Abstract

Recent research into the representability of forcing extensions within ground models has often focussed on semantic formulations; we find definable class models within V that express very closely what it means to be a forcing extension of V (such as Hamkins’ Naturalist Account of Forcing). In this paper, I argue that while this formulation appears to be a good candidate for interpreting forcing over V, it is problematic in that it is unable to interpret theorems concerning objects below a measurable cardinal whilst keeping the ultrapower well-founded, and also encounters difficulties in interpreting class forcings. Instead I suggest that the motivation of a strong class theory (such as MK) over V, combined with a syntactic approach to forcing in a strengthened logic, is able to provide an interpretation of various forcing constructions which does not encounter these difficulties.