Countable and Uncountable

On week 9, we learned how to determine whether a set is countable or uncountable. A set is countable if and only if it is listable. For example, rational numbers are countable because they can be listed. Also, the number of elements that a countable set has is no larger than the size of natural numbers. Interesting enough, the size of even natural numbers is equal to the size of natural numbers.  However, the size of real numbers between 0 and 1 is beyond the size of natural number. No matter how you list them, I can find a number that does not appear on your list. This means that the real numbers between 0 and 1 are not listable, therefore they are uncountable.