Note on ${\mathrm{\Pi }}_{n+1}^0$-LEM, ${\mathrm{\Sigma }}_{n+1}^0$-LEM and ${\mathrm{\Sigma }}_{n+1}^0$-DNE

We show that results of Akama, Berardi, Hayashi and Kohlenbach, on the relative independence of certain arithmetical principles over intuitionistic arithmetic HA, hold also over Kleene and Vesley’s system FIM of intuitionistic analysis, which extends HA and is consistent with classical arithmetic but not with classical analysis. The double negations of the universal closures of these principles are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions are left open.