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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: quantifier shift
Date: Sat, 5 Oct 2024 20:49:07 +0200
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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