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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Mon, 25 Nov 2024 13:11:26 +0100
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On 24.11.2024 19:16, Jim Burns wrote:
> 
>>>>> On 11/23/2024 3:45 PM, WM wrote:

>> for every interval (0,n]
>> the relative covering is 1/10,
>> independent of how the hats are shifted.
>> This cannot be remedied in the infinite limit
>> because
>> outside of all finite intervals (0, n]
>> there are no further hats available.
> 
> All the hats for which
>   if there are G.many then there are G^G^G.many
> are enough to "remedy" the 1/10.relative.covering

Not at all!
> 
> If each G can match G^G^G

That is a false assumption.
If there are enough hats to cover G then there are enough hats to cover 
G^G^G. G does not cover G^G^G.

> Matching.a.proper.subset is
> the sort of behavior which permits Bob to disappear
> with enough room.swapping _inside_ the Hotel.
> 
> You (WM) treat that behavior as proof that
> we are wrong and you (WM) are right.

I accept logic. Exchange of two elements never leads to loss of one of 
them. You do not.

> What it is  is proof that
> not all sets behave like finite sets.

Nonsense. Logic is also prevailing in infinite sets.

> If there are enough hats for G natural numbers,
> then there are also enough for G^G^G natural numbers.
> 
> The number G^G^G is not.first for which
> there are NOT enough hats.

There is no such number because the set of definable hats is potentially 
infinite.

> Therefore,
> the number G^G^G is not.first for which
> there are NOT enough hats.

We do not disagree. Therefore you need not prove a difference for G and 
G^G^G.
> 
> A similar argument can be made for
>   each natural number.

No, it can be made for each definable natural number, i.e., for a number 
belonging to a tiny finite initial segment which is followed bay almost 
all numbers.

>>> Consider the set of natural numbers for which
>>> there are NOT enough hats.
>>
>> It is dark.
> 
> It is not dark what we mean by 'natural number'.
> A natural number is countable.to from.0

That is a definable number.
> The natural numbers "fail" at
> being finitely.many.
> It is nothing more than that.

If they are infinitely many but complete, then they and their number 
don't vary. |ℕ| - 1 < |ℕ| < |ℕ| + 1.

Regards, WM