Math Investigation eTwinning. Mission: Infinity...Affairs! Volume: Infinity | Page 11

Countability
If we want to go even further, stretching our imagination
and the meaning of infinity, we can’t use hotels. We have to
develop a new language to express our ideas. This language
is called Set Theory.
A set is simply a collection of any objects – fruit, buildings,
or numbers – often written inside {} brackets. We say that
two sets have the same size if we can pair up their elements
so that no element is left out or connected to two elements.
Such a pairing is called a bijection.
This applies to finite sets as well as infinite sets, and we say that a set is countable if it can be
paired up with the natural numbers = {1, 2, 3, 4, 5, …}. In that case, we can write down all
elements in a numbered list.
For example, we can add one more
number to , say 0, and see whether we
can pair up the two sets:

Since the black and the yellow set can be paired up without gaps or overlaps, they must have the
same size. Infinity plus one is still infinity. This is precisely the same principle as in Hilbert’s Hotel
above, where we paired up the infinitely many room numbers with the infinitely many guests.
We can do even better: let us try to pair up the positive integers = {1, 2, 3, 4, 5, …}
with all integers (these are written as = {…,–3 ,–2, –1, 0, 1, 2, 3, …}). Now we have to jump
back and forth between the positive and negative ends of the number line and we can be sure,
counting the integers ( ), not to forget any element.
Infinity plus infinity is still infinity, and it seems that there are as many positive integers as there are
integers in total.
We can even try to pair up the positive integers with the positive rational numbers . Rational
numbers are all fractions, such as 1/2, 7/40, 13/17 and so on. To do this, we first write down all
fractions in an infinite grid where the columns are the numerators and rows the denominator. Some
fractions appear several times: 1/2, for example, is the same as 2/4 or 3/6 and so. Therefore we
delete all the duplicates and only keep the smallest version:

l

LYKEIO SCHIMATARIOY, GREECE

- VALLAURI FOSSANO, ITALY

Σελίδα 11