Re: A question for set-theorists

On 20.11.2024 19:06, joes wrote:

Am Wed, 20 Nov 2024 18:02:40 +0100 schrieb WM:

1) Let every unit interval after a natural number on the real axis be
coloured white with exception of the powers of 2 which are coloured
black. Is it possible to shift the black intervals so that the whole
real axis becomes black?

Are the intervals closed or not?

Irrelevant, but assume closed.

 

2) Let every unit interval after a natural number on the real axis be
coloured as above with exception of the intervals after the odd prime
numbers which are coloured red. Is it possible to shift the red
intervals so that the whole real axis becomes red?

with the exception of = but instead

Well understood.

What colour has the real axis after you have solved both tasks?

If you have "solved" them, I suppose it is black if you do the
second one first.

No. The density of coloured intervals within the first n intervals is a sequence converging to zero. Every positive eps is undercut. The limit cannot be 1.
Regards, WM