Abstract: ContentsI Introduction to inner models 2by William J. Mitchell1 The constructible sets . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Relative constructibility . . . . . . . . . . . . . . . . . . . 41.2 Measurable Cardinals . . . . . . . . . . . . . . . . . . . . 51.3 0, and sharps in general . . . . . . . . . . . . . . . . . . 91.4 Other sharps . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 From sharps to the core model . . . . . . . . . . . . . . . 132 Beyond one ...

We define organic sets and organically stationary sequences, which generalize tight sets and tightly stationary sequences respectively. We show that there are stationary many inorganic sets (Theorem 3) and stationary many sets that are organic but not tight (Theorem 4). Working in the Constructible Universe, we give a characterization of organic and tight sets in terms of fine structure (Theorem 7). We answer a related question posed in [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: Part two, Ann. Pure Appl. Logic 142 (2006) 55–75] about the combinatorial principle Coherent Squares (Corollary 9).

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals.

Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT's on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos-von Neumann theorem classifying ergodic MPT's with pure point spectrum. We treat both the classical case of Z as well as the case of general countable amenable groups.

From May 1 to May 6, 2004 24 set theorists met at the Banff International Research Station to discuss Singular Cardinal Combinatorics. Descriptions of the contents of their talks will be published in a Proceedings that will appear in the Notre Dame Journal of Symbolic Logic. During the workshop, several important new results were announced and explained, and there were problem sessions held (some with significant amounts of prize money attached to particular problems, see the last section for details). To summarize the direction of the conference we will present here an annotated collection of representative problems with some references. Where the problems were novel, attribution is attempted and it is noted where there is money attached to particular problems. Three closely related themes dominated the discussion: stationary sets and stationary set reflection, variations of square and approachability and the singular cardinals hypothesis. Underlying most of the discussion were ideas from Shelah’s PCF theory. Important subthemes were mutual stationarity, Aronszajn trees and superatomic Boolean Algebras.

This note proves two theorems. The first is that it is consistent to have for every n, but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such an ordinal.This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ωn, then there is a square type sequence on the points of cofinality ωk, k > 1 in ℵω+1. In particular at points of cofinality greater than ω1 there is a strongly non-reflecting stationary set of points of countable cofinality.The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.

This is a very brief survey of some results in partition theory for infinite cardinals. It is intended for a general mathematical audience. The latter portions present some recent joint work of A. Hajnal and the author.

Banach showed in 1923 that Lebesgue measure is not the unique rotation invariant finitely additive probability measure on the measurable subsets of S1. Margulis and Sullivan (for n ≥ 4) and Drinfield (for n = 2, 3) independently showed that Lebesgue measure is the unique isometry invariant finitely additive probability measure on Sn. These results all used special properties of the group action. Rosenblatt asked whether an amenable group can uniquely determine an invariant mean. Using techniques from set theory we obtain information on this question and give a complete solution in the case of locally finite groups acting on the integers.

In 1924 Banach and Tarski, using ideas of Hausdorff, proved that there is a partition of the unit sphere S2 into sets A1, . . . , A(k), B1, . . . , B(l) and a collection of isometries {sigma1, ..., sigma(k), rho1, ..., rho(l)} so that (sigma1A1, . . . , sigma(k)A(k)} and {rho1B1,...,rho(l)B(l)} both are partitions of S2. The sets in these partitions are constructed by using the axiom of choice and cannot all be Lebesgue measurable. In this note we solve a problem of Marczewski from 1930 by showing that there is a partition of S2 into sets A1, . . . , A(k), B1, . . . , B(l) with a different strong regularity property, the Property of Baire. We also prove a version of the Banach-Tarski paradox that involves only open sets and does not use the axiom of choice.

In this paper we show that the Axioms of Zermelo-Fraenkel set theory together with the Hahn-Banach theorem imply the existence of a non-Lebesgue measurable set. Our construction does not make any use of the Axiom of Choice.

In 1874, Cantor [Cl] showed that every set has cardinality strictly smaller than the cardinality of its power set. Cantor asked [C2] whether for infinite sets X there is a set Y of cardinality strictly between cardinality X and cardinality 2X. The special case of X = Z (the integers) was Hilbert's first problem in his famous list [Hi] (the continuum hypothesis). In this paper we show that it is consistent with Zermelo-Frankel set theory with the full Axiom of Choice (modulo large cardinals) that for every set X there is a set Y such that the cardinality of Y lies strictly between the cardinality of X and the cardinality of the power set of X. It was previously shown in GCdel [G] that the generalized continuum hypothesis (G.C.H) was consistent; i.e., for every infinite cardinal X the cardinal successor to X was 2X. In 1963, Cohen [Co] showed that it was consistent that the continuum hypothesis failed. Easton [E] showed that subject to relatively mild restrictions (Kdnig's theorem) essentially arbitrary behavior of the power set operation could occur at regular cardinals. Singular cardinals presented a significantly more difficult matter. The first work on them was done by Prikry and Silver [P] and [Sil] who showed that the G.C.H. can fail at singular strong limit cardinals. Magidor [Ml] showed that it was consistent that it fail at the first singular strong limit and even that one could have the first failure of the G.C.H. at a singular strong limit. After Magidor's work it was generally believed that arbitrary behavior was possible. Silver, however, showed [Si] that if the G.C.H. holds below a singular cardinal K of uncountable cofinality then it holds at K. Galvin and Hajnal [G-H] showed that under more general conditions the behavior of the power set below a singular cardinal K of uncountable cofinality strongly affects the power set at K. These results are all local results. The consistent global behavior of the power set operation was not settled. We prove the following theorem.

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of small cardinality. Part II is logically independent of Part I.