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

On 11/29/2024 10:08 AM, Jim Burns wrote:

On 11/27/2024 4:33 PM, WM wrote:

On 27.11.2024 20:47, Jim Burns wrote:

>

Finite cardinals can change by 1
The cardinal |ℕᶠⁱⁿ| cannot change by 1

>
Small wonder.
Fuzzy properties like "many" cannot change by 1.

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⎛ ℕᶠⁱⁿ is the set of finite cardinals.
⎜ Bob is not a cardinal.
⎜
⎜ ∀ᶜᵃʳᵈξ:  ξ ∈ ℕᶠⁱⁿ  ⇔
⎜ ⟦0,ξ⦆∪{Bob} ≠ ⟦0,ξ⟧  ∧  |⟦0,ξ⦆∪{Bob}| ≠ |⟦0,ξ⟧|
⎜
⎝ ℕᶠⁱⁿ∪{Bob} ≠ ℕᶠⁱⁿ  ∧  |ℕᶠⁱⁿ∪{Bob}| = |ℕᶠⁱⁿ|
>

Yes,
the sets can change membership by 1
However,
the cardinalities of those sets cannot change by 1

>
This proves that cardinality is a fuzzy property.

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The whole ℕᶠⁱⁿ×ℕᶠⁱⁿ matrix can fit in
its first column ℕᶠⁱⁿ×{0}
>
⎛ ℕᶠⁱⁿ×ℕᶠⁱⁿ ⇉ ℕᶠⁱⁿ×{0} ⇉ ℕᶠⁱⁿ×ℕᶠⁱⁿ
⎜ ⟨i,j⟩ ↦ ⟨n,0⟩ ↦ ⟨i,j⟩
⎜ n = (i+j)⋅(i+j+1)/2+j
⎜ (i+j) = ⌊(2⋅n+¼)¹ᐟ²-½⌋
⎜ j = n-(i+j)⋅((i+j)+1)/2
⎝ i = (i+j)-j
>
The fuzzy cardinality property
  predicts that it can.
Your crisp cardinoid property
  predicts otherwise
  and is incorrect.
>
After all the swaps
  (of which no swap is a change in cardinality)
what remains is a proper subset
  (which is not a change in cardinality).
>
'Bye, Bob.
>

This proves that cardinality is a fuzzy property.

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⎛
⎜ An hungry Fox with fierce attack
⎜ Sprang on a Vine, but tumbled back,
⎜ Nor could attain the point in view,
⎜ So near the sky the bunches grew.
⎜ As he went off, "They're scurvy stuff,"
⎜ Says he, "and not half ripe enough--
⎜ And I 've more rev'rence for my tripes
⎜ Than to torment them with the gripes."
⎜ For those this tale is very pat
⎜ Who lessen what they can't come at.
⎝
http://mythfolklore.net/aesopica/phaedrus/43.htm
>
>

Hmm, "fuzzy cardinals" and "crisp cardinoids", ...,
those are new ones.
Reading Copleston the other day, who once debated
Russell on a broadcast and got Russell to admit
that he was an agnostic about teleology when
confronted by philosophers yet that Russell
was irreligious because Russell though that
usual people were stupid, sort of expressing
his hypocrisy and inconstancy, anyways reading
Volume IV "From DesCartes to Leibniz", there are
some great quotes from Leibniz.
"I am so much in favour of the actual infinite
that instead of admitting that Nature abhors it,
as is commonly said, I hold that it affects it
everywhere in order better to work the perfections ...".
-- Leibniz
Then that's given to some sort of "perfection principle"
yet here it's given an account why it simply follows
from "principle of sufficient reason".
"At first, when I had freed myself from the yoke of Aristotle,
I occupied myself with consideration of the void and atoms."
-- Leibniz
A usual greater account of Aristotle of course has
that he weighs atoms, and that there's Aristotle-linear
and Aristotle-circular, and greater Aristotle, that
Leibniz there reflects on a half-account of Aristotle.
So anyways, I know there was some mention of "super-cardinals"
here the other day, like "the cardinal of '1' the cardinal",
yet, what is one to make of, "cardinoids" or furthermore
"crisp cardinoids"? Given that "futzy cardinals" are just
"don't demand how it's done just do it", vis-a-vis, "no can do".
The idea that '1' the cardinal has the greatest cardinal,
has also the idea that the sets with about half the elements
in the universe have the greatest cardinals, these sorts
of super-cardinals, that start twice, about the universe
of set theory.
Anyways your cardinoids seem pretty much about after
"pair-wise swaps in a full Hilbert hotel", and your
futzy cardinals about "infinite-union, the illative,
the univalency", which are, _not_ usually considered
part of "set theory", because, from that can be derived
contradictions, in ordinary set theory.