### 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 , 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 In unpublished work, Krajewski extended these ideas to construct chains and antichains of finitely axiomatizable theories conservative over . I will present his results. Namely, I will show (1) that there are length ascending and descending -chains of finitely axiomatizable theories conservative over . We will also get chains of such theories of order . I will additionally show (2) that there are countable -antichains of finitely axiomatizable theories conservative over .

[…] Last week, I gave an application—due to Krajewski—of iterated full satisfaction classes. As part of that we saw that if is resplendent then admits an iterated full satisfaction class of any length. But in general, these need not be inductive, as even a fragment of induction in the language with the satisfaction class is enough to prove . This week, we will consider inductive iterated full satisfaction classes. […]