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 12:16:52 -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 12:16:52 -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: 56 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-lJxCCHrZSu4YEEVzMpDzsqgnR+HukKnv+Cq4ewVSQvamAY60WN4xVoNZodsG8VpeZNgiXMyzbQRXCzJ!llxlKegwPmbSyuBElX/EKiBMZmJsvzQ5/XHfkctZYIcpC+YpjFS6/WUL3XLnoQpwPqsYZHjRfE/V 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: 4342 Xref: csiph.com comp.theory:21549 comp.ai.philosophy:21865 comp.ai.nat-lang:2302 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: 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)) -- Copyright 2020 Pete Olcott