Re: how

Am 13.06.2024 um 18:12 schrieb Moebius:

Am 13.06.2024 um 18:04 schrieb Moebius:

Am 13.06.2024 um 17:55 schrieb WM:

 

Daher würde ich es lieber so formulieren (um das klar zu stellen):
>
| Every number has ℵo numbers as successors.
| If only numbers having ℵo successors are removed (i.e. _all_ numbers)
| then nothing remains.

 Hier anhand einiger Beispiele im Detail erklärt: [...]
 D. h. _alle_ Elemente in IN haben ℵo Elemente in IN als "Nachfolger".
 Wenn man nun die Differenzmenge von IN und der Menge {n e IN : n hat ℵo Elemente in IN als "Nachfolger"} betrachtet, dann ist das IN \ IN = { }.
 -> nothing remains.

Maybe a _mathematical_ proof is helpful here.
An e IN: card({m e IN : m > n}) = ℵo (*)
"Every number has ℵo numbers as successors."
Theorem: IN \ {n e IN : card({m e IN : m > n}) = ℵo)} = { }
"If only numbers having ℵo numbers as successors are removed then nothing remains."
Proof: Since IN \ IN = { } all we have to show is that {n e IN : card({m e IN : m > n}) = ℵo)} = IN. But this follows immediately from (*).
After all, for all n, if n e IN then card({m e IN : m > n}) = ℵo (from (*)), hence n in {n e IN : card({m e IN : m > n}) = ℵo)}. And (trivially) if n in {n e IN : card({m e IN : m > n}) = ℵo)} then n in IN (by definition of {n e IN : card({m e IN : m > n}) = ℵo)}). Hence IN c {n e IN : card({m e IN : m > n}) = ℵo)} and {n e IN : card({m e IN : m > n}) = ℵo)} c IN, and hence IN = {n e IN : card({m e IN : m > n}) = ℵo)}. qed