Potential Cardinality for Countable First Order Theories
University of Maryland
Give a theory $T$, understanding the countable model theory of $T$ has long been a topic of research. The number of countable models of $T$ is a classical but very coarse invariant, and this was refined significantly by Friedman and Stanley with the notion of Borel reductions.
Given theories $T_1$ and $T_2$, it is often straightforward to show that $T_1$ is Borel reducible to $T_2$. However, there are few tools to show that no such Borel reduction exists. Most of the existing tools only work when the isomorphism relation of one or both is particularly simple, or at least Borel.
We define the notion of “potential cardinality” of $T$, denoted $|T|$, as the number of formally consistent, possibly uncountable Scott sentences which imply $T$. It turns out that if $T_1$ Borel reduces to $T_2$, then $|T_1|$ is less than or equal to $|T_2|$. Additionally, it turns out that very frequently, $|T|$ can be computed and is not a proper class.
We use this idea to give a new class of examples of first-order theories whose isomorphism relations are neither Borel nor Borel complete. Along the way we answer an old question of Koerwien and new question of Laskowski and Shelah.
This is joint work with Douglas Ulrich and Chris Laskowski.