“Only a hypothesis? This Cantor of yours hasn’t proved it?”
“No one has managed to. This hypothesis was the first of the problems set by David Hilbert to secure the foundations of mathematics.”
“The famous program whose second problem you solved with your incompleteness theorem? You’re so organized, why didn’t you start with the first?”
Cantor had died mad, I later learned. He, too, had endured many bouts of depression during his life. Why had Kurt chosen this same dark path?
“Cantor’s work was based on a controversial axiom, the ‘axiom of choice.’ ”
“You once told me that an axiom is an immutable truth!”
He raised an eyebrow.
“I’m surprised at your recall, Adele. You’re partly right, but this truth belongs in a very particular box of mathematical tools. I don’t have the energy to explain its subtleties to you. All you need to know is that using certain of these axioms leads to insoluble logical paradoxes. Which casts their legitimacy in doubt.”
“And you hate paradoxes.”
“I’m trying to establish the decidability of the continuum hypothesis. How can we show, using noncontroversial axioms, whether it is true or false?”
“You proved it yourself. All mathematical truths are not subject to proof!”
“That’s an incorrect statement of my theorem. The problem isn’t there. If these axioms are ‘false,’ we have to invalidate other theorems that build on them.”
“Is it so very serious, Dr. Gödel?”
“You can’t build a cathedral on flimsy foundations. We must know, and we will know.”9
I erased the figures at our feet. Grains of sand lodged under my fingernails. I would bring bits of infinity back to the hotel with me.
“This continuum concept is just mud soup. Can you think of a simple image that would help me understand it?”
“If the world could be explained in images, we would have no need for mathematics.”
“Nor of mathematicians! Poor darling!”
“It will never happen.”
“How would you explain it to a child?”
The real question was: “How would you have explained it to our child?” Would Kurt have had the patience to describe his universe to a more innocent reflection of himself? An inexact reflection. Would he have agreed to reformulate what for a long time now he no longer bothered to articulate to himself?
“The sand on this beach, Adele, could represent a countable infinity. You could count all its grains one by one. Now look at the wave. Where does the sand start, where does the wave end? If you look closely, you’ll see a smaller wave, and then another even smaller. There’s no simple boundary between the sand and the sea foam. Maybe we would find a similar edge between the cardinality of 
and of 
. Between the infinity of natural numbers and the infinity of real numbers.”
“Why do you spend your nights thinking about it? Why does it make you forget to eat?”
“I’ve already explained. The question is a fundamental one. It’s almost metaphysical. Hilbert put it at the top of his program for mathematics.”
“That Mr. Hilbert thinks it’s important doesn’t tell me why it is!”
“My intuition tells me, Adele, that the continuum hypothesis is false. We are missing axioms to make a correct definition of infinity.”
“Why count the ocean with a teaspoon?”
“I need to prove that the system is consistent and unflawed. I need to know whether the infinity I am exploring is a reality or a decision. I want to push us forward into an ever more decipherable universe. I need to find out whether God created the whole numbers and man all the rest.”10
He tossed the pebbles he had used in his illustration into the water with the angry gesture of a little boy.
“This proof will tell me if an order, a divine model, exists. If I am devoting my life to understanding its language and not juggling alone in the desert. It will tell me whether all this means anything.”
Raising his voice, he made an army of seagulls take flight. I put my hands on his shoulders to calm him down. He pushed me away.
I picked up the coverlet, folded it in four, and waited for his instructions.
“Let’s go back to the hotel. I’m cold.”
We left in silence. A few yards from the hotel entrance, I tried to break the awkwardness.
“Is it because of being alone? If we were in Vienna …”
“Adele, everything I need is in Princeton.”
“Will we go home someday?”
“I don’t see the point.”
I’d asked the question whose answer I had been afraid to learn. Yet even today I still believe that he left part of himself in Vienna. He quit an environment made rich by the encounters and the atmosphere it afforded: those cafés where musicians, philosophers, and writers rubbed shoulders. In Princeton, he had access to the greatest living mathematicians, but he walled himself off. Inside his closed system he went around in circles. I, too, captured by his gravity, looked for a meaning in this endless dance. We returned to Princeton frustrated: I, by this shadowy half life; he, by his partial proof, which was not elegant enough by his standards to publish. At the hotel in Blue Hill he had said, “I’m having problems.” He implied an unspoken list, the list of his defeats. He took pains to protect himself from others but didn’t know how to insulate himself from the disappointment of his own limitations. In that summer of 1942, he disappointed himself; I disappointed myself; we disappointed each other. Two people, three possibilities — living with someone teaches you to count all of your frustrations.
27
Anna waited in the hallway while the nurse fussed over Adele. Bored, the young woman closed her eyes and tried to identify the owners of the footsteps she was hearing: the staccato heels of an administrator, the rubbery squeak of a health worker’s clogs, the swish of slippers.
Before entering the room, Anna tucked in her shirt. It had worked free of her tweed skirt, which now floated loose around her hips, as did most of her clothes. Mrs. Gödel was buried under the sheets and seemed despondent. The contrast with her exuberance of a week before was striking. Anna chose to see it as a sign of health. In her multicolored scarf and flowered pajamas, with her piercing gaze, Adele had something of a wild gypsy air. Where had her turban gone? Someone had finally sent it to the cleaner’s. Unless she had decided to let it molder in a drawer.
The young woman had to set down her bag and sit: her legs were trembling. Her concern over Mrs. Gödel had left her exhausted. She couldn’t even remember how she had gotten to Pine Run.
“You have lovely circles under your eyes, my dear. Boarding at this house of the dying is not doing you any good. I can see that you are growing thinner and thinner. It’s time to call the nurse to take your blood pressure.”
Anna leapt to her feet a little too quickly. She felt an onset of dizziness. A black veil came down over eyes. She heard a distant voice, then nothing.
“Just what we needed!”
She woke up in Mrs. Gödel’s bed, her feet raised and a cold compress on her forehead. Anna recognized the lavender scent of Adele’s cologne. The old woman, wrapped in her usual scruffy bathrobe, was sitting beside her. She patted Anna’s hand. “Are we getting the vapors now?” Anna tried to sit up, but Adele firmly held her down. Gladys appeared in the doorway with a squadron of octogenarians at her back. Adele swung her head menacingly toward them.
“No need to cluster like that! We need peace and quiet. Raus!”
They left sheepishly, but not before depositing an offering of sugary treats. Adele stuffed a cookie into Anna’s mouth.