Hilbert's tenth problem, G\"odel's incompleteness, Halting problem, a unifying perspective

TL;DR

This paper presents a unifying framework based on a property P that encapsulates key theorems like Hilbert's tenth problem, G"odel's incompleteness, and the halting problem, highlighting their common roots in self-reference and diagonalization.

Contribution

It introduces a general property P that unifies several fundamental theorems and extends to various rings, providing new insights into their interconnectedness and underlying principles.

Findings

Theorems on Hilbert's tenth problem, G"odel's incompleteness, and halting problem are unified under property P.

Strengthening property P yields Tarski's definability theorem.

The framework emphasizes self-reference and diagonalization as common themes.

Abstract

We formulate a property $P$ on a class of relations on the natural numbers, and formulate a general theorem on $P$, from which we get as corollaries the insolvability of Hilbert's tenth problem, G\"odel's incompleteness theorem, and Turing's halting problem. By slightly strengthening the property $P$, we get Tarski's definability theorem, namely that truth is not first order definable. The property $P$ together with a "Cantor's diagonalization" process emphasizes that all the above theorems are a variation on a theme, that of self reference and diagonalization combined. We relate our results to self referential paradoxes, including a formalisation of the Liar paradox, and fixed point theorems. We also discuss the property $P$ for arbitrary rings. We give a survey on Hilbert's tenth problem for quadratic rings and for the rationals pointing the way to ongoing research in main stream mathematics involving recursion theory, definability in model theory, algebraic geometry and number theory.