Cantor’s Diagonal Argument. Recall that • A set S is finite iff there is a bijection between S and {1, 2,,n} for some positive integer n, and infinite otherwise. Not too long ago, while surfing the TV channels, you could lean back, press the remote, and suddenly you found a show about Georg Cantor (pronounced. The Cantor diagonal method, also called the Cantor diagonal argument or Cantor’s diagonal slash, is a clever technique used by Georg Cantor to show that the.

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Cantor’s Diagonal Argument – A Most Merry and Illustrated Example

The Cantor diagonal method, also called the Cantor diagonal argument or Cantor’s diagonal slash, is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence i. However, Cantor’s diagonal method is completely general and applies to any set as described below. Given any setconsider diagoalization power set consisting of all subsets of.

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Cantor’s diagonal method can be used to show that is larger thani.

Cantor’s Diagonal Argument

Finding an injection is trivial, as can be seen by considering the function from to which maps an element of to the singleton set. Suppose there exists diagonalozation bijection from to and consider the subset of consisting of the elements of such that does not contain. Since is a bijection, there must exist an element of such that.

But by cabtor definition ofthe set contains if and only if does not contain. This yields a contradiction, so there cannot exist a bijection from to. Cantor’s diagonal method applies to any setfinite or infinite.

Comparing infinite lists

If is a finite set of cardinalitythen has cardinalitywhich is larger than. If is an infinite set, then is a bigger infinite set.

In particular, the cardinality of the real numberswhich can be shown to be isomorphic towhere is the set of natural numbers, is larger than the cardinality of. By applying this argument infinitely many times to the same infinite set, it is possible to obtain an infinite hierarchy of infinite cardinal numbers.

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An Elementary Approach to Ideas and Methods, 2nd ed.

Oxford University Press, pp. The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics.

Explore thousands of free applications across science, mathematics, engineering, technology, diagonalizatio, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.

Cantor’s Diagonal Proof

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Mon Dec 31 Referenced on Wolfram Alpha: Contact the MathWorld Team.