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NNTP-Posting-Date: Mon, 30 Dec 2024 02:00:42 +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, 29 Dec 2024 18:00:54 -0800
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On 08/03/2024 06:14 PM, Ross Finlayson wrote:
> 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.
>
> Rationals are equivalence classes of reduced 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.
>
>