## Mathematics Illuminated

# How Big Is Infinity?

## It takes courage to push beyond the boundaries of understanding, to both explore and explain the boundlessness of the infinite. Numbers and counting are real — intrinsic to our everyday life. But acknowledging their existence ties us to the existence of the infinitude.

Throughout the ages, the notion of infinity has been a source of mystery and paradox, a philosophical question to ponder. As a mathematical concept, infinity is at the heart of calculus, the notion of irrational numbers — and even measurement. This unit explores how mathematics attempts to understand infinity, including the creative and intriguing work of Georg Cantor, who initiated the study of infinity as a number, and the role of infinity in standardized measurement.

### Unit Goals

- Ideas of infinity come to light when considering number and geometry, the worlds of the discrete and the continuous.
- Incommensurability is the idea that there is no measurement unit that fits into some two quantities a whole number of times.
- Incommensurability led to the discovery of irrational numbers.
- Irrational numbers have decimal expansions that never end and never repeat.
- Two sets are the same size if their elements can be put into one-to-one correspondence with one another.
- The size of a set is its cardinality.
- There is more than one type of infinity.
- The sets of rational and real numbers are examples of two different sizes of infinity.
- To properly describe the different sizes of infinity, a new definition of number is required.
- Given a set of any size, one can create a larger set by taking the subsets of the original set.

### Unit Glossary

### Additional Unit Resources: Bibliography

# Bibliography

## WEBSITES

http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html

http://personal.bgsu.edu/~carother/pi/Pi3a.html

Aristotle. (Edited by: Richard McKeon, Introduction by C.D. Reeve) *The Basic Works of Aristotle*. New York: Modern Library, 2001.

Benjamin, Arthur T and Jennifer J. Quinn. *Proofs that Really Count: The Art of Combinatorial Proof *(Dolciani Mathematical Expositions). Washington, D.C.: Mathematical Association of America, 2003.

Berlinghoff, William P. and Fernando Q. Gouvea. *Math Through the Ages : A Gentle History for Teachers and Others*. Farmington, ME: Oxton House Publishers, 2002.

Berlinghoff, William P. and Kerry E. Grant. *A Mathematics Sampler: Topics for the Liberal Arts*, 3^{rd} ed. New York: Ardsley House Publishers, Inc., 1992.

Boyer, Carl B. (revised by Uta C. Merzbach). *A History of Mathematics*, 2^{nd} ed. New York: John Wiley and Sons, 1991.

Burton, David M. *History of Mathematics: An Introduction*, 4^{th} ed. USA : WCB/ McGraw-Hill, 1999.

Conway, John H. and Richard K. Guy. *The Book of Numbers*. New York: Copernicus/ Springer-Verlag, 1996.

Du Sautoy, Marcus. *The Music of the Primes: Searching To Solve the Greatest Mystery in Mathematics.* New York: Harper Collins, 2003.

Gazale, Midhat. *Number: From Ahmes to Cantor*. Princeton, NJ: Princeton University Press, 2000.

Gross, Benedict and Joe Harris. *The Magic of Numbers*. Upper Saddle River, NJ: Pearson Education, Inc./ Prentice Hall, 2004.

Henle, J.M. “Non-nonstandard analysis: Real infinitesimals,” *Mathematical Intelligencer,* vol. 21 Issue 1 (Winter 1999).

Joseph, George Gheverghese.* Crest of the Peacock: The Non-European Roots of Mathematics*. Princeton, NJ: Princeton University Press, 2000.

Mueckenheim, W. “On Cantor’s Important Proofs.” Cornell University Library.

http://arxiv.org/abs/math/0306200 (accessed 2007).

Mueckenheim, W. “The Meaning of Infinity.” Cornell University Library.

http://arxiv.org/abs/math/0403238 (accessed 2007).

Newman, James R. *Volume 1 of The World of Mathematics: A Small Library of the Literature of Mathematics from A’h-mose the Scribe to Albert Einstein. *New York: Simon and Schuster, 1956.

Poonen, Bjorn. “Infinity: Cardinal Numbers.” Berkeley Math Circle, UC Berkeley. http://mathcircle.berkeley.edu/bmcarchivepages/handouts/1998_1999.html (accessed 2007).

Schechter, Eric. “Potential Versus Completed Infinity: Its History and Controversy.” Department of Mathematics, Vanderbilt University. http://www.math.vanderbilt.edu/~schectex/ http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html (accessed 2007).

Schumacher, Carol. *Chapter Zero: Fundamental Notions of Abstract Mathematics.* Reading, MA: Addison-Wesley Higher Mathematics, 1996.

Stewart, Ian. *From Here to Infinity: A Guide to Today’s Mathematics*. New York: Oxford University Press, 1996.

Tannenbaum, Peter. *Excursions in Modern Mathematics*, 5^{th} ed. Upper Saddle River, NJ: Pearson Education, Inc., 2004.

Tanton, James. “Arithmetic, Algebra and Abstraction,” Text in preparation, to appear 2009.

Weisstein, Eric W. “Newton’s Iteration.”* Wolfram Research* http://mathworld. wolfram.com/NewtonsIteration.html (accessed 2007).

Weisstein, Eric W. “Pythagoras’s Constant.” *Wolfram Research*. http:// mathworld.wolfram.com/PythagorassConstant.html (accessed 2007).

White, Michael. “Incommensurables and Incomparables: On the Conceptual Status and the Philosophical Use of Hyperreal Numbers,” *Notre Dame Journal of Formal Logic*, vol. 40, no. 3 (Summer 1999).

Zeno, of Elea. [translated by H.D.P. Lee] *Zeno of Elea*. A Text, with translation from the Greek and notes. Amsterdam: A. M. Hakkert, 1967.

## LECTURES

Allen, G. Donald. “Lectures on the History of Mathematics: The History of Infinity.” Department of Mathematics, Texas A&M University. http://www.math.tamu.edu/~dallen/masters/index.htm http://www.math.tamu.edu/~don.allen/history/m629_97a.html (accessed 2007).