The Goddess of Small Victories (22 page)

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 bijection
⁸
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.”