Ordinal analysis of partial combinatory algebras

Abstract

For every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca's, i.e. the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene's first model is $\omega_1^\textit{CK}$ and that the closure ordinal of Kleene's second model is $\omega_1$. We calculate the exact complexities of the extensionality relations in Kleene's first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca's.