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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: Sun, 24 Nov 2024 12:06:16 +0100
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On 24.11.2024 06:37, Jim Burns wrote:
> On 11/23/2024 5:01 PM, WM wrote:

> 
> If there are enough hats for G natural numbers,
> then there are also enough for G^G^G natural numbers.

So it is. But that does not negate the fact that 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.
> 
> If there are NOT enough for G^G^G natural numbers,
> then there are also NOT enough for G natural numbers.
> 
> G precedes G^G^G.
> If, for both G and G^G^G, there are NOT enough hats,
> G^G^G is not first for which there are not enough.
> 
> That generalizes to
> each natural number is not.first for which
> there are NOT enough hats.

It seems so but the sequence 1/10, 1/10, 1/10, ... has limit 1/10 with 
no doubt. This dilemma is the reason why dark numbers are required.
> 
> ----
> Consider the set of natural numbers for which
> there are NOT enough hats.

It is dark.
> 
> Since it is a set of natural numbers,
> there are two possibilities:
> -- It could be the empty set.
> -- It could be non.empty and hold a first number.

Both attempts fail. That is the reason why dark numbers are required.
> 
> Its first number, if it existed, would be
> the first natural number for which
> there are NOT enough hats.
> 
> However,
> the FIRST natural number for which
> there are NOT enough hats
> does not exist.

That however does not negate the fact that for all intervals (0, n] and 
also for all intervals (10n, 10(n+1)] with no exception from which 
another load of hats could be acquired there are not enough hats to 
cover all n.
> Therefore,
> for each natural number,
> there are enough hats.

Then mathematics fails. I don't accept that a constant sequence has 
another limit than this constant.

Regards, WM