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

On 11/24/2024 10:09 AM, Ross Finlayson wrote:

On 11/23/2024 09:37 PM, Jim Burns wrote:

On 11/23/2024 5:01 PM, WM wrote:

On 23.11.2024 22:48, Jim Burns wrote:

On 11/23/2024 3:45 PM, WM wrote:

>

⎜ Assume that there are
⎜ enough red hats for the first 𝔊 numbers
⎜ but not enough for the 𝔊+1ᵗʰ

>
That is a mistake.
If there are enough hats for G natnumbers,
then there are also enough for G^G^G natnumbers.

>
Thank you.
>

Alas they leave G^G^G unit intervals without hats.
That is the catch!

>
After all hat.shifts,
there is no first number without a hat.

>
But almost all numbers are without hat
because the number of hats has not increased.

>
If there are enough hats for G natural numbers,
then there are also enough for G^G^G natural numbers.
>
If there are NOT enough for G^G^G natural numbers,
then there are also NOT enough for G natural numbers.
>
G precedes G^G^G.
If, for both G and G^G^G, there are NOT enough hats,
G^G^G is not first for which there are not enough.
>
That generalizes to
each natural number is not.first for which
there are NOT enough hats.
>
----
Consider the set of natural numbers for which
there are NOT enough hats.
>
Since it is a set of natural numbers,
there are two possibilities:
-- It could be the empty set.
-- It could be non.empty and hold a first number.
>
Its first number, if it existed, would be
the first natural number for which
there are NOT enough hats.
>
However,
the FIRST natural number for which
there are NOT enough hats
does not exist.
>
⎛ Recall that it does not exist
⎜ because,
⎜ if there are enough hats for G natural numbers,
⎝ then there are also enough for G^G^G natural numbers.
>
The set of natural numbers
for which there are NOT enough hats
does not hold its first number.
No number exists that can be the first number.
>
It can only be the first case, that
the set of natural numbers
for which there are NOT enough hats
is empty.
>
Its complement,
the set of natural numbers
for which there ARE enough hats
is the complete set of natural numbers.
>
----
Therefore,
for each natural number,
there are enough hats.
>
>
>

>
Read an article the other day about computer algorithms
and machines in their space and time, about 'garbage
collection', which is an approach in some computer systems
to arrive at for the issue of allocation, of an own resource
from a common resource, and de-allocation, to detect when
an own resource is released or un-used, to result it reclaimed,
and then furthermore with regards to re-organizing or
"de-fragmenting" a serial range, to result that what's
retained is in-use, and furthermore, that as large as
possible serial ranges remain, to allocate serial ranges
of given sizes among them. (Allocators, "slabs", "arenas",
"free-lists", "pools", are usual sorts of concepts.)
>
So anyways in Hilbert's Hotel, when all the rooms are full,
and each has a natural number, and for each of the occupants
to have their original room number as their tracking number name,
to move each n'th even name to each n'th room, has a clear
result their destination, yet not how they get there, with
regards to that the n'th odd number, moves out, as much as
each n'th even number, moves in.
>
So, the ideal lazy forgetful mathematician, may aver that
once it's all done, these moves, which can only be swaps,
as the place is already full, will have occurred infinitely-many
times.
>
Then, it's so that the density of the evens 1/2 n is greater than
the density of the primes log n and that the density of the
squares root n, so an initial segment may finish earlier,
being entirely full and that though the only way to maintain
that each remaining swap _in_, to make a swap _out_, makes
a push _out_, that for the n'th must push the n+1'th and
the n+2'th and so on ad infinitum, that any occupant that
results entirely eliminated as not existing in the range
of the function the mapping, makes an infinite supertask
in anything that isn't merely "I don't care how it's done,
just do it", that of course one can arrive at, "no can do",
as easily as, "fait accompli".
>
>
So, for example, lazy forgetful computer programmers that
rely on allocators and schedulers to say what, where, and when,
know that in the bounded and when there aren't infinitely-many
resources that may be furthermore re-done infinitely-many
times at zero time and space, that lazy forgetful computer
programmers sometimes see lazy forgetful mathematicians
as a bit less than practical.
>
While that is so, it's well-known since Galileo that there
exist functions relating countable domains like naturals
and evens though before that was just called aliquot and
modularity, then Galileo came up with "and squares work also",
what with regards to that the Pythagoreans though have
"everything is rational here", and would point out that
Archimedes may arrive at the extended contrivance to _change_
something does not exist as simply as that its destination
exists, and, inference can't arrive that it ever gets done,
so, you must be using deductive inference to declare the
accomplishment of an inductive-impasse-refuted supertask.
>
>

You might find it easier to re-organize and satisfy the
'otel's goals if you have another, empty, infinite hotel
next to the full, infinite hotel which is full and infinite.
The idea here is that you can allocate new rooms in the
empty hotel, the evens, if starting at zero, say, for those
that are to fill the n'th rooms of the full hotel, and the odds,
say, of those that are displaced. Then it's only two infinite
supertasks to move all the stayers, as it were, into the
evens, and all the letters, as they are, into the odds,
then only a third supertask to move all the stayers into
the now infinite and empty original hotel, then as
with regards to what remains the let-ters, that those
who let, their space, all still reside in what was an empty,
infinite hotel, all the even spaces. Then, you might move
those into thirds rooms, say, another supertask, so that
it's simple that what was the empty hotel, still has a simple
layout for an even/odd pair to get/let in the modularity of reset
2 + S, where S is how many supertasks effected any sort
or reorganization that isn't finite swaps, and that then
the swap hotel as it were is (S-2)/S full in this manner,
that though you'd need another one to ever swap those
down, ....
Knuth in TAoCP has quite an extended section on
switching magnetic tapes when looking to sort a
large magnetic tape made of many magnetic tapes.
It's a classical exposition of course well known to
anybody who ever read "The Art of Computer Programming".
Anyways though what seems clear is that while it's
obvious to demonstrate bijections among countable sets,
it's simple, affecting a re-organization, involves supertasks,
that inductive inference readily arrives at do not exist.
So, maybe when brick-batting WM, think if he's talking
about the actual act of re-organization, beyond the
merely ideal imagining it were so.
You're going to need a little room, ....
So, are we "pure" or "applied" mathematicians?
Yes, indeed.
Sometimes though applied mathematicians though
are like "wake up you lazy forgetful pure mathematical
fool: your trashcan fire smoke-damaged an entire
infinite hotel. And inductively that's never going to complete".
So anyways, if you can't beat him, fix him.