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:A. Ivanov; B. Majcher-Iwanow Pages: 749 - 775 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): A. Ivanov, B. Majcher-Iwanow We extend the generalised model theory of H. Becker from [2] to the case of Polish G-spaces when G is an arbitrary Polish group. Our approach is inspired by logic actions of Polish groups which arise in continuous logic.

Authors:Bjørn Kjos-Hanssen; Frank Stephan; Sebastiaan A. Terwijn Pages: 804 - 823 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Bjørn Kjos-Hanssen, Frank Stephan, Sebastiaan A. Terwijn We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

Authors:Stephen Flood Pages: 824 - 839 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Stephen Flood The theory of simplicial graph decompositions studies the infinite graphs that are built from a sequence of irreducible graphs which are attached together at complete subgraphs. In this paper, we study the logical complexity of deciding if a graph is prime decomposable. A large part of this analysis involves determining which ordinals must appear in these types of decompositions. A result of Diestel says that every countable simplicial tree decomposition can be rearranged to have length at most ω. We show that no such ordinal bound can be found for the lengths of non-tree decompositions. More generally, we show that for each ordinal σ, there is a decomposable graph whose shortest simplicial decomposition has length exactly σ. Adapting this argument, we show that the index set of decomposable computable graphs DECOMP is Π 1 1 hard by showing that WO is 1-reducible to DECOMP .

Authors:Zvonko Iljazović; Lucija Validžić Pages: 840 - 859 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Zvonko Iljazović, Lucija Validžić We examine conditions under which a semicomputable set in a computable metric space contains computable points. We prove that computable points in a semicomputable set S are dense if S is a manifold (possibly with boundary) or S has the topological type of a polyhedron. Moreover, we find conditions under which a point in some set has a computable compact neighbourhood in that set. In particular, we show that a point x in a semicomputable set has a computable compact neighbourhood if x has a neighbourhood homeomorphic to Euclidean space.

Authors:Rupert Hölzl; Christopher P. Porter Pages: 860 - 886 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Rupert Hölzl, Christopher P. Porter We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2 ω , the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure μ, some sequence in this family fails to be random with respect to μ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2 ω , namely diminutive measures and trivial measures.

Authors:Murdoch J. Gabbay; Michael Gabbay Pages: 501 - 621 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Murdoch J. Gabbay, Michael Gabbay We give a semantics for the λ-calculus based on a topological duality theorem in nominal sets. A novel interpretation of λ is given in terms of adjoints, and λ-terms are interpreted absolutely as sets (no valuation is necessary).

Authors:Gabriel Conant Pages: 622 - 650 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Gabriel Conant Let R = ( R , ⊕ , ≤ , 0 ) be an algebraic structure, where ⊕ is a commutative binary operation with identity 0, and ≤ is a translation-invariant total order with least element 0. Given a distinguished subset S ⊆ R , we define the natural notion of a “generalized” R -metric space, with distances in S. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of S. We first construct an ordered additive structure S ⁎ on the space of quantifier-free 2-types consistent with the axioms of R -metric spaces with distances in S, and show that, if A is an R -metric space with distances in S, then any model of Th ( A ) logically inherits a canonical S ⁎ -metric. Our primary application of this framework concerns countable, universal, and homogeneous metric spaces, obtained as generalizations of the rational Urysohn space. We adapt previous work of Delhommé, Laflamme, Pouzet, and Sauer to fully characterize the existence of such spaces. We then fix a countable totally ordered commutative monoid R , with least element 0, and consider U R , the countable Urysohn space over R . We show that quantifier elimination for Th ( U R ) is characterized by continuity of addition in R ⁎ , which can be expressed as a first-order sentence of R in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.

Authors:Sebastien Vasey Pages: 651 - 692 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Sebastien Vasey In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness. We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: Theorem 0.1 Let K be an AEC and let LS ( K ) ≤ λ < θ be cardinals. If K has a type-full good [ λ , θ ] -frame and K is categorical in both λ and θ + , then K is categorical in all μ ∈ [ λ , θ ] . We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶ ℶ ( 2 LS ( K ) ) + to ℶ ( 2 LS ( K ) ) + assuming that the AEC is LS ( K ) -tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M ∪ { a } or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.

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:Junhua Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Junhua Yu Justification logics serve as “explicit” modal logics in a way that, formula ϕ is a modal theorem if and only if there is a justification theorem, called a realization of ϕ, gained by replacing modality occurrences in ϕ by (justification) terms with structures explicitly explaining their evidential contents. In justification logics, terms stand for justifications of (propositions expressed by) formulas, and as a kind of atomic terms, constants stand for that of (justification) axioms. Kuznets has shown that in order to realize (i.e., offer a realization of) some modal theorems, it is necessary to employ a self-referential constant, that is, a constant that stands for a justification of an axiom containing an occurrence of the constant itself. Based on existing works, including some of the author's, this paper treats the collection of modal theorems that are non-self-referentially realizable as a fragment (called non-self-referential fragment) of the modal logic, and verifies: (1) that fragment is not closed in general under modus ponens; and (2) that fragment is not “conservative” in general when going from a smaller modal logic to a larger one.

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.