Ordinal analysis of partial combinatory algebras

TL;DR

This paper introduces a hierarchy of extensionality relations for partial combinatory algebras (pca) using ordinals, analyzing their closure ordinals and complexities, especially in Kleene's models.

Contribution

It defines and investigates the hierarchy of extensionality relations for pca's, calculating their closure ordinals and complexities, including exact results for Kleene's models.

Findings

Closure ordinal of Kleene's first model is ω₁^CK.

Closure ordinal of Kleene's second model is ω₁.

Extensionality relations in Kleene's first model cover the hyperarithmetical hierarchy.

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.