This will be a talk for the workshop conference Mathematical Logic and Its Applications, which will be held at the Research Institute for Mathematical Sciences, Kyoto University, Japan, September 26-29, 2016, organized by Makoto Kikuchi. The workshop is being held in memory of Professor Yuzuru Kakuda, who was head of the research group in logic at Kobe University during my stay there many years ago.
Abstract. Set-theoretic potentialism is the ontological view in the philosophy of mathematics that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-$V_\beta$, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true. For example, a Grothendieck-Zermelo universe $V_\kappa$, with $\kappa$ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from $V_\kappa$ just in case $\kappa$ is a $\Sigma_3$-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting $V_\kappa\prec V$.
This is joint work with Øystein Linnebo, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing. Our research article is currently in progress.
Isn’t this exactly the same as good old creative sets from recursion theory, just rebranded to provide for new publishing opportunities?
No, this is not the same. The inspiration of set-theoretic potentialism is to be found rather, first, in the ancient Greek ideas about potential as opposed to actual infinity; and second, the early 20th century ideas of Zermelo, who had explicitly conceived of the rank-initial segments of the set-theoretic universe as having a potential character. The new contribution of our work is to identify exactly the modal theory of the corresponding modalities, and particularly to identify the various maximality principles for set-theoretic potentialism. None of this has anything to do with creative sets, but is rather connected with large cardinals and forcing. The idea that set-theoretic potentialism is “exactly the same as” creative sets, “rebranded,” is absurd.