A set is countable if and only if each of its elements can be associated with a different positive integer. Every finite set is countable. For example, the set {2, 3, 5, 7, 11} is countable.

1↔2
2↔3
3↔5
4↔7
5↔11

Infinite sets can also be countable. For example, the set of all prime numbers is countable.

1↔2
2↔3
3↔5
4↔7
5↔11
6↔13
7↔17
8↔19
9↔23
10↔29
...

1. Is the set of all integers countable?
2. Is the set of all positive rational numbers countable?
3. Is the set of all rational numbers countable?
4. Is the set of all positive real numbers countable?
5. Is the set of all real numbers countable?

(In reply to re: Interesting by Brian Smith)

Thanks, you are right. I think that disposes of one of the points I was trying to make. It's I*I, not I^I.

Because if, for example, we have infinitely many fleets arriving each day, then that is still only I*I*I, and if that number of fleets arrive every day forever, that is just I*I*I*I.

I can see how each of these can be countable.

WolframAlpha assures me that 2^I is in fact the same as I; if that is correct then I think my other point still stands. Of course it may not be right.

Edited on March 16, 2016, 11:04 am

Posted by broll on 2016-03-16 10:58:52