David Milovich

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In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2 ω = ω 2 and that L(P(ω 1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that (κ) fails for all(More)
Motivated by a question of Isbell, we show that ♦ implies there is a non-P-point U ∈ βω \ ω such that neither U , ⊇ nor U , ⊇ * is Tukey equivalent to [c] <ω , ⊆. We also show that U , ⊇ * ≡ T [c] <κ , ⊆ for some U ∈ βω \ ω, assuming cf(κ) = κ ≤ p = c. We also prove two negative ZFC results about the possible Tukey classes of ultrafilters on ω. 1. Tukey(More)
Remark. The notes which follow reflect the content of a two day tutorial which took place at the Fields Institute on 5/29 and 5/30 in 2009. Most of the content has existed in the literature for some time (primarily in the original edition of [10]) but has proved difficult to read and digest for various reasons. The only new material contained in these(More)
Extending some results of Malykhin, we prove several independence results about base properties of βω \ ω and its powers, especially the Noetherian type N t(βω \ ω), the least κ for which βω \ ω has a base that is κ-like with respect to containment. For example, N t(βω \ ω) is at least s, but can consistently be that ω 1 , c, c + , or strictly between ω 1(More)