October 17
Elliot Glazer, Harvard University
Coin flipping on models of arithmetic to define the standard cut

We will discuss the following claim: 'The standard cut of a model $M$ of PA (or even Q) is uniformly definable with respect to a randomly chosen predicate.' Restricting our consideration to countable models, this claim is true in the usual sense, i.e. there is a formula $\varphi$ such that for any countable model of arithmetic $M,$ the set $S_M^{\varphi} := \{P \subset M: \omega = \{x \in M: (M, P) \models \varphi(x)\} \}$ is Lebesgue measure 1. However, if $M$ is countably saturated, then there is no $\varphi$ such that $S_M^{\varphi}$ is measured by the completed product measure on $2^M.$ We will identify various combinatorial ideals on $2^M$ that can be used to formalize the original claim with no restriction on the cardinality of $M,$ and discuss the relationship between closure properties of these ideals and principles of choice.