Forcing and the Universe of Sets: Must We Lose Insight?
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions.
KeywordsFoundations of mathematics Set theory Forcing Potentialism Multiversism
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The author wishes to thank Carolin Antos, Tim Button, Andrés Eduardo Caicedo, Monroe Eskew, Sy Friedman, Vera Flocke, Victoria Gitman, Bob Hale, Joel Hamkins, Toby Meadows, Sandra Müller, Alex Paseau, Ian Rumfitt, Chris Scambler, Sam Roberts, Jonathan Schilhan, Daniel Soukup, Kameryn Williams, and three anonymous reviewers for insightful and helpful comments, as well as audiences in Cambridge, Konstanz, Vienna, and Toulouse for the opportunity to present and subsequent discussion. He is also very grateful for the generous support of the UK Arts and Humanities Research Council (as a PhD student), FWF (Austrian Science Fund, Project P 28420), and VolkswagenStiftung (through the project Forcing: Conceptual Change in the Foundations of Mathematics).
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