Infinity: Big and Bigger

On the Internet, you can never go wrong by quoting the The Hitchhiker’s Guide to the Galaxy.

Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.

Now, it’s kind of a cheat, because i’m not going to talk about that kind of space, I’m going to talk about spaces in a mathematical sense. But I’m offering something in exchange, because I’m going to talk about spaces that are much bigger than mere physical space.

The point of this is really to talk about (echo effect) infinity. And beyond.

Mathematically, space is much simpler than the thing in which your coffee cup is located just out of reach and that keeps your cat from being exactly where you’re sitting, no matter how much he tries. In mathematics, a space is simply a set of some sort with some kind of additional structure. (A set is just some collection of things with no duplicates, like {1, 2, 3, 4, 5}. By convention, we put sets into braces like that example.)

So far, that’s not a space — we haven’t said anything further about it than there is a bag full of things. But — since I’ve chosen a set we conveniently already know a lot about — we know that the set is ordered because we agree that 5 is bigger than 4. And we have a space.

Okay, it’s a pretty boring space, but it’s a space.

There are some other rules we think we know, like addition — 1+2=3. But in our little space, we immediately run into trouble, because 3+4 equals what? Oh, 7, but 7 isn’t in the set. To take care of 3+4, we need to expand the set to be at least {1,2,3,4,5,6,7} and then we’re immediately going to have the problem of 4+5, or for that matter, 7+1.

Now, with nothing more than the idea of addition (we talked about ordering, but we can define an order in terms of addition) we’ve run into our first experience with infinity. There is a set N that we can define like this:

• 0 is part of N
• For anything that is part of N, which we’ll call n, n+1 is also in N.

We call N the natural numbers.

Now, N is pretty big. After all, no matter what n we pick, there’s always something bigger. This is what we call infinite. And all is well, until we think about subtraction: we know 3-1=2, and we know 2-1=1, and we know 1-1=0, but 0-1 isn’t in our set. So we define a new set called the integers which has new elements -1, -2, -3, and so on. We can throw in multiplication now, and all is good, but when we put in division we’re in trouble again: 2÷3 and 1÷2 aren’t in there. So we define another set called the rational numbers, Q.

Now, we’ve pretty much defined all the numbers anyone had any use for until the Greeks and Egyptians screwed it all up by trying to measure fields and distances.

The Pythagoreans in Greece were a sort of a religious cult that believed the entire Universe was made of whole numbers and rational numbers. Of course, famously the Pythagoreans proved the Pythagorean Theorem — usually we say “Pythagoras proved” but actually one of the rules of the Pythagoreans was that everything they did had to have Pythagorus’ name on it because he was “head of the lab,” a tradition that continues in some laboratories to this very day — anyway, they proved the Pythagorean Theorem, which showed the length of the hypotenuse of a right triangle was the square root of the sum of the squares of the other two sides.

Because the Pythagoreans knew everything was described by whole numbers and their ratios, they then spent much effort trying to figure out what ratio that square root was — until a fellow named Hippasus proved that number was irrational, it couldn’t be represented exactly by any ratio.

For which discovery the Pythagoreans promptly drowned him. He made the fundamental mistake of explaining it to the other Pythagoreans while on a boat trip; they threw him overboard.

Barbie was right.

We’re going to fast-forward a bit now — we’re now looking at something with enough parts to talk about anything we’d describe with the normal geometry of the Greeks, what we call Euclidian geometry. Which gives me an excuse to quote one of my favorite poems, an Italian sonnet by Edna St. Vincent Millay:

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

The set we have now, of all the rational and irrational numbers, is pretty much all that we need for anything real, and so it’s called the real numbers, R. Oh, there was a little flurry of excitement when people started asking about the square root of negative numbers, but that was pretty easily disposed of with complex numbers.

But around the turn of the last century, there was a big effort on to sort of fill in the holes in mathematics and make everything axiomatic — reduce everything in mathematics to the sort of formal proofs that would satisfy logicians.

One question that turned out to be hard to answer was this: was the infinite set of real numbers “bigger” than the infinite set of integers? Was the set of rational number bigger than the set of integers? And what the devil would it mean to talk about the size of an infinite set at all? I mean, it’s clearly infinite.

Then along came a Russian-German guy, Georg Cantor. He started thinking about these collections of things, starting out by just asking what it meant to talk about the size of a set at all.

Finite sets were easy — just count them, and you get a number called the cardinality of the set. {1,2,3,4,5} has a cardinality of 5. No problem.

Obviously you can’t come up with a number for the size of an infinite set — the thing you see called “infinity”, ∞, doesn’t act like a number in important ways — but he did realize there was a way to tell if two sets were the same cardinality, the “same size.” This is called the pigeonhole argument.

Imagine you have something like mailboxes, and you have a bunch of junk mail. Start putting the junk mail into the slots, one at a time. If you run out before you’ve annoyed everyone in the building, the set of junk mail is smaller than the set of slots; if you have leftover junk mail, the set of junk mail is bigger; and if you come out exactly even, the two sets were the same size. And you’re Goldilocks.

Mathematically, coming out even that way is called establishing a one to one correspondence. And there’s where the trouble started.

Cantor came up with a kind of proof called a diagonalization proof. It works like this: We lay out a set that is the same size as the natural numbers as a matrix like shown in the figure here:

Each of these has an infinite binary representation. Cantor used m and w, but it could be 0 and 1 as we’re used to looking at binary numbers now. So, the first row there is 0, which has an infinite binary representation pf ‘0000…’. 1 is ‘1000…’, 2 is ‘0100…’ and so on.

Now go down the diagonal, and construct a new number, call it E, where the first digit is not 0 but 1; the second digit is 1; and so on. All we do is make sure the the n-th digit in our new number is always different from the n-th digit of the n-th row.

Our matrix of numbers, we know, has all the natural numbers in it, and all of those numbers have their representation in binary. But now we have this new number, E, which also has a binary representation, but which we know can’t be equal to any of the numbers in our list.

(Why? By definition. If it were equal to some number in the list, say 32,260,309, then it would have the same digit at place 32,260,309. But since we constructed it by getting to digit 32,260,309 and inserting something that did not match, it can’t be equal. Pick any number you like; E can’t equal that number.)

This argument — there’s a longer explanation that I like here, and another here — and the definitions Cantor came up with to go with it, form what is now called set theory.

Cantor’s Theorem had a somewhat larger implication, though. What it meant was that any set you could construct had a bigger set that could be constructed from it. So, we had the set of natural numbers, and we knew there was a bigger set than that because we showed how to construct it. But if we can construct that bigger set, then we can use the same tricks to construct an even bigger set from that, and a bigger set from that, and so on. So, since we can always construct the next bigger set, we know that this set is infinite as well. There are infinitely many infinite sets.

Set theory is now seen as one of the foundations, perhaps even the foundation of mathematics, and Cantor’s method of diagonalization was later used by Kurt Gödel in proving his famous Incompleteness theorem.

It was very controversial at the time though. Very. Leopold Kronecker, a famous professor of mathematics at the University of Berlin (where Cantor had gotten his doctorate), said Cantor was a “scientific charlatan,” a “renegade,” and a “corrupter of youth.” Henrí Poincaré, another great mathematician of the time, called set theory a “grave disease” infecting mathematics. Wittgenstein later said set theory was “laughable” and “wrong.”

This was so tough on Cantor that he started to suffer periods of depression; these resulted in his repeated hospitalization and increasing debilitation until he died.

Lucky that Cantor didn’t first explain it while he was on a boat.