# Beta models of ATR_0

This is a talk for the CUNY MoPA seminar on 29 November 2017.

Recall that $\mathsf{ATR}_0$ is the subsystem of second-order arithmetic axiomatized by (basic axioms plus) arithmetical comprehension plus arithmetical transfinite recursion. Also recall that a beta-model of arithmetic is a model $(\omega,\mathcal{X})$ of second-order arithmetic which is correct about which of its set relations are well-founded. I will present a theorem, due to Simpson, that the intersection of all beta-models of $\mathsf{ATR}_0$ is the omega-model whose sets are the hyperarithmetical sets.