September 27
Victoria Gitman, CUNY
Baby measurable cardinals

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.