Incompleteness Ex Machina

TL;DR

This paper provides two proofs of G"odel's incompleteness theorems, one classical and one computational, clarifying the distinction between the transport of computation into arithmetic and the theorems themselves.

Contribution

It offers a novel perspective by splitting G"odel's work into two conceptual parts and provides dual proofs, including a computational approach.

Findings

Proofs of G"odel's incompleteness theorems in two different frameworks

Clarification of the conceptual distinction in G"odel's work

Insight into the relationship between computation and arithmetic

Abstract

In this essay we'll prove G\"odel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that G\"odel's work, rightly viewed, needs to be split into two parts: the transport of computation into the arena of arithmetic on the one hand and the actual incompleteness theorems on the other. After we're done there will be cake.