18 ERRETT A. BISHOP because M and No, for instance, are fixed positive integers, defined explic- itly by the proof of the inequality a b. Of course we could decide to substi- tute other values for the original values of M and N if we desired, so 0 ' some choice is possible should we wish to exercise it. If we do not explicitly state what choice we wish to make, it will be assumed that the values of M and No given by the proof of a b are chosen. The number H, which is constructively a well-defined real number, is classically rational, because if the Goldbach conjecture is true then H = 0, and if the conjecture is false then H = 2- n t l , where n is the first even integer for which it fails. We are not entitled to assert constructively that H is rational: if it is rational, then either H = 0 or H # 0, meaning that either Goldbachls conjecture is true or else it is false and we are not entitled to assert this constructively, until we have a method for deciding which. We are not entitled to assert H is irrational either, because if H is irrational, then H f 0, therefore Goldbachls conjecture is false, therefore H is the rational -nt1 number 2 , a contradiction! Thus H cannot be asserted to be rational, although its irrationality is contradictory. (I am indebted to Halsey Royden for this amusing observation. ) It is easy to prove the existence of many irrational numbers, by proving the uncountability of the real numbers, as a corollary of the Baire category theorem. For the present, let us merely remark that h/Z is irrational. Of course, fi can be defined by constructing successive decimal approxima- tions. It is therefore constructively well-defined. The classical proof of the 2 2 irrationality of fi shows that if p/q is any rational number then p /q f 2. Since both p2/q2 and 2 can be written with denominator q2 , it follows that Since clearly p/q f fi in case p/q 0 or p/q 7 2 , to show that p/q # fi we may assume 0 s p/q 5 2 . Then Therefore, h/ ?T # p/q. Thus is (constructively) irrational. The failure of the usual form of trichotomy means that we must be care- ful in defining absolute values and maxima and minima of real numbers. For
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