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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable
Date: Wed, 7 May 2025 10:57:02 +0300
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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