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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge
Date: Wed, 2 Apr 2025 23:09:15 -0400
Organization: i2pn2 (i2pn.org)
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On 4/2/25 10:51 PM, olcott wrote:
> On 4/2/2025 8:56 PM, Richard Damon wrote:
>> On 4/2/25 9:30 PM, olcott wrote:
>>> 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
>>> they are essentially nothing more than an ordered
>>> set of finite strings of digits.
>>
>> No, but any logic system that can support them 
> 
> Can be defined in screwy that has undecidability
> or not defined in this screwy way.
> 
> Basic facts and expressions semantically entailed
> by the basic facts cannot have undecidability[math].
> 

Wrong, Godel shows that having the properties of the Natural numbers is 
enough.

Show what property he uses that you can withhold and still have a 
reasonably usable mathematics.

Your problem is you don't understand the power that basic logic gets 
from the basic nature of the Natural Numbers.