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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.

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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.

Additional Unit Resources: Bibliography




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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, 3rd ed. New York: Ardsley House Publishers, Inc., 1992.

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

Burton, David M. History of Mathematics: An Introduction, 4th 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. (accessed 2007).

Mueckenheim, W. “The Meaning of Infinity.” Cornell University Library. (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. (accessed 2007).

Schechter, Eric. “Potential Versus Completed Infinity: Its History and Controversy.” Department of Mathematics, Vanderbilt University. (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. (accessed 2007).

Weisstein, Eric W. “Pythagoras’s Constant.” Wolfram Research. http:// (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.


Allen, G. Donald. “Lectures on the History of Mathematics: The History of Infinity.” Department of Mathematics, Texas A&M University. (accessed 2007).