Path: ...!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Mon, 30 Dec 2024 02:05:27 +0000 Subject: Re: Replacement of Cardinality (natural infinity) Newsgroups: sci.logic,sci.math References: <227e12c2862e139d022279d3ae5bdd34427bafae@i2pn2.org> <1587b53ce632061f593a3880f94ddc20f4638662@i2pn2.org> <7d6cbcd7eda17e5fe5793af70eaccb117657fed5@i2pn2.org> From: Ross Finlayson Date: Sun, 29 Dec 2024 18:05:40 -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: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 70 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-5WRZEdk56xFTrC+SI3T6mMbAMMLFc3pgCjaxT01iz0jeB6nb/51dINaOuaGH1u/CuP+Tm0jdjpXBv+a!tdMBiTdtr4egTfSZC/Ozyq3n9iRpT6XHk3Zt1AiliqHZREfv4zcmJLhUnNQCUtpTcUn4OKmNtrA= 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: 4533 On 08/22/2024 07:04 PM, Ross Finlayson wrote: > On 08/22/2024 06:23 PM, Richard Damon wrote: >> On 8/22/24 8:19 AM, WM wrote: >>> Le 22/08/2024 à 02:10, Richard Damon a écrit : >>>> On 8/21/24 8:32 AM, WM wrote: >>> >>>>> No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all >>>>> and only reciprocals of natural numbers. >>> >>>> Can't be, because if it WAS 1/n, then 1/(n+1) would be before it, >>> >>> That is tadopted from definable numbers. It is not true for all dark >>> numbers. >>> >>> Regards, WM >> >> But you claim the Natural Numbers, which define the whole infinite >> sequence. >> >> Every Natural Number has, by its definition, a successor, so there is >> not last one. >> >> And, by your own definitions, if you can use the number individually, >> which you did for 1/n and thus n, you can use "normal mathematics" on >> it, that that says that if n exists, so does n+1 as we have a definition >> for that number, thus there is not last definable number. >> >> Yes, if 1/n was a "dark number" we might not be able to find the n+1 in >> the dark numbers, but none of those are Natural Numbers, but must be >> some beyond-finite set of numbers. > > One might imagine that the definition of "natural" > numbers is exactly insofar as what exist, "natural" > in the sense of being an entire model of integers. > Then, "whole" numbers are usually the word for > integers, the counting integer or whole numbers, > that "natural" integers, for example, often include > zero, then as with regards to whether they include > infinity, or not. > > So, some have for something like extra-ordinary sets, > that N = N+, that is to say, being "merely infinite" is a > big enough ordinal that it contains itself, and that > that's automatically "natural" because there's not > even anything that can be done about it, it arises > from naive and thus natural quantification over > the elements, there is no rule number one barring it, > so, "naturals" might have infinitely-grand members. > > Then, for that infinity caps the naturals as much as > as infinity extends the naturals, is about how by > various usual simple definitions, that the naturals > are _compact_, the naturals make a space, and N the > point-at-infinity is a one-point compactification of > the space the naturals. > > > So, really, some have that the naturals, always > have these properties, being that they are compact, > containing their compactification, and, that they > are extra-ordinary, containing their order-type, > either and both of those being infinite. > > Then, you'll welcome to aver that you've made > no mention at all of infinity in your definition > of finite, whole counting numbers. > > The natural numbers though are infinite, and infinite. >