The Goddess of Small Victories (21 page)

“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.”