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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
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Wed, 2 Oct 2024 19:01:49 -0700
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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′<q}: q∈ℚ₀ }
> ⎝ 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
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