 For the same reason, two sets of thoughts that are structurally similar in the sense that they can be expressed, under different interpretations, by the same set of sentences, can differ with respect to Frege-consistency. As applied to the geometric context, the central idea, on this account of Frege’s objection to Hilbert, is that the kinds of re-interpretation in which Hilbert engages can take one from a consistent set of thoughts (e.g., \(\textit{AX}_{R}\)) to an inconsistent one (e.g., \(\textit{AX}_{G}\)) because of the shift in subject-matter, hence invalidating the inference from the consistency of the first to the consistency of the second. 