This was a talk for the Kobe Set Theory Workshop, held on the occasion of Sakaé Fuchino’s retirement, 9-11 March 2021.

**Abstract.** I shall discuss 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 of set theory $\langle M,\in^M\rangle$, for example, one may in various senses compute $M$-generic filters $G\subset P\in M$ and 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 $M$ that lead by the computational process to non-isomorphic forcing extensions $M[G]\not\cong M[G’]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

This is joint work with Russell Miller and Kameryn Williams.

Forcing as a computational process

- J. D. Hamkins, R. Miller, and K. J. Williams, “Forcing as a computational process,” Mathematics arXiv, 2020. (Under review)
`@ARTICLE{HamkinsMillerWilliams:Forcing-as-a-computational-process, author = {Joel David Hamkins and Russell Miller and Kameryn J. Williams}, title = {Forcing as a computational process}, journal = {Mathematics arXiv}, year = {2020}, volume = {}, number = {}, pages = {}, month = {}, note = {Under review}, abstract = {}, keywords = {under-review}, source = {}, doi = {}, url = {http://jdh.hamkins.org/forcing-as-a-computational-process}, eprint = {2007.00418}, archivePrefix = {arXiv}, primaryClass = {math.LO}, }`