 Second, and more fundamentally, the intermediate Wittgenstein wrestles with irrationals in such detail because he opposes foundationalism and especially its concept of a “gapless mathematical continuum”, its concept of a comprehensive theory of the real numbers (Han 2010), and set theoretical conceptions and ‘proofs’ as a foundation for arithmetic, real number theory, and mathematics as a whole. Indeed, Wittgenstein’s discussion of irrationals is one with his critique of set theory, for, as he says, “[m]athematics is ridden through and through with the pernicious idioms of set theory”, such as “the way people speak of a line as composed of points”, when, in fact, “[a] line is a law and isn’t composed of anything at all” (PR §173; PR §§181, 183, &amp; 191; PG 373, 460, 461, &amp; 473).