Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

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
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