8 Author(s): Yurii Khomskii, Giorgio Laguzzi We investigate two closely related partial orders of trees on ω ω : the full-splitting Miller trees and the infinitely often equal trees, as well as their corresponding σ-ideals. The former notion was considered by Newelski and Rosłanowski while the latter involves a correction of a result of Spinas. We consider some Marczewski-style regularity properties based on these trees, which turn out to be closely related to the property of Baire, and look at the dichotomies of Newelski–Rosłanowski and Spinas for higher projective pointclasses. We also provide some insight concerning a question of Fremlin whether one can add an infinitely often equal real without adding a Cohen real, which was recently solved by Zapletal.

Authors:Sonia L'Innocente; Carlo Toffalori; Gena Puninski Pages: 1507 - 1516 Abstract: Publication date: August 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 8 Author(s): Sonia L'Innocente, Carlo Toffalori, Gena Puninski We will prove that the theory of all modules over the ring of algebraic integers is decidable.

Authors:Anton Freund Pages: 1361 - 1382 Abstract: Publication date: July 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 7 Author(s): Anton Freund As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k , m , n there is an N which satisfies the statement PH ( k , m , n , N ) : For any k-coloring of its n-element subsets the set { 0 , … , N − 1 } has a large homogeneous subset of size ≥m. At the same time very weak theories can establish the Σ 1 -statement ∃ N PH ( k ‾ , m ‾ , n ‾ , N ) for any fixed parameters k , m , n . Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀ m ∃ N PH ( k ‾ , m , n ‾ , N ) has a natural and short proof (relative to n and k) by Σ n − 1 -induction. In contrast, we show that there is an elementary function e such that any proof of ∃ N PH ( e ( n ) ‾ , n + 1 ‾ , n ‾ , N ) by Σ n − 2 -induction is ridiculously long. In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ 1 -reflection is related to a function that has a considerably lower growth rate than F ε 0 but dominates all functions F α with α < ε 0 in the fast-growing hierarchy.

Authors:Matthew Harrison-Trainor Pages: 1396 - 1405 Abstract: Publication date: July 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 7 Author(s): Matthew Harrison-Trainor A set A is coarsely computable with density r ∈ [ 0 , 1 ] if there is an algorithm for deciding membership in A which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least r. To any Turing degree a we can assign a value Γ T ( a ) : the minimum, over all sets A in a, of the highest density at which A is coarsely computable. The closer Γ T ( a ) is to 1, the closer a is to being computable. Andrews, Cai, Diamondstone, Jockusch, and Lempp noted that Γ T can take on the values 0, 1/2, and 1, but not any values in strictly between 1/2 and 1. They asked whether the value of Γ T can be strictly between 0 and 1/2. This is the Gamma question. Replacing Turing degrees by many-one degrees, we get an analogous question, and the same arguments show that Γ m can take on the values 0, 1/2, and 1, but not any values strictly between 1/2 and 1. We will show that for any r ∈ [ 0 , 1 / 2 ] , there is an m-degree a with Γ m ( a ) = r . Thus the range of Γ m is [ 0 , 1 / 2 ] ∪ { 1 } . Benoit Monin has recently announced a solution to the Gamma question for Turing degrees. Interestingly, his solution gives the opposite answer: the only possible values of Γ T are 0, 1/2, and 1.

Authors:Fan Yang; Jouko Väänänen Pages: 1406 - 1441 Abstract: Publication date: July 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 7 Author(s): Fan Yang, Jouko Väänänen We consider team semantics for propositional logic, continuing [34]. In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula ϕ based on finitely many propositional variables the set 〚 ϕ 〛 of teams that satisfy ϕ. We define a maximal propositional team logic in which every set of teams is definable as 〚 ϕ 〛 for suitable ϕ. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the maximal propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.

Authors:Gabriel Conant Pages: 1442 - 1471 Abstract: Publication date: July 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 7 Author(s): Gabriel Conant Given a countable, totally ordered commutative monoid R = ( R , ⊕ , ≤ , 0 ) , with least element 0, there is a countable, universal and ultrahomogeneous metric space U R with distances in R . We refer to this space as the R -Urysohn space, and consider the theory of U R in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free nth roots of the complete graph (e.g. the random graph when n = 2 ), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of Th ( U R ) by algebraic properties of R , many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOP n -hierarchy. Using the submonoid of idempotents in R , we also characterize superstability, supersimplicity, and weak elimination of imaginaries. Finally, we give necessary conditions for elimination of hyperimaginaries, which further develops previous work of Casanovas and Wagner.

Authors:Eric P. Astor; Damir D. Dzhafarov; Reed Solomon; Jacob Suggs Pages: 1153 - 1171 Abstract: Publication date: June 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 6 Author(s): Eric P. Astor, Damir D. Dzhafarov, Reed Solomon, Jacob Suggs The principle ADS asserts that every linear order on ω has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore [16]. We introduce the principle ADC , which asserts that every such linear order has an infinite ascending or descending chain. The two are easily seen to be equivalent over the base system RCA 0 of second order arithmetic; they are even computably equivalent. However, we prove that ADC is strictly weaker than ADS under Weihrauch (uniform) reducibility. In fact, we show that even the principle SADS , which is the restriction of ADS to linear orders of type ω + ω ⁎ , is not Weihrauch reducible to ADC . In this connection, we define a more natural stable form of ADS that we call General - SADS , which is the restriction of ADS to linear orders of type k + ω , ω + ω ⁎ , or ω + k , where k is a finite number. We define General - SADC analogously. We prove that General - SADC is not Weihrauch reducible to SADS , and so in particular, each of SADS and SADC is strictly weaker under Weihrauch reducibility than its general version. Finally, we turn to the principle CAC , which asserts that every partial order on ω has an infinite chain or antichain. This has two previously studied stable variants, SCAC and WSCAC , which were introduced by Hirschfeldt and Jockusch [16], and by Jockusch, Kastermans, Lempp, Lerman, and Solomon [19], respectively, and which are known to be equivalent over RCA 0 . Here, we show that SCAC is strictly weaker than WSCAC under even computable reducibility.

Authors:Ludovic Patey Pages: 1172 - 1209 Abstract: Publication date: June 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 6 Author(s): Ludovic Patey The Erdős–Moser theorem ( EM ) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs ( RT 2 2 ) by providing an alternate proof of RT 2 2 in terms of EM and the ascending descending sequence principle ( ADS ). In this paper, we study the computational weakness of EM and construct a standard model (ω-model) of simultaneously EM , weak König's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdős–Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.

Authors:William Chan Pages: 1224 - 1246 Abstract: Publication date: June 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 6 Author(s): William Chan Let F ω 1 be the countable admissible ordinal equivalence relation defined on 2 ω by x F ω 1 y if and only if ω 1 x = ω 1 y . Some invariant descriptive set theoretic properties of F ω 1 will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed F ω 1 is not the orbit equivalence relation of a continuous action of a Polish group on 2 ω . Becker stengthened this to show F ω 1 is not even the orbit equivalence relation of a Δ 1 1 action of a Polish group. However, Montalbán has shown that F ω 1 is Δ 1 1 reducible to an orbit equivalence relation of a Polish group action, in fact, F ω 1 is classifiable by countable structures. It will be shown here that F ω 1 must be classified by structures of high Scott rank. Let E ω 1 denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E ≤ a Δ 1 1 F denotes the existence of a Δ 1 1 function f : X → Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies E ω 1 ≤ a Δ 1 1 F ω 1 . However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), E ω 1 ≤ a Δ 1 1 F ω 1 is false. Lastly, the techniques of the previous result will be used to show that in L (and set gener... PubDate: 2017-04-02T17:28:22Z DOI: 10.1016/j.apal.2016.12.002

Authors:Paulo Oliva; Thomas Powell Pages: 887 - 921 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Paulo Oliva, Thomas Powell We introduce a new, demand-driven variant of Spector's bar recursion in the spirit of the Berardi–Bezem–Coquand functional of [4]. The recursion takes place over finite partial functions u, where the control parameter ω, used in Spector's bar recursion to terminate the computation at sequences s satisfying ω ( s ˆ ) < s , now acts as a guide for deciding exactly where to make bar recursive updates, terminating the computation whenever ω ( u ˆ ) ∈ dom ( u ) . We begin by exploring theoretical aspects of this new form of recursion, then in the main part of the paper we show that demand-driven bar recursion can be directly used to give an alternative functional interpretation of classical countable choice. We provide a short case study as an illustration, in which we extract a new bar recursive program from the proof that there is no injection from N → N to N , and compare this with the program that would be obtained using Spector's original variant. We conclude by formally establishing that our new bar recursor is primitive recursively equivalent to the original Spector bar recursion, and thus defines the same class of functionals when added to Gödel's system T .

Authors:Thomas Gilton; John Krueger Pages: 922 - 1016 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Thomas Gilton, John Krueger Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of ω 2 ∩ cof ( ω 1 ) in the approachability ideal I [ ω 2 ] . In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.

Authors:Paul Shafer Pages: 1017 - 1031 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Paul Shafer An element a of a lattice cups to an element b > a if there is a c < b such that a ∪ c = b . An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0 < E a < E b that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees.

Authors:Robert Goldblatt; Ian Hodkinson Pages: 1032 - 1090 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Robert Goldblatt, Ian Hodkinson There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈 t 〉 (in fact 〈 t 〉 can express ◇); (iii) ◇ , ∀ ; (iv) ◇ , ∀ , 〈 t 〉 ; (v) the derivative operator 〈 d 〉 ; (vi) 〈 d 〉 and the associated tangled closure operators 〈 d t 〉 ; (vii) 〈 d 〉 , ∀ ; (viii) 〈 d 〉 , ∀ , 〈 d t 〉 . Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈 d 〉 , X validates the well-known axiom G 1 . For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.

Authors:Vincent Guingona; Cameron Donnay Hill; Lynn Scow Pages: 1091 - 1111 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Vincent Guingona, Cameron Donnay Hill, Lynn Scow We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.

Authors:Sean Cox; Philipp Lücke Pages: 1112 - 1131 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Sean Cox, Philipp Lücke Given an uncountable regular cardinal κ, a partial order is κ-stationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in P κ ( P ) . We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial orders are κ-stationarily layered implies that κ is a Mahlo cardinal and every stationary subset of κ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

Authors:Leonardo Manuel Cabrer; Daniele Mundici Pages: 1132 - 1151 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Leonardo Manuel Cabrer, Daniele Mundici Differently from boolean logic, in Łukasiewicz infinite-valued propositional logic Ł∞ the theory Θ max , v consisting of all formulas satisfied by a model v ∈ [ 0 , 1 ] n is not the only one having v as its unique model: indeed, there is a smallest such theory Θ min , v , the germinal theory at v, which in general is strictly contained in Θ max , v . The Lindenbaum algebra of Θ max , v is promptly seen to coincide with the subalgebra of the standard MV-algebra [ 0 , 1 ] generated by the coordinates of v. The description of the Lindenbaum algebras of germinal theories in two variables is our main aim in this paper. As a basic prerequisite of independent interest, we prove that for any models v and w the germinal theories Θ min , v and Θ min , w have isomorphic Lindenbaum algebras iff v and w have the same orbit under the action of the affine group over the integers.

Authors:Jan Dobrowolski; Byunghan Kim; Junguk Lee Abstract: Publication date: Available online 4 July 2017 Source:Annals of Pure and Applied Logic Author(s): Jan Dobrowolski, Byunghan Kim, Junguk Lee Let p be a strong type of an algebraically closed tuple over B = acl eq ( B ) in any theory T. Depending on a ternary relation Image 1 satisfying some basic axioms (there is at least one such, namely the trivial independence in T), the first homology group H 1 ⁎ ( p ) can be introduced, similarly to [3]. We show that there is a canonical surjective homomorphism from the Lascar group over B to H 1 ⁎ ( p ) . We also notice that the map factors naturally via a surjection from the ‘relativised’ Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of p is independent from the choice of Image 1 , and can be written simply as H 1 ( p ) . As consequences, in any T, we show that H 1 ( p ) ≥ 2 ℵ 0 unless H 1 ( p ) is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

Authors:Enrique Casanovas; Luis Jaime Corredor Abstract: Publication date: Available online 30 June 2017 Source:Annals of Pure and Applied Logic Author(s): Enrique Casanovas, Luis Jaime Corredor Let M be the monster model of a complete first-order theory T. If D is a subset of M , following D. Zambella we consider e ( D ) = { D ′ ( M , D ) ≡ ( M , D ′ ) } and o ( D ) = { D ′ ( M , D ) ≅ ( M , D ′ ) } . The general question we ask is when e ( D ) = o ( D ) ' The case where D is A-invariant for some small set A is rather straightforward: it just means that D is definable. We investigate the case where D is not invariant over any small subset. If T is geometric and ( M , D ) is an H-structure (in the sense of A. Berenstein and E. Vassiliev) we get some answers. In the case of SU-rank one, e ( D ) is always different from o ( D ) . In the o-minimal case, everything can happen, depending on the complexity of the definable closure. We also study the case of lovely pairs of geometric theories.

Authors:L.B. Ostrovsky; M.E. Zhukovskii Abstract: Publication date: Available online 21 June 2017 Source:Annals of Pure and Applied Logic Author(s): L.B. Ostrovsky, M.E. Zhukovskii We study asymptotical probabilities of first order and monadic second order properties of Bernoulli random graph G ( n , n − a ) . The random graph obeys FO (MSO) zero-one k-law (k is a positive integer) if, for any first order (monadic second order) formulae with quantifier depth at most k, it is true for G ( n , n − a ) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a < 1 . We obtain new zero-one k-laws (both for first order and monadic second order languages) when a > 1 . Proofs of these results are based on the earlier studies of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.

Authors:Anton Freund Abstract: Publication date: Available online 20 June 2017 Source:Annals of Pure and Applied Logic Author(s): Anton Freund We describe a “slow” version of the hierarchy of uniform reflection principles over Peano Arithmetic (PA). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of PA plus slow reflection is provable in PA + Con ( PA ) . We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely ε 0 stages between PA and PA + Con ( PA ) (where Con ( PA ) refers to the usual consistency statement).

Authors:Saskia Chambille; Pablo Cubides Kovacsics; Eva Leenknegt Abstract: Publication date: Available online 9 June 2017 Source:Annals of Pure and Applied Logic Author(s): Saskia Chambille, Pablo Cubides Kovacsics, Eva Leenknegt We prove that in a P-minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.

Authors:Ari Meir Brodsky; Assaf Rinot Abstract: Publication date: Available online 8 June 2017 Source:Annals of Pure and Applied Logic Author(s): Ari Meir Brodsky, Assaf Rinot We propose a parameterized proxy principle from which κ-Souslin trees with various additional features can be constructed, regardless of the identity of κ. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent κ-Souslin tree that applies also for κ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known ⋄-based constructions of κ-Souslin trees may be redirected through this new proxy principle.

Abstract: Publication date: Available online 7 June 2017 Source:Annals of Pure and Applied Logic Author(s): M. Hrušák, D. Meza-Alcántara, E. Thümmel, C. Uzcátegui We study several combinatorial properties of (mostly definable) ideals on countable sets. In several cases, we identify critical ideals for such properties in the Katětov order. In particular, the ideal R generated by the homogeneous subsets of the random graph is critical for the Ramsey property. The question as to whether there is a tall definable Ramsey ideal is raised and studied. It is shown that no tall F σ ideal is Ramsey, while there is a tall co-analytic Ramsey ideal.

Authors:Panagiotis Rouvelas Abstract: Publication date: Available online 26 May 2017 Source:Annals of Pure and Applied Logic Author(s): Panagiotis Rouvelas We introduce the notion of pseudo-increasing sentence, and prove that all such sentences are decidable by a weak subtheory of Simple Type Theory with infinitely many zero-type elements. We then present the consequences of this result to Quine's theory of “New Foundations” (NF). In particular, we prove the decidability of certain universal-existential sentences, and establish the consistency of a subtheory of NF.

Authors:Samuel van; Gool George Metcalfe Constantine Tsinakis Abstract: Publication date: Available online 26 May 2017 Source:Annals of Pure and Applied Logic Author(s): Samuel J. van Gool, George Metcalfe, Constantine Tsinakis Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown, following related results of Ghilardi and Zawadowski, that a combination of these properties provides a sufficient condition for the first-order theory of the variety to admit a model completion.

Authors:Jacob Davis Abstract: Publication date: Available online 3 May 2017 Source:Annals of Pure and Applied Logic Author(s): Jacob Davis Starting from a supercompact cardinal we build a model in which 2 ℵ ω 1 = 2 ℵ ω 1 + 1 = ℵ ω 1 + 3 but there is a jointly universal family of size ℵ ω 1 + 2 of graphs on ℵ ω 1 + 1 . The same technique will work for any uncountable cardinal in place of ω 1 .

Abstract: Publication date: July 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 7 Author(s): András Pongrácz Let ( H n , E ) denote the Henson graph, the unique countable homogeneous graph whose age consists of all finite K n -free graphs. In this note the reducts of the Henson graphs with a constant are determined up to first-order interdefinability. It is shown that up to first-order interdefinability ( H 3 , E , 0 ) has 13 reducts and ( H n , E , 0 ) has 16 reducts for n ≥ 4 .

Authors:Vasco Brattka; Andrea Cettolo; Guido Gherardi; Alberto Marcone; Matthias Schröder Abstract: Publication date: Available online 29 April 2017 Source:Annals of Pure and Applied Logic Author(s): Vasco Brattka, Andrea Cettolo, Guido Gherardi, Alberto Marcone, Matthias Schröder The purpose of this addendum is to close a gap in the proof of [1, Theorem 11.2], which characterizes the computational content of the Bolzano–Weierstraß Theorem for arbitrary computable metric spaces.

Authors:Samaria Montenegro Abstract: Publication date: Available online 21 April 2017 Source:Annals of Pure and Applied Logic Author(s): Samaria Montenegro The main result of this paper is that if M is a bounded PRC field, then T h ( M ) eliminates imaginaries in the language of rings expanded by constant symbols. As corollary of the elimination of imaginaries and the fact that the algebraic closure (in the sense of model theory) defines a pregeometry we obtain that the complete theory of a bounded PRC field is superrosy of U th -rank 1.

Authors:Kaisa Kangas Abstract: Publication date: Available online 6 April 2017 Source:Annals of Pure and Applied Logic Author(s): Kaisa Kangas We show that if M is a Zariski-like structure (see [6]) and the canonical pregeometry obtained from the bounded closure operator (bcl) is non locally modular, then M interprets either an algebraically closed field or a non-classical group.

Authors:David Cerna; Alexander Leitsch; Giselle Reis; Simon Wolfsteiner Abstract: Publication date: Available online 5 April 2017 Source:Annals of Pure and Applied Logic Author(s): David Cerna, Alexander Leitsch, Giselle Reis, Simon Wolfsteiner In this paper we present a procedure allowing the extension of a CERES-based cut-elimination method to intuitionistic logic. Previous results concerning this problem manage to capture fragments of intuitionistic logic, but in many essential cases structural constraints were violated during normal form construction resulting in a classical proof. The heart of the CERES method is the resolution calculus, which ignores the structural constraints of the well known intuitionistic sequent calculi. We propose, as a method of avoiding the structural violations, the generalization of resolution from the resolving of clauses to the resolving of cut-free proofs, in other words, what we call proof resolution. The result of proof resolution is a cut-free proof rather than a clause. Note that resolution on ground clauses is essentially atomic cut, thus using proof resolution to construct cut-free proofs one would need to join the two proofs together and remove the atoms which where resolved. To efficiently perform this joining (and guarantee that the resulting cut-free proof is intuitionistic) we develop the concept of proof subsumption (similar to clause subsumption) and indexed resolution, a refinement indexing atoms by their corresponding positions in the cut formula. Proof subsumption serves as a tool to prove the completeness of the new method CERES-i, and indexed resolution provides an efficient strategy for the joining of two proofs which is in general a nondeterministic search. Such a refinement is essential for any attempt to implement this method. Finally we compare the complexity of CERES-i with that of Gentzen-based methods.

Authors:Nenad Abstract: Publication date: June 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 6 Author(s): Miloš S. Kurilić, Nenad Morača We investigate the interplay between several similarities of relational structures: the condensational equivalence (defined by X ∼ c Y iff there are bijective homomorphisms f : X → Y and g : Y → X ), the isomorphism, the equimorphism (bi-embedability), the elementary equivalence and the similarities of structures determined by some similarities of their self-embedding monoids. It turns out that the Hasse diagram describing the hierarchy of these equivalence relations restricted to the set Mod L ( κ ) of all L-structures of size κ collapses significantly for a finite cardinal κ or for a unary language L, while for infinite structures of non-unary languages we have a large diversity.

Authors:F. Delon; P. Simonetta Abstract: Publication date: Available online 29 March 2017 Source:Annals of Pure and Applied Logic Author(s): F. Delon, P. Simonetta We classify abelian C-minimal valued groups up to pure and elementary extensions and small modifications. We define first the notion of almost regularity for abelian valued groups. Then we introduce on the chain of valuations the structure which enables us to characterize C-minimal abelian valued groups as the almost regular ones with an o-minimal chain.

Authors:Will Boney; Rami Grossberg; Monica M. VanDieren; Sebastien Vasey Abstract: Publication date: Available online 27 March 2017 Source:Annals of Pure and Applied Logic Author(s): Will Boney, Rami Grossberg, Monica M. VanDieren, Sebastien Vasey Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for nonsplitting, a particular notion of independence. We generalize their result as follows: given any abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah–Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah–Villaveces proof.

Authors:Sebastien Vasey Abstract: Publication date: Available online 22 March 2017 Source:Annals of Pure and Applied Logic Author(s): Sebastien Vasey We prove: Theorem 0.1 Let K be a universal class. If K is categorical in cardinals of arbitrarily high cofinality, then K is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: Theorem 0.2 Let K be an AEC with amalgamation. Assume that K is fully LS ( K ) -tame and short and has primes over sets of the form M ∪ { a } . Write H 2 : = ℶ ( 2 ℶ ( 2 LS ( K ) ) + ) + . If K is categorical in a λ > H 2 , then K is categorical in all λ ′ ≥ H 2 .

Authors:John Goodrick; Byunghan Kim; Alexei Kolesnikov Abstract: Publication date: Available online 18 March 2017 Source:Annals of Pure and Applied Logic Author(s): John Goodrick, Byunghan Kim, Alexei Kolesnikov We give an explicit description of the homology group H n ( p ) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H i ( q ) are trivial for 2 ≤ i < n . The group H n ( p ) turns out to be isomorphic to the automorphism group of a certain part of the algebraic closure of n independent realizations of p; it follows from the authors' earlier work that such a group must be abelian. We call this the “Hurewicz correspondence” by analogy with the Hurewicz Theorem in algebraic topology.

Authors:Andrey Bovykin; Andreas Weiermann Abstract: Publication date: Available online 14 March 2017 Source:Annals of Pure and Applied Logic Author(s): Andrey Bovykin, Andreas Weiermann In this article, we conduct a model-theoretic investigation of three infinitary Ramseyan statements: the Infinite Ramsey Theorem for pairs and two colours ( RT 2 2 ), the Canonical Ramsey Theorem for pairs ( CRT 2 ) and the Regressive Ramsey Theorem for pairs ( RegRT 2 ). We approximate the logical strength of these principles by the strength of their first-order iterated versions, known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of the classical article by Paris and Kirby, hereby re-proving old results model-theoretically. The article is concluded by a discussion of two further outreaches of densities. One is a further investigation of the strength of the Ramsey Theorem for pairs. The other deals with the asymptotics of the standard Ramsey function R 2 2 .

Authors:Rutger Kuyper; Joseph S. Miller Abstract: Publication date: Available online 10 March 2017 Source:Annals of Pure and Applied Logic Author(s): Rutger Kuyper, Joseph S. Miller For a class C of sets, let us say that a set A is C stabilising if A △ X ∈ C for every X ∈ C . We prove that the Martin-Löf stabilising sets are exactly the K-trivial sets, as are the weakly 2-random stabilising sets. We also show that the 1-generic stabilising sets are exactly the computable sets.

Authors:Grigory K. Olkhovikov Abstract: Publication date: Available online 9 March 2017 Source:Annals of Pure and Applied Logic Author(s): Grigory K. Olkhovikov The paper continues the line of [6], [7], and [8]. This results in a model-theoretic characterization of expressive powers of arbitrary finite sets of guarded connectives of degree not exceeding 1 and regular connectives of degree 2 over the language of bounded lattices.

Authors:Daniel Perrucci; Marie-Françoise Roy Abstract: Publication date: Available online 8 March 2017 Source:Annals of Pure and Applied Logic Author(s): Daniel Perrucci, Marie-Françoise Roy We describe a new quantifier elimination algorithm for real closed fields based on Thom encoding and sign determination. The complexity of this algorithm is elementary recursive and its proof of correctness is completely algebraic. In particular, the notion of connected components of semialgebraic sets is not used.

Authors:Will Boney; Rami Grossberg Abstract: Publication date: Available online 2 March 2017 Source:Annals of Pure and Applied Logic Author(s): Will Boney, Rami Grossberg We develop a notion of forking for Galois-types in the context of Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1 Let M 0 ≺ N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M 0 , written A ⫝ M 0 N , iff for all small a ∈ A and all small N − ≺ N , we have that Galois-type of a over N − is realized in M 0 . Assuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a “big cardinal”. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [10], it is established that, if this notion is an independence notion, then it is the only one.

Authors:Dan Hathaway Abstract: Publication date: Available online 1 March 2017 Source:Annals of Pure and Applied Logic Author(s): Dan Hathaway For each a ∈ ω ω , we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show that for each Borel g : ω ω → ω ω , f a ∩ g = ∅ implies a ∈ Δ 1 1 ( c ) where c is any code for g. We generalize this theorem for g in a larger pointclass Γ. Specifically, when Γ = Δ 2 1 , a ∈ L [ c ] . Also for all n ∈ ω , when Γ = Δ 3 + n 1 , a ∈ M 1 + n ( c ) .

Authors:Carmi Merimovich Abstract: Publication date: Available online 28 February 2017 Source:Annals of Pure and Applied Logic Author(s): Carmi Merimovich The extender based Magidor-Radin forcing is being generalized to supercompact type extenders.

Authors:Somayyeh Tari Abstract: Publication date: Available online 28 February 2017 Source:Annals of Pure and Applied Logic Author(s): Somayyeh Tari Continuous extension cells, or CE-cells, are cells whose defining functions have continuous extensions on closure of their domains. An o-minimal structure has the CE-cell decomposition property if any cell decomposition has a refinement by CE-cells. If the o-minimal structure M has the CE-cell decomposition property, then it has the open cell property. In other words, every definable open set in M is a finite union of definable open cells. Here, we show that the open cell property does not imply the CE-cell decomposition property. Also, after introducing an existence of limit property, we show that the CE-cell decomposition property is equivalent to the open cell property and the existence of limit property.

Authors:Luck Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Luck Darnière We introduce topological notions of polytopes and simplexes, the latter being expected to fulfil in p-adically closed fields the function of real simplexes in the classical results of triangulation of semi-algebraic sets over real closed fields. We prove that the faces of every p-adic polytope are polytopes and that they form a rooted tree with respect to specialisation. Simplexes are then defined as polytopes whose faces tree is a chain. Our main result is a construction allowing to divide every p-adic polytope in a complex of p-adic simplexes with prescribed faces and shapes.

Authors:Yasuo Yoshinobu Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Yasuo Yoshinobu We introduce a property of posets which strengthens ( ω 1 + 1 ) -strategic closedness. This property is defined using a variation of the Banach–Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from ( ω 1 + 1 ) -operational closedness, which we introduced in our previous work, by observing which portions of MA + ( ω 1 -closed ) are preserved or destroyed under forcing over posets with either property.

Authors:Erin Caulfield Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Erin Caulfield We construct a class of finite rank multiplicative subgroups of the complex numbers such that the expansion of the real field by such a group is model-theoretically well-behaved. As an application we show that a classification of expansions of the real field by cyclic multiplicative subgroups of the complex numbers due to Hieronymi does not even extend to expansions by subgroups with two generators.

Authors:Ali Enayat; Tin Lok Wong Abstract: Publication date: Available online 2 January 2017 Source:Annals of Pure and Applied Logic Author(s): Ali Enayat, Tin Lok Wong We develop machinery to make the Arithmetized Completeness Theorem more effective in the study of many models of I Δ 0 + B Σ 1 + exp , including all countable ones, by passing on to the conservative extension WKL 0 ⁎ of I Δ 0 + B Σ 1 + exp . Our detailed study of the model theory of WKL 0 ⁎ leads to the simplification and improvement of many results in the model theory of Peano arithmetic and its fragments pertaining to the construction of various types of end extensions and initial segments.