Path: news.eternal-september.org!eternal-september.org!feeder3.eternal-september.org!news.quux.org!news.nk.ca!rocksolid2!i2pn2.org!.POSTED!not-for-mail From: Richard Damon Newsgroups: sci.logic Subject: Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of True(L,x) Date: Tue, 25 Feb 2025 23:21:28 -0500 Organization: i2pn2 (i2pn.org) Message-ID: <8da4ac4d2392bcaca8dd6ff4491961f0db81e5e1@i2pn2.org> References: <7e532aaf77653daac5ca2b70bf26d0a3bc515abf@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Wed, 26 Feb 2025 04:21:28 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="1816923"; mail-complaints-to="usenet@i2pn2.org"; posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg"; User-Agent: Mozilla Thunderbird In-Reply-To: Content-Language: en-US X-Spam-Checker-Version: SpamAssassin 4.0.0 On 2/25/25 12:41 PM, olcott wrote: > On 2/25/2025 9:46 AM, Mikko wrote: >> On 2025-02-24 22:53:06 +0000, olcott said: >> >>> On 2/24/2025 3:13 AM, Mikko wrote: >>>> On 2025-02-22 18:27:00 +0000, olcott said: >>>> >>>>> On 2/22/2025 3:18 AM, Mikko wrote: >>>>>> On 2025-02-21 23:19:10 +0000, olcott said: >>>>>> >>>>>>> On 2/20/2025 2:54 AM, Mikko wrote: >>>>>>>> On 2025-02-18 03:59:08 +0000, olcott said: >>>>>>>> >>>>>>>>> On 2/12/2025 4:21 AM, Mikko wrote: >>>>>>>>>> On 2025-02-11 14:07:11 +0000, olcott said: >>>>>>>>>> >>>>>>>>>>> On 2/11/2025 3:50 AM, Mikko wrote: >>>>>>>>>>>> On 2025-02-10 11:48:16 +0000, olcott said: >>>>>>>>>>>> >>>>>>>>>>>>> On 2/10/2025 2:55 AM, Mikko wrote: >>>>>>>>>>>>>> On 2025-02-09 13:10:37 +0000, Richard Damon said: >>>>>>>>>>>>>> >>>>>>>>>>>>>>> On 2/9/25 5:33 AM, Mikko wrote: >>>>>>>>>>>>>>>> Of course, completness can be achieved if language is >>>>>>>>>>>>>>>> sufficiently >>>>>>>>>>>>>>>> restricted so that sufficiently many arithemtic truths >>>>>>>>>>>>>>>> become inexpressible. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> It is far from clear that a theory of that kind can >>>>>>>>>>>>>>>> express all arithmetic >>>>>>>>>>>>>>>> truths that Peano arithmetic can and avoid its >>>>>>>>>>>>>>>> incompletness. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> WHich, it seems, are the only type of logic system that >>>>>>>>>>>>>>> Peter can understand. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> He can only think in primitive logic systems that can't >>>>>>>>>>>>>>> reach the complexity needed for the proofs he talks >>>>>>>>>>>>>>> about, but can't see the problem, as he just doesn't >>>>>>>>>>>>>>> understand the needed concepts. >>>>>>>>>>>>>> >>>>>>>>>>>>>> That would be OK if he wouldn't try to solve problems that >>>>>>>>>>>>>> cannot even >>>>>>>>>>>>>> exist in those systems. >>>>>>>>>>>>> >>>>>>>>>>>>> There are no problems than cannot be solved in a system >>>>>>>>>>>>> that can also reject semantically incorrect expressions. >>>>>>>>>>>> >>>>>>>>>>>> The topic of the discussion is completeness. Is there a >>>>>>>>>>>> complete system >>>>>>>>>>>> that can solve all solvable problems? >>>>>>>>>>> >>>>>>>>>>> When the essence of the change is to simply reject expressions >>>>>>>>>>> that specify semantic nonsense there is no reduction in the >>>>>>>>>>> expressive power of such a system. >>>>>>>>>> >>>>>>>>>> The essence of the change is not sufficient to determine that. >>>>>>>>> >>>>>>>>> In the same way that 3 > 2 is stipulated the essence of the >>>>>>>>> change is that semantically incorrect expressions are rejected. >>>>>>>>> Disagreeing with this is the same as disagreeing that 3 > 2. >>>>>>>> >>>>>>>> That 3 > 2 need not be (and therefore usually isn't) stripualted. >>>>>>> >>>>>>> The defintion of the set of natural numbers stipulates this. >>>>> >>>>> If NOTHING ever stipulates that 3 > 2 then NO ONE can >>>>> possibly know that 3 > 2 leaving the finite string >>>>> "3 > 2" merely random gibberish. >>>> >>>> A formal language of a theory of natural numbers needn't define "2" or >>>> "3". Those concepts can be expressed as "1+1" and "1+1+1" or as "SS0" >>>> and "SSS0" depending on which symbols the language has. >>> >>> If nothing anywhere specifies that "3>2" then no one >>> ever has any way of knowing that 3>2. >> >> Of course there is. From definitions and psotulates one can prove >> that 3 > 2, at least in some formulations. Or that 1+1+1 > 1+1 if >> the language does not contaion "3" and "2". >> > > In other words you don't know what "nothing anywhere" means. > The problem is you can't write your expression and have nothing anywhere. Unless you undefine your system so that the symbol '3', the symbol '2', and the symbol '>' don't have meaning, from their basic definitions that results follow. Just more of you not understanding how logic works, and you working on hypotheticals of fantasy.