October 31
Corey Switzer,
University of Vienna
Adding Isomorphisms between Dense Sets of Reals
In this talk we will discuss some problems regarding adding linear order isomorphisms between dense sets of reals via forcing. A set of reals $A \subseteq \mathbb R$ is called $\aleph_1$-dense just in case it has intersection size $\aleph_1$ with every nonempty open interval. Famously Baumgartner showed in the 70s that consistently all $\aleph_1$-dense sets of reals are order isomorphic, thus establishing the consistency of the natural analogue of Cantor's categoricity theorem for countable dense linear orders at the uncountable. He later showed the same statement follows from PFA. The statement 'all $\aleph_1$-dense sets of reals are isomorphic' is now known as Baumgartner's axiom and denoted BA.
Baumgartner's argument is famously tricky and has several interesting features. Given $\aleph_1$-dense sets $A$ and $B$ he shows that under the continuum hypothesis there is always a ccc partial order for making them isomorphic - but here the CH is important (though it must fail in the final model). Obviously it is therefore natural to ask whether the CH is necessary and, similarly whether BA follows already from MA and not just PFA. Avraham and Shelah showed the answer is 'no' to both: they produced a model of MA in which there is an $\aleph_1$-dense set of reals $A$ so that no ccc forcing notion can add an isomorphism between $A$ and its reverse ordering. In the first part of this talk we will strengthen this result by showing that MA is in fact consistent with an $\aleph_1$-dense linear order $A$ so that any partial order of size $\aleph_1$ which adds an isomorphism between $A$ and its reverse ordering must collapse $\aleph_1$. Thus it is consistent with MA that no such order can even be proper. This part is joint work with Pedro Marun and Saharon Shelah.
In another direction there is a natural generalization of BA to non-ordered spaces, most notably higher dimensional Euclidean spaces $\mathbb R^n$ for $n > 1$ as well as compact $n$-dimensional manifolds. Curiously, in these cases Steprans and Watson showed that the corresponding BA statements do indeed follow from MA so the case of dimension one is unique. In particular BA for $\mathbb R^n$ with $n > 1$ does not imply the one dimensional case. They then conjectured that conversely BA implies its higher dimensional analogues. In the second part of the talk we will introduce some intrigue to this conjecture by showing any 'reasonable' way of forcing BA - a very general adjective that includes all known methods - must necessarily force the higher dimensional versions and much more including large fragments of MA.