Preface to the Cantor Set

Steven R. Dunbar
Department of Mathematics and Statistics
University of Nebraska-Lincoln

David Fowler
Department of Curriculum and Instruction
University of Nebraska-Lincoln

ã Copyright  Steven R. Dunbar, David Fowler, 1992, All rights reserved.  T

Cantor Time Line

Show[Graphics[{
	Line[{{1845,0},{1918,0}}],
	Text["born,",{1845,-1}],Text["1845",{1845,-2}],Text["died,",{1918,-1}],
	Text["1918",{1918,-2}],Text["Goedel born,",{1900,2}],Text["1906",{1900,1}],
	Text["Kronecker born,",{1820,2}],Text["1823",{1820,1}],
	Text["Hilbert problems,",{1900,4}],Text["1900",{1900,3}],
	Text["The Cantor Set,",{1883,-3}],Text["1883",{1883,-4}],
	Text["Non-denumerability Theorem,",{1874,-5}],Text["1874",{1874,-6}],
	Text ["Georg Cantor",{1870,-1}]},PlotRange->{{1800,1930},{-10,10}}]]

The illustration of Cantor was digitized from a photo in the 
book Journey Through Genius. The Great Theorems of Mathematics, 
William Dunham, Wiley Science Editions, 1990.page 280. The chapter on 
Cantor is a good  nontechnical account of his work.

From Georg Cantor, His Mathematics and Philosophy of the Infinite,
Joseph Warren Dauben, Princeton University Press.

"The question uppermost in Cantor's mind: when was a given set P of arithmetic points in [an n-dimensional space] to be considered a continuum?

When Cantor first defined the real numbers in his paper on trigonometric series of 1872, using fundamental sequences of rational numbers, the reals could be characterized as the first derived set of the rational numbers.  Clearly the continuum was a perfect set, but perfect sets could be constructed which were dense in no interval, however small.  To illustrate this point, he offered the example of his famous ternary set."  [Page 109]

"Ueber unendliche lineare Punktmannichfaltigkeiten."  Georg Cantor.  Math Ann. 21, 1883. p590.

"Als ein Beispiel einer perfecten Punctmenge, die in keinem noch so kleinen Intervall öberall dicht ist, föhre ich den Inbegriff aller reellen Zahlen an, die in der Formel:

				z=c1/3 + c2/3^2 +...+ cv/3^v + ...
																		
enthalten sind, wo die Coefficienten cv nach Belieben die beiden Werthe 0 and 2 anzunehmen haben und die Reihe sowohl aus einer endlichen, wie aus einer unendlichen Anzahl von Gliedern bestehen kann."