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

Am Sat, 23 Nov 2024 21:45:06 +0100 schrieb WM:

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.

No. That other place where we take the hats from, larger primes, are of
course covered by even larger primes. We don't stop at some arbitrary
finite number, but continue forever.
>

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.

Not true. The first n prime numbers are obviously in bijection with the
numbers 1 to n.
>

Assume that there are enough red hats for the first ? numbers but
not enough for the ?+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!

Right, that is your mistake. There are still hats left over.
>

There are too few prime numbers.

No, there being too few primes leads to contradiction.

Shall unit intervals disappear like Bob?

Like who?