What If Turing Had Preceded G\"odel?

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.