Re: Does the number of nines increase?

Am 06.07.2024 um 03:58 schrieb Moebius:

Am 05.07.2024 um 20:08 schrieb Jim Burns:
 

The other way around.
It's set.inclusion which stops working as a guide to size.

 Right. How would we be able to compare, say, the sets {1, 2, 3, ...} and {-1, -2, -3, ...} concerning "size" by relying on "set inclusion"? Or, say, {1, 2, 3, ...} and {1.5, 2.5, 3.5, ...} etc.
 Or even {1, 2, orange} and {1, 2, 3}.

Or let's compare the size of, say,
{0, 1, 2, 3, ...} with the size of {(0, x_0), (1, x_1), (2, x_2), (3, x_3), ...} (for some x_0, x_1, x_2, x_3...).
It seems that in this case the size of these two sets should be the same, I'd say.
Now let's compare the size of, say, {1, 2, 3, ...} with the size of {(y_1, 1), (y_2, 2), (y_3, 3), ...} (for some y_1, y_2, y_3, ...).
Again, it seems that in this case the size of these two sets should be the same, I'd say.
So what's the size of the set {(0, 1), (1, 2), (2, 3), (3, 4), ...}?
The same as the size of {0, 1, 2, 3, ...} and/or the same as the size of {1, 2, 3, ...}?
The "conclusion" seems to be that {0, 1, 2, 3, ...} and {1, 2, 3, ...} have the same size, even though {1, 2, 3, ...} c {0, 1, 2, 3, ...}.