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

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".
Any sentence that is not "provably true"
is construed as untrue. When its negation
is also not "provably true" then the sentence
is rejected as a semantically unsound proposition.
The system that I propose can handle full natural
language semantics as well all of the notations of
the formal languages.

And don't come back with the arrogantly ignorant falsehood that somehow
Gödel's theorem doesn't apply to your "system".  He proved it applies to
any system that you can do anything at all non-trivial with.
>

Within the way that formal system are currently defined
the full semantics of an expression are strictly segregated
from its syntax.
When semantics is formalized syntactically  (with extended
Montague Grammar) such that semantics and syntax are fully
integrated into a single formal language then the notion
of semantically unsound can be directly measured.
"What time is it?" // Is not a semantically sound proposition.
"This sentence is not true" // Is not a semantically sound proposition.
Every expression and its negation that are not provably true
are rejected as not a semantically sound proposition.

[ .... ]
 

-- 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