 We add nothing that is needed to the differential and integral calculi by ‘completing’ a theory of real numbers with pseudo-irrationals and lawless irrationals, first because there are no gaps on the number line (PR §§181, 183, &amp; 191; PG 373, 460, 461, &amp; 473; WVC 35) and, second, because these alleged irrational numbers are not needed for a theory of the ‘continuum’ simply because there is no mathematical continuum. As the later Wittgenstein says (RFM V, §32), “[t]he picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers”. We have gone awry by misconstruing the nature of the geometrical line as a continuous collection of points, each with an associated real number, which has taken us well beyond the ‘natural’ picture of the number line in search of a “general theory of real numbers” (Han 2010).