Path: ...!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott Newsgroups: comp.theory Subject: Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct Date: Sat, 9 Nov 2024 13:50:01 -0600 Organization: A noiseless patient Spider Lines: 46 Message-ID: References: <4b24331953934da921cb7547b6ee2058ac9e7254@i2pn2.org> <2a5107f331836f388ad259bf310311a393c00602@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 09 Nov 2024 20:50:02 +0100 (CET) Injection-Info: dont-email.me; posting-host="529658128fc1f19cc0ff32f79f31d785"; logging-data="4141691"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/1kbEmMTu90wEsyBTlXJ9o" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:nYF9LcmIeKc8daPhuNu+OUc+RWU= X-Antivirus: Norton (VPS 241109-4, 11/9/2024), Outbound message X-Antivirus-Status: Clean Content-Language: en-US In-Reply-To: Bytes: 3239 On 11/9/2024 1:32 PM, Alan Mackenzie wrote: > olcott wrote: > >> The assumption that ~Provable(PA, g) does not mean ~True(PA, g) >> cannot correctly be the basis for any proof because it is only >> an assumption. > > It is an assumption which swifly leads to a contradiction, therefore must > be false. You just said that the current foundation of logic leads to a contradiction. Too many negations you got confused. When we assume that only provable from the axioms of PA derives True(PA, g) then (PA ⊢ g) merely means ~True(PA, g) THIS DOES NOT LEAD TO ANY CONTRADICTION. > But you don't understand the concept of proof by > contradiction, and you lack the basic humility to accept what experts > say, so I don't expect this to sink in. > >>> We know, by Gödel's Theorem that incompleteness does exist. So the >>> initial proposition cannot hold, or it is in an inconsistent system. > >> Only on the basis of the assumption that >> ~Provable(PA, g) does not mean ~True(PA, g) > > No, there is no such assumption. There are definitions of provable and > of true, and Gödel proved that these cannot be identical. > *He never proved that they cannot be identical* The way that sound deductive inference is defined to work is that they must be identical. A conclusion IS ONLY true when applying truth preserving operations to true premises. It is very stupid of you to say that Gödel refuted that. -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer