Re: Incompleteness of Cantor's enumeration of the rational numbers (conserved infinity)

On 11/20/2024 05:00 PM, Chris M. Thomasson wrote:

On 11/20/2024 4:02 PM, Ross Finlayson wrote:

On 11/20/2024 12:23 PM, FromTheRafters wrote:

on 11/20/2024, Ross Finlayson supposed :

On 11/20/2024 12:05 PM, FromTheRafters wrote:

WM wrote on 11/20/2024 :

On 20.11.2024 15:14, FromTheRafters wrote:

WM formulated on Wednesday :

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It does not make it wrong, but it unmasks it at imprecise. That's
why I don't like it. We can do better.

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It works well enough.

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Really? Then you can answer the following questions:
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Let every unit interval after a natural number on the real axis be
coloured white with exception of the powers of 2 which are coloured
black. Is it possible to shift the black intervals so that the whole
real axis becomes black?

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No, of course not.
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Or: Let every unit interval after a natural number on the real
axis be
coloured as above with exception of the intervals after the odd prime
numbers which are coloured red. Is it possible to shift the red
intervals so that the whole real axis becomes red?

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No, of course not.
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What colour has the real axis after you have solved both tasks?

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Depending on the order of the tasks. I think half red or half black.

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Well you have to reference academic reference and describe "supertask"
besides "asymptotics" about where "the asymptotic density of black or
red respectively is 1 in the limit", that you point to "supertask"
instead of mumbling like it's not already considered by proper minds,
not just ketonic neck-flap gaspers of having failures altogether
of any sort of related-rates problems.
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This is mathematics: humor is irrelevant, and so is what
anybody "thinks", or, "feels".
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It only matters what "is", and there's a language of it,
so use it. (Or lose it.)
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Good sir

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If painted black and then red, it will be red. If painted red and then
black, it will be black. These are real intervals, and as such I assume
real powers of two. In both scenarios, none of the negative real axis is
at all affected.

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"Restricted Sequence Element Interchange" is an idea that
is a sort of "conservation principle" about things in an
Integer Continuum or Linear Continuum, here for example
an Integer Continuum. The idea is that any switch, as much
as it changes a plain 0101 to 0011, happens once-at-a-time
or the pair-wise, about basically, "after so much time given
to find an offset to exchange and another for its place,
and to update the state of the data structure that it is so,
that it's a matter of book-keeping and related-rates or
a system of algorithmic resources in numerical resources,
and time", that it's not merely giving x_infinity when
"at time 0 < Sum 1/n^2 < 1 that element n changes from
0 to 1" that at t_oo at n = oo that it's all 1's,
that it's so asymptotically, or that the density as
always filling in closer to the origin has that any
first different is arbitrarily far away, still has
that it's an honest account of book-keeping to make
that into a structure as if you had to implement it
and more than merely a lazy, forgetful mathematician's
exercise in induction that can easily arrive at
from 010101... to 00000... or 111111....
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Anyways there's a theory about these things that
basically make for cases besides those that just
shove off the end and put it off forever, besides
the "asymptotics" is what's called "supertasks".
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These may include for systems that are merely
"very, very large" when not "actually infinite",
that some practical or effective infinity, yet
results as a "point at infinity" which is a critical
or accumulation point, for the swapped-out items.
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Like a "point at infinity", a "prime at infinity". Or not,
it's among things entirely independent standard number
theory, which some have as that the integers don't actually
have a standard model anyways, only fragments and extensions.
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Anyways these sorts of things make for reasonings when
things exchange and conserve besides one-sidedly shove off.
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A prime at infinity? Keep in mind that there is an infinite number of
primes. So, are you talking about perspective as in a point at infinity?

What about it?
Number theorists have various ways to define a point
at infinity, vis-a-vis geometry's usual notion as
of a perspective point-at-infinity, and projective
geometry's projective point-at-infinity, and number
theorist's compactification of the naturals or variously
with regards to the fundamental theorem of arithmetic,
whether it's so at infinity, or not.
It is rather regarded that these notions are _significant_
and _relevant_ and as well that they're _independent_,
usual enough fragments of theories of fragments of models
of numbers, or fixed views, and these kinds of things.
So, there are models of integers with a "prime at infinity",
i.e. its only multiplicative factors are itself and 1,
and it's defined. There are others where it's composite,
for example being a product of each of the primes, there
are others, there are each the others. A "prime" at infinity
decides some things and makes an _opinion_, it's a _singular_
of what's a _multiplicity_ of models.
It's usually said to be "number theory's" and no other
theories of mathematics thusly get any say at all about it,
unless said theory includes all the multiplicity of its
considerations. In this sense a usual nominalist fictionalist
"we can't say nothing" may be a good thing, as they can't say
anything wrong about it either, yet "you can't say nothing, either"
is considered objectionable, because number theorists can and do.
Consider double primes: two primes, separated by two, for
example, 11 and 13, or, 17 and 19. Now, there are many, many
less, of the double primes, than there are of primes. Then,
for triple primes, there's only 2, 3, 5. There are no other
(known) finite triple primes. Yet at infinity, it could
be in or among double, triple, even quadruple primes,
even n-tuple primes, in various models of integers.
So, anyways, the projective and perspective are very
relevant geometry beyond renderings, and about it.