Re: quantifier shift

Am Sat, 05 Oct 2024 20:49:07 +0200 schrieb WM:

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.

Of course. ω is infinite, and the naturals are finite.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.