Forcing as a computational process
Abstract
We investigate how settheoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute $M$generic filters $G\subseteq\mathbb{P}\in M$ and the corresponding forcing extensions $M[G]$. Specifically, from the atomic diagram one may compute $G$, from the $\Delta_0$diagram one may compute $M[G]$ and its $\Delta_0$diagram, and from the elementary diagram one may compute the elementary diagram of $M[G]$. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial. For any such process, it will always be possible to have different isomorphic presentations of a model of set theory $M$ that lead to different nonisomorphic forcing extensions $M[G]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.00418
 Bibcode:
 2020arXiv200700418H
 Keywords:

 Mathematics  Logic
 EPrint:
 26 pages. Inquiries and commentary can be made at http://jdh.hamkins.org/forcingasacomputationalprocess. Minor updates with version 2