Path: ...!local-1.nntp.ord.giganews.com!local-4.nntp.ord.giganews.com!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Thu, 03 Oct 2024 02:01:36 +0000 Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness) Newsgroups: sci.math References: <67d492c9-5b13-404c-80a1-7aa0b70f12a6@att.net> <9b2ffafe-78a1-4854-a27c-362a8d3a3552@att.net> <4030e5ac-0d5d-49ee-a387-da6828d600e8@att.net> <8c378b28-d9cd-42f8-bae9-5f38f4351611@att.net> <425ae3f7-fd09-4c62-8c2d-64708c727a47@att.net> <7VqdneXuS_2rpGH7nZ2dnZfqnPadnZ2d@giganews.com> <96ee1465-cf82-42e2-aeda-1117498e2b63@att.net> <65f8c6be-071c-44eb-a171-25f63fdf6c04@att.net> From: Ross Finlayson Date: Wed, 2 Oct 2024 19:01:49 -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: <65f8c6be-071c-44eb-a171-25f63fdf6c04@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 239 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-XRMw3GuccWf9AlFXjxStv5Dxkh3TUoXgBCLT/3HbtlJ9VJ+VzNs4P3LO4B8IjpYCcm5gkd+vfirw4F3!crtlo+aCQ0txugFxNf9cfOaY6aw6bM3pi0GVOvFAlMDcJBvp3Ndx1BQPmWZ+8fGnQ8/nnlUF85Ix 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: 10049 On 10/02/2024 11:57 AM, Jim Burns wrote: > On 10/1/2024 6:13 PM, Ross Finlayson wrote: >> On 10/01/2024 01:13 PM, Jim Burns wrote: >>> On 10/1/2024 2:02 PM, Ross Finlayson wrote: > >>>> Here it's that "Eudoxus/Dedekind/Cauchy is >>>> _insufficient_ to represent the character >>>> of the real numbers". >>>> >>>> Then, that there are line-reals and signal-reals >>>> besides field-reals, has that of course there are >>>> also models of line-reals and signal-reals in the >>>> mathematics today, like Jordan measure and the ultrafilter, >>>> and many extant examples where a simple deliberation >>>> of continuity according to the definitions of >>>> line-reals or signal-reals, results any contradictions >>>> you might otherwise see as arriving their existence. >>>> >>>> Then, besides noting how it's broken, then also >>>> there's given a reasoning how it's repaired, >>>> resulting "less insufficient", or at least making >>>> it so that often found approaches in the applied, >>>> and their success, make the standard linear curriculum, >>>> unsuited. >>>> >>>> Then, I think it's quite standard how I put it, >>>> really very quite standard. >>> >>> I hope this will help me understand you better. >>> Please accept or reject each claim and >>> -- this is important -- >>> replace rejected claims with >>> what you _would_ accept. >>> >>> ⎛ ℝ, the complete ordered field, is >>> ⎝ the consensus theory in 2024 of the continuum. >>> >>> ⎛ ℝ contains ℚ the rationals and >>> ⎜ the least upper bound of >>> ⎝ each bounded nonempty subset of ℚ and of ℝ >>> >>> ( The greatest lower bound of ⅟ℕ unit fractions is 0 >>> >>> ⎛ A unit fraction is reciprocal to a natural>0 >>> ⎜ >>> ⎜ A set≠{} ⊆ ℕ naturals holds a minimum >>> ⎜ A natural≠0 has a predecessor.natural. >>> ⎜ A natural has a successor.natural. >>> ⎜ >>> ⎜ The sum of two naturals is a natural >>> ⎝ the product of two naturals is a natural. >>> >>> ⎛ There are no points in ℝ >>> ⎜ between 0 and all the unit fractions >>> ⎝ (which is what I mean here by 'infinitesimal'). >>> >>> Thank you in advance. >> >> Well, first of all there's a quibble that >> R is not usually said to contain Q as much as that >> there's that in real-values that >> there's a copy of Q embedded in R. > > I take your lack of an explicit rejection of > the Dedekind.complete continuum.consensus > to be an implicit acceptance of > the Dedekind.complete continuum.consensus. > > A quibble for your quibble: > A set isomorphic to ℚ is usually said to _be_ ℚ > Each model of ℚ is ℚ > > A model of complete.ordered.field ℝ supersets > a model of rational.ordered.field ℚ, > which is to say, ℝ contains ℚ > > I think that you (RF) are pointing to this: > > ⎛ Consider a model ℚ₀ of the rationals > ⎜ which has only urelements. > ⎜ > ⎜ Using ℚ₀ construct > ⎜ a model of ℝ complete ordered field > ⎜ in any of several known ways: > ⎜ a partition of Cauchy sequences of rationals, > ⎜ open.foresplits of rationals, > ⎜ or something else. > ⎜ > ⎜ The set ℝₛ of open.foresplits of ℚ₀ > ⎜ { S⊆ℚ₀: {}≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ₀\S≠{} } > ⎜ is a model of ℝ > ⎜ It holds an open foresplit for each irrational point, > ⎜ and an open foresplit for each rational point > ⎜ > ⎜ The set ℚₛ of open.foresplits for rationals > ⎜ { {q′∈ℚ₀:q′ ⎝ is not ℚ₀ > > Yes, > ℚₛ ≠ ℚ₀ > However, > both ℚₛ and ℚ₀ are models of ℚ, > we say both ℚₛ and ℚ₀ we are ℚ, > Each theorem we prove for ℚ, > for example, that no element of ℚ is √2, > is true of both ℚₛ and ℚ₀, > and that's enough for (consensus) us. > >> The, "1/N unit fractions", what is that, >> that does not have a definition. > > Read a bit more and you'll see a definition. > >> Is that some WM-speak? >> I suppose that >> if it means the set 1/n for n in N >> then the g-l-b is zero. > > Thank you. > My motivation has been to find out if you accept that. > The rest is to make sure we're talking about > the same things. > > Because > g.l.b of ⅟ℕ (⅟n for n in ℕ) is 0 > there is no positive lower bound of ⅟ℕ > > A point between 0 and ⅟ℕ would be > a positive lower bound of ⅟ℕ > Such a point doesn't exist. > > When I say infinitesimals don't exist, > I mean points between 0 and set ⅟ℕ > in the complete.ordered.field > don't exist. > > When I say that, and then you name.check > various other systems which have infinitesimals, > it _sounds to me_ as though > you object to my claim. > All of this has been my attempt to sort out > _what you're saying_ > >> Then otherwise what you have there appear facts >> about N and R. > > They're facts which identify ℕ and ℝ from among > a host of other possible things.called ℕ or ℝ > I take your lack of an explicit rejection > to be an implicit acceptance, and > I take you and I to be talking about > the same ℕ and the same ℝ > >> Then, >> where there exists a well-ordering of R, >> then to take the well-ordering it results that ========== REMAINDER OF ARTICLE TRUNCATED ==========