Set Theory Seminar

October 19
Sam Sanders, TU Darmstadt/University of Leeds
From zero to second-order arithmetic via metastability or uniformity

Tao’s metastability, going back to Kreisel and Goedel, is a notion of convergence with nice computational properties. The trade-off involved in metastability is that one obtains uniform and effective information, but only about a finite (but arbitrarily large) domain. We apply this metastability trade-off, i.e. introducing finite domains to yield uniform and effective results, to concepts other than convergence. This results in theorems requiring full second order arithmetic for a proof. We obtain similar results for another notion inspired by proof theory, namely uniform theorems, in which the objects claimed to exist depend on few of the parameters in the theorem.