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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: quantifier shift
Date: Sat, 5 Oct 2024 23:30:33 -0400
Organization: i2pn2 (i2pn.org)
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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.