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 14:04:27 -0700 Organization: None to speak of Lines: 105 Message-ID: <87k0z85tt0.fsf@nosuchdomain.example.com> References: <87k0zc8ps5.fsf@nosuchdomain.example.com> <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="12e88733a1da72d393d7ea4b6ab6d050"; logging-data="19313"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX195lRl7qr+mR0B5gX0qyejh" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:4LJRQqzDukYcjyY8MvyFsxjZqxk= sha1:PKDWfyejkFolLX9XUNYSaDzXizU= Xref: csiph.com comp.theory:21591 comp.ai.philosophy:21920 comp.ai.nat-lang:2339 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. A system is by definition "incomplete" if and only there is at least one statement such that neither the statement nor its negation can be proven. Nothing you say about incompleteness that doesn't refer to that definition (or a more correct and rigorous version of it) is relevant. >>> Theories can be called "incomplete" yet if nothing is missing then >>> this use is a misnomer. >> >> Are you saying you accept that Q is incomplete or not? >> I can't tell. > > Q is incomplete relative to the commutativity of addition. Incompleteness is not "relative to" something. A statement about the commutativity of addition can be used to demonstrate that Q is incomplete, but that incompleteness is an unqualified characteristic of Q. Do you accept that Q is incomplete? Please start your answer with either "Yes." or "No.". Feel free to add whatever discussion you like after that. >> You quoted a definition of incompleteness above. >> Can you make clear if you are happy with that definition? > > I am unhappy with that definition because it would decide that a > formal system is incomplete (in some cases) based on the fact that the > formal system can neither prove nor disprove self-contradictory > expressions of its own language. The definition of incompleteness says nothing about self-contradictory statements. In Q, the statement ∀x ∀y (x + y = y + x) is not self-contradictory. [...] >> You accept that Q ⊬ φ. >> >> What about Q ⊬ ¬φ) ? > > Is the sentence: "cows are animals" true in mathematics, or is it > neither true nor false in mathematics and true in biology? That's non-responsive. "cow are animals" refers to things outside the scope of mathematics. "∀x ∀y (x + y = y + x)" refers to things entirely within the scope of Q. [SNIP] It's occurred to me that this raises a (perhaps) interesting question -- or maybe I'm missing something obvious. In a system that includes Euclid's first four postulate but not the fifth (the parallel postulate), neither the parallel postulate nor its negation can be proven. We can construct a consistent system with the parallel postulate as an axiom. We can *also* construct a consistent system with the negation of the parallel postulate as an axiom. 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. Can we construct a consistent system based on Robinson Arithmetic in which addition is provably *not* commutative? -- 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 */