Discussion:
A question for set-theorists
(too old to reply)
WM
2024-11-20 17:02:40 UTC
Permalink
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?

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?

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

Regards, WM
joes
2024-11-20 18:06:40 UTC
Permalink
Post by 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?
Post by WM
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
or do you mean in addition to?
Post by WM
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.
--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.
WM
2024-11-20 18:32:38 UTC
Permalink
Post by joes
Post by 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.
Post by joes
Post by WM
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.
Post by joes
Post by WM
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
joes
2024-11-21 00:18:22 UTC
Permalink
Post by WM
Post by joes
Post by 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.
You forget that you can push open intervals closer together.
Post by WM
Post by joes
Post by WM
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.
Badly written.
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.
WM
2024-11-21 11:54:22 UTC
Permalink
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?

Regards, WM
joes
2024-11-21 21:03:29 UTC
Permalink
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?
I don't believe in falsehoods. How do you derive the above? Both the
denominator and numerator diverge. The expression oo/oo is undefined.
--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.
WM
2024-11-21 21:44:19 UTC
Permalink
Post by joes
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?
I don't believe in falsehoods. How do you derive the above? Both the
denominator and numerator diverge. The expression oo/oo is undefined.
like lim n->oo n/n^3?

Regards, WM
joes
2024-11-22 12:28:51 UTC
Permalink
Post by WM
Post by joes
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?
I don't believe in falsehoods. How do you derive the above? Both the
denominator and numerator diverge. The expression oo/oo is undefined.
like lim n->oo n/n^3?
Right. What even is oo^3.
--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.
sobriquet
2024-11-22 13:48:47 UTC
Permalink
Post by joes
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?
I don't believe in falsehoods. How do you derive the above? Both the
denominator and numerator diverge. The expression oo/oo is undefined.
lim n->oo n/(2^n) = lim n->oo 1/(2^(n-(ln(n)/ln(2)))) = 0

lim n->oo n/(n^3) = lim n->oo 1/n^2 = 0


https://www.desmos.com/calculator/ulookjqjcq


https://www.wolframalpha.com/input?i=lim+n+to+infinity+1%2F%282%5E%28n-%28ln%28n%29%2Fln%282%29%29%29%29
sobriquet
2024-11-22 13:52:06 UTC
Permalink
Post by sobriquet
Post by joes
Post by WM
Post by joes
Post by WM
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.
Formally: lim n->oo n/(2^n) = 0
But you don't believe it?
I don't believe in falsehoods. How do you derive the above? Both the
denominator and numerator diverge. The expression oo/oo is undefined.
lim n->oo n/(2^n) = lim n->oo 1/(2^(n-(ln(n)/ln(2)))) = 0
lim n->oo n/(n^3) = lim n->oo 1/n^2 = 0
https://www.desmos.com/calculator/ulookjqjcq
https://www.wolframalpha.com/input?i=lim+n+to+infinity+1%2F%282%5E%28n-%28ln%28n%29%2Fln%282%29%29%29%29
Uh.. that desmos link should be:

https://www.desmos.com/calculator/etllgpm3bo

Chris M. Thomasson
2024-11-20 21:42:51 UTC
Permalink
Post by 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?
1->(black)->2->(white)->3->(black)->4->(white)->...

That? Notice how the odd numbers are black. ;^)
Post by WM
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?
What colour has the real axis after you have solved both tasks?
Regards, WM
Chris M. Thomasson
2024-11-20 21:46:35 UTC
Permalink
Post by Chris M. Thomasson
Post by 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?
1->(black)->2->(white)->3->(black)->4->(white)->...
That? Notice how the odd numbers are black. ;^)
Powers of 2?

2^1, 2^2, 2^3, ...

2->(black)->4->(white)->8->(black)->...
Post by Chris M. Thomasson
Post by WM
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?
What colour has the real axis after you have solved both tasks?
Regards, WM
Chris M. Thomasson
2024-11-20 21:50:09 UTC
Permalink
Post by Chris M. Thomasson
Post by 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?
1->(black)->2->(white)->3->(black)->4->(white)->...
Ahhh! I failed to read your details for some reason. Sorry about that.
How about this:

1->(white)->2->(black)->3->(white)->4->(black)->5->(white)->6->(white)->7->(white)->8->(black)->9->(white)->...

?

I can make a graph on it to see it visually.
Moebius
2024-11-20 23:47:34 UTC
Permalink
Post by Chris M. Thomasson
I can make a graph on it to see it visually.
The inside of his asshole? :-)
Chris M. Thomasson
2024-11-21 00:01:59 UTC
Permalink
Post by Moebius
Post by Chris M. Thomasson
I can make a graph on it to see it visually.
The inside of his asshole? :-)
Mostly black after toilet time? lol! Grosss. Sorry. ;^o
Moebius
2024-11-20 23:46:31 UTC
Permalink
Post by WM
What colour has the real axis after you have solved both tasks?
It's dark (black) just like the inside of your asshole, Mückenheim.
Loading...