Path: csiph.com!xmission!news.snarked.org!border2.nntp.dca1.giganews.com!border1.nntp.dca1.giganews.com!nntp.giganews.com!buffer1.nntp.dca1.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Fri, 10 Jul 2020 08:41:11 -0500 Subject: =?UTF-8?Q?Re=3a_Simply_defining_G=c3=b6del_Incompleteness_and_Tarsk?= =?UTF-8?B?aSBVbmRlZmluYWJpbGl0eSBhd2F5IFYyNCDiiIPPhiAoz4Yg4oaUIFQg4oqsIM+G?= =?UTF-8?Q?=29?= Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics References: <0_CdnZNahqmDrZjCnZ2dnUU7-RHNnZ2d@giganews.com> From: olcott Date: Fri, 10 Jul 2020 08:41:09 -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: 64 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-F4HshbMjDXNovZDMwFsWKn205lfym3pcFxkFmz0+qwUrDG820hCn2AY6UjfDYgAmLFqbWobe3KaKtln!yACZn16/FlKIxkwyRp7zR8ef/AeHiRte2ZvugIPXY7AiqiGtJrwlwNEGq9oQUzopCKWYyiZ6joVR 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: 4600 Xref: csiph.com comp.theory:21539 comp.ai.philosophy:21854 comp.ai.nat-lang:2292 On 7/10/2020 4:39 AM, Alan Smaill wrote: > olcott writes: > >> On 7/8/2020 1:27 PM, Jeff Barnett wrote: > [..] >>>> Every sentence of any language that asserts its own unprovability >>>> cannot possibly be proved to be satisfied. >>>> >>>> If you dodge this question then I will no longer respond to you. >>> >>> It might be provable in a meta language! >> >> That is great, you are proving to be reasonable. >> >>> Since you don't seem to understand language levels and logical >>> referencing schemes, how can you possibly understand completeness >>> issues. >> >> Not only do I understand meta-language I understand that the theory / >> meta-theory distinction is the source of Tarski and Gödel's key >> mistake. >> >> Here is Tarski's actual proof >> http://www.liarparadox.org/Tarski_Proof_275_276.pdf >> >> It is much easier to see his error than it is to see the same error of >> Gödel unless you understand that when this expression: ∃φ (φ ↔ T ⊬ φ) >> is unsatisfiable in every model this necessarily includes that it is >> unsatisfiability in every model of arithmetic. > > To follow up on the example of geometry, where there is no derivation > of the 5th Euclidean axiom from the earlier axioms: Take T to be > the first 4 axioms. > > You are telling us that > > Ax5 <-> ( T ⊬ Ax5 ) > > has no model. > "T ⊬ Ax5" is true -- you accept that, don't you? > > So, you are telling us there is no model where Ax5 is true. > So, you are telling us that there is no model for Euclidean > geometry. (if there were, then Ax5 would be true in that model) > So, there is no such thing as a consistent > mathematics of 3-dimensional space ............ > I find that if I don't stay 100% focused on perfectly resolving one key point before moving on to another point that discussion is pointless because no key points ever get resolved. The following concrete example is more analogous to my key point than your example. After we 100% perfectly resolve my example then we can get back to your example. 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. Is the above expression unsatisfiable in Q? -- Copyright 2020 Pete Olcott