03/27/2012 05:32 pm ET Updated Dec 06, 2017

To Hold Infinity and Beyond

In Auguries of Innocence, William Blake beckons the reader to "...hold infinity in the palm of your hand." If we can't see the infinite, how might we touch and explore it? Let's begin by watching the video below, which places this concept into a physical context.

In Toy Story, Buzz Lightyear exclaims, "To infinity and beyond." What could possibly be beyond infinity? In the video above, we see that an infinite amount (such as a rope going off forever in only one direction) can be added to an infinite amount and result in an infinite amount (a rope that goes on forever in both directions). Easy enough, infinity + infinity = infinity.

Infinity measures the size of a set like the set of natural numbers N = {1, 2, 3, 4, ...}. Two sets have the same size if you can describe an exact pairing of their elements. In the picture below, our set of jellybeans has the same size as the set {1, 2, 3} since the arrows describe such an exact pairing.


Now, let's visit the Mega-Motel which has infinitely many rooms and is booked solid. A trucker arrives at the front desk and the attendant states, "You'll have room 1." Over the intercom, everyone is asked to move down one room. Does everyone fit? If not, who doesn't? In fact, everyone knows where to go. As such, the sets {0, 1, 2, 3, ...} and {1, 2, 3, ...} have the same size. Notice, the motel can still house 100, 1,000 or 1,000,000,000 travelers. Is the only thing beyond infinity the infinite itself?

Infinity often refers to sets containing numbers of increasing size. In the earlier video, the mime cuts the rope in half, in half again and finally in a manner meant to suggest cutting the rope infinitely many times. There are an infinite amount of real numbers between 0 and 1. Is this set the same size as {1, 2, 3, ...}?

Consider what it means for two sets not to have the same size. Let J be the set of red, orange, green, and yellow jellybeans and T = {1, 2, 3}. These two sets are not the same size since any pairing of the sets will always leave an element of J unpaired with an element in T.


To delve deeper into this concept, let's play Dodge Ball, math style, with two players as described in Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas by E. Burger and M. Starbird. Player 1 places six O's and X's in the first row of Table 1. Then, Player 2 (the Dodger) puts an X or an O in the first column of Table 2. Player 1 then places six X's and O's in the second row of Table 1. Player 2 chooses an X or an O for the second column of Table 2. After Table 1 and Table 2 are filled, Player 2 wins if the row in Table 2 does not match any row in Table 1. Otherwise, Player 1 wins. Play the game a few times. Notice any optimal strategies?

Table 1


Table 2


If I were the Dodger, I would make the choices for the game seen below.

Table 1


Table 2


For my first column, I look at the first element in row 1 of Table 1 and choose the opposite symbol for my play. For play 2, I look at the second column of the second row of Table 1 and again choose the opposite symbol. If I continue this strategy, Player 1 cannot win -- ever.

In the late 1800's, Georg Cantor analyzed the infinite with this strategy. Suppose the set of real numbers between 0 and 1 are the same size as the set of natural numbers. Then, we can define an exact pairing between all the elements of the sets. Assume this exists. Then, we can't produce a real number between 0 and 1 that isn't paired with an integer in N. However, the Dodger knows how to produce such a number!

Suppose our pairing is

  • 1 ↔ 0.765242....
  • 2 ↔ 0.632314....
  • 3 ↔ 0.134210....
  • 4 ↔ 0.024525....
  • 5 ↔ 0.302931....

The Dodger will choose the number 0.30330.... Why? Dodger chose the first decimal digit, 3, by looking at the first decimal digit of the number paired with 1 and seeing it is not a 3. The second decimal digit of the number paired with 2 is a 3 so Dodger chose a 0. The third decimal digit of the number paired with 3 is not a 3 so Dodger chose a 3. This pattern would continue for all the digits in the list producing a number that cannot be contained in the pairing.

This is Dodge Ball on an infinite board. Regardless of a pairing between the set of natural numbers and the set of real numbers between 0 and 1, Dodger can create a decimal number not in the pairing. While both sets are infinite, one is more than the other!

Buzz Lightyear was right. We can go to infinity and beyond -- to another sized infinity. What lies beyond that size? There is, indeed, another sized infinity. How? Look it up in a book or on the Internet. Then, work to create a model of the concept that enables you to hold the idea, if only to explore a question.

Mathematics allows us to study abstract ideas in ways that might initially defy intuition. Does more than one size of infinity do that for you? It might. It did for colleagues of Cantor. In fact, Henri Poincaré, a leading mathematician of the day, called Cantor's ideas a "disease." Cantor's work was embraced in time. Accomplished mathematician David Hilbert, who originated the story of an infinite hotel, stated, "No one will drive us from the paradise which Cantor created for us."

Mathematics continually pushes the boundaries of its knowledge... in a way, to infinity and beyond.