Re: The Law of the Excluded Middle again (long)

On Sat, 01 Dec 2007 18:38:44 +0000, I wrote:

In the present case, the LEM is used in at least two places. [...]

We first argue that if any of x, y, or z is >= 1, then the sum
x^y + y^z + z^x is > 1. So we say that it suffices to consider
only the case where all of x, y and z are in (0, 1) [...]
we are appealing to the LEM, by saying "either
(x >= 1 or y >= 1 or z >= 1) or (x < 1 and y < 1 and z < 1)" [...]

We then make a further division into cases [...]
One way to phrase the argument is "either 0 < x <= y <= z <= 1
or 0 < z <= y <= x <= 1 or ..." (four other cases, which can
then be reduced to these two "by symmetry", i.e. by renaming
the variables). [...]

This application of the Law of the Excluded Middle [incidentally,
I've already mentioned my dislike of the metaphor implied by the
use of the word "law", in this sort of context - but we're stuck
with it] has a characteristic which I didn't notice at the time,
and which no-one else has pointed out either: it is applied only
to propositions containing only free variables.

I find the LEM particularly hard to disbelieve in this stark form.

There might be a variety of different "strengths" of the LEM, of
which a slightly "stronger" variant might also allow one to assert
for example that Euler's constant gamma either is or isn't rational.

(I also can't bring myself to doubt /that/ - without also doubting my
own sanity, and the entire worthwhileness of mathematics! - but again,
I am interested in learning how it might be possible so to doubt it.)

Call this Test Case I. I now need to supply (or be supplied with)
another relatively simple, everyday, realistic (and preferably real)
example of a mathematical argument, which uses a stronger form of
the LEM - so that I can see whether it is possible for me, sanely,
to doubt it - to be called Test Case II.

I'm not in a hurry. The most important thing for me is that the
argument be a real (or at least realistic) piece of mathematics.

The end result might be to have trained myself to disbelieve six
certain things before breakfast! But I would prefer instead to
strengthen my belief that mathematics makes sense, and is worth
doing.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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