Types of Infinities

One surely would have heard the word infinity many times in his life, but what most of the people dont know is that there are different kinds of infinities.

This post tries to throw some light on this.

You can see that there are infinite types of infinity via Cantor’s theorem which shows that given a set A, its power set P(A) is strictly larger in terms of infinite size (the technical term is “cardinality”).

http://en.wikipedia.org/wiki/Cantor%27s_theorem

A very nice introduction to the many different notions of infinity in mathematics is Rudy Rucker’s book: Infinity and the Mind. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Moreover, Rucker has gone to great lengths to make the presentation faithful to the mathematics but still accessible to an educated layperson. Below is an excerpt on the Alephs.

Now coming to the concept of countable and uncountable infinities, here’s an explanation that i have found really interesting:

Suppose no one every taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there’s a problem, we don’t know how to count the bushels or the sheep! So what do we do?

We form a “bijection” between the two sets. That’s just fancy language for saying you pair things up by putting one bushel next to each of the sheep. When we’re done we swap. We’ve just proved that the number of sheep is the same as the number of bushels without actually counting.

We can try doing the same thing with infinite sets. So suppose you have the set of positive numbers and I have the set of rational numbers and you want to trade me one positive number for each of my rationals. Can you do so in a way that gets all of my rational numbers?

Perhaps surprisingly the answer is yes! You make the rational numbers into a big square grid with the numerator and denominators as the two coordinates. Then you start placing your “bushels” along diagonals of increasing size, see wikipedia.

This says that the rational numbers are “countable” that is you can find a clever way to count them off in the above fashion.

The remarkable fact is that for the real numbers there’s no way at all to count them off in this way. No matter how clever you are you won’t be able to scam me out of all of my real numbers by placing a natural number next to each of them. The proof of that is Cantor’s clever “diagonal argument.”
For those interested in the origin of this above discussion can go here: http://math.stackexchange.com/questions/1/different-kinds-of-infinities/9#9

More on this in my next post.

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