 Though Wittgenstein stresses differences where others see similarities (LFM 15), in his intermediate attacks on pseudo-irrationals and foundationalism, he is not just emphasizing differences, he is attacking set theory’s “pernicious idioms” (PR §173) and its “crudest imaginable misinterpretation of its own calculus” (PG 469–70) in an attempt to dissolve “misunderstandings without which [set theory] would never have been invented”, since it is “of no other use” (LFM 16–17). Complex and imaginary numbers have grown organically within mathematics, and they have proved their mettle in scientific applications, but pseudo-irrationals are inorganic creations invented solely for the sake of mistaken foundationalist aims. Wittgenstein’s main point is not that we cannot create further recursive real numbers—indeed, we can create as many as we want—his point is that we can only really speak of different systems (sets) of real numbers (RFM II, §33) that are enumerable by a rule, and any attempt to speak of “the set of all real numbers” or any piecemeal attempt to add or consider new recursive reals (e.g., diagonal numbers) is a useless and/or futile endeavour based on foundational misconceptions. Indeed, in 1930 manuscript and typescript (hereafter MS and TS, respectively) passages on irrationals and Cantor’s diagonal, which were not included in PR or PG, Wittgenstein says: “The concept ‘irrational number’ is a dangerous pseudo-concept” (MS 108, 176; 1930; TS 210, 29; 1930).  As we shall see in the next section, on Wittgenstein’s account, if we do not understand irrationals rightly, we cannot but engender the mistakes that constitute set theory.