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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Every sufficiently competent C programmer knows --- Very Stupid
 Mistake and Liars
Date: Thu, 13 Mar 2025 07:13:10 -0400
Organization: i2pn2 (i2pn.org)
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On 3/13/25 12:17 AM, olcott wrote:
> On 3/12/2025 10:55 PM, Richard Damon wrote:
>> On 3/12/25 8:28 PM, olcott wrote:
>>> On 3/12/2025 5:46 PM, Richard Damon wrote:
>>>> On 3/12/25 10:50 AM, olcott wrote:
>>>>> On 3/12/2025 8:03 AM, André G. Isaak wrote:
>>>>>> On 2025-03-11 20:29, Richard Heathfield wrote:
>>>>>>
>>>>>>> Look up "infinite". You keep using that word. I do not think it 
>>>>>>> means what you think it means.
>>>>>>
>>>>>> If you continue to engage with Olcott, you will discover that a 
>>>>>> great many words don't mean what he thinks they mean.
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> Incomplete[0] (base meaning)
>>>>> Not having all the necessary or appropriate parts.
>>>>>
>>>>> Provable[0] (base meaning)
>>>>> Can be shown to be definitely true by some means.
>>>>>
>>>>>
>>>>>
>>>>
>>>> Right, and the appropriate part for logic that it is missing is the 
>>>> proofs of some of the statements.
>>>>
>>>> Proofs, to SHOW something, must be finite, as we can not see 
>>>> something that is infinite, as we are finite.
>>>>
>>>
>>> So then we know that G <is> TRUE because meta-math proves this.
>>> If we are stupid enough to define a system that does not know
>>> this then we are stupid.
>>>
>>>
>>
>> Yes, we know that G is true, as from the additional information 
>> provide in the meta-system we can reduce the infinite sequence that 
>> can be built in the base system to a finite proof.
>>
>> Note, PROOF belong to a system, and if they try to use something not 
>> known to be true in the system, they fail to be a proof, as they don't 
>> SHOW the needed result.
>>
>> Note, the "System" can't contain all of the Meta-System, as one 
>> important added part of the meta-system, as a set of axioms, was an 
>> enumeration of all axioms in the base system. If we try to put that 
>> enumeration into the base system, it becomes self-referential and we 
>> have an infinite set of axioms, something not allowed in normal formal 
>> logic.
>>
>> It is possible to do in a meta-system, as the system has a finite 
>> axiomization, so we just need to develope a numbering of that list of 
>> axioms.
>>
>> This seems to be part of your problem of not understanding why meta- 
>> systems can exist and not be part of the original system, but does 
>> describe it,
>>
>>
> 
> The only reason why the meta-system and the system
> cannot be the same system is that semantically incorrect
> expressions are not rejected as not truth bearers in
> the system.
> 

Nope,  guess you are just demonstrating why you don't understand the proofs.

No finite system can number all its symbols and axioms as axioms in the 
system, as the numbering axioms can't be given a number without 
increasing the number of numbers used, which causes the count to grow to 
infinity.


If you think it can be done, try to do it for even a simple system.

Try it.