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

On 06/05/2025 16:38, Alan Mackenzie wrote:

These aren't particularly difficult things to comprehend.  As I keep
saying, you ought to show a lot more respect for people who are
mathematically educated.

I concur.
As someone who is not particularly mathematically educated (I have an A-level in the subject, but that's all), I tend to steer well clear of mathematical debates, although I have occasionally dipped a toe.
I have *enormous* respect for those who know their tensors from their manifolds and their conjectures from their eigenvalues, even though it's all Greek to me.
But to understand the Turing proof requires little if any mathematical knowledge. It requires only the capacity for clear thinking.
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) 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.
In the present case, X is the proposition that a computer can answer any question that we can present to it.
Turing constructed the Halting Problem to illustrate that IF X were true it would necessarily be false - a contradiction. Conclusion: X is bollocks.
The proof couldn't be simpler. If Messrs Flibble and Olcott don't understand it by now, they never will.
--
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