Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Keith Thompson Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics Subject: Re: Simply defining =?utf-8?Q?G=C3=B6del?= Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Date: Sun, 12 Jul 2020 17:22:43 -0700 Organization: None to speak of Lines: 77 Message-ID: <87d0505kmk.fsf@nosuchdomain.example.com> References: <87k0zc8ps5.fsf@nosuchdomain.example.com> <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> <87k0z85tt0.fsf@nosuchdomain.example.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="4a04b42c90924d8b69f548a8105eba02"; logging-data="6655"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX199emfSTEyWn7mC2qlwoKLw" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:cv9xE2As8a1CDq2W5CUQpZtnXBc= sha1:h/2zOmPlqY1Qz17q6Al40qepFE0= Xref: csiph.com comp.theory:21599 comp.ai.philosophy:21931 comp.ai.nat-lang:2347 olcott writes: > On 7/12/2020 4:04 PM, Keith Thompson wrote: >> olcott writes: >>> On 7/12/2020 8:10 AM, Alan Smaill wrote: >>>> olcott writes: >>>>> On 7/11/2020 6:25 AM, Alan Smaill wrote: >>>>>> olcott writes: >>>> [...] >>>>>> You miss the point of my comment -- >>>>>> try thinking before you give your stock response. >>>>>> >>>>>> Let's go back to the start and try again: >>>>>> >>>>>>>>> Q ⊬ φ // This is true in Q >>>>>> >>>>>> Agreed. >>>>>> >>>>>>>>> ∴ φ ↔ Q ⊬ φ is not true in Q >>>>>> >>>>>> What makes you think this? >>>>> >>>>> The conventional definition of incompleteness: >>>>> A theory T is incomplete if and only if there is some sentence φ such >>>>> that (T ⊬ φ) and (T ⊬ ¬φ). >>>>> >>>>> Q is incomplete relative to the commutativity of addition: >>>>> φ = (∀x ∀y (x + y = y + x)) >>>>> (Q ⊬ φ ∧ Q ⊬ ¬φ) >>>>> >>>>> https://math.stackexchange.com/questions/998359/robinson-arithmetic-and-its-incompleteness >>>>> >>>>> Nothing can actually be incomplete unless something is missing. In the >>>>> case of Q the commutativity of addition is missing. >> >> You insist on interpreting the word "incomplete" to mean "something >> is missing". That's just not what "incomplete" means. > > You know that is not true. You know that any use of the term: > "incomplete" that does not mean that something is missing is a > misnomer. Nonsense. It's a technical term. I've talked about technical terms before, and how they don't have to match the common English meaning of a word. [...] > Any use by anyone of the term "incomplete" without anything missing > is a misnomer. Nonsense. [...] > The use of the term of the art "incomplete" is incongruous with its > common meaning thus incorrect on that basis. Incongruous? Sure. Incorrect on that basis? Nonsense. [...] >> Robinson Arithmetic cannot prove or disprove commutativity >> of addition. We can construct a consistent system based on >> Robinson Arithmetic in which addition is provably commutative. > > Sure just add an axiom: ∀x ∈ ℕ ∃y ∈ ℕ (x + y = y + x) > >> Can we construct a consistent system based on Robinson Arithmetic >> in which addition is provably *not* commutative? > > Not within the conventional semantics of the meaning of those terms. OK. Can you prove that? -- Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com Working, but not speaking, for Philips Healthcare void Void(void) { Void(); } /* The recursive call of the void */