Logic Workshop

(One version of) Cantor's second best theorem states that every pair of countable, dense sets of reals are isomorphic as linear orders. From the perspective of set theory it's natural to ask whether some variant of this theorem can hold consistently when 'countable' is replaced by 'uncountable'. This was shown in the affirmative by Baumgartner in 1973 who showed the consistency of 'all $\aleph_1$-dense sets of reals are order isomorphic' where a set is $\kappa$-dense for a cardinal $\kappa$ if its intersection with any open interval has size $\kappa$. The above became known as Baumgartner's axiom, denoted BA, and is an important axiom in both combinatorial set theory and set theoretic topology. BA has natural higher dimensional analogues - i.e., statements with the same relation to $\mathbb R^n$ that BA has to $\mathbb R$. It is a long standing open conjecture of Steprāns and Watson that BA implies its higher dimensional analogues.

In the talk I will describe some attempts to break the ice on this open problem mostly by looking at a family of weaker and stronger variants of BA and investigating their combinatorial, analytic and topological consequences. We will show that while some weak variants of BA have all the same consequences as BA, even weaker ones do not. Meanwhile a strengthening of BA for Baire and Polish space gives much more information.

Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in $\textbf{ZFC}$. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.

Measurable cardinals and other large cardinals on the larger side of things are characterized by the existence of elementary embeddings $j:V\to \mathcal M$ from the universe $V$ of sets into a transitive submodel $\mathcal M$. The clear pattern the large cardinals in that region follow is that the closer the submodel $\mathcal M$ is to $V$ the stronger the large cardinal notion. Smaller large cardinals, such as weakly compact or Ramsey cardinals, are known chiefly for their combinatorial properties, such as the existence of large homogeneous sets for colorings. But, it turns out that they too have elementary embeddings characterizations with embeddings on the correspondingly small models $M$ of (a fragment) of set theory (usually ${\rm ZFC}^-$, the theory ${\rm ZFC}$ with powerset axiom removed). Elementary embeddings of $V$ are often by-definable with the existence of certain ultrafilters or systems of ultrafilters. The classical example is that $\kappa$ is measurable if and only if there is a $\kappa$-complete ultrafilter on $\kappa$. The model $\mathcal M$ is then the transitive collapse of the ultrapower of $V$ by $U$. The connection between elementary embedding and ultrafilters also exists in the case of the small elementary embeddings. A typical elementary embedding characterization of a small large cardinal $\kappa$ follows the following template: for every $A\subseteq\kappa$, there is a (technical condition) model $M$, with $A\in M$, for which there is an $M$-ultrafilter $U$ on $\kappa$ with (technical properties). A subset $U\subseteq P(\kappa)\cap M$ is an $M$-ultrafilter if the structure $\langle M,\in, U\rangle$, with a predicate for $U$, satisfies that $U$ is a $\kappa$-complete ultrafilter on $\kappa$, meaning that $U$ measures all the sets in $M$ and its completeness applies to sequences that are elements of $M$. The reason we need to add a predicate for $U$ is that in most interesting case, and in contrast to the situation with measurable cardinals, $U$ is not an element of $M$ (indeed in most cases, $P(\kappa)$ does not exist in $M$). While the structure $M$ usually satisfies some large fragment of ${\rm ZFC}$, once, we add a predicate for the $M$-ultrafilter $U$, the structure $\langle M,\in, U\rangle$ can fail to satisfy even $\Sigma_0$-separation. In this talk, I will discuss how smaller large cardinals follow the pattern that the more set theory the structure $\langle M,\in, U\rangle$ satisfies the stronger the resulting large cardinal notion. I will use these observations to introduce a new hierarchy of large cardinals between Ramsey and measurable cardinals. This is joint work with Philipp Schlicht, based on earlier work by Bovykin and McKenzie.