To Infinity and (not) Beyond: Part II

photo from the nytimes archives

 
I'm sorry that this post is so belated, but the creative side of my brain has been dead, and understanding Maths  takes a lot of creativity.

everything maths

I promised you a discussion of Mr Cantor and his contribution to infinity... Cantor realised that there is more to understand about infinity than previously thought.

He started by thinking about different sets that we use everyday and think about as "infinite", such as the natural numbers (0,1, 2, 3, 4...). And the real numbers (0, 0.1, 0.0000000002, -0.0547829 etc...). In fact he thought about a lot of sets, and these are just two simple ones. But they will suffice for my highly simplified explanation.

Most of us would agree that there are infinitely many natural numbers. You could keep counting forever (though why you'd want to I'm not sure) , and you would never "run out" of natural numbers - even if what you were counting was the number of stars in the sky!

To briefly introduce some terminology, Cantor called the number of elements in a set the set's cardinality. He called the cardinality of the natural numbers "aleph null" (HINT: the squiggly N is the Hebrew letter aleph).

Well so far so good. All we've really done is make another label for infinity. Who cares?

The really exciting part comes now! Cantor went on to observe that the real numbers behave...differently...to the natural numbers. In fact, he demonstrated that there are infinitely many real numbers between every two natural numbers.

This is the core of the Cantor's insight, and it is demonstrated visually by Cantor sets, which you can learn how to draw here.

The idea is that if you keep dividing the interval between 0 and 1 into thirds, deleting the middle third each time, you could keep dividing forever. Since the real numbers between 0 and 1 essentially represent every possible fractional value between 0 and 1, they can in some sense be represented by the infinitely many "segments" you would have at the end of this process. If there was an end, which there obviously wouldn't be!

And yes, the Cantor set is a kind of fractal - go here for another (probably more rigorous) discussion!

Cantor went on to say (very reasonably in my view) that you can't say that the real numbers have the same cardinality (number of elements) as the natural numbers if there are infinitely many real numbers between 0 and 1 and further between any two natural numbers. It just doesn't make sense! So he labelled the cardinality of the real numbers aleph, with the rider that aleph is bigger than aleph null.

Um, aren't we forgetting something here??? We agreed that both the natural numbers and the real numbers had an infinite number of members. And do you remember Galileo: ...“it is wrong to speak of infinite quantities as being the one greater or less than or equal to the other”?

Cantor disagrees. According to him, there are at least two types of infinity. The first has the same cardinality as the natural numbers. This he called a "countable" infinity. The second has the same cardinality as the real numbers. This he called an "uncountable" infinity.

As you can imagine, this has some interesting consequences...but that's a story for another time.

from here