Chains and antichains of finitely axiomatizable theories conservative over Peano arithmetic

by kamerynwilliams

This will be a talk at the CUNY Models of Peano Arithmetic Seminar on Wednesday, October 18.

It is well-known that there are finitely axiomatizable theories which are conservative over Peano arithmetic. Perhaps the best known of these is \mathsf{ACA}_0, the third of the “big five” subsystems of second-order arithmetic from reverse mathematics. Other examples of such theories come from the addition of satisfaction classes, as a consequence of a famous result by Kotlarski, Krajewski, and Lachlan. In unpublished work, Krajewski extended these ideas to construct chains and antichains of finitely axiomatizable theories conservative over \mathsf{PA}. I will present his results. Namely, I will show (1) that there are length \omega ascending and descending \subseteq-chains of finitely axiomatizable theories conservative over \mathsf{PA}. We will also get chains of such theories of order \mathbb Q. I will additionally show (2) that there are countable \subseteq-antichains of finitely axiomatizable theories conservative over \mathsf{PA}.