Path: ...!3.eu.feeder.erje.net!feeder.erje.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Undecidability based on epistemological antinomies V2
 --Mendelson--
Date: Tue, 30 Apr 2024 11:08:08 -0500
Organization: A noiseless patient Spider
Lines: 102
Message-ID: <v0r518$2hb7o$9@dont-email.me>
References: <uvq0sg$21m7a$1@dont-email.me> <uvq359$1doq3$4@i2pn2.org>
 <uvrbvs$2acf7$1@dont-email.me> <uvs70t$1h01f$1@i2pn2.org>
 <uvsgcl$2i80k$1@dont-email.me> <uvsj4v$1h01e$1@i2pn2.org>
 <uvsknc$2mq5c$1@dont-email.me> <uvvrj6$3i152$1@dont-email.me>
 <v00r07$3oqra$1@dont-email.me> <v02ggt$6org$1@dont-email.me>
 <v03866$bitp$1@dont-email.me> <v056us$rmqi$1@dont-email.me>
 <v08i2i$1m5hp$2@dont-email.me> <v0akj8$28ghd$1@dont-email.me>
 <v0bada$2defp$2@dont-email.me> <v0d42v$2tclm$1@dont-email.me>
 <v0dp8c$31vd9$1@dont-email.me> <v0fpdc$3j50e$1@dont-email.me>
 <v0gh69$3oudg$1@dont-email.me> <v0iccd$8odv$1@dont-email.me>
 <v0iv1p$cu99$1@dont-email.me> <v0l56g$vmnj$1@dont-email.me>
 <v0ljn0$12q0o$1@dont-email.me> <v0no3u$1ldmf$1@dont-email.me>
 <v0oafd$1pbn5$4@dont-email.me> <v0oder$1qgbm$1@dont-email.me>
 <v0oe6v$1qgpk$3@dont-email.me> <v0qmo2$2epvb$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Tue, 30 Apr 2024 18:08:09 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="e18070faf38e3938218949b4b017f26c";
	logging-data="2665720"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/tVAO8FLfyfPxkzqVUsfJy"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:/4FaeAMnE5LbqtPOHTK6wigZiTg=
In-Reply-To: <v0qmo2$2epvb$1@dont-email.me>
Content-Language: en-US
Bytes: 6056

On 4/30/2024 7:04 AM, Mikko wrote:
> On 2024-04-29 15:26:23 +0000, olcott said:
> 
>> On 4/29/2024 10:13 AM, Mikko wrote:
>>> On 2024-04-29 14:22:36 +0000, olcott said:
>>>
>>>> On 4/29/2024 4:09 AM, Mikko wrote:
>>>>> On 2024-04-28 13:41:50 +0000, olcott said:
>>>>>
>>>>>> On 4/28/2024 4:34 AM, Mikko wrote:
>>>>>>> On 2024-04-27 13:36:56 +0000, olcott said:
>>>>>>>
>>>>>>>> On 4/27/2024 3:18 AM, Mikko wrote:
>>>>>>>>> On 2024-04-26 15:28:08 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 4/26/2024 3:42 AM, Mikko wrote:
>>>>>>>>>>> On 2024-04-25 14:27:23 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> On 4/25/2024 3:26 AM, Mikko wrote:
>>>>>>>>>>>>> epistemological antinomy
>>>>>>>>>>>>
>>>>>>>>>>>> It <is> part of the current (thus incorrect) definition
>>>>>>>>>>>> of undecidability because expressions of language that
>>>>>>>>>>>> are neither true nor false (epistemological antinomies)
>>>>>>>>>>>> do prove undecidability even though these expressions
>>>>>>>>>>>> are not truth bearers thus not propositions.
>>>>>>>>>>>
>>>>>>>>>>> That a definition is current does not mean that is incorrect.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> ...14 Every epistemological antinomy can likewise be used for 
>>>>>>>>>> a similar
>>>>>>>>>> undecidability proof...(Gödel 1931:43-44)
>>>>>>>>>>
>>>>>>>>>>> An epistemological antinomy can only be an undecidable sentence
>>>>>>>>>>> if it can be a sentence. What epistemological antinomies you
>>>>>>>>>>> can find that can be expressed in, say, first order goup theory
>>>>>>>>>>> or first order arithmetic or first order set tehory?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> It only matters that they can be expressed in some formal system.
>>>>>>>>>> If they cannot be expressed in any formal system then Gödel is
>>>>>>>>>> wrong for a different reason.
>>>>>>>>>
>>>>>>>>> How is it relevant to the incompleteness of a theory whether an
>>>>>>>>> epistemological antińomy can be expressed in some other formal
>>>>>>>>> system?
>>>>>>>>
>>>>>>>> When an expression of language cannot be proved in a formal 
>>>>>>>> system only
>>>>>>>> because it is contradictory in this formal system then the 
>>>>>>>> inability to
>>>>>>>> prove this expression does not place any actual limit on what 
>>>>>>>> can be
>>>>>>>> proven because formal system are not supposed to prove 
>>>>>>>> contradictions.
>>>>>>>
>>>>>>> The first order theories of Peano arithmetic, ZFC set theory, and
>>>>>>> group theroy are said to be incomplete but you have not shown any
>>>>>>> fromula of any of them that could be called an epistemoloigcal
>>>>>>> antinomy.
>>>>>>>
>>>>>>
>>>>>> The details of the semantics of the inference steps are hidden behind
>>>>>> arithmetization and diagonalization in Gödel's actual proof.
>>>>>
>>>>> The correctness of a proof can be checked without any consideration of
>>>>> semantics. If the proof is fully formal there is an algorithm to check
>>>>> the correctness.
>>>>>
>>>>>> ($)   ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
>>>>>>
>>>>>> Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
>>>>>> ⌜G⌝) says that there is no natural number that is the Gödel number 
>>>>>> of a
>>>>>> proof in K of the wf G, which is equivalent to asserting that 
>>>>>> there is
>>>>>> no proof in K of G.
>>>>>
>>>>> The standard interpretation of artihmetic does not say anything about
>>>>> proofs and Gödel numbers.
>>>>>
>>>>
>>>> That was a direct quote from a math textbook, here it is again:
>>>
>>> That quote didn't define "standard semantics".
>>>
>>
>> It need not define standard semantics once is has summed of the 
>> essence of that whole proof as: G says “I am not provable in K”.
> 
> In the standard semantics of arithmetic nothing means "I am not provable
> in K". That simply is not an arithmetic statement about numbers.
> 

Mendelson says that Gödel's actual proof is equivalent to:
G says “I am not provable in K”.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer