Different sized infinities?

Infinity is really vague concept. It is hard to imagine because it goes on forver. So, one might think that nothing can be bigger than it, right?
Well as it turns out, in Maths the answer is NO.

Some infinities are bigger than others. It was first proven by Georg Cantor, a German mathematician for which he received enormous criticism at the time from the mathematical community but his proof was later granted.

Georg Cantor

Proof:

There are an infinite number of numbers between 0 and 1 and also an infinite number of Natural numbers.

So, he started his proof by saying," Since there an infinite number of numbers between 0 and 1, I will start by first writing a number between 0 and 1 that has an infinite number of "digits". " 

Like: 

0.2334212342344672...
0.6000000000000000...

Now what he does that he assigns a natural number index to every possible number between 0 and 1 each having infinite digits and each unique. 

Like:

#1  -->   0.974274623747...
#2  -->   0.557647264823...
#3  -->   0.763849284465...
#4  -->   0.564758274875...
#5  -->   0.450000000000...
 .
 .
 .

Now if the list is complete then we have assigned a natural number for every possible number between 0 and 1 , each unique.

Then he makes an ingenious move.

He says that," Now if I change the first digit of the first number by adding 1 to it and then change the second digit of the second number by again adding 1 to it and go on forever and make a new number with these digits then this number is guarenteed to never show up in the list."

{If the digit is 9 change it to 0}

New number  ==>  0.06481 ...

                       ==>  0.[9+1][5+1][3+1][7+1][0+1] ...

#1  -->   0. [9] 74274623747...
#2  -->   0.5 [5] 7647264823...
#3  -->   0.76 [3] 849284465...
#4  -->   0.564 [7] 58274875...
#5  -->   0.4500 [0] 0000000...
 .
 .
 .

Doing this the new number will have unique digits for a unique place making it a number that will never show up in the list.   

But since the list is complete, therefore this new number will have no natural number assigned to it !!

Therefore, we have proved that there are more numbers between 0 and 1 than there are between 1 and infinity!

This discovery was ground-breaking and even surprised Cantor himself. He described the infinity between 0 and 1 to be "uncountably infinite" and the infinity between 1 and infinity to be "countably infinite".

By this proof Hilbert (another great mathematician of the time), was inspired and he went to make mathematics rigorous by setting up axioms that all mathematical statements must follow. But then he was proven terribly wrong and mathematics is still not a rigorous subject. But that's a story for another time.