Re: Replacement of Cardinality

sci.logic and sci.math
On 7/26/2024 12:31 PM, WM wrote:

_The rule of subset_ proves that
every proper subset has
less elements than its superset.

⎛ Each non.{}.set A of ordinals holds min.A
⎜
⎜ Ordinal j = {i:i<j} set of ordinals before j
⎜
⎜ Finite ordinal j has fewer elements than j∪{j}
⎜
⎝ ℕⁿᵒᵗᐧᵂᴹ is the set of ALL finite ordinals.
ℕⁿᵒᵗᐧᵂᴹ breaks the rule of subset.
If ℕ has fewer elements than ℕ∪{ℕ}
then
|ℕ| ∈ ℕ
|ℕ∪{ℕ}| ∈ ℕ
ℕ ≠⊂ {i:i<ℕ∪{ℕ}} ⊂ ℕ
ℕ ≠⊂ ℕ
and
ℕ has fewer elements than ℕ
Because ℕ does not have fewer elements than ℕ
ℕ does not have fewer elements than ℕ∪{ℕ}
and the rule of subsets is broken.

_The rule of subset_ proves that

To make a claim
  is not sufficient
to make a proof.
To make a finite sequence of claims
  such that no claim is first.false
  is sufficient
to make a proof.
The most that is true here is that
the rule of subset _claims_ without proof that

every proper subset has
less elements than its superset.

⎛ In English, grammatically speaking,
⎜ it is never correct to say "less <plural.noun>"
⎜
⎜ English has mass nouns (Stoffnamen)
⎜ "less rock" ...
⎜ and count nouns (zählbare Substantive)
⎜ "one rock", "fewer rocks" ...
⎜ Only count nouns have a plural.
⎜ Only mass nouns are modified by "less".
⎝ "Less rocks" and "lescs elements" are never correct.
⎛ ...as you (WM) know, since you've been told.
⎜ Why you say what you say is often more mysterious
⎜ than what you say.
⎜
⎜ Perhaps you see a mass noun as more appropriate
⎜ to your darkᵂᴹ numbers -- to number purée?
⎜ That's not the way it would be said.
⎜
⎜⎛ The smaller bucket holds fewer rocks and less rock.
⎜⎝ Every proper subset has less element than its superset.
⎜
⎜ Perhaps you intentionally insert small errors
⎜ in order to draw attention away from large errors:
⎜ dropping verbal chaff.
⎜ https://en.wikipedia.org/wiki/Chaff_(countermeasure)
⎜
⎝ Perhaps looking ignorant is your kink.