What If Turing Had Preceded G\"odel?

TL;DR

This paper reinterprets mathematical logic as fundamentally about computation, simplifying proofs of key theorems, eliminating G"odel numbers, and connecting logic with computational complexity and oracles.

Contribution

It introduces a novel notation for representable functions, simplifies proofs of the Incompleteness Theorems, and links logical incompleteness to computational complexity concepts.

Findings

Peano Arithmetic can represent all computable functions

Simplified proofs of the Incompleteness Theorems

Established connections between logic and computational complexity

Abstract

The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will show the following. * First, we'll all but eliminate the need for G\"odel numbers. * Next, we'll introduce a novel notational device for representable functions and walk through a condensed demonstration that Peano Arithmetic can represent every computable function. It has achieved Turing completeness. * Continuing, we'll derive the Diagonal Lemma and First Incompleteness Theorem using significantly simplified proofs. * Approaching the Second Incompleteness Theorem, we'll be able to use some self-referential trickery to avoid much of the technical morass surrounding it; arriving at three separate versions. * Extending the analogy between the First Incompleteness Theorem and the Unsolvability of the Halting Problem produces an equivalent of the Nondeterministic Time Hierarchy Theorem from the field of computational complexity. * Lastly, we'll briefly peer into the realm of the uncomputable by connecting our ideas to oracles.