Path: ...!local-2.nntp.ord.giganews.com!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 17 Nov 2024 05:30:07 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-standard) Newsgroups: sci.math References: <84d9831f-d23a-4937-8333-4029c6c1f4a9@att.net> <263d223c-c255-4158-aa08-84ed11a48f20@att.net> <488cfa55-d881-4097-9825-d7630d7221eb@att.net> <5281d9d5-b6e8-4952-8cb3-8308957a497f@att.net> <7da62e1a-4e04-444a-9a3e-b9f6312d14d0@att.net> From: Ross Finlayson Date: Sat, 16 Nov 2024 21:30:19 -0800 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: <7da62e1a-4e04-444a-9a3e-b9f6312d14d0@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 163 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-FPjWTMIDJ5jjIDlIy0E9kM1ZSV0WUv/gTumQ9QG1s+sNQMn9ISxuQkhyW8FwpPf6Y6H2v86o48otkt2!bH4IT/qAPJ797sACZsoyA2lwdCIum4BgVGV65HzltxcIqc6HHDIpqSVdbXZwBKj7d1nVOOwmBg== X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 7818 On 11/16/2024 08:57 PM, Jim Burns wrote: > On 11/16/2024 11:35 PM, Ross Finlayson wrote: >> On 11/16/2024 08:18 PM, Jim Burns wrote: >>> On 11/16/2024 7:17 PM, Ross Finlayson wrote: >>>> On 11/16/2024 02:46 PM, Jim Burns wrote: >>>>> On 11/16/2024 5:31 PM, Ross Finlayson wrote: >>>>>> On 11/16/2024 12:29 PM, Jim Burns wrote: >>>>>>> On 11/16/2024 12:07 PM, Ross Finlayson wrote: >>>>>>>> On 11/16/2024 08:58 AM, Ross Finlayson wrote: >>>>>>>>> On 11/16/2024 02:22 AM, Jim Burns wrote: >>>>>>>>>> On 11/15/2024 9:52 PM, Ross Finlayson wrote: >>>>>>>>>>> On 11/15/2024 02:37 PM, Jim Burns wrote: >>>>>>>>>>>> On 11/15/2024 4:32 PM, Ross Finlayson wrote: > >>>>>>>>>>>>> Ah, yet according to Mirimanoff, >>>>>>>>>>>>> there do not exist standard models of integers, >>>>>>>>>>>> >>>>>>>>>>>> If it is true that >>>>>>>>>>>> our domain of discourse is a model of ST+PQ >>>>>>>>>>>> then it is true that >>>>>>>>>>>> our domain of discourse holds a standard integer.model. >>>>>>>>>>>> What is Mirimanoff's argument that >>>>>>>>>>>> it doesn't exist? >>>>>>>>>>> >>>>>>>>>>> Mirimanoff's? Russell's Paradox. >>>>>>>>>> >>>>>>>>>> ST+PQ does not suffer from claiming >>>>>>>>>> that the set of all non.self.membered sets >>>>>>>>>> is self.membered or claiming it isn't. >>>>>>> >>>>>>>>>> I don't say "infinity" is an axiom >>>>>>>>>> primarily because >>>>>>>>>> "infinity" is not an axiom of ST+PQ >>>>>>>>>> ST+PQ: >>>>>>>>>> ⎛ set {} exists >>>>>>>>>> ⎜ set x∪{y} exists >>>>>>>>>> ⎜ set.extensionality >>>>>>>>>> ⎜ plurality ⦃z:P(z)⦄ exists >>>>>>>>>> ⎝ plurality.extensionality >>>>>>>>>> >>>>>>>>>> "Infinity exists" == >>>>>>>>>> "the minimal inductive plurality exists" >>>>>>>>>> is a theorem of those axioms. >>>>>>> >>>>>>>>> ^- Fragment >>>>>>> >>>>>>> No. >>>>>>> The minimal inductive plurality is >>>>>>> a standard model of the integers. >>>>>>> >>>>>>>> Let's recall an example geometrically of what's >>>>>>>> so inductively and not so in the limit. >>>>>>>> >>>>>>>> Take a circle and draw a diameter, then bisect >>>>>>>> the diameter resulting diameters of common circles, >>>>>>>> all sharing a common diameter, vertical, say. >>>>>>>> >>>>>>>> Then, notice the length of the circle, is >>>>>>>> same, as the sum of the lengths of the half-diameter >>>>>>>> circles, their sum. >>>>>>>> >>>>>>>> So, repeat his dividing ad infinitum. In the limit, >>>>>>>> the length is that of the diameter, not the perimeter, >>>>>>>> while inductively, it's the diameter. >>>>>>>> >>>>>>>> Thusly, a clear example "not.first.false" being >>>>>>>> "ultimately.untrue". >>>>>>> >>>>>>> A finite sequence of claims, each claim of which >>>>>>> is true.or.not.first.false is >>>>>>> a finite sequence of claims, each claim of which >>>>>>> is true. >>>>>>> >>>>>>> The reason that that's true is that >>>>>>> THE SEQUENCE OF CLAIMS is finite. >>>>>>> >>>>>>> Whatever those CLAIMS refer to, >>>>>>> none of those CLAIMS are first.false. >>>>>>> (They're each not.first.false.) >>>>>>> >>>>>>> Since none of those CLAIMS are first.false, >>>>>>> none of those CLAIMS are false. >>>>>>> (That sequence is finite.) >>>>>>> >>>>>>> What those claims are ABOUT doesn't affect that. >>>>>>> For example, >>>>>>> being ABOUT an indefinite one of infinitely.many >>>>>>> doesn't affect that. >>>>>>> >>>>>>> Discovering >>>>>>> a finite sequence of claims, each claim of which >>>>>>> is true.or.not.first.false >>>>>>> in which there IS an untrue claim >>>>>>> is akin to >>>>>>> counting the eggs in a carton and >>>>>>> discovering that, there, in that carton, >>>>>>> 7 is NOT between 6 and 8. >>>>>>> There is a problem, but not with mathematics. >>>>>>> >>>>>>>> Then, with regards to your fragment, >>>>>>> >>>>>>> ...the minimal inductive plurality... >>>>>>> >>>>>>>> congratulations, >>>>>>> >>>>>>> Thank you. >>>>>>> >>>>>>>> you have ignored Russell his paradox and so on >>>>>>> >>>>>>> No. >>>>>>> >>>>>>> Selecting axioms which do not suffer from claiming >>>>>>> that the set of all non.self.membered sets >>>>>>> is self.membered or claiming it isn't >>>>>>> is not ignoring Russel, >>>>>>> it is responding to Russell. >>>>>>> >>>>>>> Russell points out that >>>>>>> _we do not want_ to claim >>>>>>> that the set of all non.self.membered sets >>>>>>> is self.membered or claiming it isn't. >>>>>>> >>>>>>> We respond: Okay, we'll stop doing that. >>>>>>> >>>>>>>> and quite fully revived Frege and given yourself >>>>>>>> a complete theory and consistent as it may be, and >>>>>>>> can entirely ignore all of 20'th century mathematics. >>>>>>>> >>>>>>>> It's small, .... Fragment >>>>>>> >>>>>>> The minimal inductive plurality. >>>>>>> Big or small, that's the thing, >>>>>>> the whole thing, and nothing but the thing. >>>>>> >>>>>> Bzzt, flake-out. >>>>>> It's not pretty the act of making lies. >>>>> >>>>> Tell me what you think is a lie: >>>> >>> >>> Quote what I wrote which you think is a lie: >>> >> >> Well, it's among what you clipped because it was un-answerable, > > I didn't clip any quote of me by you. > Anyway, do you need help finding earlier posts in a thread? > > ⎛ Noun > ⎜ lie (plural lies) > ⎜ > ⎜ 1. An intentionally false statement; an intentional falsehood. > ⎜ 2. A statement intended to deceive, even if literally true. > ⎝ 3. (by extension) Anything that misleads or disappoints. > > https://en.wiktionary.org/wiki/lie > ========== REMAINDER OF ARTICLE TRUNCATED ==========