What If Turing Had Preceded Gödel?
Sebastian Oberhoff
TL;DR
By reframing incompleteness as a computation problem, the paper replaces Gödel-number coding with a direct arithmetic encoding and demonstrates that $PA$ can represent every computable function, supporting its computational completeness. It then provides simplified proofs of the Diagonal Lemma and the First Incompleteness Theorem, derives three variants of the Second Incompleteness Theorem, and connects these results to the Halting Problem and oracle concepts. The work clarifies the computational content underlying foundational logic and links incompleteness to broader themes in computability and complexity. Overall, it advances a computation-centric perspective on logic that yields more accessible proofs and a richer dialogue between logic and theory of computation.
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 Gödel 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.