Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: sci.logic Subject: Re: Mathematical incompleteness has always been a misconception --- Tarski Date: Thu, 20 Feb 2025 10:54:03 +0200 Organization: - Lines: 70 Message-ID: References: <7e532aaf77653daac5ca2b70bf26d0a3bc515abf@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 20 Feb 2025 09:54:05 +0100 (CET) Injection-Info: dont-email.me; posting-host="a14a055df37b635f93113254737f4d81"; logging-data="2915212"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+5ytnqRZrPE9ZK4n4unF8e" User-Agent: Unison/2.2 Cancel-Lock: sha1:fyWfwZSvnvMpVxVWa2h8Q56k8xo= Bytes: 4625 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. It follows from the traditional meanings of "3", "2", and ">". Therefore the above statement is meaningless. >> The >> result depends on all of the change. But as long as we don't even >> know whether that kind of change is possible at all the details are >> impossible to determine. > > LP := ~True(LP) has never been more than nonsense. More specifically, your nonnsense. The symbol ":=" usually means definition but requires that the symbol on the left side (in this case "LP") is not used on the right side (and also that it is not used in the definition of any of the symbols on the right side). Usually languages of formal logic are constructed so that symbol that is defined with an expression that starts with a negation operator cannot be used as an argument to a function or a predicate. > Tarski (although otherwise quite brilliant) had a blind spot. Tarski did not use your nonsense. -- Mikko