Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic

TL;DR

This paper extends classic incompleteness results to hierarchical frameworks within arithmetic, demonstrating that such generalizations are achievable through hierarchical principles and providing new versions of key theorems.

Contribution

It introduces hierarchical versions of fundamental incompleteness theorems, expanding their applicability to arithmetically definable extensions of arithmetic fragments.

Findings

Hierarchical versions of Mostowski's and Kripke's theorems established.

The formula expressing '$T$ is $\Sigma_n$-ill' is shown to be a canonical $\Sigma_{n+1}$ formula.

Hierarchical principles enable the derivation of generalized incompleteness results.

Abstract

There has been a recent interest in hierarchical generalisations of classic incompleteness results. This paper provides evidence that such generalisations are readibly obtainable from suitably hierarchical versions of the principles used in the original proof. By collecting such principles, we prove hierarchical versions of Mostowski's theorem on independent formulae, Kripke's theorem on flexible formulae, and a number of further generalisations thereof. As a corollary, we obtain the expected result that the formula expressing "$T$ is $\Sigma_n$-ill" is a canonical example of a $\Sigma_{n+1}$ formula that is $\Pi_{n+1}$-conservative over $T$.