It is increasingly popular to think that the mathematical universe is inherently potential. One of the main reasons comes from the method of forcing in set theory. Given a universe of sets, we can use the method of forcing to characterize universes extending it. What more could be needed for such universes to exist? There is thus no ultimate universe of sets, no ultimate V: every universe can be extended via forcing. In this talk I argue against this view. I start by showing that given mild metaphysical assumptions, there is an ultimate universe of sets, where sets are thought of as structures. I then argue that this result is significant even for non-structuralists. In particular, I show that if we combine those assumptions with the claim that any sets could have formed a set, there is an ultimate universe of sui generis sets.
See the seminar webpage http://users.ox.ac.uk/~philmath/pomseminar.html for titles and abstracts of other speakers as available.
Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Volker Halbach