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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 20 Nov 2024 16:53:11 -0800
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On 11/20/2024 4:08 PM, joes wrote:
> Am Wed, 20 Nov 2024 19:42:41 +0100 schrieb WM:
>> On 20.11.2024 19:18, joes wrote:
>>> Am Wed, 20 Nov 2024 13:04:04 +0100 schrieb WM:
>>
>>>> Try to count to a natural number that has fewer successors than
>>>> predecessors. Impossible.
>>> Because there are no such numbers.
>> All successors are natural numbers.
> So?
> 
>> If all can be counted, then no successors remain.
> All at once or every single one?
> There are no successors "after" all of the other numbers.
> 
>>>> But set theory claims that all natural numbers can be counted to such
>>>> that no successors remain.
>>> et your quantifiers in order:
>> That is a foolish excuse.
> You have shown that you don't understand them.
> 
>>> every single natural number is very clearly finite;
>> Every number that can be counted to is finite.
> There are countably infinite numbers, but ok.

A fun part is that if we artificially restrict ourselves to any real 
number that is also a natural number, well, they are countable now? Fair 
enough? All naturals are reals, not all reals are naturals... ;^)



> 
>> But every number that can
>> be counted to has more successors than predecessors.
> Every number, period. There is no number without successors.
> 
>> Therefore not every number can be counted to.
> Well, the ordinal numbers less than epsilon_0 are called countably
> infinite.
> 
>>> the cardinal number corresponding to the set of all of them is
>>> countably infinite.
>> The  set of all numbers that can be counted to is finite, namely a
>> number that is counted to. This cannot change by counting.
> WTF there is no largest number. How do you think counting changes
> anything?
>