Download 2, Infinity, and Beyond

2, , and Beyond
Debra S. Carney
Mathematics Department
University of Denver
April 11, 2008
Sonya Kovalevsky Day - CCA
Dodge Ball
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
Lets play Dodge Ball!
Mathematical Dodge Ball, that is.
Rules of the Game
Is there a winning
strategy for
Player 1? That is,
can Player 1
always win the
game if she plays
by her strategy?
How about Player
2?
Winning Strategy?
Player 2 can
always win
Dodge Ball!
Today, we will see
how the winning
strategy for
Dodge Ball is
related to sizes of
infinity.
Same Size

Do these two collections of smiley faces have the
same size?
1
☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺
2
☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺☺

Notice that we can (quickly) say “yes” without having
to count the number of smiley faces in each row.
This exactly the idea of 1-1 correspondence.
1-1 Correspondence
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Two collections that can be paired evenly
(with no leftovers) are said to be in 1-1
correspondence.
{♠, ♣, ♥, ♦}
{a, b, c, d}
There may be many ways to make the
pairing, however we only need to find one.
Same Size Oath
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We need to agree on the following.
Two sets have the same size if there exists a
1-1 correspondence between both sets.
Remember, we are not “counting”, we are
making a pairing. If we can make the pairing
then two sets must have the same size.
The Natural Numbers N
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N={1, 2, 3, 4, 5, …….} is called the set of
natural (or counting) numbers.
This is an infinite set. We will ask do all
infinite sets have the same size?
We will compare the size of some infinite sets
to the natural numbers and observe some
interesting behavior along the way.
Examples with N={1, 2, 3, 4, 5, …}


Does {2, 3, 4, 5, 6, ……} have the same
size as N?
Yes! We can find a 1-1 correspondence.
{1, 2, 3, 4, 5, ……}
{2, 3, 4, 5, 6, ……}
Examples with N={1, 2, 3, 4, 5, …}


Does {2, 4, 6, 8, ……} have the same
size as N?
Yes! We can find a 1-1 correspondence.
{1, 2, 3, 4, 5, ……}
{2, 4, 6, 8, 10, ……}
The Integers
Z={…, -3, -2, -1, 0, 1, 2, 3, …}
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Do the Integers have the same size as N?
Is this a good 1-1 correspondence?
{ 1, 2, 3, 4, 5, ……}
{…, -3, -2, -1, 0, 1, ……}

No: There are (infinitely) many integers without
a partner.
The Integers
Z={…, -3, -2, -1, 0, 1, 2, 3, …}
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Does that mean the integers are not the
same size as N?
Not necessarily. Perhaps we did not find the
correspondence yet and in fact that is
happening here.
Consider this “rearrangement” of the
integers:
Z={0, 1, -1, 2, -2, 3, -3, …}
Z={0, 1, -1, 2, -2, 3, -3, …}

Can we know find a 1-1 correspondence
between N and Z? Yes!
{1, 2, 3, 4, 5, ……}
{0, 1, -1 ,2, -2, ……}

Thus N and Z have the same size!
The Rational Numbers
Q={a/b : a,b are in Z and b≠0}
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The rational numbers are the infinite set of
fractional numbers.
Examples: 0/3, 99/7, -5/3, 15/-2, 5/5, …
Do the rationals have the same size as N?
(Surprisingly?) Yes!
To find the correspondence we need to
“list” the rational numbers in the right way.
Q={a/b : a,b are in N and b≠0}
We have found a
1-1
correspondence
between Q and N
Are all Infinities the same?
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So far it seems as if all infinite
sets are the same size as the
natural numbers.
In 1891, Georg Cantor proved
the contrary. He showed that
the real numbers have larger
size than the natural numbers
The Real Numbers (R)
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The set of real numbers refers to all
possible infinite decimal
representations.
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5/1 = 5.000000000……
7/11 = 0.63636363…..
3/2 = 1.50000000….
=3.1415926535 ….
Sqrt(2)=1.414213562…..
Reals (R) versus Rationals
(Q)
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The rational numbers correspond exactly to
the decimals that repeat.
For example: 7/11 and 3/2 are rational
numbers (and real numbers as well).
There are decimal expansions that do not
repeat. Those numbers are real (but not
rational) numbers.
For example:  and Sqrt(2).
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Cantor’s Theorem (1891): The size
of the real numbers R is larger than
that of the natural numbers N.
At the time of its publication the idea
was quite shocking to most
mathematicians of the day.
Cantor’s Revolutionary Idea
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Assume that we do have a 1-1
correspondence between R and N.
Then find a real number M that cannot
appear on the list of real numbers. (M
for “missing”)
Cantor’s Revolutionary Idea
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Since M cannot be on the list of real
numbers, we cannot have a true 1-1
correspondence. (No leftovers!)
Since this will work for any potential 1-1
pairing, then no such pairing can exist!
Cantor’s Diagonal Argument.

Here is a potential 1-1 correspondence
between N and R. Play “dodge ball” to find
M.
Cantor’s Diagonal Argument
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M cannot be on the list of real numbers
and thus we did not have a 1-1
correspondence.
This is true of any potential 1-1
correspondence.
Thus R is “bigger” than N



There is no limit to 
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There are sets that are “bigger” than R.
Cantor in fact showed how given any
infinite set to create a new set of a
larger size of infinity.
Conclusion: There are infinitely many
sizes of infinity!