View Proposal #283

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ID	283
First Name	Samuel
Last Name	Ferguson
Institution	University of Iowa
Speaker Category	graduate student
Title of Talk	Reals Revisited: NO SUP FOR YOU!
Abstract	Traditionally, first courses in analysis have started with certain axioms and then, in the course of deducing the consequences of these axioms, they prove the major theorems of calculus. The chief among these axioms is the "sup/least upper bound axiom," which seems obscure to most beginners. Where did such a thing come from, and how do we know that such a number system, satisfying such axioms, actually exists? Are the "reals" real? If teachers and students leave such questions unasked, they risk getting the impression that mathematics is just what happens when a somebody writes down a set of axioms and uses them to go on, in the words of Steven G. Krantz, "a magical mystery tour." Fortunately, in 1872 Dedekind and Cantor, independently and with different approaches, which have come to be known as the "Dedekind cut" approach to the "sup" and the "Cauchy sequence" approach to "completeness," constructed such real number systems, but their approaches are considered too complicated to present in their entirety at the beginning of most courses in analysis. In this talk, assisted by the intuition of Cauchy, Weierstrass, Courant, and others, we will give another (new?) construction of the reals, which has the advantages of both of the other constructions discussed and the complications of neither. Time permitting, the number "e" will be defined with this approach, or the Intermediate Value Theorem will be proved.
Subject area(s)	Analysis, Teaching, Foundations
Suitable for undergraduates?	Yes
Day Preference
Computer Needed?	N
Bringing a laptop?	N
Overhead Needed?	N
Software requests
Special Needs	I need a chalk or marker and a board, plus an eraser.
Date Submitted	10/8/2010
Year	2010