“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.’ ”