Amsterdam Workshop on
Set Theory 2014
Generalized Baire Space
3 & 4 November 2014
Amsterdam (The Netherlands)
The Amsterdam Set Theory Workshop 2014 is organized by the Institute for Logic, Language and Computation (ILLC) and
the logic group of the University of Hamburg
with the aim to provide the platform for exchange for the researchers active in the field of the set theory of the generalized Baire space.
The two days of the workshop will consist of three tutorials, several contributed talks and discussion sessions.
One of the outputs of this meeting is a planned paper consisting of the open problems in the generalized Baire space.
Everyone is cordially invited to attend.
Organizers:
Giorgio Laguzzi,
Benedikt Löwe,
Hugo Nobrega,
Ilya Sharankou.
Location:
Room C1.13 (Belle van Zuylen Hall),
Singel 425,
1012 WP Amsterdam,
Netherlands.
How to get there
Participants
Andrew BrookeTaylor (Bristol),
Sziráki Dorottya (Helsinki),
Peter Holy (Bristol),
Yurii Khomskii (Vienna),
Vadim Kulikov (Vienna),
Giorgio Laguzzi (Freiburg),
Benedikt Löwe (Amsterdam & Hamburg),
Philipp Lücke (Bonn),
Diana Carolina Montoya Amaya (Vienna),
Miguel Moreno (Helsinki),
Luca Motto Ros (Freiburg),
Hugo Nobrega (Amsterdam),
Philipp Schlicht (Bonn),
Ilya Sharankou (Hamburg),
Jouko Väänänen (Helsinki & Amsterdam).
Program
03.11.2014, Monday
 
9:4511:15

Jouko Väänänen
(Helsinki & Amsterdam)
Games, trees and models—a tutorial in Generalized Baire Spaces, (slides)

11:1511:40
 Coffee Break

11:4012:40

Philipp Schlicht
(Bonn)
Introduction to generalized descriptive set theory, (slides)

12:4014:00
 Lunch Break

14:0014:45
 Luca Motto Ros
(Freiburg)
The Hurewicz dichotomy for generalized Baire spaces, (slides)

15:0015:45
 Philipp Lücke
(Bonn)
The influence of closed maximality principles on generalized Baire space, (slides)

15:4516:15
 Coffee Break

16:1517:00
 Peter Holy
(Bristol)
Δ^{1}_{1}
subsets of κ^{κ}
, (slides)

17:1518:00
 Yurii Khomskii (Vienna)
Regularity properties on the generalized reals, (slides)
 

04.11.2014, Tuesday
 
10:1511:15

Andrew BrookeTaylor
(Bristol)
Building κcomplete filters for supercompact κ, (slides)

11:1511:45
 Coffee Break

11:4512:30
 Vadim Kulikov (Vienna)
Orbit Equivalence Relations and Borel Reducibility on the Generalized Baire Space, (slides)

12:3014:00
 Lunch Break

14:0014:45
 Giorgio Laguzzi (Freiburg)
Generalized random forcing

15:0016:00
 Final discussion session

Abstracts
Jouko Väänänen
(Helsinki & Amsterdam)
Games, trees and models—a tutorial in Generalized Baire Spaces
As a background to Generalized Baire Spaces I first present a gametheoretic approach to identifying uncountable structures,
covering EFgames on uncountable models, trees as clocks of games, and the relevant ordering of trees. We then delve into the topic
of Generalized Baire Spaces and higher descriptive set theory. Here the focus is on the topological aspects of uncountable models,
arising from our gametheoretic approach. Essentially, we generalize descriptive set theory from the classical Baire space to higher
Baire spaces, such as the space of models of a fixed uncountable cardinality. Finally we return to the motivating question of finding
invariants for uncountable models, and investigate this question for models of countable complete first order theories. We relate the
existence of trees as invariants of uncountable models of a given first order theory to stability theoretic properties of the theory.
Philipp Schlicht (Bonn)
Introduction to generalized descriptive set theory
We give an overview over the theory of definable subsets of generalized Baire spaces which will cover results of MeklerVäänänen, FriedmanHyttinenKulikov, LückeSchlicht and Motto Ros.
Andrew BrookeTaylor
(Bristol)
Building κcomplete filters for supercompact κ
When trying to generalize arguments from the omega case in which
filters are constructed inductively through a forcing iteration, a
common problem is to prove that κcompleteness is preserved at
limit stages. If κ is supercompact, however, a technique of
Dzamonja and Shelah may sometimes be used to overcome this problem. I
will describe this method, focusing on the case of constructing a
model with u(κ) strictly less than 2^{κ}.
Luca Motto Ros
(Freiburg)
The Hurewicz dichotomy for generalized Baire spaces (joint with Philipp Schilicht and Philipp Lücke)
Hurewicz showed that every analytic subset of the Baire space ω^{ω}
is either covered by a countable union of compact sets, or else it
contains a closed set homeomorphic to ω^{ω}. We show that it
is consistent that the analogous statement for the generalized Baire space
κ^{κ}, called the Hurewicz dichotomy for κ, holds for
any given uncountable cardinal kappa such that κ^{<κ} =
κ. Moreover, it is also consistent that the Hurewicz dichotomy holds
simultaneously at all uncountable regular cardinals κ, as well as
that it fails simultaneously at all uncountable regular κ. We
further show that the combinations of the Hurewicz dichotomy with the
perfect set property and with its negation are both consistent. Finally, we
present two applications of these results concerning a topological
characterization of κCanjar filters and a combinatorial regularity
property called κMiller measurability.
Philipp Lücke
(Bonn)
The influence of closed maximality principles on generalized Baire space
Given an uncountable regular cardinal κ, we say that a set of functions from κ to κ
is a Σ^{1}_{1}subset of κ^{κ}
if it is definable over the structure H(κ^{+}) by a Σ_{1}formula
with parameters.
It is wellknown that many basic and interesting questions about such sets are not decided by the
axioms of ZFC plus large cardinal axioms. In my talk,
I want to present different extensions of ZFC that
settle many of those questions by providing a nice
structure theory for the class of Σ^{1}_{1}subsets of
κ^{κ} in the case where κ is an uncountable cardinals with κ=
κ^{<κ}.
These axioms are variants of the maximality principle
introduced by Jonathan Stavi and Jouko Väänänen
and later rediscovered by Joel Hamkins.
Peter Holy
(Bristol)
Δ^{1}_{1} subsets of κ^{κ} (joint with Philipp Lücke)
I will present a notion of forcing that, for some
particular choices of definition φ for a class of subsets of
κ^{κ} (our standard examples for φ will specify the
class of wellorders of κ^{κ} or the class of Bernstein
subsets of κ^{κ}), is capable of producing a generic
extension in which some subset of κ^{κ} satisfying φ
becomes Σ^{1}_{1} (or sometimes Δ^{1}_{1})definable, and will
discuss some applications of this technique.
Yurii Khomskii (Vienna)
Regularity properties on the generalized reals (joint with SyDavid Friedman and Vadim Kulikov)
We investigate regularity properties derived from treelike
forcing notions in the setting of generalized descriptive set theory.
Unlike the classical situation, generalised analytic sets typically do not
satisfy regularity properties of this kind, and the generalised Δ^{1}_{1}
level reflects some (but not all) properties of the classical Δ^{1}_{2}
level. We present an abstract theory and apply it to a number of examples.
There are still many open questions in this field, including the basic
question of whether we are looking at the right kind of regularity
properties.
Vadim Kulikov (Vienna)
Orbit Equivalence Relations and Borel Reducibility on the Generalized Baire Space (joint with SyDavid Friedman and Tapani Hyttinen)
We outline some of our recent results concerning Borel reducibility of relevant equivalence
relations to each other, in particular orbit equivalence relations.
We show that when κ^{κ}
=κ > ω, the analogue of E_{0} is universal among all orbit equivalence relations
induced by an action of a group of size at most κ. Moreover it has been observed
that the analogue of E_{1} is reducible to E_{0}
(in the case of κ = ω, E_{1} is not even reducible to
any orbit equivalence relation induced by a Borel Polish group action).
Further, we state without proofs (or with sketches, if time permits)
that the jump of identity, and hence any Borel isomorphism relation,
is Borel reducible to the equivalence relation modulo the
nonstationary ideal. Also, unlike in the case κ = ω,
not all small equivalence relation are induced by a group action,
in fact there is a smooth equivalence relation with classes of size
two which is not induced by a Borel action of a group of size at most κ.
We conclude with some applications to model
theory and open questions.
Giorgio Laguzzi (Freiburg)
Generalized random forcing (joint with SyDavid Friedman)
A nontrivial issue concerning treelike forcings in the generalized framework is to introduce a randomlike forcing,
where randomlike means to be κ^{κ}bounding, <κclosed and
κ^{+}cc simultaneously.
Shelah managed to do that for κ
weakly compact. In this talk we aim at introducing a forcing satisfying these
three properties for κ inaccessible,
and not necessarily weakly compact.
How to get there:
From Hotel Citadel (~12 min walk):
From Rho Hotel (~9 min walk):
