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.
  [Bibtex]

  @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},
  }