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

On 25.11.2024 19:08, Jim Burns wrote:

On 11/25/2024 7:11 AM, WM wrote:

On 24.11.2024 19:16, Jim Burns wrote:

On 11/24/2024 6:06 AM, WM wrote:

On 24.11.2024 06:37, Jim Burns wrote:

 

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.

 Thus
it is not dark that
what we mean by 'natural number' is
countable.to from.0  and is
what you call a definable number.

Consider the black hats at every 10 n and white hats at all other numbers n. It is possible to shift the black hats such that every interval (0, n] is completely covered by black hats. There is no first n discernible that cannot be covered by  black hat. But the origin of each used black hat larger than n is now covered by a white hat. Without deleting all white hats it is not possible to cover all n by black hats. But deleting white hats is prohibited by logic. Exchanging can never delete one of the exchanged elements. Therefore we have here, like in all Cantor-pairings, the same impediment and further disussion is futile: You must deny logic. I do not.
Regards, WM