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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: quantifier shift
Date: Sun, 6 Oct 2024 12:37:54 -0700
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On 10/6/2024 4:23 AM, joes wrote:
> 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.
> 

WM needs to keep in mind that there are infinitely many naturals. They 
are unbounded, and there is no largest one... :^)