 The difficulties multiply from here. When Hilbert provides a specific reinterpretation of the geometric terms en route to proving the relative consistency of a set AX of sentences, Frege notes that we now have two different sets of thoughts in play: the set we might call “\(\textit{AX}_{G}\)” of thoughts expressed when AX’s terms take their ordinary geometric meanings (e.g., on which “point” means point) and the set we might call “\(\textit{AX}_{R}\)” of thoughts expressed when AX’s terms take the meanings assigned by Hilbert’s re-interpretation (on which, e.g., “point” means pair of real numbers). Hilbert’s reinterpretation strategy involves, from Frege’s point of view, simply shifting our attention from the set \(\textit{AX}_{G}\) of thoughts ordinarily expressed by the sentences AX (and in whose consistency we are interested) to the new set \(\textit{AX}_{R}\) of thoughts expressed by AX under the reinterpretation. And the fact that the reinterpreted sentences express truths about real numbers has little to do, from Frege’s perspective, with the consistency and independence questions that arise for the original thoughts about points, lines, and planes.