Matthew Foreman

Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

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.

A very weak square principle

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L . This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ -balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Changs Conjecture: In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

Large cardinals and definable counterexamples to the continuum hypothesis

Abstract In this paper we consider whether L( R ) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L( R ) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Changs conjecture.

A new Löwenheim-Skolem theorem

This paper establishes a refinement of the classical Lowenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.

Canonical structure in the universe of set theory: part two

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 (1–3) (2004) 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. c

THE CLUB GUESSING IDEAL: COMMENTARY ON A THEOREM OF GITIK AND SHELAH

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

On invariants for ₁-separable groups

We study the classiflcation of !1-separable groups us- ing Ehrenfeucht-Fra˜‡ssgames and prove a strong classiflcation re- sult assuming PFA, and a strong non-structure theorem assuming }.

An ℵ1-dense ideal on ℵ2

This paper establishes the consistency of a countably complete, uniform, ℵ1-dense ideal on ℵ2. As a corollary, it is consistent that there exists a uniform ultrafilterD on ω2 such that |ω1ω2D|=ω1. A general “transfer” result establishes the consistency of countably complete uniform ideal K on ω2 such thatP(ω2)/K≅P(ω1)/ {countable sets}.

A Mixture of Gaussians Approach to Mathematical Portfolio Oversight: The EF3M Algorithm

An analogue can be made between: (a) the slow pace at which species adapt to an environment, which often results in the emergence of a new distinct species out of a once homogeneous genetic pool and (b) the slow changes that take place over time within a fund, mutating its investment style. A funds track record provides a sort of genetic marker, which we can use to identify mutations. This has motivated our use of a biometric procedure to detect the emergence of a new investment style within a funds track record. In doing so, we answer the question: What is the probability that a particular PMs performance is departing from the reference distribution used to allocate her capital? The EF3M algorithm, inspired by evolutionary biology, may help detect early stages of an evolutionary divergence in an investment style and trigger a decision to review a funds capital allocation.

Organic and tight

Abstract 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).