Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 17 Sep 2024 19:18:25 +0000 Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness) Newsgroups: sci.math References: <405557f7289631d63264c712d137244c940b9926@i2pn2.org> <13c08e96ad635f8142b38d89863a80caf17a32a8@i2pn2.org> <4faa63d0ff8c163f01a38736aeb5732184218a29@i2pn2.org> <7-ycnVjnAIKXynr7nZ2dnZfqn_qdnZ2d@giganews.com> <3906cb72-4bad-4a2a-97c7-4da857adc7a4@att.net> <05bf9c39-b715-41d6-b349-87ddc67941ff@att.net> From: Ross Finlayson Date: Tue, 17 Sep 2024 12:18:27 -0700 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: 131 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-DUbt7VIOaHh1wwyH0lb/MrKwhKT6ZNo4pzJ1buaMhf64sGSrlYvB7aR+H4vsN5UvT2lg07tSgRn8mJQ!kqSVus11M/QC4EJPIOEuFugwrg/16+aceUWtmXexA9AZyseM8kbVHjiBME8HYLlySWRHcypJi3tb!dw== 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: 6720 On 09/17/2024 11:57 AM, Ross Finlayson wrote: > On 09/17/2024 10:59 AM, Jim Burns wrote: >> On 9/16/2024 10:55 PM, Ross Finlayson wrote: >>> On 09/16/2024 11:24 AM, Jim Burns wrote: >>>> On 9/16/2024 2:13 PM, Jim Burns wrote: >>>>> On 9/15/2024 9:31 PM, Ross Finlayson wrote: >>>>>> On 09/15/2024 03:07 PM, FromTheRafters wrote: >> >>>>>>> What is the successor function on the reals? >>>>>>> Give me that, and maybe we can find the >>>>>>> 'next' number greater than Pi. >>>>>> >>>>>> Ah, good sir, then I'd like you to consider >>>>>> a representation of real numbers as >>>>>> with an integer part and a non-integer part, >>>>>> the integer part of the integers, and >>>>>> the non-integer part a value in [0,1], >>>>>> where the values in [0,1], are as of >>>>>> this model of (a finite segment of a) continuous domain, >>>>>> these iota-values, line-reals, >>>>>> as so established as according to the properties of >>>>>> extent, density, completeness, and measure, >>>>>> fulfilling implementing the Intermediate Value Theorem, >>>>>> thus for >>>>>> if not being the complete-ordered-field the field-reals, >>>>>> yet being these iota-values a continuous domain [0,1] >>>>>> these line-reals. >>>>> >>>>> As n → ∞, (ι=⅟n), ⟨0,ι,2⋅ι,...,n⋅ι⟩ → ℚ∩[0,1] >> >> Here follows the meaning of >> my.best.guess.at.what.you.mean. >> ⎛ I invite you to either >> ⎜ continue implicitly accepting my guess or >> ⎝ clarify what.you.mean. >> >> Define >> f: ℕ → 𝒫(ℝ) >> f(n) = ⟨0/n,1/n,2/n,...,n/n⟩ >> >> Define >> ran(f) = limⁿᐧᐧᐧ f(n) >> >> >> lim.infⁿᐧᐧᐧ f(n) ⊆ limⁿᐧᐧᐧ f(n) ⊆ lim.supⁿᐧᐧᐧ f(n) >> >> https://en.wikipedia.org/wiki/Set-theoretic_limit >> >> lim.infⁿᐧᐧᐧ f(n) = >> ⋃⁰ᑉⁿ ⋂ⁿᑉʲ f(j) = >> ⋃⁰ᑉⁿ {0,1} = >> {0,1} >> >> lim.supⁿᐧᐧᐧ f(n) = >> ⋂⁰ᑉⁿ ⋃ⁿᑉʲ f(j) = >> ⋂⁰ᑉⁿ ℚ∩[0,1] = >> ℚ∩[0,1] >> >> {0,1} ⊆ limⁿᐧᐧᐧ f(n) ⊆ ℚ∩[0,1] >> >> ---- >> ⋂ⁿᑉʲ f(j) = {0,1} >> ⋃ⁿᑉʲ f(j) = ℚ∩[0,1] >> >> ⎛ ⋂ⁿᑉʲ f(j) ⊆ >> ⎜ ⟨0/j,1/j,...,j⋅/j⟩ ∩ ⟨0/j⁺¹,1/j⁺¹,...,j⁺¹⋅/j⁺¹⟩ = >> ⎜ {0,1} >> ⎜ >> ⎝ ⋂ⁿᑉʲ f(j) ⊇ {0,1} >> >> ⎛ ∀ᴺj>n: f(j) ⊆ ℚ∩[0,1] >> ⎜ ⋃ⁿᑉʲ f(j) ⊆ ℚ∩[0,1] >> ⎜ >> ⎜ ∀p/q ∈ ℚ∩[0,1]: >> ⎜ nq > n ∧ >> ⎜ np/nq ∈ f(nq) ⊆ ⋃ⁿᑉʲ f(j) ∧ >> ⎜ np/nq = p/q ∈ ⋃ⁿᑉʲ f(j) >> ⎝ ℚ∩[0,1] ⊆ ⋃ⁿᑉʲ f(j) >> >>> It's shewn that [0,1] has no points not in ran(f). >> >> Has it been shown, though? >> >> For ran(f) = limⁿᐧᐧᐧ ⟨0/n,1/n,2/n,...,n/n⟩ >> {0,1} ⊆ ran(f) ⊆ ℚ∩[0,1] ∌ ⅟√2 ∈ [0,1] >> >> If ran(f) isn't that, then what is ran(f)? >> >>> About 2014, .... >> >> Was that your last word on ran(f), in 2014? >> >> I had hoped you would answer the point I've made. >> >> > > Well what it is that it was arrived at that the > only way to define the diagonal resulted being > according to only the diagonal of this ran(f), > here that these days and here that's the "pick > one of anti-diagonal and only-diagonal, ha, they're > combined, you get both or none", that the results > were stated along the lines of that "for any x in ran(f), > that ~x in ran(f)", then also "there is no x in [0,1] > not in ran(d)". > > So, here we still have this bit of the only-diagonal > that there exists "non-Cartesian functions", meaning > that this range can't be re-ordered or sub-setted > as for its definition, that just like anti-diagonal > results "this class of functions doesn't exist, rest > given to Cantor-Schroeder-Bernstein theorem", then > this only-diagonal simply makes "this class of functions > doesn't exist, not given to Cantor-Schroeder-Bernstein > theorem", which just reflects transitivity of cardinal > identity, why it's crossed over these "bridges" or "ponts" > as with regards to the book-keeping, of these _structures_. > > Then that each of a) extent, b) density, c) completeness, > d) measure are established, I do seem to recall that > your account was around when it was set out, for example, > that least-upper-bound is nice neat trivial next, while > at least four sigma-algebras for measure 1.0 were given. > > Another great discussion of Finlayson and Burns was the "Correct Presentation of the Diagonal Argument" thread on sci.logic.