Re: Choice sequence lessons.

On Dec 27, 3:38 am, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx>
wrote:
|> such as "there Intuitionistically exists")
|
|There's no need for such a locution; "there exists an explicit..."
|would be quite helpful to both sides.

This would be a handy phrase in some cases except that
it's already acquired a different and informal connotation
in classical mathematics, conflicting with the one you're
proposing that it be given. Try looking up "explicit" in
the online mathematical reviews sometime for examples
of how it's typically used.

Given a continuous function f, where f(0)<0<f(1), the
smallest zero of f is an explicit example of a zero of
f, but does not serve to show constructively that there
exists a zero of f. Lots of nonconstructive constructions
are termed "explicit", in what I think is the intended
meaning of the word.

On the other hand, once it was shown that the sign of
pi(n)-li(n) changed (and it was constructively shown, by
the same token), then "the first change of sign" is a
validly constructed example of such a sign-change.
However, when people ask for an "explicit" example,
they usually intend for it to be more completely
computed in some loose sense, like 2^2^23+5, rather
than "the first example". There are Nim-like games
where it's possible to prove that the first player has
a winning strategy, and it is constructive due to the
game being finite, but an "explicit" strategy would
ordinary mean one spelled out more concretely than
"search the move tree".

|I'm sure such brief and
|simple expressions could be found in most cases.
|I already mentioned "inhabited set".

What substitute for "real number" would you recommend?
It'd better be rather good, since you're asking for it to be
used throughout a field.

Keith Ramsay
.

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