ERRETT A. BISHOP Another source of Brouwerian counter -examples is the statement LLPO (for the "lesser limited principle of omniscience"), that if Cqc 3 is any se- quence of integers, then either the first non-zero term, if one exists, is even or else the first non-zero term, if one exists, is odd. Clearly LPO + LLPO, but there seems to be no way to prove that LLPO + LPO. Nevertheless, we are just as sceptical that LLPO will ever be proved. Thus A + LLPO is another type of Brouwerian counter-example for A. As an instance, the statement that "either x 2 0 or x 0 for each real number x" implies LLPO, in fact is equivalent to it. Thus we are so sceptical that the statements LPO, LLPO, and their ilk will ever be proved that we use them for building counter-examples. The strongest counter-example to A would be to show that a proof of A is in- conceivable, in other words to prove 'hot A, " but proving "A + LPO" or "A + LLPO" is almost as good. In fact, I personally find it inconceivable that LPO (or LLPO for that matter) will ever be proved. Nevertheless I would be reluctant to accept 'hot LPO" as a theorem, because my belief in the impossibility of proving LPO is more of a gut reaction prompted by exper- ience than something I could communicate by arguments I feel would be sure to convince any objective, well-informed, and intelligent person. The accept- ance of 'hot LPO" as a theorem would have one amusing consequence, that the theorems of constructive mathematics would not necessarily be classically valid (on a formal level) any longer. It seems we are doomed to live with "LPO" and "there exists a function from [O,1] to IR that is not uniformly continuous" and similar statements, of whose impossibilities we are not quite sure enough to assert their negations as theorems. The classical paradoxes are equally viable constructively, the simplest perhaps being "this statement is false. " The concept of the set of all sets seems to be paradoxical (i. e., lead to a contradiction) constructively as well as classically. Informed common sense seems to be the best way of avoiding these paradoxes of self reference. Their spectre will always be lurking over both classical and constructive mathematics. Hermann Weyl made the meticu- lous avoidance of self reference the basis of a whole new development of the real number system (see Weyl [32]) that has since become known as predica- tive mathematics. Weyl later abandoned his system in favor of intuitionism. I see no better course at present that to recognize that certain concepts are
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