Journal of Symbolic Logic Articles (Project Euclid)

The two-cardinal problem for languages of arbitrary cardinality

Thu, 05 Aug 2010 15:41 EDT

Luis Miguel Villegas Silva

Source: J. Symbolic Logic, Volume 75, Number 3, 785--801.

Abstract:
Let ℒ be a first-order language of cardinality κ ++ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ⁺,κ) ⇒ (κ ++ ,κ⁺) for this language. This problem was posed by Chang and Keisler more than twenty years ago.

Term extraction and Ramsey's theorem for pairs

Mon, 13 Aug 2012 08:49 EDT

Alexander P. Kreuzer, Ulrich Kohlenbach

Source: J. Symbolic Logic, Volume 77, Number 3, 853--895.

Abstract:
In this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA 0 + COH + Π 0 1 -CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL 0 +Π 0 1 -CP+COH is Π 0 2 -conservative over PRA. Recent work of the first author showed that Π 0 1 -CP + COH is equivalent to a weak variant of the Bolzano—Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors (RT 2 2 ) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA 0 +Σ 0 2 -IA + RT 2 2 are in T 1 . This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T 1 -bounds from proofs that use RT 2 2 . Moreover, this yields a new proof of the fact that WKL 0 +Σ 0 2 -IA + RT 2 2 is Π 0 3 -conservative over RCA 0 +Σ 0 2 -IA. The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π 0 1 -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for RT 2 2 ).

The Bernays—Schönfinkel—Ramsey class for set theory: decidability

Mon, 13 Aug 2012 08:49 EDT

Alberto Policriti, Eugenio Omodeo

Source: J. Symbolic Logic, Volume 77, Number 3, 896--918.

Abstract:
As proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays—Shönfinkel—Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.

Small representations of SL 2 in the finite Morley rank category

Mon, 13 Aug 2012 08:49 EDT

Gregory Cherlin, Adrien Deloro

Source: J. Symbolic Logic, Volume 77, Number 3, 919--933.

Abstract:
We study definable irreducible actions of SL 2 (𝕂) on an abelian group of Morley rank ≤ 3rk(𝕂) and prove they are rational representations of the group.

The tree property and the failure of the Singular Cardinal Hypothesis at ℵ ω 2

Mon, 13 Aug 2012 08:49 EDT

Dima Sinapova

Source: J. Symbolic Logic, Volume 77, Number 3, 934--946.

Abstract:
We show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω 2 +1 and the SCH fails at ℵ ω 2 .

A sound and complete axiomatization for Dynamic Topological Logic

Mon, 13 Aug 2012 08:49 EDT

David Fernández-Duque

Source: J. Symbolic Logic, Volume 77, Number 3, 947--969.

Abstract:
Dynamic Topological Logic (𝒟𝒯ℒ) is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been model-theoretic. In particular, the problem of finding a complete axiomatization for the full language of 𝒟𝒯ℒ over the class of all dynamical systems has proven to be quite elusive. Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different context. We then provide a sound axiomatization for 𝒟𝒯ℒ over this extended language, and prove that it is complete. The polyadic modality is used in an essential way in our proof.

Non-finitely axiomatisable two-dimensional modal logics

Mon, 13 Aug 2012 08:49 EDT

Agi Kurucz, Sérgio Marcelino

Source: J. Symbolic Logic, Volume 77, Number 3, 970--986.

Abstract:
We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.

Non-genericity phenomena in ordered Fraïssé classes

Mon, 13 Aug 2012 08:49 EDT

Konstantin Slutsky

Source: J. Symbolic Logic, Volume 77, Number 3, 987--1010.

Abstract:
We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the ordered rational Urysohn space.

Σ 1 1 -definability at uncountable regular cardinals

Mon, 13 Aug 2012 08:49 EDT

Philipp Lücke

Source: J. Symbolic Logic, Volume 77, Number 3, 1011--1046.

Abstract:
Let κ be an infinite cardinal. A subset of ( κ κ) n is a Σ 1 1 -subset if it is the projection p[T] of all cofinal branches through a subtree T of ( < κ κ) n+1 of height κ. We define Σ 1 k -, Π 1 k - and Δ 1 k -subsets of ( κ κ) n as usual. Given an uncountable regular cardinal κ with κ=κ < κ and an arbitrary subset A of κ κ, we show that there is a < κ-closed forcing ℛ that satisfies the κ + -chain condition and forces A to be a Δ 1 1 -subset of κ κ in every ℛ-generic extension of V. We give some applications of this result and the methods used in its proof. i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L(𝒫(κ)). ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κ κ whose graph is a Δ 1 2 -subset of κ κ× κ κ. iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T. iv) We will show that generic absoluteness for Σ 1 3 ( κ κ)-formulae (i.e., formulae with parameters which define Σ 1 3 -subsets of κ κ) under < κ-closed forcings that satisfy the κ + -chain condition is inconsistent. In another direction, we use methods from the proofs of the above results to show that Σ 1 1 - and Δ 1 1 -subsets have some useful structural properties in certain ZFC-models.

On ω-categorical, generically stable groups

Mon, 13 Aug 2012 08:49 EDT

Jan Dobrowolski, Krzysztof Krupiński

Source: J. Symbolic Logic, Volume 77, Number 3, 1047--1056.

Abstract:
We prove that each ω-categorical, generically stable group is solvable-by-finite.

On algebraic closure in pseudofinite fields

Mon, 15 Oct 2012 11:39 EDT

Özlem Beyarslan, Ehud Hrushovski

Source: J. Symbolic Logic, Volume 77, Number 4, 1057--1066.

Abstract:
We study the automorphism group of the algebraic closure of a substructure $A$ of a pseudo-finite field $F$. We show that the behavior of this group, even when $A$ is large, depends essentially on the roots of unity in $F$. For almost all completions of the theory of pseudofinite fields, we show that over $A$, algebraic closure agrees with definable closure, as soon as $A$ contains the relative algebraic closure of the prime field.

Jump degrees of torsion-free abelian groups

Mon, 15 Oct 2012 11:39 EDT

Brooke M. Andersen, Asher M. Kach, Alexander G. Melnikov, Reed Solomon

Source: J. Symbolic Logic, Volume 77, Number 4, 1067--1100.

Abstract:
We show, for each computable ordinal $\alpha$ and degree $\mathbf{a} > \mathbf{0}^{(\alpha)}$, the existence of a torsion-free abelian group with proper $\alpha^{th}$ jump degree $\mathbf{a}$.

Definable well-orders of $H(\omega _2)$ and $GCH$

Mon, 15 Oct 2012 11:39 EDT

David Asperó, Sy-David Friedman

Source: J. Symbolic Logic, Volume 77, Number 4, 1101--1121.

Abstract:
Assuming $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$, we build a partial order that forces the existence of a well—order of $H(\omega_2)$ lightface definable over $\langle H(\omega_2), \in\rangle$ and that preserves cardinal exponentiation and cofinalities.

Homogeneously Suslin sets in tame mice

Mon, 15 Oct 2012 11:39 EDT

Farmer Schlutzenberg

Source: J. Symbolic Logic, Volume 77, Number 4, 1122--1146.

Abstract:
This paper studies homogeneously Suslin (hom) sets of reals in tame mice. The following results are established: In $0$ ¶ the hom sets are precisely the $\underset{\widetilde{}}{\Pi^1_1}$ sets. In $M_n$ every hom set is correctly $\underset{\widetilde{}}{\Delta^1_{n+1}}$, and $(\delta+1)$-universally Baire where $\delta$ is the least Woodin. In $M_\omega$ every hom set is $< \lambda$-hom, where $\lambda$ is the supremum of the Woodins.

Topological differential fields and dimension functions

Mon, 15 Oct 2012 11:39 EDT

Nicolas Guzy, Françoise Point

Source: J. Symbolic Logic, Volume 77, Number 4, 1147--1164.

Abstract:
We construct a fibered dimension function in some topological differential fields.

A generalization of Sierpiński's paradoxical decompositions: Coloring semialgebraic grids

Mon, 15 Oct 2012 11:39 EDT

James H. Schmerl

Source: J. Symbolic Logic, Volume 77, Number 4, 1165--1183.

Abstract:
A structure ${\mathcal A} = (A;E_0,E_1, \dots, E_{n-1})$ is an $n$- grid if each $E_i$ is an equivalence relation on $A$ and whenever $X$ and $Y$ are equivalence classes of, respectively, distinct $E_i$ and $E_j$, then $X \cap Y$ is finite. A coloring $\chi \colon A \longrightarrow n$ is {\it acceptable} if whenever $X$ is an equivalence class of $E_i$, then $\{x \in X \colon \chi(x) = i\}$ is finite. If $B$ is any set, then the $n$-cube $B^n = (B^n;E_0,E_1, \dots, E_{n-1})$ is considered as an $n$-grid, where the equivalence classes of $E_i$ are the lines parallel to the $i$-th coordinate axis. Kuratowski [9], generalizing the $n=3$ case proved by Sierpiński [17], proved that $\mathbb{R}^n$ has an acceptable coloring iff $2^{\aleph_0} \leq \aleph_{n-2}$. The main result is: if ${\mathcal A}$ is a semialgebraic (i.e., first-order definable in the field of reals) $n$-grid, then the following are equivalent: (1) if ${\mathcal A}$ embeds all finite $n$-cubes, then $2^{\aleph_0} \leq \aleph_{n-2}$; (2) if ${\mathcal A}$ embeds $\mathbb{R}^n$, then $2^{\aleph_0} \leq \aleph_{n-2}$; (3) ${\mathcal A}$ has an acceptable coloring.

Interpreting true arithmetic in the local structure of the enumeration degrees

Mon, 15 Oct 2012 11:39 EDT

Hristo Ganchev, Mariya Soskova

Source: J. Symbolic Logic, Volume 77, Number 4, 1184--1194.

Abstract:
We show that the theory of the local structure of the enumeration degrees is computably isomorphic to the theory of first order arithmetic. We introduce a novel coding method, using the notion of a $\mathcal{K}$-pair, to code a large class of countable relations.

The range property fails for H

Mon, 15 Oct 2012 11:39 EDT

Andrew Polonsky

Source: J. Symbolic Logic, Volume 77, Number 4, 1195--1210.

Abstract:
We work in $\lambda{\mathcal{H}}$, the untyped $\lambda$-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a $\lambda$-term $\Xi$ such that $\Xi M=\Xi {\mathtt I} \iff \Xi M\neq \Xi \Omega$. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.

Undecidability of representability as binary relations

Mon, 15 Oct 2012 11:39 EDT

Robin Hirsch, Marcel Jackson

Source: J. Symbolic Logic, Volume 77, Number 4, 1211--1244.

Abstract:
In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

On Kueker's conjecture

Mon, 15 Oct 2012 11:39 EDT

Predrag Tanović

Source: J. Symbolic Logic, Volume 77, Number 4, 1245--1256.

Abstract:
We prove that a Kueker theory with infinite $dcl(\emptyset)$ does not have the strict order property and that strongly minimal types are dense: any non-algebraic formula is contained in a strongly minimal type.

Forking in VC-minimal theories

Mon, 15 Oct 2012 11:39 EDT

Sarah Cotter, Sergei Starchenko

Source: J. Symbolic Logic, Volume 77, Number 4, 1257--1271.

Abstract:
We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model $M$ is equivalent to containment in global types definable over $M$, generalizing a result of Dolich on o-minimal theories in [4].

Reverse mathematics and a Ramsey-type König's Lemma

Mon, 15 Oct 2012 11:39 EDT

Stephen Flood

Source: J. Symbolic Logic, Volume 77, Number 4, 1272--1280.

Abstract:
In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

Kurepa trees and Namba forcing

Mon, 15 Oct 2012 11:39 EDT

Bernhard König, Yasuo Yoshinobu

Source: J. Symbolic Logic, Volume 77, Number 4, 1281--1290.

Abstract:
We show that strongly compact cardinals and MM are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding ‘regressive' $\lambda$-Kurepa trees in either case. We argue that the destruction of regressive Kurepa trees requires a non-standard application of MM. As a corollary, we find a consistent example of an $\omega_2$-closed poset that is not forcing equivalent to any $\omega_2$-directed-closed poset.

Finitely generated free Heyting algebras: the well-founded initial segment

Mon, 15 Oct 2012 11:39 EDT

R. Elageili, J. K. Truss

Source: J. Symbolic Logic, Volume 77, Number 4, 1291--1307.

Abstract:
In this paper we describe the well-founded initial segment of the free Heyting algebra ${\mathbb A}_\alpha$ on finitely many, $\alpha$, generators. We give a complete classification of initial sublattices of ${\mathbb A}_2$ isomorphic to ${\mathbb A}_1$ (called ‘low ladders'), and prove that for $2 \le \alpha < \omega$, the height of the well-founded initial segment of ${\mathbb A}_\alpha$ is $\omega^2$.

A constructive Galois connection between closure and interior

Mon, 15 Oct 2012 11:39 EDT

Francesco Ciraulo, Giovanni Sambin

Source: J. Symbolic Logic, Volume 77, Number 4, 1308--1324.

Abstract:
We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

Simultaneous reflection and impossible ideals

Mon, 15 Oct 2012 11:39 EDT

Todd Eisworth

Source: J. Symbolic Logic, Volume 77, Number 4, 1325--1338.

Abstract:
We prove that if $\mu^+\rightarrow[\mu^+]^2_{\mu^+}$ holds for a singular cardinal $\mu$, then any collection of fewer than $cf(\mu)$ stationary subsets of $\mu^+$ must reflect simultaneously.

Getting more colors I

Wed, 23 Jan 2013 09:25 EST

Todd Eisworth

Source: J. Symbolic Logic, Volume 78, Number 1, 1--16.

Abstract:
We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta < \mu$.

Getting more colors II

Wed, 23 Jan 2013 09:25 EST

Todd Eisworth

Source: J. Symbolic Logic, Volume 78, Number 1, 17--38.

Abstract:
We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,cf(\mu))$ for singular $\mu$.

On the failure of BD-ࡃ and BD, and an application to the anti-Specker property

Wed, 23 Jan 2013 09:25 EST

Robert S. Lubarsky

Source: J. Symbolic Logic, Volume 78, Number 1, 39--56.

Abstract:
We give the natural topological model for $\neg$BD-${\mathbb N}$, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-${\mathbb N}$. Also, the natural topological model for $\neg$BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-$\mathbb{N}$, it is brought out in detail how BD-$\mathbb N$ fails.

Some jump-like operations in $\mathbf \beta $-recursion theory.

Wed, 23 Jan 2013 09:25 EST

Colin G. Bailey

Source: J. Symbolic Logic, Volume 78, Number 1, 57--71.

Abstract:
In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

Fields with few types

Wed, 23 Jan 2013 09:25 EST

Cédric Milliet

Source: J. Symbolic Logic, Volume 78, Number 1, 72--84.

Abstract:
According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic $2$ is a field.

A minimal Prikry-type forcing for singularizing a measurable cardinal

Wed, 23 Jan 2013 09:25 EST

Peter Koepke, Karen Räsch, Philipp Schlicht

Source: J. Symbolic Logic, Volume 78, Number 1, 85--100.

Abstract:
Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures.

Canonizing relations on nonsmooth sets

Wed, 23 Jan 2013 09:25 EST

Clinton T. Conley

Source: J. Symbolic Logic, Volume 78, Number 1, 101--112.

Abstract:
We show that any symmetric, Baire measurable function from the complement of $\ezero$ to a finite set is constant on an $\ezero$-nonsmooth square. A simultaneous generalization of Galvin's theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on $E_0$-nonsmooth sets, this result is proved by relating $\ezero$-nonsmooth sets to embeddings of the complete binary tree into itself and appealing to a version of Hindman's theorem on the complete binary tree. We also establish several canonization theorems which follow from the main result.

Indifferent sets for genericity

Wed, 23 Jan 2013 09:25 EST

Adam R. Day

Source: J. Symbolic Logic, Volume 78, Number 1, 113--138.

Abstract:
This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in $\notGLII$ which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, $\Delta^0_2$ 1-generic sets can compute an indifferent set for themselves. We show that all $\Delta^0_2$ weakly 1-generic sets can compute an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented.

Pointwise definable models of set theory

Wed, 23 Jan 2013 09:25 EST

Joel David Hamkins, David Linetsky, Jonas Reitz

Source: J. Symbolic Logic, Volume 78, Number 1, 139--156.

Abstract:
A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

P-ideal dichotomy and weak squares

Wed, 23 Jan 2013 09:25 EST

Dilip Raghavan

Source: J. Symbolic Logic, Volume 78, Number 1, 157--167.

Abstract:
We answer a question of Cummings and Magidor by proving that the P-ideal dichotomy of Todorčević refutes ${\square}_{\kappa, \omega}$ for any uncountable $\kappa$. We also show that the P-ideal dichotomy implies the failure of ${\square}_{\kappa, < \mathfrak{b}}$ provided that $cf(\kappa) > {\omega}_{1}$.

Borel's conjecture in topological groups

Wed, 23 Jan 2013 09:25 EST

Fred Galvin, Marion Scheepers

Source: J. Symbolic Logic, Volume 78, Number 1, 168--184.

Abstract:
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. 2. If it is consistent that ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal. 3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent. 4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$. 5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality.

Mutually algebraic structures and expansions by predicates

Wed, 23 Jan 2013 09:25 EST

Michael C. Laskowski

Source: J. Symbolic Logic, Volume 78, Number 1, 185--194.

Abstract:
We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion $(M,A)$ by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct, and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.

Random reals, the rainbow Ramsey theorem, and arithmetic conservation

Wed, 23 Jan 2013 09:25 EST

Chris J. Conidis, Theodore A. Slaman

Source: J. Symbolic Logic, Volume 78, Number 1, 195--206.

Abstract:
We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$.

An analogue of the Baire category theorem

Wed, 23 Jan 2013 09:25 EST

Philipp Hieronymi

Source: J. Symbolic Logic, Volume 78, Number 1, 207--213.

Abstract:
Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.

On the decidability of implicational ticket entailment

Wed, 23 Jan 2013 09:25 EST

Katalin Bimbó, J. Michael Dunn

Source: J. Symbolic Logic, Volume 78, Number 1, 214--236.

Abstract:
The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable . Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$.

Universal sets for pointsets properly on the n th level of the projective hierarchy

Wed, 23 Jan 2013 09:25 EST

Greg Hjorth, Leigh Humphries, Arnold W. Miller

Source: J. Symbolic Logic, Volume 78, Number 1, 237--244.

Abstract:
The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.

Model theoretic connected components of finitely generated nilpotent groups

Wed, 23 Jan 2013 09:25 EST

Nathan Bowler, Cong Chen, Jakub Gismatullin

Source: J. Symbolic Logic, Volume 78, Number 1, 245--259.

Abstract:
We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

Atomic polymorphism

Wed, 23 Jan 2013 09:25 EST

Fernando Ferreira, Gilda Ferreira

Source: J. Symbolic Logic, Volume 78, Number 1, 260--274.

Abstract:
It has been known for six years that the restriction of Girard's polymorphic system $\text{\bfseries\upshape F}$ to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each $\beta$-reduction step of the full intuitionistic propositional calculus translates into one or more $\beta\eta$-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to $\eta$-conversions.

Unbounded and dominating reals in Hechler extensions

Wed, 23 Jan 2013 09:25 EST

Justin Palumbo

Source: J. Symbolic Logic, Volume 78, Number 1, 275--289.

Abstract:
We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real with two ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and Löwe (Conjecture 15 of [4]). We also answer a question due to Laflamme.

Centralisateurs génériques

Wed, 23 Jan 2013 09:25 EST

Bruno Poizat

Source: J. Symbolic Logic, Volume 78, Number 1, 290--306.

Abstract:
We comment on an early and inspiring remark of an Omskian mathematician concerning the Cherlin—Zilber Conjecture, meeting in passing some well-known properties of algebraic groups whose generalization to arbitrary groups of finite Morley rank seems to be very uncertain. This paper assumes a familiarity with the model theoretic tools involved in the study of the groups of finite Morley rank.

Characterizing quantifier extensions of dependence logic

Wed, 23 Jan 2013 09:25 EST

Fredrik Engström, Juha Kontinen

Source: J. Symbolic Logic, Volume 78, Number 1, 307--316.

Abstract:
We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.

Strong tree properties for small cardinals

Wed, 23 Jan 2013 09:25 EST

Laura Fontanella

Source: J. Symbolic Logic, Volume 78, Number 1, 317--333.

Abstract:
An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.

Uniform distribution and algorithmic randomness

Wed, 23 Jan 2013 09:25 EST

Jeremy Avigad

Source: J. Symbolic Logic, Volume 78, Number 1, 334--344.

Abstract:
A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random . Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

Satisfaction relations for proper classes: applications in logic and set theory

Wed, 15 May 2013 10:11 EDT

Robert A. Van Wesep

Source: J. Symbolic Logic, Volume 78, Number 2, 345--368.

Abstract:
We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate ($\models^*$) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension $\Theta$ of ZF there is a finitely axiomatizable extension $\Theta'$ of GB that is a conservative extension of $\Theta$. We also prove a conservative extension result that justifies the use of $\models^*$ to characterize ground models for forcing constructions.

Extensions of ordered theories by generic predicates

Wed, 15 May 2013 10:11 EDT

Alfred Dolich, Chris Miller, Charles Steinhorn

Source: J. Symbolic Logic, Volume 78, Number 2, 369--387.

Decidability for some justification logics with negative introspection

Wed, 15 May 2013 10:11 EDT

Thomas Studer

Source: J. Symbolic Logic, Volume 78, Number 2, 388--402.

Abstract:
Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

Canonical measure assignments

Wed, 15 May 2013 10:11 EDT

Steve Jackson, Benedikt Löwe

Source: J. Symbolic Logic, Volume 78, Number 2, 403--424.

Abstract:
We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal $\kappa < \aleph_{\varepsilon_0}$ in a recursive definition of a canonical measure assignment . We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).

A fixed point for the jump operator on structures

Wed, 15 May 2013 10:11 EDT

Antonio Montalbán

Source: J. Symbolic Logic, Volume 78, Number 2, 425--438.

Abstract:
Assuming that $0^\#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $\mathcal A$ such that \[ \textit{Sp}({\mathcal A}) = \{{\bf x}'\colon {\bf x}\in \textit{Sp}({\mathcal A})\}, \] where $\textit{Sp}({\mathcal A})$ is the set of Turing degrees which compute a copy of $\mathcal A$. More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.

Borel reductions and cub games in generalised descriptive set theory

Wed, 15 May 2013 10:11 EDT

Vadim Kulikov

Source: J. Symbolic Logic, Volume 78, Number 2, 439--458.

Abstract:
It is shown that the power set of $\kappa$ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on $2^\kappa$ under Borel reducibility. Here $\kappa$ is an uncountable regular cardinal with $\kappa^{<\kappa}=\kappa$.

Partial impredicativity in reverse mathematics

Wed, 15 May 2013 10:11 EDT

Henry Towsner

Source: J. Symbolic Logic, Volume 78, Number 2, 459--488.

Abstract:
In reverse mathematics, it is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on how to weaken the assumption while preserving the conclusion (other than reducing all the way to the tautology of assuming the conclusion). A main cause of this phenomenon is the proof of a $\Pi^1_2$ sentence from the theory $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$. Using methods based on the functional interpretation, we introduce a family of weakenings of $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$ and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.

Ample thoughts

Wed, 15 May 2013 10:11 EDT

Daniel Palacín, Frank O. Wagner

Source: J. Symbolic Logic, Volume 78, Number 2, 489--510.

Abstract:
Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

Nonexistence of minimal pairs for generic computability

Wed, 15 May 2013 10:11 EDT

Gregory Igusa

Source: J. Symbolic Logic, Volume 78, Number 2, 511--522.

Abstract:
A generic computation of a subset $A$ of $\mathbb{N}$ consists of a computation that correctly computes most of the bits of $A$, and never incorrectly computes any bits of $A$, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.

Unexpected imaginaries in valued fields with analytic structure

Wed, 15 May 2013 10:11 EDT

Deirdre Haskell, Ehud Hrushovski, Dugald Macpherson

Source: J. Symbolic Logic, Volume 78, Number 2, 523--542.

Abstract:
We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

Models of transfinite provability logic

Wed, 15 May 2013 10:11 EDT

David Fernández-Duque, Joost J. Joosten

Source: J. Symbolic Logic, Volume 78, Number 2, 543--561.

Abstract:
For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary $\Lambda$. More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough.

On colimits and elementary embeddings

Wed, 15 May 2013 10:11 EDT

Joan Bagaria, Andrew Brooke-Taylor

Source: J. Symbolic Logic, Volume 78, Number 2, 562--578.

Abstract:
We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as $\alpha$-strongly compact and $C^{(n)}$-extendible cardinals.

Probabilistic algorithmic randomness

Wed, 15 May 2013 10:11 EDT

Sam Buss, Mia Minnes

Source: J. Symbolic Logic, Volume 78, Number 2, 579--601.

Abstract:
We introduce martingales defined by probabilistic strategies, in which randomness is used to decide whether to bet. We show that different criteria for the success of computable probabilistic strategies can be used to characterize ML-randomness, computable randomness, and partial computable randomness. Our characterization of ML-randomness partially addresses a critique of Schnorr by formulating ML randomness in terms of a computable process rather than a computably enumerable function.

Independence, dimension and continuity in non-forking frames

Wed, 15 May 2013 10:11 EDT

Adi Jarden, Alon Sitton

Source: J. Symbolic Logic, Volume 78, Number 2, 602--632.

Abstract:
The notion $J$ is independent in $(M,M_0,N)$ was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal $\lambda$ and has a non-forking relation, satisfying the good $\lambda$-frame axioms and some additional hypotheses. Shelah uses independence to define dimension. Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved. As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

A quasi-order on continuous functions

Wed, 15 May 2013 10:11 EDT

Raphaël Carroy

Source: J. Symbolic Logic, Volume 78, Number 2, 633--648.

Abstract:
We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

On the definability of radicals in supersimple groups

Wed, 15 May 2013 10:11 EDT

Cé{d}ric Milliet

Source: J. Symbolic Logic, Volume 78, Number 2, 649--656.

Abstract:
If $G$ is a group with a supersimple theory having a finite $SU$-rank, then the subgroup of $G$ generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If $H$ is any group with a supersimple theory, then the subgroup of $H$ generated by all of its normal soluble subgroups is definable and soluble.

Topological dynamics and definable groups

Wed, 15 May 2013 10:11 EDT

Anand Pillay

Source: J. Symbolic Logic, Volume 78, Number 2, 657--666.

Abstract:
We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of $G(M)$ for measure-stable groups. We also point out that for $G$ definably amenable (in a NIP theory) $G/G^{00}$ can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global $f$-generic types.

On skinny stationary subsets of $\mathcal {P}_\kappa \lambda $

Wed, 15 May 2013 10:11 EDT

Yo Matsubara, Toschimichi Usuba

Source: J. Symbolic Logic, Volume 78, Number 2, 667--680.

Abstract:
We introduce the notion of skinniness for subsets of $\mathcal{P}_\kappa \lambda$ and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or $2^\lambda$-saturation of $\mathrm{NS}_{\kappa\lambda}\mid X$, where $\mathrm{NS}_{\kappa\lambda}$ denotes the non-stationary ideal over $\mathcal{P}_\kappa \lambda$, implies the existence of a skinny stationary subset of $X$. We also show that if $\lambda$ is a singular cardinal, then there is no skinnier stationary subset of $\mathcal{P}_\kappa \lambda$. Furthermore, if $\lambda$ is a strong limit singular cardinal, there is no skinny stationary subset of $\mathcal{P}_\kappa \lambda$. Combining these results, we show that if $\lambda$ is a strong limit singular cardinal, then $\mathrm{NS}_{\kappa\lambda}\mid X$ can satisfy neither precipitousness nor $2^\lambda$-saturation for every stationary $X \subseteq \mathcal{P}_\kappa \lambda$. We also indicate that $\diamondsuit_\lambda(E^{\lambda}_{<\kappa})$, where $E^{\lambda}_{<\kappa} \stackrel{\mathrm{def}}{=} \{\alpha < \lambda \mid \mathrm{cf}(\alpha) < \kappa\}$, is equivalent to the existence of a skinnier (or skinniest) stationary subset of $\mathcal{P}_\kappa \lambda$ under some cardinal arithmetical hypotheses.