Toronto Set Theory

Event Information The Abraham-Rubin-Shelah Open Coloring Axiom with a Large Continuum
13:30 on Thursday December 03, 2020
15:00 on Thursday December 03, 2020
Virtual
Thomas Daniells Gilton

University of Pittsburgh

The Open Coloring Axioms may be viewed as a consistent generalizations of Ramsey's Theorem to $\omega_1$ in which topological restrictions are placed on the colorings. The first of these, denoted $\mathsf{OCA}_{ARS}$, appeared in the 1985 paper by Abraham, Rubin, and Shelah. There, the authors showed that $\mathsf{OCA}_{ARS}$ is consistent with $\mathsf{ZFC}$. To ensure that the posets which add the homogeneous sets satisfy the c.c.c., they construct a type of ``diagonalization" object (for a continuous coloring $\chi$) called a preassignment of colors, which guides the forcing to add the $\chi$-homogeneous sets.

However, the only known constructions of effective preassignments require the $\mathsf{CH}$. Since a forcing iteration of $\aleph_1$-sized posets all of whose proper initial segments satisfy the $\mathsf{CH}$ results in a model in which $2^{\aleph_0}$ is at most $\aleph_2$, this leads naturally to the question of whether $\mathsf{OCA}_{ARS}$ is consistent, say, with $2^{\aleph_0}=\aleph_3$.

In joint work with Itay Neeman, we answer this question in the affirmative. In light of the $\mathsf{CH}$ obstacle, we only construct names for preassignments with respect to a small class $\mathcal{A}$ of $\mathsf{CH}$-preserving iterations. However, our preassignments are powerful enough to work even over models in which the $\mathsf{CH}$ fails.

Our final forcing is built by combining the members of $\mathcal{A}$ into a new type of forcing, called a partition product. A partition product is a type of restricted memory iteration with isomorphism and coherent-overlap conditions on the memories. In particular, each ``memory" is isomorphic to a member of $\mathcal{A}$.

In this talk, we will describe in some detail the definition of a Partition Product. We will then discuss how to construct more general preassignments than those used by Abraham, Rubin, and Shelah, gesturing at the end towards the full construction which we use for our theorem.

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More information can be found at http://www.fields.utoronto.ca/activities/20-21/set-theory-seminar