We are all familiar with the notion of infinities. Something which is uncountable, unbelievably massive in magnitude is an infinity. For example: number of all numbers, number of points on a line, disappointment of my parents etc. are infinities. But how do we say which infinity is larger? Is the number of all numbers greater than number of points on a line? Can we compare Infinities?

The same question was asked by the famous mathematician Georg Cantor and he also answered it.

Take this situation for example-: A man found the treasure. But he doesn’t know how to count. For that person, the number of coins in that treasure seems to be infinite. The man also want to know if he has more pearls than coins or more coins than pearls. Should he give up because he cannot count them? No. He compares the coins with pearls one by one, piece by piece. If he runs out of pearls first, he knows that he has more coins than pearls and if he runs out of coins first, he will know that he has more pearls than coins.

Exactly the same method was proposed by Cantor for comparing infinities. If we could pair the objects of two infinite groups such that each object of first infinite group pairs with each object of the other infinite group such that no object is left unpaired, we can say that the two Infinities are equal. If, however, this arrangement was not possible and objects in one of the two infinities are left unpaired, then we can say that the infinity with unpaired objects is larger than the one whose all objects are already paired up.

Lets apply it on odd and even numbers. We are going to find out if the infinity of odd numbers is larger than even or is it the other way around.

As you can see, there is one-to-one correspondence between all even and odd numbers. Since every odd number can be paired with every even number, we can say that the 2 Infinities are equal.

But wait, let us now compare number of all numbers with number of all odd numbers. Which do you think is larger? Well, you might think that of course, the infinity of all numbers is larger than infinity of all even number but that conclusion might be wrong. Lets apply the above method to find out the correct answer.

Again, we can make a table with one-to-one correspondence between all numbers and all even numbers. So by the rule, the both Infinities are equal. But that seems paradoxical. Since even numbers are just a part of all numbers but we have to remember that here we are dealing with Infinities and that we will encounter different properties. A part can be equal to the whole.

There is a great video by Ted-Ed on one of these strange properties of Infinities. You might want to check it out-

But doesn’t all of these mean that all infinities are equal and that there is no larger or smaller Infinities?

Thats actually not true. Some infinities are in fact larger than the others. Like the numbers of points on a line are more than number of whole numbers. Let us try to make the one-to-one correspondence between the two. We can number the points on a line as the distance of it from the end of the line. For example we take a 1 meter long line, hence the last point on this line will be denoted as 1 and middle point will be denoted as 0.5

Now its easy for us to compare them

As you can see in the above table, the numbers are paired up with points on the table. After all whole numbers are paired up, one can come and write coordinates of a new point and have an unpaired point left. He can do that by selecting 1st decimal place from 1st point , 2nd decimal place from 2nd point and so on and can argue that there is no point with same coordinates as this new number and hence its an unpaired point. Hence proving that numbers of points on a line are more than numbers of whole numbers.