Path: csiph.com!weretis.net!feeder8.news.weretis.net!reader5.news.weretis.net!news.solani.org!.POSTED!not-for-mail From: polcott Newsgroups: sci.logic,comp.theory,sci.math Subject: Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Date: Sat, 27 Jun 2026 10:47:26 -0500 Message-ID: <111orae$1kcvi$3@solani.org> References: <110v2oa$25osr$1@dont-email.me> <1119bav$10b1s$1@dont-email.me> <1119g43$2m31$1@news.muc.de> <1119ibn$12eb6$1@dont-email.me> <1119jrq$v9fi$4@dont-email.me> <1119qho$14js0$1@dont-email.me> <111alun$1avuu$1@dont-email.me> <111b8nj$1ghve$2@dont-email.me> <111d7ol$22057$1@dont-email.me> <111e6cm$2b34r$2@dont-email.me> <111g9sf$2u0b2$1@dont-email.me> <111heic$39se8$3@dont-email.me> <111ikue$3jb8p$1@dont-email.me> <111jk5s$3tg4p$1@dont-email.me> <111l6s0$b5nm$1@dont-email.me> <111ltoc$i1vi$1@dont-email.me> <111luau$hlri$2@dont-email.me> <111lvq9$il98$1@dont-email.me> <111m0g5$hlri$5@dont-email.me> <111m230$jdo3$1@dont-email.me> <111m85r$lde0$1@dont-email.me> <111mcgq$ms4u$1@dont-email.me> <111mfi9$ns29$1@dont-email.me> <111mhh6$oha7$1@dont-email.me> <111mho7$lde0$6@dont-email.me> <111mild$osp8$1@dont-email.me> <111mj8p$lde0$7@dont-email.me> <111mkfl$pefu$1@dont-email.me> <111mlfh$lde0$8@dont-email.me> <111msrp$rtlv$1@dont-email.me> <111o07k$149pv$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 27 Jun 2026 15:47:26 -0000 (UTC) Injection-Info: solani.org; logging-data="1717234"; mail-complaints-to="abuse@news.solani.org" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:XLsW8DKTLpsiUD1jDKYw1Jamo+c= X-Antivirus: Norton (VPS 260627-2, 6/27/2026), Outbound message X-User-ID: eJwNycEBwCAIA8CViAa04wiS/Uew9z2fgajF8KDLtSkaeljvGEBfHOHfwVw4n1XeTvkswWllDw67EOM= Content-Language: en-US X-Antivirus-Status: Clean In-Reply-To: <111o07k$149pv$1@dont-email.me> Xref: csiph.com sci.logic:347085 comp.theory:142034 sci.math:645742 On 6/27/2026 3:05 AM, Mikko wrote: > On 27/06/2026 01:01, olcott wrote: >> On 6/26/2026 2:55 PM, dbush wrote: >>> On 6/26/2026 3:38 PM, olcott wrote: >>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>> >>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>> >>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>> the statement "no number is equal to its >>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>> >>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>> >>>>>>>>> André >>>>>>>>> >>>>>>>> >>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>> >>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>> through inference steps. This simple sentence seems >>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>> >>>>>>> >>>>>>> So "out-of-scope" is merely a synonym for unprovable.  Then to >>>>>>> put things in words you can understand: >>>>>>> >>>>>> >>>>>> "I am driving to Walmart to buy a carton of >>>>>>   Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>> In both cases the semantics in not represented in PA. >>>>> >>>>> Not applicable, as that is not a sentence in PA. >>>>> >>>> >>>> It is expressed in PA >>> >>> False.  The above is not a sentence of PA. >>> >>>> to the same degree that G is expressed >>>> in PA has a huge natural number. The semantics of it and >>>> the semantics of G are neither expressible in PA. >>> >>> False.  G is simply a sentence like ~∃x x>10 v x<5 but much more >>> complex. >>> >>>> >>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>> scope"). >>>>> >>>> >>>> If its semantics is not expressible in Q (What RA is called) >>>> then it is not actually expressible in Q. >>> >>> "No number is equal to its successor" is a sentence in the language >>> of Q.  More formally, it is this: >>> >>> ~∃x x=S(x) >>> >>> And this sentence is not provable from the axioms of Q (or, in terms >>> you would understand, the above is "out-of-scope" of Q). >>> >> >> OK I checked the details so I need to make my >> language more precise. >> >> Within proof theoretic semantics any expression >> that cannot be proven in Q is not semantically >> grounded in Q. > > Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and > that way in the theory. > Colorless green ideas sleep furiously was composed by Noam Chomsky in his 1957 book Syntactic Structures as an example of a sentence that is grammatically well-formed, but semantically nonsensical. Proving that syntax is not enough. -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).