Re: Incompleteness of Cantor's enumeration of the rational numbers

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.

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?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.