The success of the movie Toy Story 3, along with the character Buzz Lightyear's "To infinity and beyond" statement, reminded me of a book by Eli Maor that has sat on my shelf now for several years. Published by Princeton University Press, the book "To Infinity and Beyond: A Cultural History of the Infinite," delves into a number of complex mathematical theories that I have yet to fully comprehend, but it's a book I've cherished for the depth of the subject matter alone and the fact that I can pick it up after just about any length of time and learn something from it. See: Princeton University Press. (1)
Just the phrase, "To infinity and beyond" is a remarkable one in itself, if you think about it. How does one go "beyond" infinity? And where does one end up if he or she does accomplish the feat? The writers of Toy Story deserve credit for attaching such a crafty phrase to one of their star characters.
The preface of Maor’s book begins with an interesting story which he attributes to the mathematician David Hilbert. The story involves a man who walks into a hotel one night looking for a room. He is at first told that they don’t have any available rooms, however, the owner thinks about it and tells him that maybe they have one after all. He then reluctantly wakes up each of his guests and asks them, one by one, to move one room over. The guest in room one would move to room two. The guest in room two would move to room three. And so on.
Amazingly, the man is then shown to room number one, which of course is now vacant. In fact, every guest has a room because, as the story goes, the man had unknowingly checked into Hilbert's Hotel -- the "one hotel in town known to have an infinite number of rooms!" Maor writes.
An idea of just how strange things can become when dealing with concepts like infinity is seen in Chapter 10, entitled simply "Beyond Infinity." Using studies first published by George Cantor at the University of Halle in Germany in 1874, Maor discusses the concept of denumeration, and certain sets which contain elements so dense that it is impossible to count every element, one being the "set of points along an infinite line, the number line."
Such points -- which reflect the real number system and all their corresponding decimal forms -- form what Cantor called the infinity of the continuum. "They are not denumerable; they contain more elements -- vastly more- - than a denumerable set," Maor writes. (2)
Then comes the remarkable opening statement of Chapter 10:
"To show that the real numbers cannot be counted, Cantor first established a fact which, if anything, seems to be almost beyond belief: There are as many points along an infinite straight line as there are on a finite segment of it." Think about that for minute.
The rest of the chapter is used to expound on Cantor's Continuum Hypothesis, which Maor says remained unsettled for 60 years, until 1963. "The hypothesis turned out to be both true and false -- depending on what assumptions one starts from," he writes. In other words, the hypothesis sets apart from the standard axioms of set theory and can be rejected or accepted accordingly.
(1) Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540. Copyright 1987 by Birkhauser Boston, 765 Massachusetts Ave., Cambridge, Mass. 02139.
(2) "To Infinity and Beyond," page 60.