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

On 5/5/2025 1:19 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 11:05 AM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 5:47 AM, Alan Mackenzie wrote:

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

 

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

 

Do you believe in the tooth fairy, too?

 

Counter-examples to my claim seem to be categorically impossible.

 

Arrogantly wrong in the extreme.

 

That you could not find one seems to prove my point.

 

Follow the details of the proof of Gödel's Incompleteness Theorem, and
apply them to your "system".  That will give you your counter example.

 

My system does not do "provable" instead it does "provably true".

 I don't know anything about your "system" and I don't care.  If it's a
formal system with anything above minimal capabilities, Gödel's Theorem
applies to it, and the "system" will be incomplete (in Gödel's sense).
 

I reformulate the entire notion of "formal system"
so that undecidability ceases to be possible.
When True(x) means semantic logical entailment
from basic facts (stipulated to be true) then
there is no place for undecidability to exist.
True("This sentence is true") == FALSE
True("This sentence is not true") == FALSE
True("This sentence can be proven") == FALSE
True("This sentence has five words") == TRUE

[ Irrelevant nonsense snipped. ]
 

-- 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