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What is a choice sequence ? How a solution of Troelstra ’ s paradox shows the way to an answer to this question

In 1907 L.E.J.Brouwer, then an unknown Dutch mathematician, stepped into the running debate on the origin and certainty of mathematics with his thesis On the foundations of mathematics (Brouwer 1907). In his view mathematics consists of mental constructions only; there are no mathematical truths outside the human mind. The constructions are built up from the natural numbers N and an intuitively given continuum. With N Brouwer constructed the integers Z and the rational numbers Q, but he had no satisfactory way to introduce the real numbers. It would take more than ten years before Brouwer would present a solution of this problem. In 1918 Brouwer, now a famous topologist, presented a complete new reconstruction of mathematics along lines set out in his thesis (Brouwer 1918). He called his reconstruction intuitionism, the existing mathematical practice classical. His foundational problem, reaching the power of the continuum from the discrete, he solved by the introduction of sequences not completely determined by a law: choice sequences. In reaching the power of the continuum Brouwer used only global properties of a continuum with choice sequences. He did not use a particular choice sequence. The question what exactly is a choice sequence turned out to be an intriguing one. Brouwer added notes to his definitions, sometimes extending a former note, sometimes changing or withdrawing a former one, and may never have been fully satisfied. The standard work on the subject, Troelstra’s Choice Sequences concentrated on global properties. This corresponds to the fact that no instances of particular choice sequences had been found in the work of Brouwer, except one possible exception: his proof of the negative continuity theorem in Brouwer 1927. But that proof is very vague and allows many possible reconstructions. When Brouwer returned to the proof later he clearly uses a determined sequence. Our claim is that at this later time he was fully exploiting particular choice sequences in the method of the creating subject. The method of the creating subject characterizes Brouwer’s papers after 1945, when after a long delay he started to publish again. The method has