Path: ...!weretis.net!feeder6.news.weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: sci.logic Subject: Re: Undecidability based on epistemological antinomies V2 --Mendelson-- Date: Wed, 24 Apr 2024 12:49:28 +0300 Organization: - Lines: 78 Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 24 Apr 2024 11:49:28 +0200 (CEST) Injection-Info: dont-email.me; posting-host="7316bde8859540be15fe71b9e804f57a"; logging-data="2376237"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/DLfdgbMfGcZXhy9llFuAd" User-Agent: Unison/2.2 Cancel-Lock: sha1:YFXEhPvvX4NgaDJbmXMQkxVddVs= Bytes: 4311 On 2024-04-23 14:54:09 +0000, olcott said: > On 4/22/2024 3:26 AM, Mikko wrote: >> On 2024-04-21 14:34:44 +0000, olcott said: >> >>> On 4/21/2024 2:50 AM, Mikko wrote: >>>> On 2024-04-20 16:37:27 +0000, olcott said: >>>> >>>>> On 4/20/2024 2:41 AM, Mikko wrote: >>>>>> On 2024-04-19 02:25:48 +0000, olcott said: >>>>>> >>>>>>> On 4/18/2024 8:58 PM, Richard Damon wrote: >>>>>> >>>>>>>> Godel's proof you are quoting from had NOTHING to do with undecidability, >>>>>>> >>>>>>> *Mendelson (and everyone that knows these things) disagrees* >>>>>>> >>>>>>> https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf >>>>>>> >>>>>> >>>>>> On questions whether Gödel said something or not the sumpreme authority >>>>>> is not Mendelson but Gödel. >>>>>> >>>>> >>>>> When some authors affirm that undecidability and incompleteness >>>>> are the exact same thing then whenever Gödel uses the term >>>>> incompleteness then he is also referring to the term undecidability. >>>> >>>> That does not follow. Besides, a reference to the term "undecidability" >>>> is not a reference to the concept 'undecidability'. >>>> >>> >>> In other words you deny the identity principle thus X=X is false. >> >> It is not a good idea to lie where the truth can be seen. >> > > It is not a good idea to say gibberish nonsense and > expect it to be understood. > >>> a reference to the term "undecidability" > >>> is not a reference to the concept 'undecidability'. That is how a sentence must be quoted. The proof that the quoted sentence can be understood is that Richard Damon undesstood it. >>> An undecidable sentence of a theory K is a closed wf ℬ of K such that >>> neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and >>> not-⊢K ¬ℬ. (Mendelson: 2015:208) >> >> So that is what "undecideble" means in Mendelson: 2015. Elsewhere it may >> mean something else. > It usually means one cannot make up one's mind. > In math it means an epistemological antinomy expression > is not a proposition thus a type mismatch error for every > bivalent system of logic. No, it doesn't. There is no reference to an epistemological anitnomy in "undecidable". > not-⊢K ℬ and not-⊢K ¬ℬ. (Mendelson: 2015:208) > K ⊬ ℬ and K ⊬ ¬ℬ. // switching notational conventions > >>> Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x)) >> >> So not the same. > > When an expression cannot be proved or refuted is a formal system > this is exactly the same as an expression cannot be proved or refuted > in a formal system. To say about an expression that neither it nor its negation cannot be proven is not the same as to say about a formal system that it contains expressions that can neither be proven or disproven. -- Mikko