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

On 23.11.2024 21:18, Jim Burns wrote:

On 11/23/2024 5:30 AM, WM wrote:

On 22.11.2024 22:50, Jim Burns wrote:

 

ℙ covers ℕ, and ℕ covers ℙ

>
Let every unit interval on
the infinite real axis
be coloured white. Cover the unit intervals of prime numbers
by red hats.
It is impossible to shift the red hats

 Yes, because we are finite beings,
and there are infinitely.many red hats.

No, the reason is that every shift removes the hat from its place and requires an other hat, taken from wherever, but with certainty leaving an uncovered interval. That does never change.

 

It is impossible to shift the red hats
so that all unit intervals of
the whole real axis get red hats.
There are too few prime numbers.

 No.
 ⎛ Assume that that is so.

That need not be assumed but that is obviously so for every part of the real axis.

⎜ 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. Alas they leave G^G^G unit intervals without hats. That is the catch!

There are too few prime numbers.

 No,
there being too few primes
leads to contradiction.

Shall unit intervals disappear like Bob?
Regards, WM