Paul B. Larson

Katetov’s problem

In 1948 Miroslav Katetov showed that if the cube X 3 of a compact space X satisfies the separation axiom T 5 then X must be metrizable. He asked whether X 3 can be replaced by X 2 in this metrization result. In this note we prove the consistency of this implication.

THE FILTER DICHOTOMY AND MEDIAL LIMITS

The \emph{Filter Dichotomy} says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A function between topological spaces is \emph{universally measurable} if the preimage of %every open subset of the codomain is measured by every Borel measure on the domain. A \emph{medial limit} is a universally measurable function from

Separating Stationary Reflection Principles

\mathcal{P}(\omega)

Forcing over Models of Determinacy

to the unit interval [0,1] which is finitely additive for disjoint sets, and maps singletons to 0and

ON THE HEREDITARY PARACOMPACTNESS OF LOCALLY COMPACT, HEREDITARILY NORMAL SPACES

\omega

An Ω-logic Primer

to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. We show that the Filter Dichotomy implies that there are no medial limits.

Bounding by canonical functions, with ch

We present a variety of (ω, ∞)-distributive forcings which when applied to models of Martins Maximum separate certain well known reflection principles. In particular, we do this for the reflection principles SR, SR α (α ≤ ω 1 ), and SRP.

ABSOLUTENESS FOR UNIVERSALLY BAIRE SETS AND THE UNCOUNTABLE II

A theorem of Woodin states that the existence of a proper class of Woodin cardinals implies that the theory of the inner model L(ℝ) cannot be changed by set forcing. The Axiom of Determinacy is part of this fixed theory for L(ℝ). The partial order ℙmax is a forcing construction in L(ℝ) which lifts the absoluteness properties of L(ℝ) to models of the Axiom of Choice. The structure H(ω 2) in the ℙmax extension of L(ℝ) (assuming AD L(ℝ)) satisfies every Π2 sentence φ for H(ω 2) which is forceable from a proper class of Woodin cardinals. Furthermore, the partial order ℙmax can be easily varied to produce other consistency results and canonical models.

The canonical function game

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.

The stationary set splitting game

In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ω-logic and the Ω-conjecture.