BSU Set Theory Seminar

July 2, 2009 (2:00 p.m. - 2:50 p.m.)
Thomas E. Forster, Cambridge University, UK

Wellfounded transitive inner models of stratified fragments of ZF + ⌐AC

We work in models of ZFC. Let G be a group of permutations of V_ω. Let us say that a set x is n-symmetric (wrt G understood) if, for every π in G, x is fixed by the action of π ``acting n levels down''. A set is plain symmetric iff it is n-symmetric for all suff large n. HS is the inner model of all hereditarily symmetric sets. This construction can be tweaked in various ways, and a wide variety of structures obtained. They are all models of the stratified fragment of ZF, and they all refute AC - for Fraenkel-Mostowski-like reasons, despite the fact that they are both transitive and extensional unlike all the other non-forcing construction of models of ⌐ - AC. The original article is to be found at http://www.dpmms.cam.ac.uk/~tf/strZF.pdf.
The fact that all stratified comprehension is preserved gives hope that this might eventually shed light on the continuing (and shameful) mystery that is the open question of the consistency of the Quine systems. I have two Ph.D. students working on this material and I shall be reporting on the progress that we have made since 2005.

May 4, 2009
Liljana Babinkostova
Haver property, screnability property and products

Abstract: We show when the product of two metric spaces with the Haver (screnability) property is a space with the Haver (screnability) property.


April 27
Liljana Babinkostova
Infinite versions of covering dimension

Abstract: We define a game related to a selection principle and use it to give an infinite version of covering dimension in separable metric spaces. Then we prove that the space is countable dimensional iff TWO has a winning strategy in the game that we define.


April 20
Liljana Babinkostova
Infinite versions of covering dimension

Abstract: We give a brief survey of infinite versions of covering dimension: some basic definitions and main results in dimension theory related to the covering dimension.
Then we define a game related to a selection principle and use it to give an infinite version of covering dimension in separable metric spaces.

April 13
Richard Ketchersid
A dichotomy theorem for models of AD+

This talk is a continuation of the previous talk.
Abstract: We study the countable finite game under determinacy and show that in models of AD+ this game is undetermined. The proof uses a dichotomy theorem for sets in models of AD+, namely, every set is either well ordered or the reals embed into X. I will discuss this dichotomy and explain why it holds, a crucial part was recently answered by Hugh Woodin, and I will present hi contribution as well.

April 6
Richard Ketchersid
A dichotomy theorem for models of AD+

We study the countable finite game under determinacy and show that in models of AD+ this game is undetermined. The proof uses a dichotomy theorem for sets in models of AD+, namely, every set is either well ordered or the reals embed into X. I will discuss this dichotomy and explain why it holds, a crucial part was recently answered by Hugh Woodin, and I will present hi contribution as well.

March 9 - March 16, 2009
Andres Caicedo - Richard Ketchersid

"Scheepers's Countable-Finite game and the axiom of determinacy"

Abstract: The countable-finite game, introduced by Marion Scheepers, is an infinite two-player perfect information game relative to a set S. At move n, player I plays a countable subset O_n of S and player II responds by playing a finite subset T_n of S. Player II wins iff the union of the T_n contains the union of the O_n. Clearly II has a winning strategy no matter what S is, in the presence of choice, so Scheepers restricts the notion of strategy to that of tactic. Here, we instead study what happens in the absence of choice. We show that in many models of determinacy (including L(R)) the game is determined iff S is countable. This is a consequence of a general dichotomy theorem in the presence of determinacy. For L(R), this result states that any set either embeds the reals, or else is well-orderable.

Abstract: We give Galvin's examples (in ZFC) of:

(1) A subset X of the Cantor set such that for all k with powerset of cardinality at most Continuum, the binary Gale-Stewart game for X on k boards is undetermined.

(2) A subset X of the set of irrational numbers such that for all k less than the Continuum, the integer Gale-Stewart game for X on k boards is undetermined.

Abstract: A topological space is a Baire space if the intersection of any omega-sequence of dense open sets is dense. It is well-known that for the classical Banach-Mazur game on a space X: If player TWO (also called "Nonempty") has a winning strategy then all box-topology powers of X are Baire spaces. Galvin conjectures that for all topological spaces this implication is an equivalence. It has been known since the 1970's that if it is consistent that there is a proper class of measurable cardinals, then it is consistent that Galvin's conjecture holds. It is not known if this conjecture is just outright true. This sequence of talks is intended to resurrect interest in determining the status of this conjecture.