Learning Session Four: Counting Infinity
Georg Cantor (1845-1918) caused uproar amongst mathematicians when he developed a theory that would
give a size to infinity. He started by focusing on collection of numbers which he called ‘sets’. The number
of members of a set is called it ‘cardinality’. So the set, 1,2,3 has cardinality 3 and the set,
24,2,4,29,13,15 has cardinality 6.
1. (i) What is the cardinality of the set of prime numbers less 50?
(ii) What is the cardinality of the set of odd numbers less than 100?
(iii) Find the cardinality of the set of multiples of 7 less than 100.
Cantor then invented a name for the cardinality of the set of natural numbers, 1,2,3,4,5,... . He called the
cardinality of this infinite set ‘aleph-null’ and gave it the symbol
.
If an infinite set can be matched one-to-one with the natural numbers then it also has cardinality alephnull. For example, the set of natural numbers has this one-to-one match with its squares so the cardinality
of the squared natural numbers is
.
1 2 3 4
5
6
7 ... n ...
1 4 9 16 25 36 49 ... n² ...
2. (i) List another set of numbers which has cardinality
(ii) Does the set of prime numbers have cardinality
.
?
David Hilbert (1862-1943), an influential mathematician of the time, defended Cantor against his critics. To
explain Cantor’s theory he told his students the story of a hotel with an infinite number of rooms. This
hotel has come to be known as the Hilbert Hotel. All the rooms at the Hilbert Hotel are occupied when a
lone traveller arrives seeking accommodation.
The manager says “No worries!” and frees up a room by asking all the other guests to move to the room
numbered one up from them; the guest in Room 1 moves to Room 2, Room 2 guest moves to Room 3 and
so on. This is like saying one plus infinity is infinity or 1 +
=
.
3. (i) If Thomas’s new room number is 372 what was his room number before he moved?
(ii) What is the new room number for Bella who was in Room 1,009,999?
Next minute a bus full of passengers arrives; a special bus carrying an infinite number of people. The
Manager welcomes them into Hilbert Hotel. To accommodate them she asks the guests to move to a room
that is double the number of their room. The guest in Room 7 moves to Room 14, the guest in Room 14
moves to Room 28 and so on. This frees up all the odd numbered rooms for the infinite number of
passengers in the bus. This is like saying infinity plus infinity is infinity or
+
=
(iii) Harry has moved from Room 1999. What is his new room number?
(iv) Emily Button is now in Room 992. What was her old room number?
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Next thing more buses arrive. Many more buses. An infinite number of buses arrive and each bus carries
an infinite number of passengers and all of them want rooms at the Hilbert Hotel. Unperturbed, the
manager beckons them into the hotel. She repeats her earlier strategy of moving guests to the room
numbered twice their room and freeing up all the odd numbered rooms. Now she must count the bus
passengers. This is how she does it.
Each passenger is identified by their bus number and the number of their seat on the bus. Like this:
Seat 1 Seat 2 Seat 3
Seat 4
Bus 1
Bus 2
Bus 3
Bus 4
The manager counts the passengers by using the zigzag fashion shown by the arrows in the diagram.
(v) Amiria’s code is 12/35. What are her bus and seat numbers?
(vi) Mathew is assigned Room 21. What are his bus and seat numbers?
(vii) Why doesn’t the Manager count along the rows?
So the infinite number of passengers on the infinite number of buses have all be accommodated in the
Hilbert Hotel. This is the same as saying infinity X infinity = infinity or
X
=
Write a sentence about infinity and compare it to your description at the start of the worksheet.
For a good demonstration of the Hilbert Hotel story watch the video.
For an entertaining and informative version of the Hilbert Hotel read Pages 399-404 in Alex’s Adventures in
Numberland by Alex Bellos.
This book is also a good place to learn about an even bigger infinity than aleph-null. Read about the rabble
that turns up to the Hilbert Hotel, each person wearing a t-shirt with an infinite decimal number on it. Will
they all fit into the Hilbert Hotel?
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