 Even if set theory is unnecessary, it still might constitute a solid foundation for mathematics. In his core criticism of set theory, however, the later Wittgenstein denies this, saying that the diagonal proof does not prove non-denumerability, for “[i]t means nothing to say: ‘Therefore the X numbers are not denumerable’” (RFM II, §10). When the diagonal is construed as a proof of greater and lesser infinite sets it is a “puffed-up proof”, which, as Poincaré argued (1913: 61–62), purports to prove or show more than “its means allow it” (RFM II, §21).