Re: how

Le 13/06/2024 à 16:53, Moebius a écrit :

Am 13.06.2024 um 16:39 schrieb WM:

Le 13/06/2024 à 14:50, Moebius a écrit :

Am 13.06.2024 um 14:22 schrieb Jim Burns:

On 6/12/2024 4:33 PM, WM wrote:

>

If every number is subtracted,
then no successors remain.
If only definable numbers are subtracted,
then successors remain.

>
Observation: If "definable numbers" is replaced by "finitely many numbers"

 By definition definable implies existence of a FISON.

 Ich glaube, Du verwechselst hier gerade "definable" mit "defined".
 Aber natürlich impliziert "defined" "definable". :-)
 (Denn wenn eine Zahl nicht "definable" wäre, könnte sie auch nicht "defined" sein.)
 Hinweis: Da die natürlichen Zahlen nach von Neumann FISONs SIND, sind diese also auch "definable" bzw. "defined".

Of course.

 Damit ist dann aber klar, dass die Behauptung "If only definable numbers are subtracted, then successors remain." falsch ist.

No.

 "Richtig" dagegen wäre die Behauptung: "If only finitely many numbers are subtracted, then successors remain."

All FISONs or v. Neumann ordinals are finite.
Theorem   More than finitely many finite initial segments cannot be merged. Proof: This is caused by the pigeonhole principle and the definition "finite initial segment". If each of the first n positive integers has a unary representation in form of a string, like ooooo, that is shorter than n then, by the pigeonhole principle, there must be two different positive integers defined by the same unary representation. Clearly this is absurd.
Same holds in case of ℵo finite strings. ℵo is a fixed quantity such that ∀n ∈ ℕ: n < ℵo. If each one of all ℵo positive integers has a unary representation in form of a string that is shorter than ℵo then, by the pigeonhole principle, there must be two different positive integers defined by the same unary representation. Clearly this is absurd too.
Same holds for finite initial segments. Since the order of numbers in {1, 2, 3, 4, 5} does not matter, it has the same information content as a unary representation like ooooo. Regards, WM