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Path: ...!3.eu.feeder.erje.net!feeder.erje.net!weretis.net!feeder8.news.weretis.net!reader5.news.weretis.net!news.solani.org!.POSTED!not-for-mail From: Mild Shock <janburse@fastmail.fm> Newsgroups: comp.theory,sci.logic Subject: Re: "undecidable" / "unentscheidbar" (Was Analytic Truth-makers) Date: Tue, 23 Jul 2024 23:13:06 +0200 Message-ID: <v7p6d0$8cip$1@solani.org> References: <v7m26d$nrr4$1@dont-email.me> <e41a2d324173031e1fe47acc0fd69b94b7aba55e@i2pn2.org> <v7msg0$sepk$1@dont-email.me> <3fb77583036a3c8b0db4b77610fb4bf4214c9c23@i2pn2.org> <v7much$sepk$2@dont-email.me> <9577ce80fd6c8a3d5dc37b880ce35a4d10d12a0e@i2pn2.org> <v7n3ho$t590$1@dont-email.me> <7d9b88425623e1166e358f1bce4c3a2767c36da0@i2pn2.org> <v7naae$120r5$1@dont-email.me> <2a0f9a4235d75dee94ccae62b10d3afef5a966a5@i2pn2.org> <v7og8o$17h8r$8@dont-email.me> <v7p5g1$8c1e$1@solani.org> <v7p5o6$8c1e$2@solani.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 23 Jul 2024 21:13:04 -0000 (UTC) Injection-Info: solani.org; logging-data="275033"; mail-complaints-to="abuse@news.solani.org" User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:91.0) Gecko/20100101 Firefox/91.0 SeaMonkey/2.53.18.2 Cancel-Lock: sha1:8LKujSUlsg6lDj1lUVssrB+HaRc= X-User-ID: eJwFwQkBwDAIA0BLZSUB5PAU/xJ2h0thmxJULPbVe5tH2/IrSbhYhWsZ52Pm3Zo52RMAoXJdH6qKpyN83H59LBYo In-Reply-To: <v7p5o6$8c1e$2@solani.org> Bytes: 3429 Lines: 57 That Gödel emphasized "formal" in his paper has to do that his "decidable" comes from an ontology related to syntactic derivability. "deciable" is defined on the basis of the notion of general validity embodied as provability. But Gödel wasn't that one dimensional, you find also a semantic leaning terminology in some of his papers, for example his completeness theorem. There he uses another terminology, which doesn't have a one-to-one mapping to "decidable". He uses notions such as "erfüllbar" or "statisfiable", and "unerfüllbar" or "unsatisfiable". The main theorem here is: an unprovable sentences A is satisfiable by a counter model, meaning its complement has a model. Mild Shock schrieb: > For example Gödel belongs to the generation of > logicians that use the term "undecidable". > In German the term is translated to "unentscheidbar": > > Über formal unentscheidbare Sätze der Principia Mathematica und > verwandter Systeme I" ("On Formally Undecidable Propositions of > Principia Mathematica and Related Systems I") > https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems > > > Mild Shock schrieb: >> Since generations logicians have called sentences >> which you clumsily call "not a truth-bearer", >> simple called "undecidable" sentences. >> >> A theory is incomplete, if it has undecidable >> sentences. There is a small difference between >> unprovable and undecidable. >> >> An unprovable senetence A is only a sentence with: >> >> ~True(L, A). >> >> An undecidable sentence A is a sentence with: >> >> ~True(L, A) & ~True(L, ~A) >> >> Meaning the sentence itself and its complement >> are both unprovable. >> >> olcott schrieb: >>> ~True(L,x) ∧ ~True(L,~x) >>> means that x is not a truth-bearer in L. >>> It does not mean that L is incomplete >