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NNTP-Posting-Date: Sat, 23 Nov 2024 21:52:48 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (clock hypothesis)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 23 Nov 2024 13:52:48 -0800
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On 11/23/2024 11:37 AM, Ross Finlayson wrote:
> On 11/22/2024 05:09 PM, Ross Finlayson wrote:
>> On 11/22/2024 01:08 PM, Chris M. Thomasson wrote:
>>> On 11/22/2024 12:47 PM, Ross Finlayson wrote:
>>>> On 11/22/2024 12:37 PM, Chris M. Thomasson wrote:
>>>>> On 11/22/2024 6:51 AM, WM wrote:
>>>>>> On 22.11.2024 13:32, joes wrote:
>>>>>>  > Am Fri, 22 Nov 2024 13:00:52 +0100 schrieb WM:
>>>>>>
>>>>>>  >>>>>>>> The number of ℕ \ {1} is 1 less than ℕ.
>>>>>>  >>>>>>> And what, pray tell, is Aleph_0 - 1 ?
>>>>>>  >>>>>> It is "infinitely many" like Aleph_0.
>>>>>>  >>> Thanks for agreeing with |N| = |N\{0}|.
>>>>>>  >> Of course. ℵo means nothing but infinitely many.
>>>>>>  > Good. Then we can consider those sets to have the same number.
>>>>>>  >
>>>>>> That is the big mistake. It makes you think that the sets of naturals
>>>>>> and of prime numbers could cover each other.
>>>>>
>>>>> prime numbers are a sub set of the naturals. They are both infinite.
>>>>
>>>> Finite, though large, "sets", as though all the relations
>>>> among them besides just "the set of", make it so in the
>>>> asymptotics, that it's possible to work up when the
>>>> density of primes, which is kind of known and on the
>>>> order of log n, vis-a-vis pi^2/6 and co-primes, have
>>>> it so that in some "practically" or "effectively"
>>>> "un-bounded", if that's a short-hand interchangeable
>>>> with "infinite", have only finitely many primes.
>>>>
>>>> Maybe one at infinity, ....
>>> [...]
>>>
>>> How can there be one prime at infinity? That's like saying there is a
>>> natural number at infinity. There is no largest natural just that there
>>> is no largest prime. So, if you artificially say this prime is at
>>> infinity you just went into finite mode!
>>>
>>
>> Actually, some arrive at that if there are infinitely-many,
>> then there is at least one infinitely-grand, in the same
>> structure, of the same type. Called variously compactification,
>> or fixed-point, it can be arrived at via plain comprehension
>> the extra-ordinary, according to definitions of direct sum and
>> product of copies of infinite sets, and in geometry it's
>> usually called point-at-infinity, and lots of reasons.
>>
>> Then, "finite mode" as you put it, is as mentioned about
>> variously "very, very large", yet only showing one side
>> or the other what's "finite" or "infinite", meaning merely
>> according to a definition of finite like I have, what happens
>> in ordering theory, for example, these objects of the elements
>> of discourse, il discorso.
>>
>> So, then that makes for a reading of somebody like AP,
>> who has sorts of problems being stuck in finite mode,
>> half-way, makes for a generous reading, because as is
>> often put here, various under-informed reasonings about
>> infinity, result incorrect conclusions.
>>
>> So, here's a generous reading, of your intuition,
>> I've tried plentiful times to give something like WM
>> reasons to say truthful things about things it's declared
>> to declare, yet, it seems incorrigeable and even along
>> the lines of a purposeful "soft-ball straw-man" of the
>> easily mechanized sock-puppet toy of the sort launched
>> by some childish, churlish chucklers, laughing at our
>> expense (and dismay).
>>
>> Yet, while it's so wrong, then it's still necessary to
>> shelter its bait-and-switch part of the proposition
>> the bait, that must be upheld from getting trampled
>> in the shuffle.
>>
>>
>
>
> It's kind of like, meters/second and seconds/meter.
> So, 0 meters/second is infinity seconds/meter, yet,
> the idea is that in one dimension, for example, there's
> a line with some arbitrary origin marked 0, and displacements
> about that. Then, consider displacements as integers,
> or displacements as rationals, or, displacements as
> real numbers. So, given that position is an abitrary
> function of time and to get there motion is an arbitrary
> function of time and to get there acceleration is an
> arbitrary function of time, each of the higher orders
> of acceleration is an arbitrary function of time, and
> any change at all affects a nominally non-zero, yet
> vanishing, value each of the higher-order derivatives
> of displacement (from the origin) with respect to time.
>
> So, 1 m/s = 1 s/m, with 0 m/s = infinity s/m,
> and correspondingly infinity m/s = 0 s/m, though
> it's usual the "rest" seems more likely than
> "infinite velocity".
>
> An object at rest, then, is it, 0 m/s? As long as
> it rests there, it is. Then, is it infinity s/m?
> Potentially, ..., for as long as you count it's
> 1, then 2, then 3, ..., infinity seconds / meter.
>
> While though its velocity is zero, the seconds
> per meter is no less than infinity.
>
> So, if infinity is so bad, what about an origin?
>
> Then, there's an idea that x = y = z = ... the
> identity line in all dimensions, is also an origin.
>
> Is there instantaneous anything at all?
>
>
> Related-rates are simple enough, and of course
> there is finite-element analysis and most people
> know f = ma though it's really f(t) = ma(t),
> how does anything ever change at all?
>
>
> The regular singular points of the hypergeometric:
> are: zero, one, and infinity, in mathematics.
>
>



So, infinity seconds per meter, is it zero meters per second?

Maybe not, then there's that t seconds per meter,
grows, at a rate t, for time t, from zero, for as long
as velocity equals zero meters per second.

Then, thusly, it's again the _same_ thing,
a constant velocity growing itself, while,
as with respect to what's at rest, its origin
and with all its infinitely-many higher orders
of derivatives of displacement, or here velocity,
with respect to time. It's a "moving quantity".

So, then what this arrives at is a sort "clock hypothesis",
that re-introduces t into all the otherwise instantaneous
"quantities", "moving quantities".

There's not finitely-many of those,
so once again "infinity is _in_ (the numbers)".


If you needed another reason why infinity is in,
the numbers, here's a reason that you don't
need another reason - it never left.