March 23
Mateusz Łełyk, University of Warsaw
Nonequivalent axiomatizations of PA and the Tarski Boundary: Part III

This is a continuation of the talk from Feb 16th. This time we shall study different theories of the form ${\rm CT}^-[\delta]$ which are conservative extensions of a ${\rm PA}$. In particular, we prove the following theorem.

Theorem 2 There exists a family $\{\delta_f\}_{f\in\omega^*}$ such that for all $f,g\in\omega^*$
1) ${\rm CT}^-[\delta_f]$ is conservative over ${\rm PA}$;
2) if $f\subsetneq g$, then ${\rm CT}^-[\delta_g]$ properly extends ${\rm CT}^-[\delta_f]$;
3) if $f\perp g$ then ${\rm CT}^-[\delta_g]\cup {\rm CT}^-[\delta_f]$ is nonconservative over ${\rm PA}$ (but consistent).

We will finish the proof of the theorem announced in the abstract of part II.