Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.mathDate : 24. Nov 2024, 06:37:10
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <11c85fcd-7f48-4573-ba8e-1509e7173d34@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
User-Agent : Mozilla Thunderbird
On 11/23/2024 5:01 PM, WM wrote:
On 23.11.2024 22:48, Jim Burns wrote:
On 11/23/2024 3:45 PM, WM wrote:
⎜ Assume that there are
⎜ enough red hats for the first 𝔊 numbers
⎜ but not enough for the 𝔊+1ᵗʰ
>
That is a mistake.
If there are enough hats for G natnumbers,
then there are also enough for G^G^G natnumbers.
>
Thank you.
>
Alas they leave G^G^G unit intervals without hats.
That is the catch!
>
After all hat.shifts,
there is no first number without a hat.
>
But almost all numbers are without hat
because the number of hats has not increased.
If there are enough hats for G natural numbers,
then there are also enough for G^G^G natural numbers.
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.
----
Consider the set of natural numbers for which
there are NOT enough hats.
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.
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.
⎛ Recall that it does not exist
⎜ because,
⎜ if there are enough hats for G natural numbers,
⎝ then there are also enough for G^G^G natural numbers.
The set of natural numbers
for which there are NOT enough hats
does not hold its first number.
No number exists that can be the first number.
It can only be the first case, that
the set of natural numbers
for which there are NOT enough hats
is empty.
Its complement,
the set of natural numbers
for which there ARE enough hats
is the complete set of natural numbers.
----
Therefore,
for each natural number,
there are enough hats.