# Transfinite Recursion from Gödel–Bernays to Kelley–Morse

This is a talk in the CUNY Set Theory Seminar on Friday, May 10 at 10:00am.

Gödel–Bernays set theory GB and Kelley–Morse set theory KM are two formal theories for second-order set theory, allowing both sets and proper classes as objects. GB is the weaker of the two theories, being conservative over ZF, while KM is the stronger. Set theorists have used KM in applications where GB is not strong enough; for instance, Kunen formulated his celebrated inconsistency result in the context of KM, as KM has the resources to directly allow talk of elementary embeddings of the universe of sets. But weaker theories than KM suffice for many of these applications. Between GB and KM there is a hierarchy of intermediate theories based upon restricting the logical complexity allowed in the comprehension axiom.

In this talk I will present a hierarchy of second-order set theories which refines the comprehension-based hierarchy. This hierarchy is based upon transfinite recursion principles, ordered first by the logical complexity of the properties allowed and second by the lengths of well-orders on which we may carry out the recursions. Theories in this hierarchy are separated in terms of consistency strength. The substantive new result to establish this hierarchy is the following: Let $k$ be a natural number. Suppose $(M,\mathcal{X})$ satisfies GB and that $\Gamma \in \mathcal{X}$ is a class well-order which is closed under addition. In case $k = 0$ further assume $\Gamma \ge \omega^\omega$. Then, if $(M,\mathcal{X})$ satisfies $\Pi^1_k$-Transfinite Recursion for recursions along $\Gamma$, there is $\mathcal{Y} \subseteq \mathcal{X}$ coded in $\mathcal{X}$ so that $(M,\mathcal{Y})$ satisfies GB plus the principle of $\Pi^1_k$-Transfinite Recursion for recursions along well-orders $< \Gamma$.