Path: ...!weretis.net!feeder9.news.weretis.net!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 Truth Date: Sun, 2 Mar 2025 16:25:03 -0500 Organization: i2pn2 (i2pn.org) Message-ID: <98bd133ddac2eff2f7a258b67452c36af9ab5a9b@i2pn2.org> References: <8638c66ecc1669437be5a141cfa358c8c6168cde@i2pn2.org> <83cd07284fba793a0c2865dc5f6c21a9b9788a3e@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Sun, 2 Mar 2025 21:25:03 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="2553149"; mail-complaints-to="usenet@i2pn2.org"; posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg"; User-Agent: Mozilla Thunderbird Content-Language: en-US In-Reply-To: X-Spam-Checker-Version: SpamAssassin 4.0.0 Bytes: 5297 Lines: 81 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, and thus shows that True has made an error, as the expression P HAS a connection, a finite one in fact, to its proxy truth-maker of True. This is not allowed. If True determines that P has that connection shown above, then P will be false, and thus we find that True has declared a false statement to have a truth maker. This is not allowed either. Thus, True can not exist. The problem is that P defined as ~True(P) is an expression that can be created to exist in the Theory, based on the Meta-Theory created, as long as the Theory can express the properties of the Natural Numbers, and has a True Predicate. Thus, The expression can't just be "rejected" because it was a valid statement. The answer is that such a True Predicate can't exist.