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

WM laid this down on his screen :

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.

How much paint would be needed to paint the entire cartesian plane? Too much huh? then just paint the first quadrant with one quarter the amount of paint. Your 'logic' is too easily denied.