Set-theoretic forcing as a computational process, Midwest Computability Seminar, Chicago, May 2023

This is a talk for the MidWest Computability Seminar conference held May 2, 2023 at the University of Chicago. The talk will be available via Zoom at https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09.

Abstract: I shall explore several senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.

The paper is available on the arxiv at https://arxiv.org/abs/2007.00418.

The talk took place in “The Barn” in the upper space between the Reyerson Laboratory and Eckhart Hall, where the University of Chicago Department of Mathematics is located:

The Tennenbaum phenomenon for computable quotient presentations of models of arithmetic and set theory, Shanghai, August 2021

This will be a talk for the conference Fudan Model Theory and Philosophy of Mathematics, held at Fudan University in Shanghai and online, 21-24 August 2021. My talk will take place on Zoom on 23 August 20:00 Beijing time (1pm BST).

Abstract. Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I shall discuss the Tennenbaum phenomenon as it arises for computable quotient presentations of models. Quotient presentations offer a philosophically attractive treatment of identity, a realm in which questions of identity are not necessarily computable. Objects in the presentation serve in effect as names for objects in the final quotient structure, names that may represent the same or different items in that structure, but one cannot necessarily tell which. Bakhadyr Khoussainov outlined a sweeping vision for quotient presentations in computable model theory and made several conjectures concerning the Tennenbaum phenomenon. In this talk, I shall discuss joint work with Michał Godziszewski that settles and addresses several of these conjectures.