Archives of: Barcelona Set Theory Seminar

Samuel Gomes da Silva: Reductions between certain incidenceproblems and the Continuum Hypothesis

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR
Reductions between certain incidence
problems and the Continuum Hypothesis
Samuel Gomes da Silva
UFBA, Brazil
Abstract: We consider two families of incidence problems, C 1 and C 2 ,
which are related to real numbers and countable subsets of the real
line. Instances of problems in C 1 are as follows: given a real number x,
pick randomly a countable set A of reals hoping that x is in A, whereas
instances of problems in C 2 are as follows: given a countable set A of
reals, pick randomly a real number x hoping that x is not in A. One could
arguably defend that, at least intuitively, problems in C 2 are easier to
solve than problems in C 1 . Indeed, we show that, after some suitable
formalization, one can prove (in ZFC) that, on the one hand, problems in
C 2 are at least as simple to solve as problems in C 1 . On the other hand,
the statement “Problems in C 1 have the exact same complexity as
problems in C 2 ” is equivalent to the Continuum Hypothesis.
Date: Thursday 21 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona
* Enter the University building through the door 20 meters to the right of the main
door and, as you enter the courtyard, turn left, go to the end of the corridor, and
then downstairs.

Alejandro Poveda: Prikry-type forcing and the failure of the Singular Cardinal Hypothesis

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR

Prikry-type forcing and the failure of the
Singular Cardinal Hypothesis

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals.
In this session we will describe the Prikry forcing with collapses
and present a proof of Magidor’s theorem on the consistency,
relative to appropriate large cardinal hypotheses, of the failure
of the SCH at the first singular cardinal.

Date: Thursday 7 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.

Alejandro Poveda: Prikry-type forcing: properties and applications

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR

Prikry-type forcing: properties and applications

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals,
or the Diagonal supercompact Prikry forcing with collapses, due
also to Magidor, which can be used to force the failure of SCH
at the first singular cardinal. In this session we will prove some
of the key properties of Prikry forcing and will see how it is
used to prove the consistency of the negation of SCH.

Date: Thursday 28 February 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.

Alejandro Poveda: An invitation to the world of Prikry-type forcing

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR

An invitation to the world of Prikry-type forcing

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals,
or the Diagonal supercompact Prikry forcing with collapses, due
also to Magidor, which can be used to force the failure of SCH
at the first singular cardinal. In this session we will give an
introduction from the very beginning to this family of forcings,
and if time permits we will present some easy applications.

Date: Thursday 14 February 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.

Moritz Müller: Forcing against bounded arithmetic

BARCELONA RESEARCH GROUP IN SET THEORY

BARCELONA SET THEORY SEMINAR

Forcing against bounded arithmetic

Moritz Müller

Universitat Politècnica de Catalunya

Abstract: We study the following problem. Given a nonstandard
model of arithmetic we want to expand it by a binary relation that
does something prohibitive, e.g. violates the pigeonhole principle in
the sense that it is the graph of a bijection from n+1 onto n for
some (nonstandard) n in the model. The goal is to do so while
preserving as much as possible of true arithmetic. More precisely,
we want the expansion to model the least number principle for a
class of formulas as large as possible. The problem is of central
importance in bounded arithmetic and propositional proof
complexity. It is not well understood. The talk describes a general
method of forcing to produce such expansions.

Date: Thursday 7 February 2019
Time: 16:00

Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.

Joan Bagaria: The consistency strength of simultaneous stationary reflection

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY

BARCELONA SET THEORY SEMINAR

ICREA and Universitat de Barcelona

The consistency strength of simultaneous
stationary reflection

Joan Bagaria

ICREA and Universitat de Barcelona

Abstract: We shall present M. Magidor’s beautiful proof that if a
regular cardinal k reflects simultaneously all pairs of its stationary
subsets, then k is a weakly compact cardinal in the constructible
universe L. We will then discuss some of the difficulties involved in
extending Magidor’s result to the hyperstationary case.

Date: Thursday 31 January 2019

Time: 15:30

Place: Room S-1
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and go downstairs.

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