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