Re: mathematics

Virgil wrote:
|For example, the intermediate value theorem, as I recall, is proved
|without actually coming up with all those intermediate values.

One needs to be a little careful here.

Suppose f is a continuous function on [0,1]
and f(0)<0<f(1). The greatest lower bound of
{x : 0<=x<=1 and f(x)>=0} is what I would call
an explicit example of a zero of f.

I hope the issue isn't with the term "all" here,
because it's somewhat unclear what it should mean
to "come up with" a set of reals that could be
pretty arbitrary (just a closed subset of (0,1)).

The intermediate value theorem is nonconstructive,
but for a different reason. One standard definition
of "real number" that works constructively too is
that it's an equivalence class of Cauchy-converging
sequences of rationals. In this case, we could take
for example the sequence a_n = i/n where i is the
smallest integer in 0,1,...,n such that for some
x<i/n one has f(x)>=0. This also is explicit. But
for a claim that an integer exists to be established
constructively, there must be a way in principle to
produce an example, and there's no such general way.

Nonconstructively, one says that such an integer
exists, and is explicitly given, because it's the
smallest integer having some property that 0 doesn't
have but n does have. If we assume that each integer
either has the property or does not have it, then
the existence of a smallest such integer follows by
mathematical induction. As usual, the culprit is the
law of excluded middle. One can't necessarily tell
whether f(x)>=0 holds somewhere on an interval; one
just assumes that in some abstract sense it either
does or does not.

A number of variations on the intermediate value theorem
hold constructively. The basic problem is with functions
that hover around near 0 in some subinterval. So one can
get values where the function is within an epsilon of 0.
One can also get an exact zero if for every subinterval
of [0,1] there's an N and a value of f whose absolute
value is >1/N on the subinterval. Obviously also given
some subinterval on which the function is zero, you can
get a zero!

Keith Ramsay

.

Relevant Pages

  • Re: Math, The Friend of Relativity
    ... Don't be reassured by the ranting of Uncle Bonehead, ... As if we would believe something just because Einstein said so. ... zero, either, by the same argument Uncle Ben gave for real numbers. ... I've already proven h does NOT belong to the set of reals, ...
    (sci.physics.relativity)
  • Re: in fact zero _is_ neither positive nor negative. THIS STATEMENT IS NOT TRUE
    ... So in fact zero can be negative and can be positive and it just ... which most then build the reals from. ... and agree that magnitude is much more fundamental ... and if you want to use the bourbaki "positive" (positif ?) to include ...
    (sci.math)
  • Re: The common usage of "nonnegative real number" is ludicrous.
    ... not necessarily zero, and that is usually what is meant by ... a complicated construction. ... When I want to talk about magnitude I am talking about ... reals contain zero or not is just convention. ...
    (sci.math)
  • Re: An Alternative to Standard Arithmetic
    ... You have raised some extremely interesting issues concerning zero ... Wallis number in my original post because I thought it would ... suffers that same problem of reliance upon the reals to communicate. ... of new fundamentals, and so erasing some of the accumulation is ...
    (sci.math)
  • Re: On writing negative zero - with or without sign
    ... exact zero, contrary to your claim that there is no such thing. ... that using float for something *other* than simulating continuous reals ... "exactitude" in general, but rather a very specific issue, namely ...
    (comp.lang.fortran)