Midlands Logic Seminar

Week 8, 3 March 2014 (16:00-18:00, 52 Pritchatts Road LT1)

Neil Barton (Birkbeck)

"Universism" about Set Theory (the view that there is a single, unique, maximal interpretation of set-theoretic discourse) has come under fire in recent years, partly because of the way in which Set Theory is practised. The characterisation of large cardinals in terms of proper-class-sized elementary embeddings is fairly common and provides a cohesive way of understanding higher axioms of infinity. However, proper classes in general, especially of the non-definable variety needed for interpreting certain embeddings, seem to invite paradox on the Univesist's picture. Forcing over V , on the other hand seems perfectly cogent, whilst the Universist is forced to deny its coherency. My talk explores these issues, analysing some recent presentations of technical material and using them to inform the philosophical debate. My aims are twofold:

1. Show that there is a deep relationship between representations of forcing extensions within a ground model and the class theory countenanced.
2. Show that these criticisms of Universism both depend upon a maxim concerning modality in the context of mathematics, a principle the Universist is likely to reject.