Let us focus on the second question for a moment. Consider the following analogy. Recall that a forcing Q satisfies the countable chain condition (c.c.c.) if every uncountable collection of… (More)

We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically… (More)

In [23], Todorcevic gives a survey of basis problems in combinatorial set theory, listing nine theorems and six working conjectures — all in the presence of PFA — including the following three of… (More)

The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah.

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or “open coloring axioms.” In particular I will show that the… (More)

A coloring c : [X]n → Z is said to be irreducible if for every Y ⊆ X of equal cardinality c′′[Y ]n = Z. The focus of this note will be to show that there are continuous irreducible colorings on sets… (More)

In this article I will analyze the impact of forcing with a measure algebra on various topological statements. In particular our interest will focus on the study of hereditary separability and the… (More)

A space X is called an α-Toronto space if X is scattered of CantorBendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprans by constructing a… (More)

The focus of the following lectures is on forcing axioms in the presence of the Continuum Hypothesis. Early on Jensen showed that Souslin’s Hypothesis is relatively consistent with CH. While Martin’s… (More)

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or “open coloring axioms.” In particular it will be shown that the… (More)