Ordinal analysis of partial combinatory algebras

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