(Helsinki & Amsterdam)
As a background to Generalized Baire Spaces I first present a game-theoretic approach to identifying uncountable structures, covering EF-games 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 game-theoretic 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)
We give an overview over the theory of definable subsets of generalized Baire spaces which will cover results of Mekler-Väänänen, Friedman-Hyttinen-Kulikov, Lücke-Schlicht and Motto Ros.
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 κ-completeness 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 2κ.
Luca Motto Ros
Hurewicz showed that every analytic subset of the Baire space ωω is either covered by a countable union of compact sets, or else it contains a closed set homeomorphic to ωω. We show that it is consistent that the analogous statement for the generalized Baire space κκ, 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 κ-Canjar filters and a combinatorial regularity property called κ-Miller measurability.
Given an uncountable regular cardinal κ, we say that a set of functions from κ to κ is a Σ11-subset of κκ if it is definable over the structure H(κ+) by a Σ1-formula with parameters. It is well-known 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 Σ11-subsets of κκ 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.
I will present a notion of forcing that, for some particular choices of definition φ for a class of subsets of κκ (our standard examples for φ will specify the class of wellorders of κκ or the class of Bernstein subsets of κκ), is capable of producing a generic extension in which some subset of κκ satisfying φ becomes Σ11- (or sometimes Δ11)-definable, and will discuss some applications of this technique.
Yurii Khomskii (Vienna)
We investigate regularity properties derived from tree-like 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 Δ11 level reflects some (but not all) properties of the classical Δ12 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)
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 κκ =κ > ω, the analogue of E0 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 E1 is reducible to E0 (in the case of κ = ω, E1 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 non-stationary 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)
A non-trivial issue concerning tree-like forcings in the generalized framework is to introduce a random-like forcing, where random-like means to be κκ-bounding, <κ-closed 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):
|Impressum||2014-11-07, BL, wwwmath (JMD)|