Re: Incompleteness of Cantor's enumeration of the rational numbers

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.