Counting with Infinities

The set of natural numbers 1, 2, 3, 4, 5, ..................... is infinite meaning that no matter how large a number we go, there are always larger numbers we can go further. Likewise, the set of even numbers 2, 4, 6, 8, 10, ..................... is also infinite. While both are infinite, we want to ask, "Are there more natural numbers than even numbers?"
Our answer tends to be yes because natural numbers consist of both even numbers and odd numbers. Apparently the number of natural numbers is doubled that of even numbers. This answer may be correct IF both of them are finite (i.e. of fixed quantities). But things behave differently when they are infinite. Twice of infinite quantity is still infinite quantity. Seemingly amazingly, the correct answer would be: "No, there are as many even numbers as natural numbers."

Suppose we ask, "Are there as many fingers on the right hand as the left hand?" The simple way to determine the answer is to count the number of fingers on the right hand and the left. If they both come to five, we know the answer is yes. But suppose we haven't developed the number system, and we don't have any knowledge of one, two, three and so on. Then we would not be able to count. But it doesn't mean we cannot answer the question. We can still match the fingers on the right hand to those on the left; thumb against thumb, index finger against index finger, middle finger against middle finger and so on. If all the fingers on the right hand can find a corresponding partner on the left, then we know the answer is yes.
Now suppose we have infinite number of fingers on the right and left hands (a monster!). We would not be able to count as they are infinite. But we can still use the method of matching to find the answer. Although the process of matching goes on indefinitely, if each finger on the right hand is able to find a one-to-one corresponding partner on the left, and each finger on the left hand can find a one-to-one corresponding partner on the right, then we know the answer is yes. However, if we can find some finger on the right hand that doesn't have any partner on the left while each finger on the left hand can find a partner on the right, we know there are more fingers on the right hand than the left.
Alternatively, imagine we have a very very large audience and a very very large cinema hall. In order to determine whether there are as many spectators as seats in the hall, we could ask the audience to seat themselves one by one. If there is no spectator left unseated and if there is no seat left empty, then we know there are as many spectators as seats in the cinema hall without going through the tedious process of counting with very very large numbers.

So, using the same methodology, we list the natural numbers N and the even numbers E as follows:

N: 1 2 3 4 5 ............ n ......
E: 2 4 6 8 10 ........ 2n .....

We can see that for each natural number, there is always a one-to-one corresponding even number by doubling its value. And for each even number, there is always a one-to-one corresponding natural number by halving its value. Although the process goes on indefinitely, there is no natural number nor even number left unmatched. Hence, we conclude there are as many even numbers as natural numbers.

Situation of this type is said to be countably infinite, and is the basis upon which Georg Cantor (1845-1918) developed his theory of infinite sets.