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

On 07/05/2025 18:11, Mr Flibble wrote:

On Wed, 07 May 2025 17:22:52 +0100, Richard Heathfield wrote:
 

On 07/05/2025 17:01, Ben Bacarisse wrote:

Richard Heathfield <rjh@cpax.org.uk> writes:

<snip>

>
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.

 Proofs by contradiction are only valid if the contradiction is *well-
formed*: the contradiction at the heart of the halting problem is *ill-
formed* as it is a category (type) error.

You are free to believe that, but the contradiction at the heart of the Halting Problem is /extremely/ well-formed.
I realise that you won't agree, of course. One may lead a horse to water, but one can't make it drink.
The water is here:
<https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf>
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
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