Path: ...!eternal-september.org!feeder3.eternal-september.org!i2pn.org!i2pn2.org!.POSTED!not-for-mail
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: Sat, 5 Apr 2025 16:15:28 -0400
Organization: i2pn2 (i2pn.org)
Message-ID: <03ebb956fb92c0d27959296f63dd38f5bf8809ff@i2pn2.org>
References: <vrfvbd$256og$2@dont-email.me> <vrhami$3fbja$2@dont-email.me>
 <vrj9lu$1791p$1@dont-email.me> <vrjn82$1ilbe$2@dont-email.me>
 <vrmpc1$bnp3$1@dont-email.me> <vrmteo$cvat$6@dont-email.me>
 <vru000$33rof$1@dont-email.me> <vrug71$3gia2$6@dont-email.me>
 <0306c3c2d4a6d05a8bb7441c0b23d325aeac3d7b@i2pn2.org>
 <vrvnvv$ke3p$1@dont-email.me> <vs0egm$1cl6q$1@dont-email.me>
 <vs1f7j$296sp$2@dont-email.me> <vs3ad6$2o1a$1@dont-email.me>
 <vs4sjd$1c1ja$8@dont-email.me> <vs63o2$2nal3$1@dont-email.me>
 <vs6v2l$39556$17@dont-email.me> <vs8hia$13iam$1@dont-email.me>
 <vs8uoq$1fccq$2@dont-email.me> <vsb4in$14lqk$1@dont-email.me>
 <vsb9d5$19ka5$1@dont-email.me>
 <04aa9edbe77f4e701297d873264511f820d85526@i2pn2.org>
 <vsbu9j$1vihj$1@dont-email.me> <vsdlso$3shbn$2@dont-email.me>
 <vsen5l$th5g$5@dont-email.me> <vsg1b2$2ed9k$1@dont-email.me>
 <vsh9c9$3mdkb$2@dont-email.me> <vsj073$1g8q1$1@dont-email.me>
 <vsjn4k$26s7s$3@dont-email.me> <vsqn07$1nmlv$1@dont-email.me>
 <vsrqrl$2rgr9$3@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Sat, 5 Apr 2025 20:21:15 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="3280877"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
Content-Language: en-US
In-Reply-To: <vsrqrl$2rgr9$3@dont-email.me>
X-Spam-Checker-Version: SpamAssassin 4.0.0
Bytes: 3993
Lines: 52

On 4/5/25 1:56 PM, olcott wrote:
> On 4/5/2025 2:44 AM, Mikko wrote:
>> On 2025-04-02 16:03:32 +0000, olcott said:
>>
>>> On 4/2/2025 4:32 AM, Mikko wrote:
>>>> On 2025-04-01 17:56:25 +0000, olcott said:
>>>>
>>>>> On 4/1/2025 1:33 AM, Mikko wrote:
>>>>>> On 2025-03-31 18:33:26 +0000, olcott said:
>>>>>>
>>>>>>>
>>>>>>> Anything the contradicts basic facts or expressions
>>>>>>> semantically entailed from these basic facts is proven
>>>>>>> false.
>>>>>>
>>>>>> Anything that follows from true sentences by a truth preserving
>>>>>> transformations is true. If you can prove that a true sentence
>>>>>> is false your system is unsound.
>>>>>
>>>>> Ah so we finally agree on something.
>>>>> What about the "proof" that detecting inconsistent
>>>>> axioms is impossible? (I thought that I remebered this).
>>>>
>>>> A method that can always determine whether a set of axioms is 
>>>> inconsistent
>>>> does not exist. However, there are methods that can correctly determine
>>>> about some axiom systems that they are inconsistent and fail on others.
>>>>
>>>> The proof is just another proof that some function is not Turing 
>>>> computable.
>>>
>>> A finite set of axioms would seem to always be verifiable
>>> as consistent or inconsistent.  This may be the same for
>>> a finite list of axiom schemas.
>>
>> If ordinary logic is used it is sufficient to prove that there is
>> a sentence that cannot be proven in order to prove consistency or
>> to prove two sentences that contradict each other in order to prove 
>> inconsistency. But if neither proof is known there is no method to
>> find one.
>>
> 
> We are only talking about the inability to detect
> that basic facts contradict each other. I need a
> 100% concrete example proving this that this is
> sometimes impossible.
> 

Read Godel's proof.

Note, this follows from the incompleteness proof, as a proof of 
consistency yields a proof of completeness and thus any set powerful 
enough to be incomplete also can not prove its own consistancy.