Re: quantifier shift

On 10/5/24 2:49 PM, WM wrote:

On 05.10.2024 15:00, joes wrote:

Am Sat, 05 Oct 2024 11:43:50 +0200 schrieb WM:

On 05.10.2024 10:46, Moebius wrote:
>

a quantifier shift is NOT reliable und wird daher in der Mathematik
tunlichst vermieden (und nicht nur dort).

I many cases it is correct. For instance if every definable natural
number has ℵo natural successors, then there are ℵo natural numbers
larger than all definable natural numbers. They are dark however and
cannot be specified.

Since it is logically invalid, you need to prove your deduction
independently. In general those are two different propositions.

 If every definable number has ℵo-infinitely many successors, then no definable number is closer to ω. Then there is a infinite gap between definable numbers and ω.
 Regards, WM
 

So?
Would you expect less that an infinite gap between a finite number and infinity?
Omega may be the next counting number pass the infinite set of the Natural Numbers, but that dpesn't mean the gap for that step is less than infinite. If it was only a finite step from the set, it would be part of the Natural Numbers, since that includes all the unit steps.
Note, this also comes from the fact that there is no "last" Natural Number to be the one just before Omega, and that step between types of numbers has a bigger step then between the Natural Numbers, and bigger in a qualative way, a change of order of measurement, from finite to infinite.