Giorgio Genovesi (University of Leeds) – The tree interpretation and transfinite recursion

The theory of ZFC without powerset is interpretable in the system $Z_2$ of second order arithmetic. The interpretation is usually done in two steps. The first is to interpret an intermediate theory of sets by considering well founded trees modulo an appropriate equivalence relation. The second is by defining the constructible universe $L$ in the intermediate system.

I will go over some of the ways such interpretations are suited in more general systems that may have uncountable sets and restricted separation. I will also talk about how it turns out that the weakest principle needed to carry out the tree interpretation can be characterized in terms of clopen games and transfinite recursion. This is a joint work with Emanuele Frittaion.