Some Second Order Set Theory [chapter]

Joel David Hamkins
2008 Lecture Notes in Computer Science  
Mathematical Platonism. Many set theorists hold that there is just one universe of set theory, and our task is to understand it. Paradoxically, however, the most powerful tools in set theory are actually methods of constructing alternative universes. We build new models of set theory from existing models, via forcing and ultrapowers. These other models offer us glimpses of alternative universes and alternative truths. The Multiverse View. This philosophical position accepts these alternative
more » ... verses as fully existing mathematically. This is realism, not formalism, but rejects the uniqueness of the mathematical universe. This philosophical view has guided the research on which I speak. Second-order set theory, ICLA 2009 Joel David Hamkins, New York The extension V [G] is closely related to the ground model V , but exhibits new truths in a way that can be carefully controlled. Second-order set theory, ICLA 2009 Joel David Hamkins, New York p ϕ if every V -generic G with p ∈ G has V [G] |= ϕ. Forcing Lemmas 1 The forcing extension V [G] satisfies ZFC. This leads us naturally to the modal logic of forcing. A ground model has access, via names and the forcing relation, to the objects and truths of the forcing extension. So there is a natural Kripke model lurking here. The possible worlds are the models of set theory. The accessibility relation relates a model M to its forcing extensions M[G]. Many set theorists habitually operate within this Kripke model.
doi:10.1007/978-3-540-92701-3_3 fatcat:vw3nwx7b55d2rhnjt26os2g23y