countably infinite
How can infinity be countable?
It all comes down to how you define countable.
Let N be the set of all natural numbers
(Alternatively, N = {x|x>-1, x<Infinity})
Then, any set which is a subset of N, all natural numbers, or can be paired
one to one to N is countable.
Say the set of all even numbers. {0,2,4,6,8,10,12...}
And the set of all natural numbers. {0,1,2,3,4,5,6...}
We can pair them one to one, like so:
{0,2,4,6,8,10,12...}
{0,1,2,3,4, 5, 6...}
As we can see, the cardinality of all the even numbers is the same as the
natural numbers, so the set of all even numbers is countably infinite.
What about the set of all decimal numbers?
That is uncountably infinite.
Say we have a list of decimal numbers paired to N.
+---+----------+
|1 |0.01324...|
|2 |0.25674...|
|3 |0.23242...|
|4 |0.25675...|
|5 |0.74632...|
|. |. |
|. |. |
|. |. |
+---+----------+
Now, we cna create a new number which is not in the list.
Start with 0.
Take the first number after the decimal point in the first number.
0. Then, add 1 to it.
1. Now our number is 0.1 Then, we do this to the second number and subtract 1
instead.
0.14
We do this infinitely until the end. Then, no matter which number you look at,
it will still be different.
0.14363... You don't see this in the list, do you?
Therefore, the set of all decimal numbers is uncountable infinite. That is, you
cannot pair it to the set of all natural numbers.