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NNTP-Posting-Date: Sun, 04 Aug 2024 14:51:38 +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 07:52:13 -0700
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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.