| date | speaker | title |
|---|---|---|
| 2004-01-09 | Akihiro Kanamori (Mathematics Department, Boston University) | Zermelo and Set Theory |
| Ernst Friedrich Ferdinand Zermelo (1871--1953) transformed the set theory of Cantor and Dedekind into {\it abstract} set theory in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization, and thereby tempered the ontological thrust of early set theory and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory that would have a modern resonance. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the knowledge of what has endured in the subsequent development of set theory. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. | ||
| 2004-01-13 | Sy Friedman (KGRC) | Borel and Analytic Equivalence Relations |
| 2004-01-20 | John Krueger (KGRC) | Destroying Stationary Sets |
| 2004-01-27 | John Krueger (KGRC) | Destroying Stationary Sets, Part 2 |
| 2004-02-03 | Menachem Kojman (Ben-Gurion University) | (KGS Lecture) Topology and Combinatorics of Singular Cardinals |
| In the talk I will present the pcf approach to singular cardinals and some of its applications to topology and to combinatorics. | ||
| 2004-03-09 | Boris Piwinger (KGRC) | Mind the Gap: Hyperfine Structure Theory and Morasses |
| 2004-03-16 | Boris Piwinger (KGRC) | Mind the Gap: Hyperfine Structure Theory and Morasses, Part II |
| 2004-03-23 | Andres Caicedo (KGRC) | CH and the saturation of the nonstationary ideal on omega_1 |
| The saturation of the nonstationary ideal on omega_1 was shown consistent (with ZFC) from a strong form of determinacy by Steel and VanWesep in the early 80's. Their techniques produced a model where CH fails. It has been an open question since then whether a model can be produced where the ideal is saturated and CH holds. Although this problem is still open, significant progress towards a (negative) solution was made by Woodin in the 90's. Specifically, Woodin proved that the saturation of the ideal contradicts CH, *in the presence of large cardinals*. In fact, a definable counterexample is produced. However, no such definable counterexample can exist if the large cardinals are absent from the picture, and apparently a completely new idea is necessary to settle the problem in this case. A nice side effect of Woodin's techniques is the development of the theory of P_max. In this talk I plan to present Woodin's result, together with its limitations. | ||
| 2004-04-27 | James Hirschorn (KGRC) | CCC Forcing and Splitting Reals |
| Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is provided using the P-ideal dichotomy first formulated by Abraham and Todorcevic and later extended by Todorcevic. In the process, I show that under the P-ideal dichotomy every weakly distributive ccc complete Boolean algebra carries a Maharam submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. | ||
| 2004-05-04 | James Hirschorn (KGRC) | CCC Forcing and Splitting Reals, Part 2 |
| Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is provided using the P-ideal dichotomy first formulated by Abraham and Todorcevic and later extended by Todorcevic. In the process, I show that under the P-ideal dichotomy every weakly distributive ccc complete Boolean algebra carries a Maharam submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. | ||
| 2004-05-18 | John Krueger (KGRC) | Strong Compactness and Stationary Sets |
| I will show how to construct a model in which \(\kappa\) is a strongly compact cardinal and the set \(S(\kappa,\kappa^+) = \{ a \in P_\kappa \kappa^+ : \ot(a) = (a \cap \kappa)^+ \}\) is non-stationary. | ||
| 2004-05-25 | John Krueger (KGRC) | Strong Compactness and Stationary Sets 2 |
| I will show how to construct a model in which \(\kappa\) is a strongly compact cardinal and the set \(S(\kappa,\kappa^+) = \{ a \in P_\kappa \kappa^+ : \ot(a) = (a \cap \kappa)^+ \}\) is non-stationary. | ||
| 2004-06-15 | Jean Larson (University of Florida) | Infinite paths and a plain product order type |
| 2004-06-15 | Bill Mitchell (University of Florida) | On I[\omega_2] |
| 2004-06-22 | Peter Koepke (Universität Bonn) | (missing title) |
| 2004-06-29 | Martin Zeman (U. California, Irvine) | Forcing the failure of square from a weak hypothesis |
| 2004-10-05 | Menachem Kojman (Ben Gurion University of the Negev) | Extending Baire measures to Borel measures in normal spaces below the first real-valued measurable using pcf theory |
| 2004-10-12 | Menachem Kojman (Ben Gurion University of the Negev) | Extending Baire measures to Borel measures in normal spaces below the first real-valued measurable using pcf theory, Part 2 |
| 2004-10-19 | Mirna Dzamonja (University of East Anglia) | A partition theorem for large dense linear orders |
| 2004-10-25 | Yuri Matijasevic (Steklov Institute, St. Petersburg) | (KGS Lecture) Hilbert's Tenth Problem today: Main results and open questions |
| 2004-10-27 | Grzegorz Plebanek (University of Wroclaw, Poland) | Measures defined on sigma algebras contained in Bor[0,1] |
| The talk is devoted to properties of measures defined on sigma algebras contained in Bor[0,1], or more generally, in Bor(X), where X is a Polish space. In particular, we are going to mention some open problems on measures and infinite games. | ||
| 2004-11-16 | Natasha Dobrinen (KGRC) | Games and distributive laws in Boolean algebras, Part 1 |
| 2004-11-23 | Natasha Dobrinen (KGRC) | Games and distributive laws in Boolean algebras, Part 2 |
| 2004-11-30 | Andres Caicedo (KGRC) | Bounded forcing axioms and projective well-orderings of the reals |
| In the absence of Woodin cardinals, fine srtuctural inner models for mild large cardinal hypotheses admit set forcing extensions where bounded forcing axioms hold and the reals are projectively well-ordered. | ||
| 2004-12-09 | Ilijas Farah (York University) | Absoluteness and the Uncountable |
| 2004-12-14 | Ilijas Farah (York University) | Characterizing measure algebras |
