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NNTP-Posting-Date: Thu, 20 Jun 2024 20:40:33 +0000
Subject: Re: how (quantities and units, implicits and explicits, intensional
 and extensional)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 20 Jun 2024 13:40:36 -0700
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On 06/20/2024 09:47 AM, Jim Burns wrote:
> On 6/19/2024 8:32 PM, Ross Finlayson wrote:
>> On 06/19/2024 01:29 PM, Ross Finlayson wrote:
>>> On 06/19/2024 09:43 AM, Jim Burns wrote:
>
>>>> https://en.wikipedia.org/wiki/Individual
>>>> | An individual is
>>>> | that which exists as a distinct entity.
>>>>
>>>> Nice thing about the English language:
>>>> There are separate grammatical categories for
>>>> what exists as distinct entities (count nouns)
>>>> and what doesn't (mass nouns).
>>>>
>>>> Is the continuum a count noun or a mass noun?
>>>> (Not the best question. English ≠ math)
>>>>
>>>> It seems to me that it crosses back and forth.
>>>> Points are definitely a count noun.
>>>> But the idea of a continuum seems
>>>> inescapably not.individuals.
>>>>
>>>> Perhaps that count/mass dimorphism is
>>>> why the occasional poster rejects uncountability.
>>>
>>> Well good sir,
>>> mostly it's that firstly there's that
>>> the "infinite limit" must concede that
>>> it's actually infinite
>>> and that
>>> the limit is not only "close enough"
>>> yet actually that
>>> it achieves the limit, the sum,
>>> because deduction arrives at that
>>> otherwise it's no more than half,
>>> and, not close enough.
>
> That reason confuses 'infinite' with 'humongous'.
>
> If I recall correctly,
> I have pointed this confusion out to you, and
> your riposte has been (framed non.technically)
> that, yes, that's 'infinite': 'humongous'.
>
> So, I'm wondering why you have clung so tightly to
> this specific confusion.
>
>>> Then there's
>>> for division and divisibility,
>>> the "infinite-divisibility" and
>>> for this sort of "actually complete infinite limits"
>>> the "infinitely-divided".
>
> The infinitely.divided means the continuum limit.
> The continuum limit means lattice.spacing → 0
>
> The continuum is such that, for each split,
> the foresplit holds a last point or
> the hindsplit holds a first point.
>
> The continuum limit is not the continuum.
> in part because
> the continuum limit is countable and
> the continuum is uncountable.
>
>>> Then it's pretty much exactly
>>> most people's usual notion of that
>>> an infinitude of integers,
>>> regular both in increment and in dispersion,
>>> so equi-distributed and equi-partitioning
>>> the space of integers, is
>>> the same kind of thing when shrunk to [0,1],
>>> the space of [0,1]
>>> as by the same members, that it fulfills
>>> extent, density, completeness, measure,
>>> thusly that
>>> the Intermediate Value Theorem holds,
>>> then thusly
>>> any relevant standard analysis about calculus
>>> holds, or has forms that hold.
>
> The humongous shrunk to [0,1] stays
> equi-distributed and equi-partitioning
> However, the infinite is different, and
> an analogous claim for the infinite is inconsistent.
>
> No,
> the intermediate value theorem does not hold
> for the → 0 limit.lattice.
> AKA the continuum limit.
>
>> What it is is that at one point
>> I wrote non-standard field axioms for [-1, 1],
>> so, now the usual
>> "the complete ordered field being unique
>> up to isomorphism"
>> is a distinctness result
>> instead of a uniqueness result.
>
> The complete ordered field remains
> the complete ordered field.
>
> You have the freedom to write
> non.standard field axioms.
> If they don't describe the complete ordered field,
> then they aren't complete.ordered.field axioms.
>
> It does not follow from
> not.the.complete.ordered.field being countable that
> the complete ordered field is countable.
>
>> Then, another thing is about
>> a deconstructive account of complex analysis about
>> the very definition of complex numbers a + bi and
>> the definition of the operations upon them.
>> The thing is that division, for complex numbers,
>> the definition of division, can be de-constructed,
>> left and right,
>> so that now there are non-principal branches of
>> division, in complex numbers.
>
> The complex field remains
> the complex field.
>
> The complex field has
> single.valued division for non.0 numbers.
>
> You have the freedom to describe (deconstructively?)
> something with a different division.
> It will be something different, not.the.complex.field.
>
>
> You (RF) have what I consider a non.standard use
> of the word 'hypocrisy'.
> 'Hyposcrisyᴿꟳ' seems to refer to (for example)
> the practice of not.calling cats 'elephants'.
>
> If that is what you mean, and you want
> less hypocrisyᴿꟳ (whatever reasons you have),
> you would do better by pointing out
> the advantages of calling cats 'elephants'
> (whatever advantages it has),
> as distinct from calling cats 'elephants' yourself,
> and distinct from entertaining us with stories of
> classical figures calling cats 'elephants', etc.
>
>

Field axioms for [-1,1] their real values, ....

More later and thank you for your reply,
while then though the considerations of
the "potential, practical, effective, actual"
infinity get into what law(s) of large numbers apply,
as about the indefinite, humongous, un-bounded, infinite,
infinity, that there are multiple laws of large numbers
and they largely reflect these three models of continuous
domains, line-reals, field-reals, and field-reals,
and three models of Cantor space: sparse,square, and signal,
that the point about infinite limits is that they are
indeed actual.
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