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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Formal systems that cannot possibly be incomplete except for
 unknowns and unknowable
Date: Wed, 7 May 2025 17:22:52 +0100
Organization: Fix this later
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On 07/05/2025 17:01, Ben Bacarisse wrote:
> Richard Heathfield <rjh@cpax.org.uk> writes:
> ...
>> Having been on the receiving end of lengthy Usenet diatribes by cranks in
>> my own field, I don't hold out much hope for our current culprits
>> developing either the capacity for clear thought or any measure of respect
>> for expertise any time soon.
>>
>> Nor do I believe they are capable of understanding proof by contradiction,
>> which is just about the easiest kind of proof there is. In fact, the most
>> surprising aspect of this whole affair is that (according to Mike)
> 
> It was me, but Mike may well have pointed it out recently.

He has (and I'll bet he credited you and I forgot; sorry).

>> Mr
>> Olcott seems to have (correctly) spotted a minor flaw in the proof
>> published by Dr Linz. How can he get that and not get contradiction? Proof
>> by contradiction is /much/ easier.
>>
>> Let us say we have a hypothesis X. If it is false, we might prove its
>> falsity in any number of 'positive' ways. But proof by contradiction takes
>> a different track.
>>
>> We begin by assuming that X is true.
>>
>> Then we show that IF X is true, it necessarily entails Y, where Y is
>> self-evidently a load of bollocks. From this we deduce that X is false.
>>
>> That's all there is to it.
> 
> As I am sure you know, that it not all there is to it,

It's pretty much the essence of proof by contradiction.

If you thought I was summarising HP, that would explain your 
response (because of course the details matter); but if you knew 
I was summarising proofs by contradiction, I'm curious to know 
what I omitted.

> but I digress...
> 
>> In the present case, X is the proposition that a computer can answer any
>> question that we can present to it.
> 
> That's way too vague!!

It is, yes. But it is in the wash of a discussion of the Halting 
Problem that has been going on for a very long time; we all know 
we're all talking about decidability.

>    There exists a TM, H, that computes h(n, i).

You're going for formality, which is of course admirable. I was 
going for informality, which is not always to be sneered at.

<lots of good stuff read and snipped>
-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within