DIFFERENCE BETWEEN COUNTABLE AND FINITE SET

 FINITE SET

A finite set is any set which contains a finite number of elements,or any set that is not infinite.This just means that in theory ,you could write down every elements of the set explicitly .These sets have a specific number of elements like 42,13,1267.

COUNTABLE SETS  are infinite sets.

The strict mathematical definition of a countable set is a set that is in bijective correspondance to the natural numbers, which are the integers from. Basically, this means that you can assign a natural number to every element in the set, so in essence you are "counting" the set even though it is infinite. For example, the rational numbers are a countable set since you can write a pattern which will generate all rational numbers, and then just assign the natural numbers to this pattern in order.

Countably infinite sets are the "smallest" infinite sets, there are also uncountable infinite sets such as the real numbers or complex numbers, in which it is impossible to write a pattern which will explicitly write all of the reals. So these are larger infinities.

I hope this explanation helped, although I know thinking about counting an infinite set versus not being able to count an infinite set may be difficult to grasp at first.