  Another mathematical “possible” to which Weyl gave a  great deal of thought is the continuum. During the period  1918–1921 he wrestled with the problem of providing the  mathematical continuum—the real number line—with a  logically sound formulation. Weyl had become increasingly critical of the principles underlying the set-theoretic construction of the  mathematical continuum. He had come to believe that the whole  set-theoretical approach involved vicious   circles[11]  to such an extent that, as he says, “every cell (so to speak)  of this mighty organism is permeated by contradiction.” In  Das Kontinuum he tries to overcome this by providing  analysis with a predicative formulation—not, as  Russell and Whitehead had attempted, by introducing a hierarchy of  logically ramified types, which Weyl seems to have regarded as  excessively complicated—but rather by confining the  comprehension principle to formulas whose bound variables range over  just the initial given entities (numbers). Accordingly he restricts  analysis to what can be done in terms of natural numbers with the aid of three basic logical operations, together with the operation of  substitution and the process of “iteration”, i.e.,  primitive recursion. Weyl recognized that the effect of this  restriction would be to render unprovable many of the central results of classical analysis—e.g., Dirichlet’s principle that any  bounded set of real numbers has a least upper   bound[12]—but  he was prepared to accept this as part of the price that must be paid for the security of mathematics.