"To infinity and beyond!" These were the inspired words of Buzz
Lightyear in the Disney movie Toy Story. Granted, one would not expect to find
much mathematical content in an animated film directed toward children, but
these words raise an interesting issue that mathematicians and the general
public struggled with for many years. Can one go beyond infinity? How can such a
concept be possible or even imaginable? These questions led to the development
of many new theories and even a new system of numbers.
A study of infinity must begin with an introduction to set theory. A
set is merely a collection of objects. Georg Cantor was the sole creator of set
theory; he published an article in 1874 that marks the beginning of set theory
and has come to change the course of mathematics. Cantor's theory was met with a
great deal of opposition due to its assertion of infinite numbers. The famous
mathematician Leopold Kronecker was especially opposed to Cantor's revolutionary
new way of looking at numbers. Kronecker believed only in constructive
mathematics, those objects that can be constructed from a finite set of natural
numbers. Despite this opposition from influential thinkers, set theory laid the
foundation for twentieth century mathematics. Although there were some flaws in
Cantor's theory, sets became an essential part of new mathematics and therefore
set theory was adapted to eliminate its original paradoxes [2].
Cantor's set theory incorporated infinity in the form of infinite cardinal numbers. Cardinal numbers are those which measure the number of objects in a set, as opposed to ordinal numbers, which are numbers with a fixed predecessor and successor. If two sets have equal cardinality, then they contain an equal number of objects. One way to determine this is through "one-to-one correspondence." Two sets are said to be in one-to-one correspondence if each object, or element, of the first set can be paired with exactly one element of the second set, and vice versa [1].
Cantor compared the cardinality of the set of all positive even
integers to the set of all positive integers and found them to be equal. Thus
the infinite cardinal number of these two sets is the same and is defined to be
aleph-naught. This is the first transfinite number. The set of rational numbers
also has a cardinality of aleph-naught, and thus is the same size as the set of
integers. The set of real numbers has a cardinal number that is greater than
aleph-naught, which Cantor proved through his famous "diagonalization" argument.
He called the set of real numbers aleph-1 and guessed that the cardinality of a
set cannot be anywhere in between aleph-naught and aleph-1. This conjecture is
called the "Continuum Hypothesis" and has been shown to be impossible to prove
under the constraints of set theory [1].
Cantor went on to prove that there is a hierarchy of infinities;
there are an infinite number of transfinite cardinal numbers. In other words,
the set of all transfinite numbers is described by a cardinal number that is
also transfinite [1]. This number is known as the Continuum
[9].
In my foundation seminar class at Bucknell University, Mysteries of
Mathematics, we learned about the Hotel Infinity, which was an idea of David
Hilbert's to present the complex hierarchy of transfinite numbers. Hotel
Infinity has an infinite number of rooms, and one night all the rooms are full.
The even-numbered rooms are in the east wing of the hotel, and the odd-numbered
rooms are in the west wing. This story unfolds as a weary couple knocks on the
door of the hotel office. They ask for a room for the night. The manager,
feeling sorry for the couple, does some quick thinking and comes up with a plan
to find the couple a room, despite the fact that the sign outside says "no
vacancy." The manager asks everyone in the west wing of the hotel to move into
the room next to them, one door farther down the hall. Now everyone has a room
and there is an empty room right by the lobby for the couple
[8].
A little while later a lost taxi driver pulls up and says he has an infinite number of Persian cats in his back seat. The cats are very pampered and ill-tempered, and each requires a separate room for the night. The manager does some more thinking, and then he asks all of the guests to move into the room that is double the number of the room in which they had been sleeping. That is, the inhabitants of room n will move into room 2n. In this way all of the original guests will have a room in the east wing, and there remains an infinite number of rooms in the west wing where the cats can sleep in peace [8].
The manager hardly has a chance to feel smug before an infinite
number of buses pulls up, each carrying an infinite number of tired salesmen,
and ask for rooms. The poor manager is now confronted with the problem of an
infinite number of infinities! The manager was almost stumped this time before
conjuring a final plan to keep everyone happy. He told the occupants of room n
to move to room 2^n. Then he went out to the first bus and assigned them the
rooms 3^n and to the second van the rooms 5^n and so on. (Euclid proved two
thousand years ago that there are indeed an infinite number of primes.) Each of
the infinite buses gets a prime number as a base and the nth rider in that bus
gets a room P^n. In this way all the original guests still have a room, and all
of the salesmen have a place to sleep as well. Blissful with success, the
manager walks through the halls and realizes that many of the rooms he passes
are now empty. Room 6 is empty… room 10 is empty… what is going on? All the
rooms with a number that is a product of two or more different primes are empty.
We have an infinite number of rooms that are empty from a hotel that was full a
few hours before [8].
It seems that Buzz Lightyear must have been correct. It seems that we have indeed gone beyond infinity with our analogy of the hotel. Without realizing it, the manager of the Hotel Infinity taught us an important lesson in the cardinality of infinite sets. Infinity plus one is infinity, and infinity plus infinity is still infinity. Even when we multiply infinity times infinity we get infinity. However, we have different levels of infinity, just as Cantor pointed out. This knowledge alone does not help us much; transfinite arithmetic is still a daunting task with no straightforward rules or properties [8].
Over half a century passes until a man named John H. Conway arrives
on the scene. John Horton Conway was born on December 26, 1937, in Liverpool,
England. By the time he was three or four he knew the powers of two and not long
after that he started dreaming about being a mathematician at Cambridge. He
would go on to single-handedly shaping the course of modern mathematics
[6].
Conway started his career by studying number theory at Cambridge. His professor challenged him with a very difficult number theory problem that had yet to be solved. Conway solved it and afterwards was allowed to study any subject he liked. Conway decided to study set theory and in particular transfinite numbers. He did not stay in that field long, however. In his late twenties he discovered a previously unknown group of numbers. A group is a set, such as the integers, that is associated with an operation for combining the elements of the set. Conway's discovery was of such great importance that he was admitted into the prestigious Fellows of the Royal Society of London. He thus joined the ranks of Isaac Newton, Albert Einstein, and many of the other greatest minds of the last four hundred years [6].
Conway said, "I only started doing real mathematics after I found the Conway group" [6]. At the same time he started studying game theory. He observed his colleague play Go, a Japanese board game. He then worked to find a mathematical understanding of the game. Upon analysis of Go, Conway discovered an entire set of new numbers. Later these numbers were dubbed "surreal numbers" in a book by Stanford computer scientist Donald Knuth. The discovery of surreal numbers in 1970 coincided with Conway's invention of the game of life, which is fundamentally easy to understand, having only three rules. Yet the mathematical strategies of the game are endless. Conway received inspiration for the game from his past study of Go [6].
To understand surreal numbers, we should take a look at the opening paragraph of Donald Knuth's book that gave these numbers their name.
This wordy description is hard to swallow on the first try.
Fortunately, we can get some help from Michael Conrady. He explains the quote as
"the construction of any number is defined by two sets, one called the left set
and the other called the right set. The sets must have this property: no member
of L is greater than or equal to any member of R" [3]. In this construction, it
is acceptable to use the empty set for either the right set or the left set. A
surreal number is the "simplest" number between the left set and the right set.
Zero is the simplest number, then 1 and -1, then 2 and -2, and so on
[3].
There are other, less technical ways of describing these haunting
numbers. Arithmetic with the transfinite ordinals is severely limited, but the
surreal numbers help this problem. Thinking in terms of infinite distances,
surreal numbers are chains of upward and downward movements. Each up or down is
an ordinal number in length. This means that just as each integer and rational
number is a real number, each real is an ordinal number and hence a surreal
number. However, there are many surreal numbers that are not real
[7].
The simplest transfinite ordinal is called omega. Omega is very
similar to Cantor's aleph-naught except that because omega is a transfinite
ordinal, its order is important, whereas aleph-naught is a transfinite cardinal,
so its order is not important. Both omega and aleph-naught represent the members
of the set of integers and equivalent sets. If we want a number that is
infinitely small, then we look to the surreal number called iota, which is an
infinitesimal distance above zero. Iota is the simplest number greater than zero
but less than all the positive real numbers; it is not an ordinal number
[7].
When using surreal numbers, we find that there are holes in all of our familiar number lines. The real number line and the ordinal number line both have unexpected holes in them. For example, there is a huge hole looming between the finite numbers and the positive infinites. This hole is what is meant when we say "infinity." There is also a hole between zero and the first positive number. We can find holes between every pair of numbers, and it is in these holes that the surreal numbers dwell [7].
Surreal numbers are more useful than ordinal numbers in that we can
conduct calculations with surreal numbers, but our usual rules of arithmetic do
not apply to the surreal numbers. Omega plus one is not the same expression as
one plus omega. Two times omega is equal to omega, but omega times two is not
equal to omega [6]. With this in mind, let us look at the types of surreal
numbers. Omega times any real number is a surreal integer. Even omega times the
square root of two is a surreal integer [7].
Surreal numbers work very well with normal algebraic calculations.
However, the calculus operation of integration does not work with the surreal
numbers because of the gaps in the surreal number line. For now the most useful
aspect of surreal numbers is to evaluate the rate at which a function approaches
infinity. Surreal numbers are perfect for this sort of analysis
[7].
Surreal numbers that cannot be "broken down" any further are called
travagances, of which there are two types, the extravagances and the
intravagances. An extravagance is a number that cannot be found as the result of
any number of calculations in finite arithmetic. The simplest extravagance is
omega. An intravagance is a number that is very deeply embedded between two
numbers, an "enormously medium number" [7]. The travagances are used to fill in
the holes of the real number line. To plug in a hole, one carefully selects the
appropriate travagance, called a ghost. The ghost stands in the gap of the
number line. This use of ghosts allows mathematicians study the behavior of
functions as they cross various gaps, the most compelling of which is infinity
[7].
There have been many varying philosophies toward infinity over the centuries. Cantor viewed infinity as possessing an almost divine quality, a representation of God that man could not understand [4]. A hundred years later, armed with an entire new system of numbers, we can see what happens to numbers as they cross the threshold of infinity. John Conway has built us a bridge; I hope that more determined mathematicians will cross that bridge into a world that is, well, surreal. There are ghosts in that world, but the ghosts are the most concrete things we will find. There is much power in describing our world in a new and fantastic way, if only someone with the courage and genius will lead the way.
[1] Ask Dr. Math: Questions and Answers from Our Archives.
http://forum.swarthmore.edu/dr.math/dr-math.html.
(12/4/99).
[2] The Beginnings of Set Theory. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html
(12/5/99).
[3] Conrady, Michael. Surreal Numbers. http://nadn.navy.mil/MathDept/wdj/surreal1/htm
(12/6/99).
[4] Dauben, Joseph Warren. Georg Cantor: His Mathematics and
Philosophy of the Infinite. Cambridge, Massachusetts: Harvard University
Press, 1979.
[5] Knuth, Donald E. Surreal Numbers. London:
Addison-Wesley, 1974.
[6] Seife, Charles. "Mathemagician: Presenting the
Legenadary John Horton Conway; Trickster, Group Theorist, Inventor of the Game
of Life." The Sciences. 34 (1994), 12-16. http://www.web4.infotrac.galegroup.com/
(12/2/99).
[7] Shulman, Polly. "Infinity Plus One and Other Surreal Numbers."
Discover. 16 (1995), 96-106. http://www.web4.infotrac.galegroup.com/
(12/1/99).
[8] Stacy, B. David. The Story of the Hotel Ad Infinitum.
http://www.scidiv.bcc.ctc.edu/Math/InfiniteHotel.html(12/1/99).
[9]
Welcome to the Hotel Infinity. http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html
(12/1/99).
Peer reviewed by:
Melissa Lenzi Luke Herbertson Laura Elliott Jim Leahy Jake Buchanon