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Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott <polcott333@gmail.com> Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_o?= =?UTF-8?Q?f_Logic?= Date: Tue, 23 Jul 2024 09:53:21 -0500 Organization: A noiseless patient Spider Lines: 125 Message-ID: <v7og51$17h8r$7@dont-email.me> References: <v6m7si$1uq86$2@dont-email.me> <v6mhc7$20hbo$2@dont-email.me> <v6mito$bbr$1@news.muc.de> <v6mjlg$20sio$2@dont-email.me> <v6mlfj$bbr$2@news.muc.de> <v6mlk6$21d9q$1@dont-email.me> <v6nu2n$2bepp$1@dont-email.me> <v6op7v$2fuva$5@dont-email.me> <v6qoms$2ukg7$1@dont-email.me> <v6rat7$30qtt$8@dont-email.me> <v6repr$32501$2@dont-email.me> <v6tbpe$3gg4d$1@dont-email.me> <v6traj$3imib$7@dont-email.me> <v703f7$2ooi$2@dont-email.me> <v70of6$61d8$8@dont-email.me> <v72kp6$k3b1$1@dont-email.me> <v738db$mjis$14@dont-email.me> <v756r9$15qot$1@dont-email.me> <v7614g$19j7l$11@dont-email.me> <v77qm6$1ntfr$1@dont-email.me> <v78g43$1rc43$5@dont-email.me> <v7ahpv$2arco$1@dont-email.me> <v7b5pl$2e2aq$3@dont-email.me> <v7d4mr$2svvi$1@dont-email.me> <v7dqs3$30pvh$1@dont-email.me> <v7ft98$3fbg8$1@dont-email.me> <v7gdmn$3hlc2$3@dont-email.me> <v7ikah$1hri$1@dont-email.me> <v7j1u4$3o7r$2@dont-email.me> <v7l4c9$ijpn$1@dont-email.me> <v7lr19$luh0$3@dont-email.me> <v7nobe$14dfq$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 23 Jul 2024 16:53:22 +0200 (CEST) Injection-Info: dont-email.me; posting-host="c53d2de4672c698529f342dcfedcfa3a"; logging-data="1295643"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/J3yjBcVSoW5B6Mtzj1e2L" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:J+j7j7I3xeANP4Oc68NRSNO/dVI= Content-Language: en-US In-Reply-To: <v7nobe$14dfq$1@dont-email.me> Bytes: 6919 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. ~True(L,x) ∧ ~True(L,~x) means that x is not a truth-bearer in L. It does not mean that L is incomplete >>>>>>> A proof is about sentences, not about >>>>>>> numbers. >>>>>>> >>>>>>>> The Liar Paradox: "This sentence is not true" >>>>>>> >>>>>>> cannot be said in the language of Peano arithmetic. >>>>>> >>>>>> Since Tarski anchored his whole undefinability theorem in a >>>>>> self-contradictory sentence he only really showed that sentences that >>>>>> are neither true nor false cannot be proven true. >>>>> >>>>> By Gödel's completeness theorem every consistent incomplete first >>>>> order >>>>> theory has a model where at least one unprovable sentence is true. >>>>> >>>>>> https://liarparadox.org/Tarski_247_248.pdf // Tarski Liar Paradox >>>>>> basis >>>>>> https://liarparadox.org/Tarski_275_276.pdf // Tarski proof >>>> >>>> It is very simple to redefine the foundation of logic to eliminate >>>> incompleteness. >>> >>> Yes, as long as you don't care whether the resulting system is useful. >>> Classical logic has passed practical tests for thousands of years, so >>> it is hard to find a sysem with better empirical support. >> >> When we show how incompleteness is eliminated then this also shows >> how undefinability is eliminated and this would have resulted in a >> chatbot that eviscerated Fascist lies about election fraud long >> before they could have taken hold in the minds of 45% of the electorate. > > The simplest way to elimita incompleteness is to construct a theory > where everytihing is provable. Of course such theory is not useful. > > The next simplest way is to construct a theory for a finite universe. > As the theory is complete it specifies the number of objects in the > universe. Then it is possible to evaluate every quantifier with a > simple finite loop or recursion, so the truth of every sentence is > computable. > > This kind of theory may have some use but its applicability is very > limited. In particular, a complete theory cannot be used in situations > where somthing is not known. > >> Because people have been arguing against my correct system of reasoning >> we will probably see the rise of the fourth Reich. > > Trying something impossible does not prevent anything. > -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer