Path: csiph.com!xmission!news.snarked.org!border2.nntp.dca1.giganews.com!nntp.giganews.com!buffer2.nntp.dca1.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Fri, 10 Jul 2020 15:11:41 -0500 Subject: =?UTF-8?Q?Re=3a_Simply_defining_G=c3=b6del_Incompleteness_and_Tarsk?= =?UTF-8?Q?i_Undefinability_away_V24_=28Are_we_there_yet=3f=29?= Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics References: <87k0zc8ps5.fsf@nosuchdomain.example.com> From: olcott Date: Fri, 10 Jul 2020 15:11:41 -0500 User-Agent: Mozilla/5.0 (Windows NT 10.0; WOW64; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Language: en-US Content-Transfer-Encoding: 8bit Message-ID: Lines: 139 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-IcB6rbObDo5qNgdZbamefL+pV9xRimc+iQ9oTBlzdWHABjnoBkGzXDp4X3TkR3YSD3r6WnzFTBiuVrA!zKd8MYIOeWkUMFbnh5F7QDM0c8p/z4REULmvrGa0uqxzhZnXe9IjA2xXCNQxn0ikjXmOZSK/Movx X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 X-Original-Bytes: 7998 Xref: csiph.com comp.theory:21559 comp.ai.philosophy:21876 comp.ai.nat-lang:2311 On 7/10/2020 1:12 PM, André G. Isaak wrote: > On 2020-07-10 12:04, olcott wrote: >> On 7/10/2020 12:27 PM, André G. Isaak wrote: >>> On 2020-07-10 11:16, olcott wrote: >>>> On 7/10/2020 11:55 AM, André G. Isaak wrote: >>>>> On 2020-07-10 09:54, olcott wrote: >>>>>> On 7/10/2020 10:42 AM, André G. Isaak wrote: >>>>>>> On 2020-07-10 08:29, olcott wrote: >>>>>>>> Correction >>>>>>>> >>>>>>>> On 7/10/2020 8:41 AM, olcott wrote: >>>>>>>> ∃x ∃y (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q. >>>>>>>> ∃x ∃y ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable >>>>>>>> in Q. >>>>>>>> ∴ Q is incomplete relative to the commutative property of addition. >>>>>>>> >>>>>>>> The only aspect of this that I am unsure of is whether or not my >>>>>>>> use of the technical term unsatisfiable corresponds to its >>>>>>>> conventional use. >>>>>>> >>>>>>> Since you clearly don't understand the term, why insist on using it? >>>>>>> >>>>>>> It makes sense to ask whether some proposition φ is satisfiable. >>>>>>> It makes no sense to ask whether ⊢φ is satisfiable. >>>>>>> >>>>>>> André >>>>>>> >>>>>> >>>>>> Because when I ask: >>>>>> >>>>>> Is this expression true in Q? >>>>>> ∃x ∃y (Q ⊢ "x + y = y + x") people generally tell me that I am >>>>>> saying it incorrectly as if there is no such thing as true in Q >>>>>> until we say it using model theory. >>>>> Who are 'most people'? Are you true they aren't pointing out that >>>>> what you are actually trying to say is: >>>>> >>>>> Q ⊢ (∃x ∃y (x + y = y + x)) >>>>> >>>>> (with the ⊢ actually in the correct place and without the >>>>> meaningless quotation marks)? >>>>> >>>>> Except that presumably isn't what you're trying to say because that >>>>> is trivially provable in Q, and we were talking about statements >>>>> that weren't provable in Q. >>>> >>>> This means that either your paraphrase or my statement or both does >>>> not express this meaning: >>> >>> Mine wasn't a paraphrase. It was a syntactic correction, which >>> apparently you ignored since you keep putting the Q ⊢ in the wrong >>> place. >>> >>>> https://math.stackexchange.com/questions/998359/robinson-arithmetic-and-its-incompleteness >>>> >>>> >>>> Wikipedia in Italian has a sketch-of-proof that Robinson arithmetic >>>> is not complete, since commutativity of addition is undecidable. >>>> >>>> This seems much closer: >>>> ∃x ∈ N ∃y ∈ N (Q ⊢ (x + y = y + x)) >>>> ∃x ∈ N ∃y ∈ N (Q ⊢ (x + y != y + x)) >>> >>> Why don't you actually *think* about things instead of just randomly >>> making minor changes to your formulae. If you want to express the >>> fact that commutativity is not provable in Q, the correct expression >>> would be: >>> >>> Q ⊬ (∀x ∀y (x + y = y + x)) >> Yes that makes sense. That is a good way to say it. >> I need to translate that into this form: ∃φ (φ ↔ T ⊬ φ) >> >> φ = (∀x ∀y (x + y = y + x)) >> Q ⊬ φ // This is true in Q >> ∴ φ ↔ Q ⊬ φ is not true in Q > > Here's a suggestion. You keep mentioning Mendelson. While I am not sure, > I am assuming that this is a reference to Elliot Mendelson's Intro to > Mathematical Logic, which suggests you have access to a copy of this. > > Why don't you actually try reading this book STARTING AT THE BEGINNING. > You will notice that most chapters contain exercises. When you reach > such exercises, do not continue further until you can actually > successfully do all of those exercises and are confident you have done > them correctly. (I'm sure you can find a key online to verify your > solutions). > > This way you might actually learn how to use logical notation correctly, > as well as learning some actual logic in the process. Once you have > completed the entire book, then perhaps you will be able to say exactly > what it is that you are trying to say in a way that will be > comprehensible to others. > > André > It all boils down to two key things: (1) I need to express ideas that are inexpressible in conventional notation (2) I need to express ideas of mathematical logic at a very much higher level of abstraction than can be expressed using conventional notion. These are better than Mendelson's definitions because his do not simultaneously apply to every level of logic and every notion of a formal system. Satisfiability A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true. https://en.wikipedia.org/wiki/Satisfiability Interpretation (logic) An interpretation is an assignment of meaning to the [non-logical] symbols of a formal language. https://en.wikipedia.org/wiki/Interpretation_(logic) Model theory A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. https://en.wikipedia.org/wiki/Model_theory Within the above definitions: This sentence ∃φ (φ ↔ T ⊬ φ) really is unsatisfiable in every model including every model of arithmetic. When you tried to provide a counter-example you chopped off the existential quantifier making the strawman error. strawman An intentionally misrepresented proposition that is set up because it is easier to defeat than an opponent's real argument. https://www.lexico.com/definition/straw_man It does not matter that this: φ ↔ T ⊬ φ is provable in a meta-theory. What matters it that this: ∃φ (φ ↔ T ⊬ φ) is not provable in any theory. -- Copyright 2020 Pete Olcott