 Gödel [1933e] interpreted these results as showing that intuitionistic logic and arithmetic are richer than classical logic and arithmetic, because the intuitionistic theory distinguishes formulas which are classically equivalent, and has the same  consistency strength  as the classical theory. In particular, Gödel’s incompleteness theorems apply to \(\mathbf{HA}\) as well as to \(\mathbf{PA}.\)