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Infinity: Big and Bigger


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.