The focus of this talk is the question of what a computable process can output by passing to a nonstandard model of arithmetic. It is not difficult to see that a computable process can change its output by passing to a nonstandard model, but in fact, for some processes, we can thus affect any arbitrary desired change. I will discuss and prove a theorem of Woodin, recently generalized by Blanck and Enayat, showing that for every computably enumerable theory $T$ extending ${\rm PA}$, there is a corresponding index $e$ such that ${\rm PA}\vdash “W_e$ is finite” and whenever a model $M\models T$ satisfies that $W_e$ is contained in some $M$-finite set $s$, then $M$ has an end-extension $N\models T$ in which $W_e=s$. Indeed, the hypotheses can be relaxed to say that $T$ extends ${\rm I}\Sigma_1$, but I will not discuss this in the talk.

An extended abstract can be found here on my blog.