Path: ...!feeds.phibee-telecom.net!3.eu.feeder.erje.net!feeder.erje.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: comp.theory Subject: =?utf-8?Q?Re:_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_of_Logic?= Date: Thu, 25 Jul 2024 11:55:43 +0300 Organization: - Lines: 73 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 10:55:44 +0200 (CEST) Injection-Info: dont-email.me; posting-host="02bedc76074b37eacb17f68ffc18830d"; logging-data="2338504"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+3NHMocdrP79ZPuwMOItzR" User-Agent: Unison/2.2 Cancel-Lock: sha1:+pSTsM8Fm7oeBwkeNWieVr5G3X8= Bytes: 4857 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 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. -- Mikko