2. Infinity and Induction
2.1. Countable Infinity
One of the more obvious features of the three number systems N, Z,
that were introduced in the previous chapter is that each contains infinitely
many elements. Before defining our next (and last) number system, R
, we want
to take a closer look at how one can handle 'infinity' in a mathematically precise way. We
would like to be able to answer questions like:
- Are there more even than odd numbers ?
- Are there more even numbers than integers ?
- Are there more rational numbers than negative integers ?
While most people would probably agree that there are just as many even than odd
numbers, it might be surprising that the answer to the last two questions is no as well. All
of the sets mentioned have the same number - albeit infinite - of elements. The person who
first established a rigorous 'theory of the infinite' was G. Cantor.
The basic idea when trying to count infinitely large (or otherwise difficult to count) sets
can roughly be described as follows:
- Suppose you are standing in an empty classroom, with a lot of students waiting to
get in. How could you know whether there are enough chairs for everyone? You can not
count the students, because they walk around too much. So, you simply let in the students,
one by one, and take a seat. If all the seats are taken, and no students are left standing,
then there was the same number of students as chairs.
This simple idea of matching two sets element by element is the basis for comparing two
sets of any size, finite or infinite. Since 'matching elements from one set with those in
another set ' seems related to the concept of a function, we have arrived at the following
|Definition 2.1.1: Cardinality|
- Let A and B be two sets. We say that A and B
have the same cardinality if there is a bijection f from A
to B. We write card(A) = card(B).
- If there exists a function f from A to B that is
injective (i.e. one-to-one) we say that card(A)
- If there exists a function f from A to B that is
surjective (i.e. onto) we say that card(A)
Please explain carefully what this definition has to do with the above idea of
counting students and chairs?
We can now answer questions similar to the ones posed at the beginning:
- Let E be the set of all even integers, O be the set of
odd integers. Then card(E) = card(O). What is the
- Let E be the set of even integers, Z be the set of all
integers. Again, card(E) = card(Z). Can you find the
- Let N be the set of natural numbers, Z be the set of all
integers. Which set, if any, has the bigger cardinality ?
|Definition 2.1.3: Countable and Uncountable|
If a set A has the same cardinality as N (the natural
numbers), then we say that A is countable. In other words, a
set is countable if there is a bijection from that set to N.
An alternate way to define countable is: if there is a way to
enumerate the elements of a set, then the set has the same cardinality
as N and is called countable.
A set that is infinite and not countable is called uncountable.
The second part of this definition is actually just rephrasing of what it means
to have a bijection from N
to a set A
- If a set A is countable, there is a bijection f from
N to A. Therefore, the elements f(1), f(2), f(3), ...
are all in A. But we can easily enumerate them, by putting them in
the following order: f(1) is the first element in A, f(2)
is the second element in A, f(3) is the third one, and so on...
- If a set A can be enumerated, then there is a first element, a
second element, a third element, and so on. Then the function that assigns to
each element of A its position in the enumeration process is a bijection
between A and N and thus A is countable by definition.
By the above examples, the set of even integers, odd integers, all positive
and negative integers are all countable.
Note that there is a difference between finite and countable, but we will
often use the word countable to actually mean countable or finite (even
though it is not proper). However, here is a nice result that distinguishes
the finite from the infinite sets:
Use Dedekind's Theorem to show that the set of integers Z
and the interval of real numbers between 0 and 2, [0, 2], are both infinite
(which is of course not surprising).
The surprising fact when dealing with countably infinite sets is that when
combining two countable sets one gets a new set that contains no more elements
than each of the previous sets. The next result will illustrate that.
|Proposition 2.1.6: Combining Countable Sets|
- Every subset of a countable set is again countable (or finite).
- The set of all ordered pairs of positive integers is countable.
- The countable union of countable sets is countable
- The finite cross product of countable sets is countable.
Think about these propositions carefully. It seems to be contrary to ones
beliefs. To see some rather striking examples for the above propositions,
consider the following:
The set of all rational numbers is countable.
- The collection of all polynomials with integer coefficients is countable.
To prove this, follow these steps:
Show that all polynomials of a fixed degree n (with integer coefficients) are
countable by using the above result on finite cross products.
Show that all polynomials (with integer coefficients) are countable by writing
that set as a countable union of countable sets.
A nice example that illustrates how difficult and counter-intuitive the
concept of infinity can be is provided by the story of Hilbert's Hotel.
This particular version of Hilbert's Hotel has been created by Jillian Gaglione,
a former Seton Hall student and avid Yankee (a New York baseball team) fan -
with apologies to any Red Sox (a Boston baseball team) fan.
|Example 2.1.8: Hilbert's Hotel|
Imagine two hotels in Boston, Massachusetts; the Holiday Inn, which has three
hundred rooms and Hilbertís Hotel, which contains a countable infinite number
of rooms. The New York Yankees are leaving town after defeating the Red Sox,
again! However, their tour bus breaks down and they are forced to get off the
bus and stay overnight in Boston. The Yankees, excluding their pitching coach,
decide to walk to the Holiday Inn only to find that all three hundred rooms
are already occupied. At the same time the pitching coach decides to walk over
to Hilbertís Hotel because he heard it was a nicer hotel. When the pitching
coach arrives at Hilbertís Hotel, he discovers that all of the rooms are taken
as well. However, the manager of the hotel, a Mr. Hilbert, is an extremely big
Yankee fan and says, after some thought, that he could accommodate one more person.
If all of the rooms are occupied, how is the manager of Hilbertís
Hotel going to find a room for the pitching coach?
After the pitching coach settles into his room, he calls his two best friends,
the right fielder and the shortstop, to tell them to come over to Hilbertís
Hotel for a room. When they arrive at Hilbertís Hotel the manager informs them
that all the rooms are still occupied, but, once again, he can find
accommodations for two more people.
If all of the rooms are still occupied, how can the manager of Hilbertís
Hotel find room for two more players?
The Yankees soon find out that Hilbertís Hotel was accommodating some
members of their team and the manager of the hotel finds himself trying to find
rooms for the rest of the members. There were fifty-eight people that need to
find a room and still none of the guests have checked out of their rooms. The
manager of the hotel says not to worry because he could find rooms for the
If every room is occupied, how can the manager find enough rooms for the
Throughout the night, rumors spread that the Yankees are staying in Hilbertís
Hotel. Many fans from the surrounding area travel to Hilbertís Hotel to book
a room. When Hilbert looks outside his hotel, he sees a never-ending line of
people that wish to stay for the night. But Hilbert wonders how he could
find rooms for a countable infinite number of guests ... then he looks outside
the hotel again and reassurs the people waiting in line that everyone will soon
have a room for the night.
How could the manager of the hotel accommodate a countable infinite
number of guests?