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 09:12:49 -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 09:12:50 -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: <87k0zc8ps5.fsf@nosuchdomain.example.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Language: en-US Content-Transfer-Encoding: 8bit Message-ID: Lines: 47 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-Ai68lr2AKy4XPPhy55a+c+5vbXAywKV4QsF5w4+a3yYaTbv1RiiBlXN4f+xZg/I37oESY5GEQLye59w!eaWvsMq7RfukyxM9zwWVrhEvA6kUFk/t4B4IZzr/aWsMyKRtKISzymZ2oFC3A78ExcOG8Jar2ZfS 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: 4053 Xref: csiph.com comp.theory:21541 comp.ai.philosophy:21856 comp.ai.nat-lang:2294 On 7/9/2020 2:14 PM, Keith Thompson wrote: > olcott writes: >> On 7/9/2020 8:40 AM, André G. Isaak wrote: > [...] >>> I've asked you repeatedly about Robinson's arithmetic, in which x + >>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The >>> law of the excluded middle demands that one of those be true, so >>> there exists a true statement in Q which is not provable in Q. >>> >>> And one can prove that x + y = y + x is true in Q. You just can't >>> prove it from within Q. >> >> That is the exactly same key mistake that you, Tarski and presumably >> Gödel made. How do we know that it is true IN Q when it is not >> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS >> ONLY TRUE OUTSIDE OF Q. > > If it is not true in Q, then there are values x and y in Q such that > x + y = y + x is false in Q. > > In fact there are no such values. (You could refute that if you could > provide such values.) > > I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is > false in Q" are the only possibilities (law of the excluded middle). > Do you accept that assumption? > This is my current best guess of the correct use of the term satisfiable if the term satisfiable can even be applied to a single theory: ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q. ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q. This would seem to indicate that Q is incomplete relative to commutativity. I am certain that the ideas are correct. I am uncertain if my use of the term unsatisfiable corresponds to its conventional use. I am certain that my use of the term incomplete correctly augments the conventional use of the term such that my use is more correct than the conventional use. -- Copyright 2020 Pete Olcott