 As noted above, Frege views Hilbert’s reinterpretations as involving a shift of attention from geometric thoughts (whose consistency and independence are at issue) to thoughts of a wholly different kind, those about the background theory B (whose consistency and independence are not in question). Regarding consistency proofs, his view is that Hilbert makes an illegitimate inference from the consistency of a collection \(\textit{AX}_{R}\) of thoughts about real numbers to the consistency of a collection \(\textit{AX}_{G}\) of thoughts about geometric points, lines, and planes. Frege acknowledges that Hilbert’s set AX of sentences can be understood as providing an implicit definition of an abstract relation \(R_{\textit{AX}}\), one that is satisfied by Hilbert’s constructed n-tuples, and that the consistency (i.e., satisfiability) of \(\textit{AX}_{R}\) entails the consistency of that defined relation. But here too, Frege takes it that Hilbert’s crucial inference, from the consistency of \(R_{\textit{AX}}\) to the consistency of \(\textit{AX}_{G}\), is problematic. As Frege himself puts it, referring to \(\textit{AX}_{R}\) and \(\textit{AX}_{G}\) as “special geometries”, and to \(R_{\textit{AX}}\) as the “general case:” 