Cohen Goes to Work

Paul Cohen is not famous. And he has nothing to do. He got his PhD from University of Chicago under Zygmund–the godfather of American analysis school, but has done nothing good in the area. He would’ve been stuck at Rochester if his high school and grad school pal Stein didn’t help him get short stints at MIT and IAS. Now his Stanford professorship is again in peril.

“I have nothing to do and I’m not famous…” Basking in California sunshine in the Quad, he had a revelation: he has to do something that makes him famous, and do it in 5 years before he’s too old for the Fields Medal.

During lunch, he casually asks his mates: “How can I get famous?”

“Shoot the President.”

“I’m not in CIA yet.”

“Break Dick Feynman’s safe.”

“He hasn’t got Nobel yet.”

“Solve any Hilbert’s Problem.”

“They haven’t been solved yet?”

That pesky Dilberto von Hilberto made a list of things he couldn’t figure out on his own, so his name will always be tagged along. Nobody cares if you solved something really important yet not in any of those “problems of the century” lists, but as soon as your hiccup sounds like “Hilbert 9” your picture is on the front page.

One day Cohen sneaks out of a snoring seminar to the library and found a list of Hilbert Problems. He flips to a page and sees No. 8, Riemann Hypothesis. “That will make me even more famous after I become famous. I’ll leave it for icing on the cake.”

A few more random page turns give him solved problems (“they are too trivial!”) and unsolvable problems (“they are too vague!”). “Is there anything not crappy?!” Going back to the Table of Contents, he sees a page that is almost blank.

Almost, except a single line:

No. 1. Continuum Hypothesis

Cantor Goes to Sleep

Georg Cantor wakes up in cold sweat all over his body. He cannot recall to which cardinality he had counted when he fell asleep.

It started a few days ago when he realized how to count infinity. In his definition, anyway. If he can establish an exactly one-to-one correspondence between the elements of two arbitrary sets, even if they have infinite number of elements, it means the two sets have the same “size”–cardinality as he calls it. In this sense, the set of all natural numbers is as large as the set of all integers, and–surprise!–the set of all rational numbers too.

He started to conjure up an infinite set with a larger cardinality, and he found one by brute force: the power set, which is the combination of all subsets of a set, including itself and the empty set. And it took him two sleepless nights to prove that power set has more elements than the original set.

He would have had a nice sleep had he stopped right there. But how could he stop? The power set has its own power set with an even larger cardinality. And it can go on and on and on, ad infinitum, just like Hugh Everett’s many worlds.

Is there an end? If there is, what is it?

That was what startled Cantor from his sleep, dreaming about larger and larger cardinality. Too scared to recall the dream, he goes back to the first power set of counting numbers. It doesn’t take too long to realize that power set is exactly the set of all real numbers, since any real number can be represented by a subset of counting numbers.

But soon a question looms: is there anything in between the set of counting numbers and real numbers? The real numbers is a continuum on the number axle in which the counting numbers take infinitely small space, so it seems likely that there’s a set whose cardinality is between the two.

“I’m too tired to solve it, so let’s call it a hypothesis for now.”

Cantor knew his mentor Weierstrasse would like these results, but suspected Kronecker the Red-Knecker would throw him into an asylum over such crazy ideas. “God made the integers; all else is the work of man”, so proclaimed Prof. Knecker, sitting firmly at the top of University of Berlin’s math department. “Knecker is right that the cardinality of an infinite set is the work of man. One Man, that is me.” Cantor’s conscious faded with a grin.

Gödel Goes to the Door

Knock, knock. “Who is it?”

Kurt Gödel asks even though he knows who is at the door. But he doesn’t actually know the person, and in this crazy world, it doesn’t hurt to be a little paranoid.

“Mr. Gödel? This is Paul Cohen from Stanford. The secretary said she had called you about my paper…” The door squeaks open a small slit.

“What do you want?”

“People told me you’re the only one who can put the final word on the Continuum Hypothesis.”

“What do you have?”

“You proved the hypothesis can’t be disproved within Zermelo–Fraenkel set theory, and I think I proved the other half: the hypothesis can’t be proved within ZF. So it’s a certain piece of uncertainty.”

Gödel grabs the envelope. “Quite forceful. Are you certain about the uncertainty?” He shuts the door without waiting for an answer.

Three days later, Mrs. Gödel answers the phone.

“Hello, Adele, this is Magritte from the Institute. A fella from Stanford sent a paper to Mr. Gödel a few days ago, and we’d like to know what he thinks of it. You see, we can’t accommodate the guy any longer if the paper is no good.”

“Hi Magritte. You haven’t seen Kurt yet? He went to the Institute this morning. I’m not sure if he likes it or not. He looked quite distressed for the last few days, and I heard him saying ‘I can’t let Albert know it’. He even forgot to ask me to taste his food once.”

Gödel hasn’t been to Princeton for years, so he stays in the only familiar common room and grabs any passer-by:

“Where is the famous Paul Cohen? He has done something that makes the rest of my life meaningless!”

Everyone Goes Away

Paul Cohen (1934 – 2007) won the 1966 Fields Medal for resolving the Continuum Hypothesis in 1963 with the forcing method that he invented. He died from a rare lung disease, after spending much of his late years in solving the Riemann Hypothesis with no success.

Kurt Gödel (1906 – 1978) created the two Incompleteness Theorems in 1931 and resolved part of the Continuum Hypothesis in 1940. He died from starvation when his wife was hospitalized for an extended period of time and could not taste his food for him.

Georg Cantor (1845 – 1918) created set theory and proposed the Continuum Hypothesis around 1880. His career was gravely compromised by strong opposition from renowned mathematicians like Kronecker and Poincaré. He was hospitalized in mental institutions many times, and died in one.

The basic story came from my colleague LJ, who is Cohen’s academic grand-nephew as LJ comes from the Elias Stein line. It can’t be all bogus, according to a 1st-person account of Cohen’s IAS trip within a super tedious autobiography page (search for Cohen). No wonder why most math is so hard: most mathematicians can’t write! Cohen’s 1-page memorial website provides some corroboration as well.