Wittgenstein and set theory

Abstract

Wittgenstein was hostile towards set theory; see his remark in §22 of RFM II: ‘I believe and hope that a future generation will laugh at this hocus pocus’. At the same time, he says that what philosophy owes set theory is ‘tremendous’ and that this is something ‘deep’. I want to clarify these two statements and to reconcile them. The hocus‐pocus remark is mainly directed at the temptation to talk about an own world of sets, focussing on extensions and thereby forgetting the actual mathematical manoeuvres leading to them. Wittgenstein does not shy away from using set‐theoretical symbols and expressions, but he regularly deprives them of their genuine set‐theoretical character. When speaking of the ‘depth’ we encounter with regard to set theory, what he means is the depth explained in PI §111: It arises ‘through a misinterpretation of our forms of language’, and this is particularly so in set theory.