Cantor Diagonal Method

A clever technique used by Georg Cantor to show that the Integers andReals cannot be put into a One-to-One correspondence (i.e., the Uncountably Infinite Set ofReal Numbers is ``larger'' than the Countably Infinite Set of Integers).It proceeds by first considering a countably infinite list of elements from a set , each of which is an infinite set(in the case of the Reals, the decimal expansion of each Real). A new member of is then created by arranging its th term to differ from the th term of the th member of . Thisshows that is not Countable, since any attempt to put it in one-to-one correspondence with theintegers will fail to include some elements of .

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 81-83, 1996.

Penrose, R. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 84-85, 1989.

• Chebyshev Approximation Formula
• Chebyshev Constants
• Chebyshev Deviation
• Chebyshev Differential Equation
• Chebyshev Function
• Chebyshev-Gauss Quadrature
• Chebyshev Inequality
• Chebyshev Integral
• Chebyshev Integral Inequality
• Chebyshev Phenomenon
• Chebyshev Polynomial of the First Kind
• Chebyshev Polynomial of the Second Kind
• Chebyshev Quadrature
• Chebyshev-Radau Quadrature
• Chebyshev's Theorem
• Chebyshev Sum Inequality
• Chebyshev-Sylvester Constant
• Checkerboard
• Checker-Jumping Problem
• Checkers
• Checksum
• Cheeger's Finiteness Theorem
• Chefalo Knot
• Chen's Theorem
• Chern Class