SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 19 example, if x = {x 3 is a real number, with sequence {Nn 3 of convergence n parameters, then I xl is defined to be the Cauchy sequence { 1 x n 1 3 of rational numbers (with sequence {N 3 of convergence parameters). Similar- n m ly, min{xsy3 is defined to be the Cauchy sequence {minExn9yn3~n=1 , and m max{xsy3 to be { m a x ~ x n s y n ~ ~ n = l . This definition of min , in particular, has an amusing consequence. Consider the equation Clearly 0 and the number H' are solutions. Are they the only solutions? It depends on what we mean by "only. " Clearly min {O, H') is a solution, and we are unable to identify it with either 0 or H' . Thus it is a third solution! The reader might like to amuse himself looking for others. This discussion incidentally makes the point that if the product of two real numbers is 0 we are not entitled to conclude that one of them is 0. (For example, x(x - H') = 0 does not imply that x=O or x-H'=o: set x =minf0, H'] .) The constructive real number system as I have described it is not accepted by all constructivists. The intuitionists and the recursive function theorists have other versions. For Brouwer, and his followers (the intuitionists), the constructive real numbers described above do not constitute all of the real number system. In addition there are incompletely determined real numbers, corresponding to sequences of rational numbers whose terms are not specified by a master algorithm. Such sequences are called "free-choice sequencesS1'because the creating subject, who defines the sequence, does not completely commit him- self in advance but allows himself some freedom of choice along the way in defining the individual terms of the sequence. There seem to be at least two motivations for the introduction of free- choice sequences into the real number system. First, since each constructive real number can presumably be described by a phrase in the English language, on superficial consideration the set of constructive real numbers would appear to be countable. On closer consideration this is seen not to be the case: Cantor's uncountability theorem holds, in the following version. If {xn 3 is any sequence of real numbers, there exists a real number x with x # x for n all n. Nevertheless it appears that Brouwer was troubled by a certain aura of the discrete clinging to the constructive real number system W. Second,

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