Cantor and Infinity, Part One

Only in America would someone sue someone for over a billion trillion dollars. To be exact, we’re talking $1,784,000,000,000,000,000,000,000. Kevin Houston of the University of Leeds is cited in the article as saying, “I don’t think the human brain is set to deal with those numbers.”

But are they? Mathematicians study the properties of not just very large number but the infinite. In calculus one only discusses approaching infinity, and (besides the idea of counting and counting and never having to stop) this was my first idea of infinity. Infinity isn’t a number, the teachers always said. And they’re right, at least in the numbers that most of know and love. It was just generally accepted that things got weird at infinity, but at least they got weird in a way we could understand. Despite all this talk that infinity isn’t a number, we still talked about being “at infinity” an awful lot, even though the technical term was “approaches infinity”. Heck, this infinity business wasn’t even all that rigorous to mathematicians at the time for years. Don’t like the rigor of analysis? Don’t blame Newton and Leibniz. Blame those who came after.

Then this guy named Georg Cantor gatecrashed the party and started looking at infinities. Not infinity, infinities. He didn’t mess with this language of approaching infinity not because it wasn’t his style but because he didn’t have to. It didn’t help him get too many papers published, but to be honest, it didn’t help him with too many other things either. Cantor decided to look at the set of natural numbers (yes, the ones you count with) as a set. We already know that we can start counting and never stop, so there are certainly infinitely many natural numbers. Cantor called the cardinality of this set aleph-null.

But what if we look at all the possible subsets of natural numbers (we call this the power set), like {1 2 3} or {42 56 100 2341534 8675309}? Certainly there are infinitely many of these too, but how many more? Thanks to basic set theory, we know that there have to be more subsets of natural numbers than natural numbers.

“Sushi, you’re tricking me,” I hear you say. “You just told me there are infinitely many natural numbers and now you’re pulling this on me too? Now you’re going to tell me that the moon really is made of cheese.”

If the moon’s made of cheese, sign me up for the next flight. I’m not kidding–about the math or the moon. First, let’s look at {1 2 3} and its power set.

Elements: 1, 2, 3 (cardinality 3)
Power Set: {empty set, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} (cardinality 8 )

Now try to match up an element of the original set with an element of the power set. Things go terribly wrong. Don’t believe me? Try it yourself with a different set. Besides giving an example of the fact that the power set is bigger than the original set, we also note that this can work even for infinite sets, as Cantor’s diagonal argument shows.

That’s a story for another day, though. Stay tuned! You will get your infinities.

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