Archives of: Barcelona Set Theory Seminar

Ralf Schindler: A Hamel basis for the reals without choice

BARCELONA SET THEORY SEMINAR

Date: Monday 30 October 2017

Time: 16:00

Place: IMUB*
University of Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

Speaker: Ralf Schindler

Title: A Hamel basis for the reals without choice (Universität Münster)

Abstract: The Cohen-Halpern-Levy model N has an infinite set of
reals without a countable subset. Answering a question of D.
Pincus and K. Prikry from 1975, we show that there is a Hamel
basis (i.e., a basis for R as a vector space over Q) in N. This is
joint work with Liuzhen Wu and Liang Yu, inspired by earlier joint
work with Mariam Beriashvili. The axiom of Dependent Choice
(DC) fails in N, but in later joint work with Wu and Yu we also
showed that there is a model of ZF+DC with a Hamel basis and in
which the reals cannot be wellordered.

Jinglun Cai: C(n)-Ultrastrong Cardinals

BARCELONA SET THEORY SEMINAR

Date: Wednesday 25 October 2017

Time: 16:00

Place: Room S-3
Department of Mathematics & Computer Science
University of Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

Speaker: Jinglun Cai  (Universitat de Barcelona)

Title: C(n)-Ultrastrong Cardinals

Abstract: see attached.

Alejandro Poveda: Woodin’s HOD-Dichotomy

Date: Wednesday 27 September 2017

Time: 16:00

Place: Room S-3
Department of Mathematics & Computer Science
University of Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

Speaker: Alejandro Poveda (Universitat de Barcelona)

Title: Woodin’s HOD-Dichotomy

Abstract: We shall give a complete proof of W. H.
Woodin’s remarkable result that if there exists an
extendible cardinal, then either the set-theoretic universe
V is very “close” to HOD (the class of Hereditarily Ordinal
Definable sets), or it is very “far” from it.

Juan Carlos Martínez: On the existence of pcf algebras

Speaker: Juan Carlos Martínez   (Universitat de Barcelona)

Title: On the existence of pcf algebras

Abstract: We shall give a direct proof of a result of Jech
and Shelah on the existence of a type of pcf algebras on
ω1.

Date: Thursday, 6 April 2017

Time: 15:30

Place: Room T-2*
Department of Mathematics & Computer Science
University of Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

Joan Bagaria: Berkeley Cardinals, II

Thursday, March 30, 2017, from 3:30 to 5pm
Dept. of Mathematics and Computer Science, Univ. of Barcelona

Speaker: Joan Bagaria (ICREA and UB)

Title: Berkeley Cardinals, II

Abstract:

We will continue our discussion about Berkeley cardinals by presenting some recent results (joint with P. Koellner and W. H. Woodin) on the possible cofinalities of the first Berkeley cardinal.

Joan Bagaria: Berkeley Cardinals

BCNSETS: BARCELONA RESEARCH GROUP IN SET THEORY
Thursday, 23 March 2017, 15:30

Speaker: Joan Bagaria

Title: Berkeley Cardinals

Abstract: Berkeley cardinals are  large cardinals whose existence
contradicts the Axiom of Choice. We will  present some recent
results (joint with P. Koellner and W. H. Woodin) about the relative
position of Berkeley cardinals in the large cardinal hierarchy, and
also about the possible cofinalities of the first Berkeley cardinal.

Daisuke Ikegami: Omega-logic and Boolean valued second order logic

Wednesday, September 12, from 15:30 to 17:00
Seminar Room, Department of Logic and History and Philosophy of Science, UB, Montalegre 6, 4th floor, Barcelona.

Speaker: Daisuke Ikegami (UC Berkeley).
Title: Omega-logic and Boolean valued second order logic

Abstract:
Woodin’s Omega-logic is a logic on generic absoluteness and Woodin’s Omega-conjecture states that all the possible results on generic absoluteness must be explained by looking at certain good sets of reals called “universally Baire sets”. The Omega-conjecture explains the phenomena of generic absoluteness obtained by large cardinals in the last decades.

Second order logic has two standard semantics, one is full semantics where one interprets second order quantifiers over all the subsets of a universe and the other is Henkin semantics where a second order structure satisfies full comprehension. Second order logic with full semantics can express many complicated things but does not enjoy good logical properties (e.g. completeness) while the one with Henkin semantics is very weak and enjoys basic logical properties as first order logic does.

We introduce the Boolean valued semantics for second order logic and investigate Boolean valued second order logic. We will compare this logic with Omega-logic and second order logic with full semantics in terms of basic logical properties such as validity, completeness, and compactness.

Joerg Brendle: Aspects of splitting

Barcelona Set Theory Weekly Seminar

Wednesday, February 15
from 15:30 to 17:00
Dep. of Logic, History and Philosophy of Science,
University of Barcelona, Montalegre 6, 4th Floor, Barcelona

Speaker: Joerg Brendle (Kobe University)

Title: Aspects of splitting

Barcelona Set Theory Seminar

Barcelona Set Theory Weekly Seminar

Tuesday, October 11
from 15:30 to 17:00
Dep. of Logic, History and Philosophy of Science,
University of Barcelona, Montalegre 6, 4th Floor, Barcelona

Speaker: Kostas Tsaprounis (University of Barcelona)
Title: Elementary Chains and C^(n) cardinals (II)

Barcelona Set Theory Seminar

Barcelona Set Theory Weekly Seminar

Wednesday, October 5
from 15:30 to 17:00
Dep. of Logic, History and Philosophy of Science, University of Barcelona, Montalegre 6, 4th Floor, Barcelona

Speaker: Kostas Tsaprounis (University of Barcelona)

Title: Elementary Chains