Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

Moebius <invalid@example.invalid> writes:

Am 06.12.2024 um 20:46 schrieb Chris M. Thomasson:

On 12/5/2024 11:06 PM, Moebius wrote:

Am 06.12.2024 um 01:30 schrieb Chris M. Thomasson:

On 12/5/2024 8:08 AM, WM wrote:

On 05.12.2024 13:26, Richard Damon wrote:
>

Which ones can not be "taken" or "given".

>
Those with less than infinitely many successors.

>
Mückenheim, bei Dir sind wirklich ein paar Schrauben locker.
>
Indeed! Numbers which do not exist can not be "taken" or "given"
Mückenheim is completely right here! On the other hand, "those"?!
>

Do you even know how to take any natural number, create a unique pair
and then get back to the original number from said pair?

>
I certainly don't know. Please tell me!

Take any natural number and run it through Cantor Pairing to get a unique
pair. From this pair alone we can also get back to the original number.

>
Where can I find those natural numbers? And what EXACTLY do you mean by
"run it through Cantor Pairing"? I and how does it _create_ pairs?
>
Strange things are going on.

I don't know if you objection is to CMT's rather unmathematical
language, or if it is more philosophical than that, but in case it is
just the former, CMT is probably referring to the classic bijection (and
its inverse) between N and NxN given by (in Haskell)

index_of (a, b) = (a + b)*(a + b + 1) `div` 2  +  a

pair n = (offset, sum - offset)
  where sum = floor ((sqrt(8 * fromIntegral n + 1) - 1)/2)
        offset = n - sum * (sum + 1) `div` 2

Obviously this can be written in mathematical symbols, but Haskell is
easier to post (because it's ASCII) and allows one to experiment with
the mapping:

ghci> map pair [0..10]
[(0,0),(0,1),(1,0),(0,2),(1,1),(2,0),(0,3),(1,2),(2,1),(3,0),(0,4)]
ghci> map index_of (map pair [0..10])
[0,1,2,3,4,5,6,7,8,9,10]

(The formulas are very similar if you prefer 1 to be the smallest
natural.)

--
Ben.