 The “hocus pocus” of the diagonal proof rests, as always for Wittgenstein, on a conflation of extension and intension, on the failure to properly distinguish sets as rules for generating extensions and (finite) extensions. By way of this confusion “a difference in kind” (i.e., unlimited rule vs. finite extension) “is represented by a skew form of expression”, namely as a difference in the cardinality of two infinite extensions. Not only can the diagonal not prove that one infinite set is greater in cardinality than another infinite set, according to Wittgenstein, nothing could prove this, simply because “infinite sets” are not extensions, and hence not infinite extensions. But instead of interpreting Cantor’s diagonal proof honestly, we take the proof to “show there are numbers bigger than the infinite”, which “sets the whole mind in a whirl, and gives the pleasant feeling of paradox” (LFM 16–17)—a “giddiness attacks us when we think of certain theorems in set theory”—“when we are performing a piece of logical sleight-of-hand” (PI §412; §426; 1945). This giddiness and pleasant feeling of paradox, says Wittgenstein (LFM 16), “may be the chief reason [set theory] was invented”.