Godel's Incompleteness Phenomenon - Computationally

TL;DR

This paper explores the implications of Godel's theorems, linking completeness and incompleteness to the decidability and constructiveness of theories, and establishes an analogue of Rice's Theorem for recursively enumerable theories.

Contribution

It provides a novel interpretation of Godel's theorems in terms of theory completion and undecidability, extending the understanding of logical properties.

Findings

Completeness is equivalent to the ability to complete consistent theories.

Incompleteness implies some theories cannot be extended to complete, consistent, and recursively enumerable theories.

Decidability of deduction and consistency is impossible in general logic.

Abstract

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable.