# 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.

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Ancient Mathematicians

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.

# Bibliography

## PRINT

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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.

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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, 5th 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).

### Credits

Produced by Oregon Public Broadcasting. 2008.
• Closed Captioning
• ISBN: 1-57680-886-6