Re: Amateur takes on Wiles's work

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 11:32:49 +0200

akolowski@hotmail.com (Andrzej Kolowski) writes:

> I like this idea of a "null test". Here is how I understand it.
> You take the conclusion of a given proof. You state its negation.
> You then go through the proof line by line to see if anything
> contradicts the negation.
>
> If there is *no* step that contradicts the negation, then the
> proof must be wrong.
>
> If there *is* a step that contradicts the negation, then it
> is still possible that the proof is wrong. You just have
> slight and nonconclusive evidence that it isn't.

You overestimate the utility of the null test.

Let P be the conclusion of a given proof, so that we will assume NOT P
and look for contradictions.

What does "contradict" mean here?

(1) NOT P contradicts Q iff Q is either P or NOT NOT P.

Not very useful, NOT P probably only contradicts the conclusion P.

(2) NOT P contradicts Q iff T, NOT P, Q |- R & NOT R for some R, where T
is the theory in which we work.

Not very useful. Since T |- P, clearly T, NOT P, Q |- P & NOT P for
every formula Q in the proof.

(3) NOT P contradicts Q iff NOT P, Q |- R & NOT R for some R.

Similar to (1). Not very useful.

So how is this test useful?

[Snip remainder, in which it becomes probable that Andrzej was having
James on and I've wasted my time writing this. Dagnabit.] 

-- 
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