 As we saw above, in the paper from 1908 Brouwer had given weak counterexamples to PEM. In the 1920s Brouwer developed a general technique for constructing weak counterexamples which also made it possible to widen their scope and include principles of analysis. The development began in 1921, when Brouwer gave a weak counterexample to the proposition that every real number has a decimal expansion (Brouwer 1921A). The argument proceeded by defining real numbers whose decimal developments are dependent on specific open problems concerning the decimal development of \(\pi\). Brouwer closed by observing that, should these open problems be solved, then other real numbers without decimal expansion can be defined (Brouwer 1921A: 210 [Mancosu 1998: 34]). The general technique was made explicit in a lecture from 1923 (Brouwer 1924N: 3 and footnote 4 [van Heijenoort 1967: 337 and footnote 5]) and reached its perfection with the “oscillatory number” method in the first Vienna lecture in 1928 (Brouwer 1929A: 161 [Mancosu 1998: 51]). The method involves the reduction of the validity of a mathematical principle to the solvability of an open problem of the following type: we have a decidable property \(P\) (defined on the natural numbers) for which we have as yet shown neither \(\exists xP(x)\) nor \(\forall x\neg P(x)\). This reduction is carried out in such a way that it only uses the fact that \(P\) induces an open problem of this type, and does not depend on the exact definition of \(P\); that is, if the open problem is solved, one can simply replace it by another one of the same type, and exactly the same reduction still works. This uniformity means that, as long as there are open problems of this type at all (and this is practically certain at any time), there is no intuitionistic proof of the general mathematical principle in question. In the 1920s, Brouwer constructed weak counterexamples to the following general mathematical propositions, among others (where \(\mathbf{R}\) stands for the set of intuitionistic real numbers, and \(\mathbf{Q}\) for the set of rationals):