Toronto Set Theory Seminar

Friday, June 10 from 1:30 to 3pm

Fields Institute, Room 210

Speaker: Paul Larson

Title: Choosing Ideals

Abstract :

We show that, in certain inner models of determinacy, there is a definable procedure which, given a tall ideal $latex I$ on $latex omega$ containing all finite sets, and a function from $latex I setminus mathrm{Fin}$ to a countable set, chooses a finite subset of the range of this function. In the case we are most interested in, $latex I$ is generated by a countable collection of pairwise orthogonal ideals. In this context, $latex I$ represents a $latex mathcal{P}(omega)/mathrm{Fin}$-name for an element of a countable set. Our result then says that, whenever $latex M[U]$ is a $latex mathcal{P}(omega)/mathrm{Fin}$-extension of a model $latex M$ of the type we consider here, if $latex F$ is a function in $latex M[U]$ with domain $latex X in M$, and $latex G$ is a function in $latex M$ with domain $latex X$ such that, for all $latex x in X$, $latex F(x) in G(x)$ and $latex G(x)$ is countable, then there exists a function $latex G’$ in in $latex M$ with domain $latex X$ such that, for all $latex x in X$, $latex G'(x)$ is a finite subset of $latex G(x)$. This gives another proof of a theorem of Di Prisco-Todorcevic that in $latex M[U]$ there is no function which selects a single member from each $latex E_{0}$-equivalence class, where $latex E_{0}$ is the relation of mod-finite agreement on the Baire space.

Abstract :

We show that, in certain inner models of determinacy, there is a definable procedure which, given a tall ideal $latex I$ on $latex omega$ containing all finite sets, and a function from $latex I setminus mathrm{Fin}$ to a countable set, chooses a finite subset of the range of this function. In the case we are most interested in, $latex I$ is generated by a countable collection of pairwise orthogonal ideals. In this context, $latex I$ represents a $latex mathcal{P}(omega)/mathrm{Fin}$-name for an element of a countable set. Our result then says that, whenever $latex M[U]$ is a $latex mathcal{P}(omega)/mathrm{Fin}$-extension of a model $latex M$ of the type we consider here, if $latex F$ is a function in $latex M[U]$ with domain $latex X in M$, and $latex G$ is a function in $latex M$ with domain $latex X$ such that, for all $latex x in X$, $latex F(x) in G(x)$ and $latex G(x)$ is countable, then there exists a function $latex G’$ in in $latex M$ with domain $latex X$ such that, for all $latex x in X$, $latex G'(x)$ is a finite subset of $latex G(x)$. This gives another proof of a theorem of Di Prisco-Todorcevic that in $latex M[U]$ there is no function which selects a single member from each $latex E_{0}$-equivalence class, where $latex E_{0}$ is the relation of mod-finite agreement on the Baire space.