Corey Switzer (University of Vienna)

Baumgartner's Axiom and its Higher Dimensional Versions

Note: the speaker will be joining us online.

A set of reals $A \subseteq \mathbb{R}$ is called $\aleph_1$-dense if its intersection with every nonempty open interval has size $\aleph_1$. Baumgartner's axiom (BA) is the statement that every pair of $\aleph_1$-dense set of reals are isomorphic as linear orders. BA is the most straightforward generalization of Cantor's theorem about countable dense linear orders to the uncountable. This axiom, proved consistent by Baumgartner in 1973, while seemingly innocuous is actually both very finnicky and also seems to induce a lot of structure on the reals. For instance (on the finnicky side) it is implied by PFA, but not MA, even in the presence of strong reflection principles. On the '"induces a lot of structure" side, it implies the bounding number is greater than $\aleph_1$ (Todorčević). BA also has a natural generalization to higher dimensions i.e. $\mathbb{R}^n$ for $n > 1$ and these versions do follow from MA and in fact weaker cardinal characteristic assumptions (Steprāns-Watson). In this talk we will discuss these issues and show that the higher dimensional versions however also induce a lot of structure on the reals, in particular for every natural number $n$ BA for $\mathbb{R}^n$ implies the bounding number is bigger than $\aleph_1$.

This is joint work with Juris Steprāns.