This will be a talk on November 6, 2017 for the Logic and Metaphysics workshop at the CUNY Graduate Center, run by Graham Priest.

## The modal principles of potentialism in mathematics

**Abstract.** Potentialism is the view in the philosophy of mathematics that one’s mathematical universe, whether in arithmetic or set theory, is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. As in the classical dispute between actual versus potential infinity, the potentialist holds that objects in the upper or outer reaches have potential as opposed to actual existence, in the sense that one can imagine forming or discovering always more objects from that realm, as many as desired, but the task is never completed. In this talk, I shall describe a general model-theoretic account of the modal logic of potentialism, identifying specific modal principles that hold or fail depending on features of the potentialist system under consideration. This work makes use of modal control statements, such as buttons, switches, dials and ratchets and the connection of these kinds of statements with the modal theories S4.2, S4.3 and S5. I shall take the various natural kinds of set-theoretic potentialism as illustrative cases.

This is joint work with Øystein Linnebo, University of Oslo. Our paper is available: The modal logic of set-theoretic potentialism and the potentialist maximality principles.

- J. D. Hamkins and Ø. Linnebo, “The modal logic of set-theoretic potentialism and the potentialist maximality principles.” (manuscript under review)
`@ARTICLE{HamkinsLinnebo:Modal-logic-of-set-theoretic-potentialism, author = {Hamkins, Joel David and Linnebo, \O{}ystein}, title = {The modal logic of set-theoretic potentialism and the potentialist maximality principles}, journal = {}, year = {}, volume = {}, number = {}, pages = {}, month = {}, note = {manuscript under review}, abstract = {}, keywords = {}, source = {}, eprint = {1708.01644}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/set-theoretic-potentialism}, doi = {}, }`