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NNTP-Posting-Date: Sun, 04 Aug 2024 14:58:16 +0000
Subject: Re: Replacement of Cardinality (real-valued)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 4 Aug 2024 07:59:01 -0700
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On 08/04/2024 07:52 AM, Ross Finlayson wrote:
> On 08/04/2024 03:41 AM, FromTheRafters wrote:
>> Ross Finlayson formulated the question :
>>> On 08/03/2024 02:59 PM, FromTheRafters wrote:
>>>> Chris M. Thomasson formulated on Saturday :
>>>>> On 8/3/2024 7:25 AM, WM wrote:
>>>>>> Le 02/08/2024 à 19:31, Moebius a écrit :
>>>>>>> For each and every of these points [here referred to with the
>>>>>>> variable "x"]: NUF(x) = ℵ₀ .
>>>>>>
>>>>>> I recognized lately that you use the wrong definition of NUF.
>>>>>> Here is the correct definition:
>>>>>> There exist NUF(x) unit fractions u, such that for all y >= x: u < y.
>>>>>> Note that the order is ∃ u ∀ y.
>>>>>> NUF(x) = ℵ₀ for all x > 0 is wrong. NUF(x) = 1 for all x > 0 already
>>>>>> is wrong since there is no unit fraction smaller than all unit
>>>>>> fractions.
>>>>>> ℵ₀ unit fractions need ℵ₀*2ℵ₀ points above zero.
>>>>>
>>>>> 0->(...)->(1/1)
>>>>>
>>>>> Contains infinite unit fractions.
>>>>>
>>>>> 0->(...)->(1/2)->(1/1)
>>>>>
>>>>> Contains infinite unit fractions.
>>>>>
>>>>> 0->(...)->(1/3)->(1/2)->(1/1)
>>>>>
>>>>> Contains infinite unit fractions.
>>>>>
>>>>> However, (1/3)->(1/1) is finite and only has three unit fractions
>>>>> expanded to:
>>>>>
>>>>> (1/3)->(1/2)->(1/1)
>>>>>
>>>>> Just like the following has four of them:
>>>>>
>>>>> (1/4)->(1/3)->(1/2)->(1/1)
>>>>>
>>>>>
>>>>> (0/1) is not a unit fraction. There is no smallest unit fraction.
>>>>> However, the is a largest one at 1/1.
>>>>>
>>>>> A interesting part that breaks the ordering is say well:
>>>>>
>>>>> (1/4)->(1/2)
>>>>>
>>>>> has two unit fractions. Then we can make it more fine grain:
>>>>>
>>>>> (1/4)->(1/2) = ((1/8)+(1/8))->(1/4+1/4)
>>>>>
>>>>> ;^)
>>>>
>>>> Unit fractions are ordered pairs, not infinite. :)
>>>
>>> Real numbers are equivalence classes of sequences that are Cauchy,
>>> and cardinals are equivalence classes of sets under
>>> Cantor-Schroeder-Bernstein.
>>
>> He didn't use real intervals this time, so I will treat this as dealing
>> with a subset of rationals. He often uses a term like 'infinite unit
>> fractions' when he means 'infinitely many unit fractions' instead.
>>
>>> Rationals are equivalence classes of reduced fractions.
>>
>> Need they be reduced, or are the reduced and/or proper fractions chosen
>> from all of the proper and improper fractions?
>>
>>> In ZF's usual standard descriptive set theory, ....
>>>
>>>
>>> Then, a common way to talk about this is the "real values",
>>> that, the real-valued of course makes sure that there are
>>> equivalence classes of integers, their values as rationals,
>>> and their values as real numbers, keeping trichotomy or
>>> otherwise the usual laws of arithmetic all among them,
>>> where they're totally different sets of, you know, classes,
>>> that though in the "real-valued" it's said that extensionality
>>> is free and in fact given.
>>>
>>> It's necessary to book-keep and disambiguate these things
>>> in case the ignorant stop at a definition that though is
>>> supported way above in the rest of the usual model assignment.
>>
>> My view is that the rationals as embedded in the reals should act like
>> the rationals in Q, so why not use Q's ordered pairs instead of R to
>> reduce complications. It's like simplification in chess.
>
> Yeah, the reduced fractions is a bit contrived, thanks.
>
> Here "Dedekind cuts" or "partitions of rationals by reals"
> don't exist except as "partitions of rationals by reals",
> as with regards to the rationals being HUGE and all.
>
> The other day I was reading about Cantor at Halle and Dirichlet
> and the formulation and formalism of the Fourier series in the
> Fourier-style analysis, where right before the Mengenlehre or
> set theory, Cantor arrived at a way to show that the coefficients
> of a Fourier series are unique. Then though the other day I was
> reading a collection from a symposium after the '50's and '60's
> in turbulence theory, where it's suggested that Phythian provides
> a counterexample.
>
> After Cauchy-Weierstrass then the Riemann then Lebesgue "what is
> integrable" or measure theory and the measure problem and the
> Dirichlet function (1 at rationals, 0 at irrationals, content?)
> then there are lots of developments in the measure, the content,
> the analytical character.
>
> What's of interest of formalism is to provide rigor to derivations,
> here it's so that the standard reals are equivalence classes of
> sequences that are Cauchy, and that about the HUGE rationals and
> that their real-values are trichtomous and dense in the reals,
> they yet do not have the least-upper-bound property, which
> the real numbers, of the linear continuum, do.
>
>

(Apocryphally there was already a development with regards to
the uniqueness of the coefficients of Fourier series.  Also
the anti-diagonal was discovered by du Bois-Reymond and various
other turns of thought in combinatorics and quantification were
already known.)