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NNTP-Posting-Date: Sun, 04 Aug 2024 15:54:06 +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 08:54:52 -0700
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On 08/04/2024 08:46 AM, Ross Finlayson wrote:
> On 08/04/2024 07:59 AM, Ross Finlayson wrote:
>> 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.)
>>
>>
>
>
> It's kind of like when people say "hey you know the
> initial ordinal assignment is what we can say 'are'
> cardinals", then it's like, "with the Continuum Hypothesis
> being undecide-able and all, then there are and aren't
> ordinals between what would be those cardinals by
> their cardinals the ordinals", sort of establishing that
> such a definition does and doesn't keep itself non-contradictory,
> then that's more or template boiler-plate lines to
> add to "rule 1: stop thinking and forget".
>
> So, cardinals are equivalence classes of sets according
> to function theory, which itself is a bit loose, here though
> that it's battened down that there's always the Cartesian
> courtesy comprehension, except a sort of special non-Cartesian
> example, then that above that again is the long-line of
> duBois-Reymond of all the expressions of real functions.
>
> ... Which only has the "complete" linear continuum to sit
> on, these line-reals, field-reals, and signal-reals, "real-valued".
>

Of course Eudoxus is really great about the field and complete
ordered field, in terms of Aristotle's line-reals and field-reals.

Zeno's theories, ....