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

olcott <polcott333@gmail.com> wrote:

On 5/6/2025 3:17 AM, Alan Mackenzie wrote:

[ Followup-To: set ]

In comp.theory olcott <polcott333@gmail.com> wrote:

[ .... ]

When we begin with truth and only apply truth preserving
operations then WE NECESSARILY MUST END UP WITH TRUTH.

You will necessarily end up with only a subset of truth, no matter how
shouty you are in writing it.  You'll also end up with undecidability,
no matter how hard you try to pretend it isn't there.

When we ALWAYS end up with TRUTH then we NEVER end up with
UNDECIDABILITY.

Shut your eyes, and you won't see it.

Try to provide one simple concrete example where we begin with truth
and only apply truth preserving operations and end up with
undecidability.

There is no need to provide examples for rigorously proven mathematical
theorems, in particular for Gödel's incompleteness theorem.  I don't go
around providing examples for 2 + 2 = 4 either.

Experience shows that if anybody actually did provide such an example,
you wouldn't understand it, and you'd carry on lying about nobody having
produced an example.

With the Tarski Undefinability theorem Tarski began with a falsehood.

That mis-impression of yours is due to you utterly failing to understand
the concept of proof by contradiction, and even more, utterly failing to
understand the concept of a mathematical proof.

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.

[ .... ]

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

--
Alan Mackenzie (Nuremberg, Germany).