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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!npeer.as286.net!npeer-ng0.as286.net!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_o?= =?UTF-8?Q?f_Logic?= Date: Thu, 25 Jul 2024 10:51:24 -0500 Organization: A noiseless patient Spider Lines: 112 Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 25 Jul 2024 17:51:25 +0200 (CEST) Injection-Info: dont-email.me; posting-host="43e64e6e679a39fe462151fef9da7f11"; logging-data="2475650"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18L4J7teZodJEbC5M++wrPo" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:vYnandJet9iwDoAsCylxI/zcapQ= Content-Language: en-US In-Reply-To: Bytes: 6508 On 7/25/2024 3:55 AM, Mikko wrote: > On 2024-07-23 14:53:21 +0000, olcott said: > >> On 7/23/2024 3:07 AM, Mikko wrote: >>> On 2024-07-22 14:40:41 +0000, olcott said: >>> >>>> On 7/22/2024 3:14 AM, Mikko wrote: >>>>> On 2024-07-21 13:20:04 +0000, olcott said: >>>>> >>>>>> On 7/21/2024 4:27 AM, Mikko wrote: >>>>>>> On 2024-07-20 13:22:31 +0000, olcott said: >>>>>>> >>>>>>>> On 7/20/2024 3:42 AM, Mikko wrote: >>>>>>>>> On 2024-07-19 13:48:49 +0000, olcott said: >>>>>>>>> >>>>>>>>>> >>>>>>>>>> Some undecidable expressions are only undecidable because >>>>>>>>>> they are self contradictory. In other words they are undecidable >>>>>>>>>> because there is something wrong with them. >>>>>>>>> >>>>>>>>> Being self-contradictory is a semantic property. Being >>>>>>>>> uncdecidable is >>>>>>>>> independent of any semantics. >>>>>>>> >>>>>>>> Not it is not. When an expression is neither true nor false >>>>>>>> that makes it neither provable nor refutable. >>>>>>> >>>>>>> There is no aithmetic sentence that is neither true or false. If >>>>>>> the sentnece >>>>>>> contains both existentia and universal quantifiers it may be hard >>>>>>> to find out >>>>>>> whether it is true or false but there is no sentence that is >>>>>>> neither. >>>>>>> >>>>>>>>  As Richard >>>>>>>> Montague so aptly showed Semantics can be specified syntactically. >>>>>>>> >>>>>>>>> An arithmetic sentence is always about >>>>>>>>> numbers, not about sentences. >>>>>>>> >>>>>>>> So when Gödel tried to show it could be about provability >>>>>>>> he was wrong before he even started? >>>>>>> >>>>>>> Gödel did not try to show that an arithmetic sentence is about >>>>>>> provability. >>>>>>> He constructed a sentence about numbers that is either true and >>>>>>> provable >>>>>>> or false and unprovable in the theory that is an extension of >>>>>>> Peano arithmetics. >>>>>>> >>>>>> >>>>>> You just directly contradicted yourself. >>>>> >>>>> I don't, and you cant show any contradiction. >>>>> >>>> >>>> Gödel's proof had nothing what-so-ever to do with provability >>>> except that he proved that g is unprovable in PA. >>> >>> He also proved that its negation is unprovable in PA. He also proved >>> that every consistent extension of PA has a an sentence (different >>> from g) such that both it and its negation are unprovable. >>> >> >> L is the language of a formal mathematical system. >> x is an expression of that language. >> >> When we understand that True(L,x) means that there is a finite >> sequence of truth preserving operations in L from the semantic >> meaning of x to x in L, then mathematical incompleteness is abolished. > > No, it is not. From the meaning of "formal mathematical system" follows I am overriding and superseding that. It is ridiculously stupid to ignore the semantics of a formal expression when evaluating its truth. Because truth *is* a semantic property. Haskell Curry shows the correct way to evaluate the semantic truth of an expression syntactically. http://www.liarparadox.org/Haskell_Curry_45.pdf Richard Montague's grammar of natural language semantics does this same thing. > that whether x is an expression of language L does not depend on semantics > or L is not a language of a formal mathiematical system. In addition, > the system is incomplete if there is a sentence that can be determined > to be true from the meaning of x but cannot be proven in the system. > This sentence is not true: "This sentence is not true" is only true because the indirect reference of the outer sentence escapes the self-contradiction. Tarski made this same stupid mistake: Tarski's Liar Paradox from page 248 It would then be possible to reconstruct the antinomy of the liar in the metalanguage, by forming in the language itself a sentence x such that the sentence of the metalanguage which is correlated with x asserts that x is not a true sentence. https://liarparadox.org/Tarski_247_248.pdf Formalized as: x ∉ True if and only if p where the symbol 'p' represents the whole sentence x https://liarparadox.org/Tarski_275_276.pdf In my own Minimal Type Theory the self-reference would not be so clumsy--- x := ~True(x) -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer