Journal of Symbolic Logic Articles (Project Euclid)
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The latest articles from Journal of Symbolic Logic on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 19 May 2011 09:13 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The two-cardinal problem for languages of arbitrary cardinality
http://projecteuclid.org/euclid.jsl/1278682200
<strong>Luis Miguel Villegas Silva</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 75, Number 3, 785--801.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1278682200_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
On Kueker's conjecture
http://projecteuclid.org/euclid.jsl/1350315585<strong>Predrag Tanović</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1245--1256.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1350315585_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Forking in VC-minimal theories
http://projecteuclid.org/euclid.jsl/1350315586<strong>Sarah Cotter</strong>, <strong>Sergei Starchenko</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1257--1271.</p><p><strong>Abstract:</strong><br/>
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].
</p>projecteuclid.org/euclid.jsl/1350315586_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Reverse mathematics and a Ramsey-type König's Lemma
http://projecteuclid.org/euclid.jsl/1350315587<strong>Stephen Flood</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1272--1280.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1350315587_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Kurepa trees and Namba forcing
http://projecteuclid.org/euclid.jsl/1350315588<strong>Bernhard König</strong>, <strong>Yasuo Yoshinobu</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1281--1290.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1350315588_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Finitely generated free Heyting algebras: the well-founded initial segment
http://projecteuclid.org/euclid.jsl/1350315589<strong>R. Elageili</strong>, <strong>J. K. Truss</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1291--1307.</p><p><strong>Abstract:</strong><br/>
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 $\omega^2$.
</p>projecteuclid.org/euclid.jsl/1350315589_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
A constructive Galois connection between closure and interior
http://projecteuclid.org/euclid.jsl/1350315590<strong>Francesco Ciraulo</strong>, <strong>Giovanni Sambin</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1308--1324.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1350315590_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Simultaneous reflection and impossible ideals
http://projecteuclid.org/euclid.jsl/1350315591<strong>Todd Eisworth</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 77, Number 4, 1325--1338.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1350315591_Mon, 15 Oct 2012 11:39 EDTMon, 15 Oct 2012 11:39 EDT
Getting more colors I
http://projecteuclid.org/euclid.jsl/1358951096<strong>Todd Eisworth</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 1--16.</p><p><strong>Abstract:</strong><br/>
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 $\theta
< \mu$.
</p>projecteuclid.org/euclid.jsl/1358951096_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Getting more colors II
http://projecteuclid.org/euclid.jsl/1358951097<strong>Todd Eisworth</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 17--38.</p><p><strong>Abstract:</strong><br/>
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 $\mu$.
</p>projecteuclid.org/euclid.jsl/1358951097_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
On the failure of BD-ࡃ and BD, and an application to the anti-Specker property
http://projecteuclid.org/euclid.jsl/1358951098<strong>Robert S. Lubarsky</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 39--56.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951098_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Some jump-like operations in $\mathbf \beta $-recursion theory.
http://projecteuclid.org/euclid.jsl/1358951099<strong>Colin G. Bailey</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 57--71.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951099_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Fields with few types
http://projecteuclid.org/euclid.jsl/1358951100<strong>Cédric Milliet</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 72--84.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951100_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
A minimal Prikry-type forcing for singularizing a measurable cardinal
http://projecteuclid.org/euclid.jsl/1358951101<strong>Peter Koepke</strong>, <strong>Karen Räsch</strong>, <strong>Philipp Schlicht</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 85--100.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951101_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Canonizing relations on nonsmooth sets
http://projecteuclid.org/euclid.jsl/1358951102<strong>Clinton T. Conley</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 101--112.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951102_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Indifferent sets for genericity
http://projecteuclid.org/euclid.jsl/1358951103<strong>Adam R. Day</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 113--138.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951103_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Pointwise definable models of set theory
http://projecteuclid.org/euclid.jsl/1358951104<strong>Joel David Hamkins</strong>, <strong>David Linetsky</strong>, <strong>Jonas Reitz</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 139--156.</p><p><strong>Abstract:</strong><br/>
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>projecteuclid.org/euclid.jsl/1358951104_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
P-ideal dichotomy and weak squares
http://projecteuclid.org/euclid.jsl/1358951105<strong>Dilip Raghavan</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 157--167.</p><p><strong>Abstract:</strong><br/>
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 $cf(\kappa) > {\omega}_{1}$.
</p>projecteuclid.org/euclid.jsl/1358951105_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Borel's conjecture in topological groups
http://projecteuclid.org/euclid.jsl/1358951106<strong>Fred Galvin</strong>, <strong>Marion Scheepers</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 168--184.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951106_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Mutually algebraic structures and expansions by predicates
http://projecteuclid.org/euclid.jsl/1358951107<strong>Michael C. Laskowski</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 185--194.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951107_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Random reals, the rainbow Ramsey theorem, and arithmetic conservation
http://projecteuclid.org/euclid.jsl/1358951108<strong>Chris J. Conidis</strong>, <strong>Theodore A. Slaman</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 195--206.</p><p><strong>Abstract:</strong><br/>
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$}_2$.
</p>projecteuclid.org/euclid.jsl/1358951108_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
An analogue of the Baire category theorem
http://projecteuclid.org/euclid.jsl/1358951109<strong>Philipp Hieronymi</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 207--213.</p><p><strong>Abstract:</strong><br/>
Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.
</p>projecteuclid.org/euclid.jsl/1358951109_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
On the decidability of implicational ticket entailment
http://projecteuclid.org/euclid.jsl/1358951110<strong>Katalin Bimbó</strong>, <strong>J. Michael Dunn</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 214--236.</p><p><strong>Abstract:</strong><br/>
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 $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$.
</p>projecteuclid.org/euclid.jsl/1358951110_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Universal sets for pointsets properly on the n th level of the projective hierarchy
http://projecteuclid.org/euclid.jsl/1358951111<strong>Greg Hjorth</strong>, <strong>Leigh Humphries</strong>, <strong>Arnold W. Miller</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 237--244.</p><p><strong>Abstract:</strong><br/>
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 $\alpha$.
</p>projecteuclid.org/euclid.jsl/1358951111_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Model theoretic connected components of finitely generated nilpotent groups
http://projecteuclid.org/euclid.jsl/1358951112<strong>Nathan Bowler</strong>, <strong>Cong Chen</strong>, <strong>Jakub Gismatullin</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 245--259.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951112_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Atomic polymorphism
http://projecteuclid.org/euclid.jsl/1358951113<strong>Fernando Ferreira</strong>, <strong>Gilda Ferreira</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 260--274.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951113_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Unbounded and dominating reals in Hechler extensions
http://projecteuclid.org/euclid.jsl/1358951114<strong>Justin Palumbo</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 275--289.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951114_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Centralisateurs génériques
http://projecteuclid.org/euclid.jsl/1358951115<strong>Bruno Poizat</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 290--306.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951115_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Characterizing quantifier extensions of dependence logic
http://projecteuclid.org/euclid.jsl/1358951116<strong>Fredrik Engström</strong>, <strong>Juha Kontinen</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 307--316.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951116_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Strong tree properties for small cardinals
http://projecteuclid.org/euclid.jsl/1358951117<strong>Laura Fontanella</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 317--333.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951117_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Uniform distribution and algorithmic randomness
http://projecteuclid.org/euclid.jsl/1358951118<strong>Jeremy Avigad</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 1, 334--344.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1358951118_Wed, 23 Jan 2013 09:25 ESTWed, 23 Jan 2013 09:25 EST
Satisfaction relations for proper classes: applications in logic and set theory
http://projecteuclid.org/euclid.jsl/1368627054<strong>Robert A. Van Wesep</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 345--368.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627054_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Extensions of ordered theories by generic predicates
http://projecteuclid.org/euclid.jsl/1368627055<strong>Alfred Dolich</strong>, <strong>Chris Miller</strong>, <strong>Charles Steinhorn</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 369--387.</p>projecteuclid.org/euclid.jsl/1368627055_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Decidability for some justification logics with negative introspection
http://projecteuclid.org/euclid.jsl/1368627056<strong>Thomas Studer</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 388--402.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627056_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Canonical measure assignments
http://projecteuclid.org/euclid.jsl/1368627057<strong>Steve Jackson</strong>, <strong>Benedikt Löwe</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 403--424.</p><p><strong>Abstract:</strong><br/>
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).
</p>projecteuclid.org/euclid.jsl/1368627057_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
A fixed point for the jump operator on structures
http://projecteuclid.org/euclid.jsl/1368627058<strong>Antonio Montalbán</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 425--438.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627058_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Borel reductions and cub games in generalised descriptive set theory
http://projecteuclid.org/euclid.jsl/1368627059<strong>Vadim Kulikov</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 439--458.</p><p><strong>Abstract:</strong><br/>
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 $\kappa^{<\kappa}=\kappa$.
</p>projecteuclid.org/euclid.jsl/1368627059_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Partial impredicativity in reverse mathematics
http://projecteuclid.org/euclid.jsl/1368627060<strong>Henry Towsner</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 459--488.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627060_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Ample thoughts
http://projecteuclid.org/euclid.jsl/1368627061<strong>Daniel Palacín</strong>, <strong>Frank O. Wagner</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 489--510.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627061_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Nonexistence of minimal pairs for generic computability
http://projecteuclid.org/euclid.jsl/1368627062<strong>Gregory Igusa</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 511--522.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627062_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Unexpected imaginaries in valued fields with analytic structure
http://projecteuclid.org/euclid.jsl/1368627063<strong>Deirdre Haskell</strong>, <strong>Ehud Hrushovski</strong>, <strong>Dugald Macpherson</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 523--542.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627063_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Models of transfinite provability logic
http://projecteuclid.org/euclid.jsl/1368627064<strong>David Fernández-Duque</strong>, <strong>Joost J. Joosten</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 543--561.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627064_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
On colimits and elementary embeddings
http://projecteuclid.org/euclid.jsl/1368627065<strong>Joan Bagaria</strong>, <strong>Andrew Brooke-Taylor</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 562--578.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627065_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Probabilistic algorithmic randomness
http://projecteuclid.org/euclid.jsl/1368627066<strong>Sam Buss</strong>, <strong>Mia Minnes</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 579--601.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627066_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Independence, dimension and continuity in non-forking frames
http://projecteuclid.org/euclid.jsl/1368627067<strong>Adi Jarden</strong>, <strong>Alon Sitton</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 602--632.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627067_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
A quasi-order on continuous functions
http://projecteuclid.org/euclid.jsl/1368627068<strong>Raphaël Carroy</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 633--648.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627068_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
On the definability of radicals in supersimple groups
http://projecteuclid.org/euclid.jsl/1368627069<strong>Cé{d}ric Milliet</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 649--656.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627069_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
Topological dynamics and definable groups
http://projecteuclid.org/euclid.jsl/1368627070<strong>Anand Pillay</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 657--666.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627070_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
On skinny stationary subsets of $\mathcal {P}_\kappa \lambda $
http://projecteuclid.org/euclid.jsl/1368627071<strong>Yo Matsubara</strong>, <strong>Toschimichi Usuba</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 2, 667--680.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsl/1368627071_Wed, 15 May 2013 10:11 EDTWed, 15 May 2013 10:11 EDT
On the optimality of conservation results for local reflection in arithmetic
http://projecteuclid.org/euclid.jsl/1388953992<strong>A. Cordón-Franco</strong>, <strong>A. Fernández-Margarit</strong>, <strong>F. F. Lara-Martín</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1025--1035.</p><p><strong>Abstract:</strong><br/>
Let $T$ be a recursively
enumerable theory extending Elementary Arithmetic $\textup{EA}$. L. D. Beklemishev
proved that the $\Sigma_2$ local reflection principle
for $T$, $\mathsf{Rfn}_{\Sigma_2}(T)$, is conservative over the
$\Sigma_1$ local reflection principle, $\mathsf{Rfn}_{\Sigma_1}(T)$,
with respect to boolean combinations of $\Sigma_1$-sentences; and
asked whether this result is best possible. In this work we answer
Beklemishev's question by showing that $\Pi_2$-sentences are not
conserved for $T = \textup{EA}{}+{}\textit{\mbox{“$f$ is total}}$,” where $f$
is any nondecreasing computable function with elementary graph. We also
discuss how this
result generalizes to $n > 0$ and obtain as an application that for
$n > 0$, $I\Pi_{n+1}^-$ is conservative over $I\Sigma_n$ with
respect to $\Pi_{n+2}$-sentences.
</p>projecteuclid.org/euclid.jsl/1388953992_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Decidability of definability
http://projecteuclid.org/euclid.jsl/1388953993<strong>Manuel Bodirsky</strong>, <strong>Michael Pinsker</strong>, <strong>Todor Tsankov</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1036--1054.</p><p><strong>Abstract:</strong><br/>
For a fixed countably infinite structure $\Gamma$ with finite relational signature $\tau$,
we study the following computational problem: input are quantifier-free $\tau$-formulas $\phi_0,\phi_1,\dots,\phi_n$
that define relations $R_0,R_1,\dots,R_n$ over $\Gamma$. The question is whether the relation $R_0$ is primitive positive definable from $R_1,\dots,R_n$, i.e., definable by a first-order formula
that uses only relation symbols for $R_1, \dots, R_n$, equality,
conjunctions, and existential quantification (disjunction, negation,
and universal quantification are forbidden).
We show decidability of this problem for
all structures $\Gamma$ that have a first-order definition in an ordered homogeneous structure $\Delta$ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures $\Gamma$ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal $C$-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability.
Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.
</p>projecteuclid.org/euclid.jsl/1388953993_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Computably isometric spaces
http://projecteuclid.org/euclid.jsl/1388953994<strong>Alexander G. Melnikov</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1055--1085.</p><p><strong>Abstract:</strong><br/>
We say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space $ \mathcal{C}[0,1]$ of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of $\mathbb{R}^n$, and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.
</p>projecteuclid.org/euclid.jsl/1388953994_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Homology groups of types in model theory and the computation of $H_2(p)$
http://projecteuclid.org/euclid.jsl/1388953995<strong>John Goodrick</strong>, <strong>Byunghan Kim</strong>, <strong>Alexei Kolesnikov</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1086--1114.</p><p><strong>Abstract:</strong><br/>
We present definitions of homology groups $H_n(p)$, $n\ge 0$, associated to a complete type $p$. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group $H_2(p)$ for strong types in stable theories and show that any profinite abelian group can occur as the group $H_2(p)$.
</p>projecteuclid.org/euclid.jsl/1388953995_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
The determinacy of context-free games
http://projecteuclid.org/euclid.jsl/1388953996<strong>Olivier Finkel</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1115--1134.</p><p><strong>Abstract:</strong><br/>
We prove that the determinacy of Gale-Stewart games whose winning sets
are accepted by real-time 1-counter Büchi automata is equivalent
to the determinacy of (effective) analytic Gale-Stewart games which is
known to be a large cardinal assumption. We show also that the
determinacy of Wadge games between two players in charge of
$\omega$-languages accepted by 1-counter Büchi automata is equivalent
to the (effective) analytic Wadge determinacy. Using some results of
set theory we prove that one can effectively construct a 1-counter
Büchi automaton $\mathcal{A}$ and a Büchi automaton $\mathcal{B}$
such that: (1) There exists a model of ZFC in which Player 2 has a
winning strategy in the Wadge game $W(L(\mathcal{A}),
L(\mathcal{B}))$; (2) There exists a model of ZFC in which the Wadge
game $W(L(\mathcal{A}), L(\mathcal{B}))$ is not determined. Moreover
these are the only two possibilities, i.e. there are no models of ZFC
in which Player 1 has a winning strategy in the Wadge game
$W(L(\mathcal{A}), L(\mathcal{B}))$.
</p>projecteuclid.org/euclid.jsl/1388953996_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Friedberg numbering in fragments of Peano Arithmetic and $\alpha$-recursion theory
http://projecteuclid.org/euclid.jsl/1388953997<strong>Wei Li</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1135--1163.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the existence of a Friedberg
numbering in fragments of Peano Arithmetic and initial segments
of Gödel's constructible hierarchy $L_\alpha$, where $\alpha$
is $\Sigma_1$ admissible. We prove that
(1) Over $P^-+B\Sigma_2$, the existence of a Friedberg numbering
is equivalent to $I\Sigma_2$, and
(2) For $L_\alpha$, there is a Friedberg numbering if and only
if the tame $\Sigma_2$ projectum of $\alpha$ equals the $\Sigma_2$
cofinality of $\alpha$.
</p>projecteuclid.org/euclid.jsl/1388953997_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Mad families constructed from perfect almost disjoint families
http://projecteuclid.org/euclid.jsl/1388953998<strong>Jörg Brendle</strong>, <strong>Yurii Khomskii</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1164--1180.</p><p><strong>Abstract:</strong><br/>
We prove the consistency of $\mathfrak{b} > \aleph_1$ together with the
existence of a $\PI^1_1$-definable mad family, answering a question
posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in $L$ which is an $\aleph_1$-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number $\mathfrak{a}_B$, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of $\mathfrak{a}_B < \mathfrak{b}$ (and hence, $\mathfrak{a}_B < \mathfrak{a}$).
</p>projecteuclid.org/euclid.jsl/1388953998_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
$\Sigma^1_2$ and $\Pi^1_1$ mad families
http://projecteuclid.org/euclid.jsl/1388953999<strong>Asger Törnquist</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1181--1182.</p><p><strong>Abstract:</strong><br/>
We answer in the affirmative the following question of Jörg Brendle: If there is a $\Sigma^1_2$ mad family, is there then a $\Pi^1_1$ mad family?
</p>projecteuclid.org/euclid.jsl/1388953999_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Anomalous Vacillatory Learning
http://projecteuclid.org/euclid.jsl/1388954000<strong>Achilles A. Beros</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1183--1188.</p><p><strong>Abstract:</strong><br/>
In 1986, Osherson, Stob and Weinstein asked whether two variants of anomalous vacillatory learning, TxtFex$^*_*$ and TxtFext$^*_*$, could be distinguished [3]. In both, a machine is permitted to vacillate between a finite number of hypotheses and to make a finite number of errors. TxtFext$^*_*$-learning requires that hypotheses output infinitely often must describe the same finite variant of the correct set, while TxtFex$^*_*$-learning permits the learner to vacillate between finitely many different finite variants of the correct set. In this paper we show that TxtFex$^*_*$ $\neq$ TxtFext$^*_*$, thereby answering the question posed by Osherson, et al . We prove this in a strong way by exhibiting a family in TxtFex$^*_2 \setminus \mbox{TxtFext}^*_*$.
</p>projecteuclid.org/euclid.jsl/1388954000_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Spectra of atomic theories
http://projecteuclid.org/euclid.jsl/1388954001<strong>Uri Andrews</strong>, <strong>Julia F. Knight</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1189--1198.</p><p><strong>Abstract:</strong><br/>
For a countable structure $\mathcal{B}$, the spectrum is the set of
Turing degrees of isomorphic copies of $\mathcal{B}$. For a complete
elementary first order theory $T$, the spectrum is the set of Turing
degrees of models of $T$. We answer a question from [1] by showing
that there is an atomic theory $T$ whose spectrum does not match the
spectrum of any structure.
</p>projecteuclid.org/euclid.jsl/1388954001_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Copyable structures
http://projecteuclid.org/euclid.jsl/1388954002<strong>Antonio Montalbán</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1199--1217.</p><p><strong>Abstract:</strong><br/>
We introduce the notions of copyable and diagonalizable classes of structures.
We then show how these notions are connected to two other notions that had already been studied for some particular classes of structures, namely the listability property and the low property.
The main result of this paper is the characterizations of the classes of structures with the low property, that is, the classes whose low members all have computable copies.
We characterize these classes as the ones whose structural jumps are listable.
</p>projecteuclid.org/euclid.jsl/1388954002_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
From bi-immunity to absolute undecidability
http://projecteuclid.org/euclid.jsl/1388954003<strong>Laurent Bienvenu</strong>, <strong>Adam R. Day</strong>, <strong>Rupert Hölzl</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1218--1228.</p><p><strong>Abstract:</strong><br/>
An infinite binary sequence $A$ is absolutely undecidable if it is
impossible to compute $A$ on a set of positions of positive upper density.
Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch
and Schupp [2] asked whether, unlike the case for
bi-immunity, there is an absolutely undecidable set in every non-zero
Turing degree. We provide a positive answer to this question by applying
techniques from coding theory. We show how to use Walsh—Hadamard codes
to build a truth-table functional which maps any sequence $A$ to a sequence
$B$, such that given any restriction of $B$ to a set of positive upper density,
one can recover $A$. This implies that if $A$ is non-computable,
then $B$ is absolutely undecidable.
Using a forcing construction, we show that this result cannot be
strengthened in any significant fashion.
</p>projecteuclid.org/euclid.jsl/1388954003_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
A perfect set of reals with finite self-information
http://projecteuclid.org/euclid.jsl/1388954004<strong>Ian Herbert</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1229--1246.</p><p><strong>Abstract:</strong><br/>
We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is
$$I(A:B)=\log\sum\limits_{\sigma,\tau\in 2^{<\omega}} 2^{K(\sigma)-K^A(\sigma)+K(\tau)-K^B(\tau)-K(\sigma,\tau)},$$
where $K(\cdot)$ is the prefix-free Kolmogorov complexity. A real $A$ is said to have finite self-information if $I(A:A)$ is finite. We give a construction for a perfect $\Pi^0_1$ class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals with $K(\sigma)\leq^+ K^{A}(\sigma)+f(\sigma)$ for any given $\Delta^0_2$ $f$ with a particularly nice approximation and for a specific choice of $f$ it can also be used to produce a perfect $\Pi^0_1$ set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfy $K(\sigma) \leq^+ K^A(\sigma)+f(\sigma)$ for all $f$ in a ‘nice' class of $\Delta^0_2$ functions which includes all $\Delta^0_2$ orders.
</p>projecteuclid.org/euclid.jsl/1388954004_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Extensions and applications of the S-measure construction
http://projecteuclid.org/euclid.jsl/1388954005<strong>David A. Ross</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1247--1256.</p><p><strong>Abstract:</strong><br/>
S-measures are Loeb measures restricted to the sigma algebra generated
by standard sets. This paper gives new extensions of the
S-measure machinery, with applications to standard measure theory.
</p>projecteuclid.org/euclid.jsl/1388954005_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
On the structure of finite level and $\omega$-decomposable Borel functions
http://projecteuclid.org/euclid.jsl/1388954006<strong>Luca Motto Ros</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1257--1287.</p><p><strong>Abstract:</strong><br/>
We give a full description of the structure under inclusion of all finite level Borel classes of
functions, and provide an elementary proof of the well-known
fact that not every Borel function can be written as a countable union of
$\boldsymbol{\Sigma}^0_\alpha$-measurable functions (for every fixed $1
\leq \alpha < \omega_1$). Moreover, we present some results concerning
those Borel functions which are $\omega$-decomposable into continuous functions (also
called countably continuous functions in the literature): such results should be viewed as a contribution
towards the goal of
generalizing a remarkable theorem of Jayne and Rogers to all finite levels,
and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel
functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.
</p>projecteuclid.org/euclid.jsl/1388954006_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Automatic continuity for homomorphisms into free products
http://projecteuclid.org/euclid.jsl/1388954007<strong>Konstantin Slutsky</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1288--1306.</p><p><strong>Abstract:</strong><br/>
A homomorphism from a completely metrizable topological group into a free product of groups whose image is not
contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the
range. In particular, any completely metrizable group topology on a free product is discrete.
</p>projecteuclid.org/euclid.jsl/1388954007_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Anti-complex sets and reducibilities with tiny use
http://projecteuclid.org/euclid.jsl/1388954008<strong>Johanna N. Y. Franklin</strong>, <strong>Noam Greenberg</strong>, <strong>Frank Stephan</strong>, <strong>Guohua Wu</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1307--1327.</p><p><strong>Abstract:</strong><br/>
In contrast with the notion of complexity, a set $A$ is called
anti-complex if the Kolmogorov complexity of the initial segments of $A$
chosen by a recursive function is always bounded by the identity
function. We show that, as for complexity, the natural arena for
examining anti-complexity is the weak-truth table degrees. In this
context, we show the equivalence of anti-complexity and other lowness
notions such as r.e. traceability or being weak truth-table reducible to
a Schnorr trivial set. A set $A$ is anti-complex if and only if it is
reducible to another set $B$ with tiny use , whereby we mean that
the use function for reducing $A$ to $B$ can be made to grow arbitrarily
slowly, as gauged by unbounded nondecreasing recursive functions. This notion
of reducibility is then studied in its own right, and we also investigate
its range and the range of its uniform counterpart.
</p>projecteuclid.org/euclid.jsl/1388954008_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Isomorphism of computable structures and {V}aught's {C}onjecture
http://projecteuclid.org/euclid.jsl/1388954009<strong>Howard Becker</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1328--1344.</p><p><strong>Abstract:</strong><br/>
The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle $a \in 2^\omega$, we can ask the same question relativized to $a$. A negative answer for every $a$ implies Vaught's Conjecture for $L_{\omega_1 \omega}$.
</p>projecteuclid.org/euclid.jsl/1388954009_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST
Corrigendum to: “Relation algebra reducts of cylindric algebras and complete representations”
http://projecteuclid.org/euclid.jsl/1388954010<strong>R. Hirsch</strong><p><strong>Source: </strong>J. Symbolic Logic, Volume 78, Number 4, 1345--1346.</p>projecteuclid.org/euclid.jsl/1388954010_Sun, 05 Jan 2014 15:33 ESTSun, 05 Jan 2014 15:33 EST