How There Can Be One Infinity Larger Than Another Infinity

Intuitively speaking, infinity goes on forever so it doesn't make much sense to us that you can add anything to infinity. Even if you could, isn't infinity just infinite? How can one be bigger than another? It is pretty counter intuitive but some clever stick by the name of George Cantor showed, in simple terms, how this can be so. This is actually very easy to follow and very elegant in my opinion.

Before we start, its important to set aside our preconception about what infinity is and what it is not. The majority of us learn a basic definition that our young minds can understand and this is what we tend to hold as true. However, infinity has been redefined over and over again throughout history. Along with zero, infinity remains one of the most counter intuitive concepts.

Imagine an infinite string of 2 arbitrary characters. We will use a and b.

Here is an example of such a string...

aababbaaababbbabababbabbbababab...........(and so on)

Furthermore, imagine an infinite number of these unique strings. No string is exactly the same; it is easy to comprehend that there can be an infinite number of these strings.

I will start a few for illustrative purposes. I have just randomly selected as and bs by arbitrarily, and rapidly hitting those keys.

abababababbbababbbabaaaababababababab...
bbababbababababababbabbababbababbabab...
aabbababbababababbababababaaababaabab...
aababaabababababbbabababaabbabbababab...
.............................................................................
...these strings go on "forever."

Now, lets select a string made up of the 1st character from the first string, the 2nd character from the 2nd row, the nth character from the nth row...
(nth basically means that you can pick any number you like to be n)

abababababbbababbbabaaaababababababab...
bbababbababababababbabbababbababbabab...
aabbababbababababbababababaaababaabab...
aababaabababababbbabababaabbabbababab...
......................................................................................

Now, of the string selected, lets flip the characters so that every a becomes a b and every b becomes an a.

bbababababbbababbbabaaaababababababab...
baababbababababababbabbababbababbabab...
aaabababbababababbababababaaababaabab...
aabbbaabababababbbabababaabbabbababab...
......................................................................................

It is now easy to see that this new string must be unique within our set of strings: It differs from the 1st string by its 1st character; it differs from the 2nd string by its 2nd character; it differs from the nth string by its nth character.

So we can always add another unique string to our already infinite set of strings. Therefore, we have one infinite set bigger than another infinite set! That means there must be one infinity larger than another.