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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Mon, 15 Jul 2024 17:24:23 -0700
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On 7/15/2024 5:09 PM, Moebius wrote:
> Am 16.07.2024 um 01:57 schrieb Moebius:
> 
>  > Siehe auch: https://en.wikipedia.org/wiki/Quantifier_shift
> 
> Hier wird als ein Beispiel für diesen Fehlschluss angeführt:
> 
> 3. Every natural number 𝑛 has a successor 𝑚 = 𝑛 + 1, the smallest of all 
> natural numbers that are greater than 𝑛. Therefore, there is a natural 
> number 𝑚 that is a successor to all natural numbers.
> 
>      ∀𝑛 ∃𝑚 𝑆𝑛𝑚 ⊢ ∃𝑚 ∀𝑛 𝑆𝑛𝑚.
> 
> It is fallacious to conclude that there is a single natural number that 
> is the successor of every natural number.

There are infinite successors to any finite number?

Say, 5:

5, 5 + 1, 5 + 2, 5 + 3, 5 + 4, ...

5 + 1 is a successor to 5? Fair enough?


> 
> Für Mückenheim ist es nun aber so, dass wegen seiner "Gleichsetzung" von 
> ∀𝑛 ∃𝑚 𝑆𝑛𝑚 mit ∃𝑚 ∀𝑛 𝑆𝑛𝑚 eben ∀𝑛 ∃𝑚 𝑆𝑛𝑚 ∃𝑚 ∀𝑛 𝑆𝑛𝑚 "impliziert" (besser 
> gleichbedeutend ist mit ∃𝑚 ∀𝑛 𝑆𝑛𝑚).
> 
> Im Mückenland gibt es daher eine natürliche Zahl, die "Successor" jeder 
> natürlichen Zahl ist (also auch von sich selbst); d a s ist die größte 
> natürliche Zahl!