1.
On skinny stationary subsets of $\mathcal {P}_\kappa \lambda $ - Matsubara, Yo; Usuba, Toschimichi
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...
2.
Topological dynamics and definable groups - Pillay, Anand
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...
3.
On the definability of radicals in supersimple groups - Milliet, Cé{d}ric
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.
4.
A quasi-order on continuous functions - Carroy, Raphaël
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.
5.
Independence, dimension and continuity in non-forking frames - Jarden, Adi; Sitton, Alon
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.
6.
Probabilistic algorithmic randomness - Buss, Sam; Minnes, Mia
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.
7.
On colimits and elementary embeddings - Bagaria, Joan; Brooke-Taylor, Andrew
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.
8.
Models of transfinite provability logic - Fernández-Duque, David; Joosten, Joost J.
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...
9.
Unexpected imaginaries in valued fields with analytic structure - Haskell, Deirdre; Hrushovski, Ehud; Macpherson, Dugald
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.
10.
Nonexistence of minimal pairs for generic computability - Igusa, Gregory
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.
11.
Ample thoughts - Palacín, Daniel; Wagner, Frank O.
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.
12.
Partial impredicativity in reverse mathematics - Towsner, Henry
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.
13.
Borel reductions and cub games in generalised descriptive set theory - Kulikov, Vadim
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 $\kappa^{<\kappa}=\kappa$.
14.
A fixed point for the jump operator on structures - Montalbán, Antonio
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.
15.
Canonical measure assignments - Jackson, Steve; Löwe, Benedikt
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).
16.
Decidability for some justification logics with negative introspection - Studer, Thomas
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.
18.
Satisfaction relations for proper classes: applications in logic and set theory - Van Wesep, Robert A.
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.
19.
Uniform distribution and algorithmic randomness - Avigad, Jeremy
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...
20.
Strong tree properties for small cardinals - Fontanella, Laura
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.
1.
