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

On 2025-05-06 14:36:30 +0000, olcott said:

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.

Incompleteness of every extension of Peano arithmetic is one example.
Unsolvability of Group theory is another one. Yet another one is
halting problem.
--
Mikko