Path: csiph.com!xmission!news.alt.net!feeder.usenetexpress.com!tr1.iad1.usenetexpress.com!border1.nntp.dca1.giganews.com!nntp.giganews.com!buffer1.nntp.dca1.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 12 Jul 2020 22:47:54 -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> <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> <87k0z85tt0.fsf@nosuchdomain.example.com> <87d0505kmk.fsf@nosuchdomain.example.com> <5Lmdnehh4P6hLZbCnZ2dnUU7-LdQAAAA@giganews.com> <878sfo5elp.fsf@nosuchdomain.example.com> From: olcott Date: Sun, 12 Jul 2020 22:47:53 -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: <878sfo5elp.fsf@nosuchdomain.example.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Language: en-US Content-Transfer-Encoding: 8bit Message-ID: Lines: 40 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-fRE3wNn8hwcXfedusFEI4LUbhxUaSy3wKx2bwZHGGm93LRWL7amTJwW3xKJT+jXY1gCU/6+SKmPWuAR!b4PjLCU5DnWVPGTVjlDJeBJcHlKGaP9PT/6QmgKKxdByuSI8wfAWE7qHABURk+MU0NMor2GvE7o= 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: 3723 Xref: csiph.com comp.theory:21609 comp.ai.philosophy:21937 comp.ai.nat-lang:2353 On 7/12/2020 9:32 PM, Keith Thompson wrote: > olcott writes: >> On 7/12/2020 7:22 PM, Keith Thompson wrote: >>> olcott writes: >>>> On 7/12/2020 4:04 PM, Keith Thompson wrote: > [...] >>>>> 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. >>>> >>>> Sure just add an axiom: ∀x ∈ ℕ ∃y ∈ ℕ (x + y = y + x) >>>> >>>>> Can we construct a consistent system based on Robinson Arithmetic >>>>> in which addition is provably *not* commutative? No. That would be like proving that existence never existed or finding some integer Crazy_Number such that Crazy_Number > 5 and Crazy_Number < 3. No thing of all things can be proved false that has been defined to be true. Defined to be true it the ultimate foundation of all truth. >>>> >>>> Not within the conventional semantics of the meaning of those terms. >>> >>> OK. Can you prove that? >> >> Nothing can possibly be disproved that is true by definition. > > I suppose that's a true statement, but how is it relevant? What > "definition" implies that no consistent system based on Robinson > Arithmetic can have non-commutative addition? > > Please don't bother posting a reply that doesn't actually answer my > question.. > -- Copyright 2020 Pete Olcott