Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic

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 ${\mathrm{\Sigma }}_n$-ill” is a canonical example of a ${\mathrm{\Sigma }}_{n+1}$ formula that is ${\mathrm{\Pi }}_{n+1}$-conservative over $T$.