Jing Zhang: Rado’s Conjecture and its Baire Version II

Mathematical logic seminar – Jan 30 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     Rado’s Conjecture and its Baire Version II

Abstract:

Rado’s Conjecture is a reflection/compactness principle formulated by Todorčević, who also showed its consistency relative to the existence of strongly compact cardinals. One of its equivalent forms asserts that any nonspecial tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1. Although it is incompatible with Martin’s Axiom, Rado’s Conjecture turns out to imply a lot of consequences of forcing axioms, for example Strong Chang’s Conjecture, failure of square principles, the semi-stationary reflection principle, the Singular Cardinal Hypothesis etcetera. In fact, almost all known consequences of Rado’s Conjecture are consequences of a weaker statement, the Baire version of it which asserts any Baire tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1.

We will show that in the forcing extension by countable support iteration of Sacks forcing of strongly compact length, the Baire version of Rado’s Conjecture holds. Using a classical Mitchell style model, we show Rado’s conjecture along with not-CH does not imply ω2 has the super tree property, answering a question by Torres-Pérez and Wu. We will also see that in general the Baire version of Rado’s Conjecture does not imply Rado’s Conjecture.