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C a n t o r S e t By Merlin Dargham Introduction Heroes in the history of mankind are many and their stories are still told from one to another. We remember them for their courageous and extraordinary deeds. To say that there is a hero in mathematics may seem strange, however I am humbled by the sacrifices Georg Cantor endured. No one shall expel us from the paradise Cantor has created for us! (Hilbert). The Cantor set was so startling, unbelievable and fantastic to the normal mind that it was attacked at first by mathematicians in every quarter. Today Cantor’s results are basically accepted, and they have led to a radical overhaul of the foundation of traditional logic and of classical mathematics. This paper will investigate what the Cantor set is, why it is so significant, the paradoxes mathematicians faced during Cantor’s time, pedagogical obstacles that students may face while learning it, and what it would mean for students to understand the topic. In this paper I will try my best to express the beauty that lies in the revolution of such mathematical thinking. No man would have landed on the Moon had it not been for irrational numbers. These are the result of modern math, structured on the shoulders of mathematicians like Cantor. The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing the middle third from the unit interval [0, 1], leaving [0, 1/3]  [2/3, 1]. Next, the middle thirds of all of the remaining intervals are removed. This process is continued ad infinitum. The Cantor set consists of all points in the interval [0, 1] that are not removed at any step during this process. This technique forms “holes” within the “wholes”. The holes are the open intervals that are taken out from each remaining whole. (Gordon, 2004) Paradoxes: (Has a zero measure yet it is not empty) The total length removed forms a geometric series which is: However, we started with a finite unit length of interval 1 and the intervals removed also add up to 1. The total length removed is equal to the length of the original interval. Hence mathematically speaking there is nothing left in the cantor set for which we can say there is a measure of zero (1-1=0) - but the Cantor set is not empty. We are sure that all endpoints of the intervals remain in it as follows: 0, 1, , , , , , , , … as well as families of and its friends as: , , , , … (Gullberg, 1997) References: Bradley S. Tice (2003). Cantor's Dust and Transmission Errors. Paperback http://www.authorhouse.com/BookStore/ItemDetail~bookid~18854.aspx Jan Gullberg (1997). Mathematics: From The Birth Of Numbers. W. W. Norton and Company, New York. Lesmoir-Gordon N. (2004) The Colors of Infinity: The Beauty, the Power and the Sense of Fractals Clear Books Ltd. Michael Spivak (2006). Calculus 3rd edition, Cambridge University Press Pérez-López, Ana and Moneda-Corrochano, Mercedes and Moros-Ramirez, Angel (2002). Application of the Cantor Set Theory in Making. Decisions about the Collections Development. Conference Paper. Integration of knowledge across boundaries. http://dlist.sir.arizona.edu/1258/01/ISKO.pdf Samuels P. & Samuels M. (1999), “On Some sets associated with Cantor Ternary Set.” International Journal of Mathematical Education in Science and Technology (30), p435-442 Sarah Smitherman (2004). “Chaos and Complexity Theories: A Conversation” at the American Educational Research Association Louisiana (AERA) Annual Meeting, San Diego, CA, on Thursday, April 15, 2004. http://ccaerasig.com/papers/04/Smitherman-CCT.pdf William Byers (2007). “Topics in Mathematics: Real numbers and beyond” [Course pack] Concordia University, Math 601 M/4 AA, Eastman Systems Inc. Montreal, Quebec.