Sujet : Re: Replacement of Cardinality
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logic sci.mathDate : 28. Jul 2024, 13:17:51
Autres entêtes
Organisation : Nemoweb
Message-ID : <vg44QVKbPSR4U0Tq71L-fg5yqgM@jntp>
References : 1 2
User-Agent : Nemo/0.999a
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