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

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.

⎛ 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.

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.

⎛
⎜ 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