Infinity

When I was young, I remember sitting on top of the steps on the outside of my house, lying back and gazing up into the night sky. It all seemed so vast, and made me feel small, and perhaps insignificant. I thought, "where does this end - how far is up?" I thought that somewhere out there, if you kept going far enough, you would hit a brick wall (I'm not sure why I thought it would be brick, but it just seemed right) and that would be the end. But only a few moments later, I wondered what would be on the other side of that brick wall, and how far past that. At that, I sat up, when into the house and stopped thinking about it, because I knew it was something beyond my ability to comprehend.

But one thing we do know - infinity is more than a mathematical concept, it is real, and it does exist. The concept of reality we cannot understand is a theme I'd like to delve into later.

Anyway, where I wanted to go with this post takes me to graduate school, where I studied math. My favorite courses there were in "Real Analysis", and in that course we did talk about the infinite in several different ways. We all know that the counting numbers - the positive integers - are infinite. There is no "last number". So, consider the collection ("set") of all counting numbers. There is a term called 'cardinality' which reflects the number of elements in any set. A finite set - say a set of all the members of your family - would have cardinality of seven if you had a mom, dad, and four other siblings (plus yourself). The cardinality of the set of counting numbers is, of course, infinite, since there are infinitely many numbers in the set.

Now, let's look at other infinite sets. Say, first, the set of both positive and negative whole numbers (the integers). If you can (and you can, take my word for it) create a one-to-one mapping between the set of integers and the set of counting numbers, then we can say the set of integers is 'countably' infinite. Interestingly, you can also create such a one-to-one mapping between the set of counting numbers and the set of 'rational' numbers (rational numbers are those that can be expressed as a fraction with integral numerators and denominators). So, the set of rational numbers have the same cardinality as the set of counting numbers.

So far, so good. You might know that there are numbers that are not rational - numbers that cannot be expressed as a fraction with integral numerators and denominators. Examples are numbers like 'pi' and some square roots (like the square root of 2). Numbers like this are called 'irrational' and the set of all numbers both rational and irrational are called the 'real' numbers. The curious thing is that you can prove mathematically that it is impossible to create a one to one mapping between the set of counting numbers and the set of real numbers. So, amazingly, we have two sets of infinite cardinality - yet one set is 'bigger' than the other! Thus, there is more than one infinity. In math, we call the cardinality of the counting numbers 'aleph null' and the cardinality of the reals 'aleph 1'. If we use the constructive nature of the aforementioned proof of the existence of 'aleph 1', we can construct infinite sets which have cardinality greater than 'aleph 1' (call that one 'aleph 2') - repeat and you have 'aleph 3' and so on. Thus, there are infinitely many infinities. From the way I have described it, perhaps the number of infinities is countably infinite (going from 0 to 1 to 2 ...) but I bet they aren't. At this, I think I should just get up from my stoop and go into the house and stop thinking about it...

welcome to my blog

So why am I starting a blog, anyway. Well, it just seems the thing to do. This blog will be my own reflections on computer science (I teach a few courses for a well known university), net-centric technology (my current area of interest), and mathematics (the discipline of my degree). So it will be an mix of topics. More later...