2, , and Beyond

1
2, ?, and Beyond

• Debra S. Carney
• Mathematics Department
• University of Denver
• April 11, 2008
• Sonya Kovalevsky Day - CCA

2
Dodge Ball

• Lets play Dodge Ball!
• Mathematical Dodge Ball, that is.

3
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?
4
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.
5
Same Size

• Do these two collections of smiley faces have the
  same size?
• ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
• ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
• 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
2
6
1-1 Correspondence

• 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.

7
Same Size Oath

• 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.

8
The Natural Numbers N

• N1, 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.

9
Finite Sets

• We will call a set A finite if there is a 1-1
  correspondence between A and the set 1, 2, 3, .
  , n for some natural number n.
• Example ?, ?, ?, ? is finite due to the 1-1
  correspondence with 1, 2, 3, 4 as seen
  previously.

10
Infinite Sets

• If a set is not finite we will call it infinite.
• We could 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.

11
Examples with N1, 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,

12
Examples with N1, 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,

13
The IntegersZ, -3, -2, -1, 0, 1, 2, 3,

• 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.

14
The IntegersZ, -3, -2, -1, 0, 1, 2, 3,

• 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
• Z0, 1, -1, 2, -2, 3, -3,

15
Z0, 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!

16
The Rational NumbersQa/b a,b are in Z and
b?0

• 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.

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Qa/b a,b are in N and b?0
We have found a 1-1 correspondence between Q and N
18
Are all Infinities the same?

• 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

19
The Real Numbers (R)

• The set of real numbers refers to all possible
  infinite decimal representations.
• 5/1 5.000000000
• 7/11 0.63636363..
• 3/2 1.50000000.
• ?3.1415926535 .
• Sqrt(2)1.414213562..

20
Reals (R) versus Rationals (Q)

• 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).

21

• Cantors 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.

22
Cantors Revolutionary Idea

• 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)

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Cantors Revolutionary Idea

• 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!

24
Cantors Diagonal Argument.

• Here is a potential 1-1 correspondence between N
  and R. Play dodge ball to find M.

25
Cantors Diagonal Argument

• 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

26
There is no limit to ?
?
?
?

• 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!