Re: Replacement of Cardinality

Le 27/07/2024 à 19:34, Jim Burns a écrit :

If ℕ has fewer elements than ℕ∪{ℕ}
then
|ℕ| ∈ ℕ

|ℕ| = ω-1 ∈ ℕ

ℕ has fewer elements than ℕ

ℕ has ω-1 elements.

 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.

ℕ = {1, 2, 3, ..., ω-1} = {1, 2, 3, ..., |ℕ|}

 

_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.

First false is your claim that |ℕ| is larger than all elements of ℕ. ℕ counts its elements.

 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.

The proof is easy. Since the superset has at least one more element than its proper subset, it has more elements than its proper subset.

 ⎛ 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.

Thank you, I will try to remember it.
Regards, WM