Infinite Infinities

As a kid, I was fancinated with infinity which probably is why I got an extra degree in Mathematics.  Pondering infinity can lead some into an uncomfortable zone of a great spiral of never ending logic.  In fact, it takes discipline to pause in the journey of pondering infinity.

Infinite series were intriguing in mathematics.  I remember my first revelation of "different sizes" of infinities when a professor asked the class which infinite set was larger?  The number of irrational numbers (fractions) between zero and one OR the number of positive integers from 0 to infinity.  My response - you can't quantify infinity.  Wrong (sort of).

Using the principle of matching -  match each integer ....  2, 3 ,4, 5 ...   with the following fractions  1/2, 1/3, 1/4, 1/5 .......    Consequently you have matched every number (integer) in the infinite set of integers with specific fractions in the infinite set of fractions between zero and one.  YET  you still have all the other fractions between 1/2 and 1 available (e.g.  2/3, 3/4, 4/5 etc.).  SO the infinite set of irrational numbers is a larger infinity - in fact infinitely larger. :)

Just think there are an infinite number of small and large infinities.

PS -  For the astute - I forgot  something.  You first need to match THE ONE in each set.