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From: WM <wolfgang.mueckenheim@tha.de>
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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