From: kramsay@aol.commangled (Keith Ramsay) Subject: Re: Cantor-Schroeder-Bernstein: Constructively Valid? Date: 19 Dec 1999 05:26:06 GMT Newsgroups: sci.math Keywords: Law of Excluded Middle in action In article , Virgil writes: |Let f(1/n) = n^2, for n an integer and f(x) = 0 otherwise. |Then f is a real function on [0,1], but not uniformly continuous. |Do you mean the uniform continuity of *continuous* functions? No, I'm familiar with the standard proof in nonconstructive real analysis. We can see that it is nonconstructive by considering what is required for f to be defined everywhere on [0,1]. Consider the following number x: x is the limit of the sequence x1,x2,x3,... where x_i=1/3 if there is no odd perfect number <=i, and x_i=1/3+1/k if the smallest odd perfect number is k<=i. The sequence x1,x2,x3,... is constructively a convergent sequence, and it's limit x is a perfectly well-defined real number. If f were defined on x, then either x=1/3, f(x)=9, and there does not exist an odd perfect number, or x>1/3, f(x)=0, and there does exist an odd perfect number. In order to assert that f is defined on x, then, we have to say that the answer to the question "is there an odd perfect number?" is now known to exist. In fact, the answers to a whole class of related questions has to be asserted to exist, even though we don't in general have a way to get at them. That's the essential nonconstructivity of the result. The way that it's proven, in the ordinary development of mathematics, is by implicit application of the law of excluded middle: that every proposition P is either true or false. It's been said that we have compelling reasons for using the law of excluded middle in mathematics, but I don't know of any compelling reasons to do so, certainly not all of the time and while not paying any particular attention to it. It has a way of masking out some information, and that doesn't seem very useful. Here, for instance, there's no reason why you can't study this function. You just don't lump it in with functions on "all of [0,1]" automatically; the fact of its being defined by cases (which in this simple case is obvious enough, to be sure) stays with you. In more complicated situations it seems to me this might be an improvement. A mathematician who does constructive functional analysis claims it is, and none of the arguments to the contrary seem solid. Keith Ramsay