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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic knowledge
Date: Thu, 3 Apr 2025 10:08:12 +0300
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On 2025-04-03 01:30:28 +0000, olcott said:

> On 4/2/2025 5:05 PM, Richard Damon wrote:
>> On 4/2/25 11:59 AM, olcott wrote:
>>> On 4/2/2025 4:20 AM, Mikko wrote:
>>>> On 2025-04-01 17:51:29 +0000, olcott said:
>>>> 
>>>>> 
>>>>> All we have to do is make a C program that does this
>>>>> with pairs of finite strings then it becomes self-evidently
>>>>> correct needing no proof.
>>>> 
>>>> There already are programs that check proofs. But you can make your own
>>>> if you think the logic used by the existing ones is not correct.
>>>> 
>>>> If the your logic system is sufficiently weak there may also be a way to
>>>> make a C program that can construct the proof or determine that there is
>>>> none.
>>>> 
>>> 
>>> When we define a system that cannot possibly be inconsistent
>>> then a proof of consistency not needed.
>> 
>> But you can't do that unless you limit the system to only have a finite 
>> number of statements expressible in it, and thus it can't handle most 
>> real problems
>> 
>>> 
>>> A system entirely comprised of Basic Facts and Semantic logical 
>>> entailment cannot possibly be inconsistent.
>>> 
>> 
>> Sure it can.
>> 
>> The problem is you need to be very careful about what you allow as your 
>> "Basic Facts", and if you allow the system to create the concept of the 
>> Natural Numbers, you can't verify that you don't actually have a 
>> contradition in it.
>> 
> 
> It never has been that natural numbers have
> ever actually had any inconsistency themselves

That is generally believed but not actually proven.

> they are essentially nothing more than an ordered
> set of finite strings of digits.

The "nothing more" part cannot be proven. In first order logic one cannot
even say that those strings must be finite. Higher order logics can have
multiple interpretations about what is infinite and there is no way to
specify which intepretation is intended.

-- 
Mikko