Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 29 Dec 2024 03:54:50 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively) Newsgroups: sci.math References: <7356267c-491b-45c2-b86a-d40c45dfa40c@att.net> <4bf8a77e-4b2a-471f-9075-0b063098153f@att.net> <31180d7e-1c2b-4e2b-b8d6-e3e62f05da43@att.net> <45a632ed-26cc-4730-a8dd-1e504d6df549@att.net> <15f183ae29abb8c09c0915ee3c8355634636da31@i2pn2.org> From: Ross Finlayson Date: Sat, 28 Dec 2024 19:54:49 -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: 90 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-QWWbmyxf3ceS+Wi1tEUMcTEn3uUXjNK3ujqIUTPBlp3xEikvALov4xa920g81pS/149NKWfo4ev2UFV!u15AI+8QxwsraW+nr8GOUCpAoZZGr8Z9zybygkiNM6Y+I4RKtczaB8S7mruBnrpF8KUkqKNSALaU 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: 4742 On 12/28/2024 04:22 PM, Jim Burns wrote: > On 12/28/2024 5:36 PM, Ross Finlayson wrote: >> On 12/28/2024 11:17 AM, Richard Damon wrote: > >>> [...] >> >> Consider >> a random uniform distribution of natural integers, >> same probability of each integer. > > A probability.measure maps events > (such as: the selection of an integer from set S) > to numbers in real.interval [0,1] > > For each x ∈ (0,1] > there is a finite integer nₓ: 0 < ⅟nₓ < x > > ⎛ Assume a uniform probability.measure > ⎜ P: 𝒫(ℕ) -> [0,1] > ⎜ on ℕ the non.negative integers. > ⎜ ∀j,k ∈ ℕ: P{j} = P{k} = x > ⎜ x ∈ [0,1] > ⎜ > ⎜ If x = 0, Pℕ = ∑ᵢ₌᳹₀P{i} = 0 ≠ 1 > ⎜ P isn't a probability.measure. > ⎜ > ⎜ If x > 0, 0 < ⅟nₓ < x > ⎜ P{0,…,nₓ+1} > (nₓ+1)/nₓ > 1 > ⎝ P isn't a probability.measure. > > Therefore, > there is no uniform probability.measure on > the non.negative integers. > >> Now, you might aver >> "that can't exist, because it would be >> non-standard or not-a-real-function". > > I would prefer to say > "it isn't what it's describe to be, > because what's described is self.contradictory". > >> Then it's like >> "no, it's distribution is non-standard, >> not-a-real-function, >> with real-analytical-character". > > Which is to say, > "no, it isn't what it's described to be" > > You already accept that the "natural/unit equivalency function" has range with _constant monotone strictly increasing_ has _constant_ differences, _constant_, that as a cumulative function, for a distribution, has that relating to the naturals, as uniform. And that they always add up to 1, .... That most certainly is among the definitions of a distribution, the probabilities even range between 0 and 1. So, even though you refuse that this is a real function, because it's not, yet it's also a distribution, which it is. "Standardly as a limit of functions" if you won't, like Dirac's unit impulse function, never actually so esxcept its completion, yet necessarily actually so in the derivations that depend on it, like Fourier-style analysis, "real analytical character", say. Then also that it really is "a continuous domain" and "a discrete distribution", just keeps pointing out how special it is, "the natural/unit equivalency function". One of at least three set-theoretic accounted models of continuous domains, and establishing that there are non-Cartesian functions via an anti-anti-diagonal-argument.