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NNTP-Posting-Date: Sat, 07 Sep 2024 15:28:08 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (replete large numbers)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 7 Sep 2024 08:27:59 -0700
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On 09/07/2024 03:30 AM, FromTheRafters wrote:
> Ross Finlayson presented the following explanation :
>> On 09/06/2024 03:31 PM, FromTheRafters wrote:
>>> WM has brought this to us :
>>>> On 06.09.2024 20:17, FromTheRafters wrote:
>>>>> WM submitted this idea :
>>>>
>>>>>>> What the Hell could mean "to increase at an x" ?
>>>>>>
>>>>>> Example: The function f(x) = [x] increases at every x ∈ ℕ by 1.
>>>>>
>>>>> Make up your mind, is x real or natural.
>>>>
>>>> ℕ c ℝ.
>>>
>>> So what? There is no natural number of unit fractions less than any
>>> positive real or natural number. You said x was real in another post and
>>> here you claim it is natural. The output of your function is a constant
>>> Aleph_zero not a continuum.
>>
>> Some number theorists have that there's a point at infinity
>> and it's natural that way.
>>
>> The first "counterexample in topology" in "Counterexamples in Topology"
>> is that there's a smallest non-zero iota-value.
>>
>> When there are at least three models of real numbers,
>>
>> line-reals
>> field-reals
>> signal-reals
>>
>> after an Integer Continuum a la Scotists and before a
>> Long-Line Continuum a la duBois-Reymond, and there are
>> at least three law(s) of large numbers, and at least
>> three models of Cantor Space the square, sparse, and signal,
>> in a world where Vitali and Hausdorff already proved the
>> existence of doubling-spaces and doubling-measures before
>> there were Banach and Tarski, then it gets into that
>> "natural" is of a more replete surrounds than counting numbers.
>
> However, if one desires to add two plus four, it is not necessary to use
> complex numbers as your domain and codomain just because the naturals
> are contained in the complex number system. This, a form of the K.I.S.S.
> priciple. He wants to use the reals so he can pretend that there is a
> smooth sweep across all values in the interval. In reality each of his
> values is jumped to and there are Aleph_zero of them as each is defined
> as next (successor function) to the previous.

"Keep it simple, stupid", is reasonable but at some point it's
too dumb, even though everyone always hauls out the old "according
to Einstein, things should be as simple as possible, no, even
simpler", at some point in time that's too dumb.

As a continuous domain or the linear continuum, there are only
and everywhere and all real numbers between zero and one,
and there being three models of them indicates _variously_
their construction and access.

Paucity and a natural elegance in being trim is appreciated,
this is the all of mathematics' continuous domains and the
linear continuum in all its roles, a bit of book-keeping
has that at some point it's simpler to have all three and
keep track of their differences than one that gets dumb.