CEFISES Seminar: Stéphanie Ponsar, “The Multiverse in Mathematics”

Series: Work in Progress

Abstract: The idea of the multiverse in mathematics emerged as a consequence of the forcing technique of Cohen which allows to generate independence results and hence, conflicting models of set theory. The multiverse appears as an adequate tool to represent the co-existence of these models of set theory.

Although there are several notions of the multiverse in mathematics, the multiverse view is not unanimously accepted among set theorists and philosophers of mathematics. Currently, a vivid controverse remains between the tenants of the multiverse view and the ones of the universe view. The central question of the controversy is whether forcing extensions of the universe exist i.e. whether the multiverse is an appropriate meta-mathematical object to study set theoretic interpretations or whether it is a kind of “transient phenomena”.

We will first highlight the history of the universe notion in mathematics. Then, the three main points of view in the multiverse controversy will be presented : the multiverse view of Hamkins and the one of Väänänen and the universe view of Woodin. They will be compared according to different criteria such as the aim and status of the multiverse, the set conception, the notion of truth, the foundational status of set theory and the philosophical background. We will then examine how these criteria contribute to the discussion of the central question of the controversy.