Pour-El's Landscape

TL;DR

This paper explores effective versions of incompleteness, undecidability, and inseparability, strengthening existing theorems and comparing different notions within the framework of Pour-El's insights.

Contribution

It advances the understanding of effective incompleteness and inseparability, providing new theorems and comparisons for finite extensions and hereditariness.

Findings

Strengthened Pour-El's theorem on effective essential incompleteness and inseparability.

Compared notions of effective incompleteness restricted to finite extensions.

Proved an adapted version of Pour-El's result for hereditariness and effectiveness.

Abstract

We study the effective versions of several notions related to incompleteness, undecidability and inseparability along the lines of Pour-El's insights. Firstly, we strengthen Pour-El's theorem on the equivalence between effective essential incompleteness and effective inseparability. Secondly, we compare the notions obtained by restricting that of effective essential incompleteness to intensional finite extensions and extensional finite extensions. Finally, we study the combination of effectiveness and hereditariness, and prove an adapted version of Pour-El's result for this combination.