The Goddess of Small Victories

by Yannick Grannec

  I asked for a breakfast tray to be sent up to our room. Mrs. Frederick grudgingly obliged. Folded in plain view was the headline: “Nazis Reach Canada! German Subs Found in Saint Lawrence!”

  When I went back upstairs, Kurt was still at his desk. He gulped down the coffee and pushed the toast away. I wandered around the room looking for something to do. I didn’t feel like knitting and even less like reading in the semidarkness. Kurt grew irritated at my restlessness. He took his glasses off to clean them. His eyes were red from lack of sleep.

  “Let’s go look at the ocean. You’re like a caged animal. I can’t concentrate when you’re like this.”

  In a moment I was at the door, basket in hand, but Kurt took his time and locked his papers away in our trunk. That damned woman was liable to read them as encrypted messages.

  We went downstairs without making any noise. From the radio in the pantry came snatches of the endlessly replayed patriotic song “We Must Be Vigilant.” Whenever we walked by, the landlady turned up the volume.

  Once back in Princeton, we would notice that our trunk key had disappeared. Kurt immediately wrote Mrs. Frederick to accuse her of stealing it. What a charming impression we must have left behind!

  We followed Parker Point Road, a narrow lane along the shore. Through a screen of pines, we could see magnificent Blue Hill Bay, dotted with islands. Then we took a path toward an attractive cove that we had found on an earlier walk. I spread a quilted coverlet over the rocks. Kurt valued his comfort.

  “It’s too damp to stop here.”

  “We’re at the seashore, Kurt! In town you’re always complaining of how stuffy it is.”

  He sat down unwillingly.

  “We could have lunch outdoors today. I’d like to try clam soup.”

  Three boats were moored close to shore and their halyards slapped gently against the masts to the rhythm of the waves. Gulls chased each other, skimming over the foam. In the distance, I could see a lumbering shape haul itself onto a rocky ledge. The sun warmed my shoulders. I breathed the air deeply, dazzled by the quiet splendor. The war was so far away.

  “I could look at this forever.”

  “You don’t know how to swim, Adele. You should learn.”

  Although the day was warm, he had bundled himself up in his overcoat.

  “Do you see that amazing blue where the sea and the sky meet?”

  The brim of his hat rose barely a fraction.

  “You’re not even looking! What do you think about when you’re in front of the ocean?”

  “I see a field of wave interactions whose complexity fascinates me.”

  “How sad! You should bathe your mind in all this beauty.”

  “Mathematics has true beauty.”6

  His matter-of-factness cut against the spirit of the moment.

  “What’s bothering you now? You don’t talk to me about your work anymore.”

  If sarcasm had had any effect on him, I would have added, “You haven’t talked to me about anything for a long time.” I took his hand; it was cold and tense.

  “I’m wondering about the existence of infinity.”

  He let go of my hand and stood looking out at the sea. A small wave licked the toe of his shoe. He backed away, scowling.

  “When you look at the ocean, you might have a sensation of infinity. Nonetheless you can’t measure this infinity or, rather, you can’t understand it.”

  “You might as well try to empty the sea with a teaspoon!”

  “We have made teaspoons, as you put it, to define infinity, but how can we know that these mathematical tools are not just an intellectual construct?”

  “Infinity surely existed before man invented mathematics!”

  “Do we invent mathematics, or do we discover it?”

  “Does a thing exist only if we have the words to talk about it?”

  “That’s a vast question for your small brain.”

  I drew an ∞ over my heart.

  “The infinities that occupy me at the moment relate to set theory. It’s very different.”

  “What a cockamamy idea! Infinity is infinity, there’s nothing bigger.”

  “Some infinities are greater than others.”

  He carefully lined up three pebbles he had picked up from the beach.

  “Here is a set. A pile, if you like. It makes no difference if they’re pebbles or pieces of candy, think of them as elements.”

  I stood up to show that I was paying faithful attention. He rarely made an attempt to teach me anything.

  “I can count them: one, two, three. So I have a set with three elements. I can then decide to make subpiles: the white with the gray, the white with the black, the black with the gray; then the white alone, the gray alone, the black alone; all three together; and none. I have eight possibilities, eight subsets. The set of the parts of a set always has more elements than the set itself.”

  “So far, so good.”

  “If you lived for several centuries, you could count all the pebbles on this beach. And in theory if you lived forever, you could spend the time counting, but … there is always a larger number.”

  “There is always a larger number.”

  I turned these words over in my mouth; they had a peculiar taste.

  “Even if you could count to infinity, there would still always be a bigger infinity around the corner. The set of the parts of an infinite set is greater than the infinite set itself. Just as the possible permutations of these three pebbles is greater than three.”

  “That’s a funny little game of construction!”

  “For you to understand the next part, you have to be clear on the difference between cardinal and ordinal numbers. Cardinal numbers allow you to count the elements in a set: you’ve got three pebbles. Ordinals put the elements in order: here’s the first pebble, the second, and the third. The cardinal number counts the elements up to infinity without assigning an order to the elements. The symbol used for the cardinality of an infinite set is the Hebrew letter aleph.

  He drew an esoteric sign in the sand, then wiped his finger with his handkerchief: . I handed him a stick of driftwood, which he accepted with the ghost of a smile in place of thanks.

  “The three pebbles stand for natural numbers, the numbers we all use to count normal objects: 1, 2, 3, etc. This set is called .”

  He drew an and made a big circle around it, putting the three pebbles inside.

  “Why? Are there others?”

  I liked hearing him laugh. It happened so rarely.

  “We have, among others, the integers: the set . The integers are defined with respect to zero. We add the minus sign to a whole number to show that it is below zero: -1 is less than zero; 1 is more than zero. Do you remember in the train, people were talking about a temperature of -50 degrees Celsius? To be more accurate, they should have said 50 degrees below what the Celsius scale determines as zero degrees of temperature.”

  He drew a larger circle around the first, then a third around the other two. He labeled each with a large, elegant capital letter: , then .

  “ is the set of rational numbers, which includes all the whole-number fractions like 1/3 and 4/5.”

  “N, Z, Q … My poor brain!”

  “Common sense will tell you that the set of all natural numbers is smaller than the set of all integers . The set 1, 2, 3 is smaller than the set 1, 2, 3, -1, -2, -3. In the same way, the set of all integers is smaller than the set of all rational numbers . The set 1, 2, 3, -1, -2, -3 is smaller than the set 1, 2, 3, -1, -2, -3, 1/2, 1/3, 2/3, -1/2, -1/3, -2/3, etc. All these sets are embedded one within the other. The natural numbers, you could say, form the smallest pile, and the rational numbers the largest.”

  “Like cooking pots! So they have different infinities?”

  “Wrong! They have the same cardinality. I’ll spare you the proof. Georg Cantor proved it with the help of a bijective function in the first case and using a diagonal argument in the second.”

  “It’s all Hebrew
, this cardinality business.”

  A curious gull landed on a nearby rock. It looked at me with the outraged expression that birds typically wear when someone comes too close to them.

  “You’re not listening to me, Adele!”

  “Of course I am! In the end, all infinities are equal? So it comes back to there being just one.”

  “No. Because there are others still. For example, , the set of real numbers. The real numbers include the rational numbers, all the fractions, and the irrational numbers like pi. They’re called “irrational” because they can’t be expressed as fractions of integers. The cardinality of , which is the infinity of rational and irrational numbers, is, in point of fact, bigger. Cantor proved that as well.”

  He drew an enormous circle with a dotted outline around the three others. The seagull nodded its approval before taking flight.

  “The infinity corresponding to whole numbers 1, 2, 3, etc., is called ‘aleph-naught,’ and though the terminology is inaccurate, we say that it’s a ‘countable infinity.’ ”

  “A countable infinity, isn’t that a little presumptuous?”

  “To insist on making jokes when I am trying to explain a difficult subject, that is presumptuous, Adele.”

  I struck my breast in contrition.

  “If you’ve followed from the beginning, you know that the set of the parts of aleph-naught is bigger than aleph-naught itself. You can make more different piles than you have pebbles. According to Cantor,7 this set of parts can be put into a direct bijection8 with the set of real numbers. They can be bijected—coupled, if you like—one to one, the way you might pair up dancers in a ballroom. But that’s as far as I can take the metaphor.”

  The sand in the cove was starting to be covered with esoteric symbols. I glanced around. A suspicious passerby might take us for spies.

  “To sum it up, Adele, there is no infinity … there appears to be no infinity intermediate between the infinity of natural numbers and the infinity of real numbers. If a demarcation exists, it would be between and : the smallest pile of pebbles and the one that includes them all but that can’t be represented by pebbles because it is uncountable. We ignore the intermediate sets and , as I said, since their infinity is indistinguishable from the infinity of . We go from the countable, or ‘discrete,’ to the ‘continuous’ in a single leap. That’s called the ‘continuum hypothesis.’ ”

  “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.

  “Force yourself to swallow a real meal from time to time. Not that garbage from the vending machine! If I were still living at home, I would have made you schnitzels.”