Philosophy of Mathematics seminar




Sam Roberts (Oslo)
Ultimate V
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 nonstructuralists. 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.

