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NNTP-Posting-Date: Tue, 01 Oct 2024 22:13:41 +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: Tue, 1 Oct 2024 15:13:44 -0700
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On 10/01/2024 01:13 PM, Jim Burns wrote:
> On 10/1/2024 2:02 PM, Ross Finlayson wrote:
>> On 10/01/2024 10:00 AM, Jim Burns wrote:
>
>>> 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.
>>
>> 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.

That is to say, the formalist has that "R" and "Q" are
as altogether different elements, though you might
have "partitioned the rationals by Dedekind cuts afterward",
that is not so, no. Instead there's speak of "real-valued",
and what it means is with regards to intensionality and
extensionality of for example __partial_ or _un-fulfilling_
models of some or R, like Q.

The, "1/N unit fractions", what is that, that does not
have 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.

Then otherwise what you have there appear facts
about N and R.

Then, where there exists a well-ordering of R,
then to take the well-ordering it results that
first there's a well-ordering of [0,1] for both
simplicity and necessity, and it's as the range of
the function n/d with 0 <= n < d and as d -> oo,
i.e., only in the infinite limit, that the properties
of the range of naturals, apply to the properties
of the range of [0,1].

It works in reverse, also.