2, , and Beyond Debra S. Carney Mathematics Department University of Denver April 11, 2008 Sonya Kovalevsky Day - CCA

Dodge Ball 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? ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ 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

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.

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.

The Natural Numbers N 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.

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.

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.

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, …} 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, …} 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, …}

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} 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? 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 ) The set of real numbers refers to all possible infinite decimal representations. 5/1 = …… 7/11 = ….. 3/2 = ….  = …. Sqrt(2)= …..

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).

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 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 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 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  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!   