Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: olcott Newsgroups: sci.logic Subject: Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of Truth Date: Sun, 2 Mar 2025 16:01:09 -0600 Organization: A noiseless patient Spider Lines: 86 Message-ID: References: <83cd07284fba793a0c2865dc5f6c21a9b9788a3e@i2pn2.org> <98bd133ddac2eff2f7a258b67452c36af9ab5a9b@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Sun, 02 Mar 2025 23:01:10 +0100 (CET) Injection-Info: dont-email.me; posting-host="f1890a324b06dc16a921f95b9719194f"; logging-data="999653"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX194092of0evHI3MngtEkTOw" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:bJ/zdujA0f4RtFcS6B+E0zI8rwQ= In-Reply-To: <98bd133ddac2eff2f7a258b67452c36af9ab5a9b@i2pn2.org> X-Antivirus: Norton (VPS 250302-6, 3/2/2025), Outbound message Content-Language: en-US X-Antivirus-Status: Clean Bytes: 5450 On 3/2/2025 3:25 PM, Richard Damon wrote: > On 3/2/25 4:16 PM, olcott wrote: >> On 3/2/2025 2:11 PM, dbush wrote: >>> On 3/2/2025 3:01 PM, olcott wrote: >>>> On 3/2/2025 1:27 PM, dbush wrote: >>>>> On 3/2/2025 2:21 PM, olcott wrote: >>>>>> >>>>>> When formal systems can be defined in such a way that they are not >>>>>> incomplete and undecidability cannot occur it is stupid to define >>>>>> them differently. >>>>>> >>>>> >>>>> That doesn't change the fact that Robinson arithmetic contains the >>>>> true statement "no number is equal to its successor" that has >>>>> *only* an infinite connection to the axioms >>>> >>>> If RA is f-cked up then toss it out on its ass. >>>> We damn well know that no natural number is equal to its >>>> successor as a matter of stipulation. >>> >>> We know it in RA though *only* an infinite connection to its axioms. >>> Yet the system still exists, and the axioms of the system make that >>> statement true, but *only* though an infinite connection to its axioms. >>> >>>> >>>> I have eliminated the necessity of systems that contain true >>>> statements that have *only* an infinite connection to their >>>> truthmakers. All >>>> formal systems that can represent arithmetic do not >>>> contain true statements that have *only* an infinite connection to >>>> their truthmakers unless you stupidly define them in a way that >>>> makes them contain true statements that have *only* an infinite >>>> connection to their truthmakers. >>> >>> As it turns out, any system capable of expressing all of the >>> properties of natural numbers contain at least one true statement >>> that has *only* an infinite connection to its truthmakers. >>> >>> Note also that I took the liberty of replacing "incomplete" in your >>> above statement with the accepted definition to make it more clear to >>> all what's being discussed. >>> >>> So if you only allow systems where all true statements have a finite >>> connection to their truthmakers, then you don't have natural numbers. >>> >>> So choose: do you want to have natural numbers, or do you only want >>> systems where all true statements have a finite connection to their >>> truthmaker? >> >> Tarski's True(X) is implemented by determining a finite connection >> to a truth-maker for every element of the set of human knowledge >> and an infinite connection to a truth-maker for all unknowable truths. >> >> > > Right, and thus is itself a proxy truth-maker for what it answer. > > Thus given P := ~True(P) > > If True determines that P has no connection to a truth maker, and thus > returns false, then P will be true, True(LP) determines that P is an infinite sequence, aborts its evaluation of this infinite sequence and returns false meaning not true stopping all evaluation thus not feeding false back into the evaluation sequence. The self-contradictory part of LP is unreachable in the same way as shown below. int DD() { int Halt_Status = HHH(DD); if (Halt_Status) HERE: goto HERE; return Halt_Status; } The self-contradictory part of DD emulated by HHH is unreachable code. -- Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer