Ordinal analysis via provability logics and ordinal spectra of arithmetical theories
Dept. Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona
If we start with a sound `weak’ theory –say Primitive Recursive Arithmetic– we can iterate adding consistency statements to it to get stronger and stronger theories. We call the $Pi^0_1$ ordinal of an arithmetical theory U how often one has to iterate adding consistency to PRA “before you hit on U”. We shall see how provability logics are suited for performing large part of the computation. Using different notions of consistency statements yields different ordinals giving rise to an ordinal spectrum of an arithmetical theory. These spectra can be collected into a nice structure and well-known structure: Ignatiev’s universal model for GLP_omega. We shall see how this structure can be used as a roadmap to $Pi_n$ conservation results between fragments.