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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality (real-valued)
Date: Sun, 04 Aug 2024 06:41:28 -0400
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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.