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

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

[ Followup-To: set ]
 In comp.theory olcott <polcott333@gmail.com> wrote:

On 5/5/2025 10:31 AM, olcott wrote:

On 5/5/2025 6:04 AM, Richard Damon wrote:

On 5/4/25 10:23 PM, olcott wrote:

When we define formal systems as a finite list of basic facts and
allow semantic logical entailment as the only rule of inference we
have systems that can express any truth that can be expressed in
language.

 Including the existence of undecidable statements.  That is a truth in
_any_ logical system bar the simplest or inconsistent ones.
 

Also with such systems Undecidability is impossible. The only
incompleteness are things that are unknown or unknowable.

 

Can such a system include the mathematics of the natural numbers?

 

If so, your claim is false, as that is enough to create that
undeciability.

 

It seems to me that the inferences steps that could
otherwise create undecidability cannot exist in the
system that I propose.

 

The mathematics of natural numbers (as I have already explained)
begins with basic facts about natural numbers and only applies
truth preserving operations to these basic facts.

 

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.
With the Tarski Undefinability theorem Tarski
began with a falsehood.
Tarski's Liar Paradox from page 248
    It would then be possible to reconstruct the antinomy of the liar
    in the metalanguage, by forming in the language itself a sentence
    x such that the sentence of the metalanguage which is correlated
    with x asserts that x is not a true sentence.
    https://liarparadox.org/Tarski_247_248.pdf
Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

Its not that hard, iff you pay enough attention.

 It's too hard for you.  As I've already suggested in another post, you'd
do better to show some respect to those who understand the matters you're
dabbling in.  Accept that your level of understanding is not particularly
high, and _learn_ from these other people.
 

That is factually incorrect. If you would pay much
better attention you would see this.
My simulating termination analyzer DOES correctly
determine the halt status of the Halting Problem's
counter-example input.
That you don't understand this does not make you
more knowledgeable than me.

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

 

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