Cantor Diagonal Method
المؤلف:
Courant, R. and Robbins, H.
المصدر:
What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press
الجزء والصفحة:
...
23-12-2021
1595
Cantor Diagonal Method
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.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers). However, Cantor's diagonal method is completely general and applies to any set as described below.
Given any set 
, consider the power set 
consisting of all subsets of 
. Cantor's diagonal method can be used to show that 
is larger than 
, i.e., there exists an injection but no bijection from 
to 
. Finding an injection is trivial, as can be seen by considering the function from 
to 
which maps an element 
of 
to the singleton set ![<span style=]()
{s}" src="https://mathworld.wolfram.com/images/equations/CantorDiagonalMethod/Inline12.gif" style="height:16px; width:14px" />. Suppose there exists a 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 the definition of 
, the 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 set 
, finite or infinite. If 
is a finite set of cardinality 
, then 
has cardinality 
, which is larger than 
. If 
is an infinite set, then 
is a bigger infinite set. In particular, the cardinality 
of the real numbers 
, which can be shown to be isomorphic to 
, where 
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.
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.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 220-223, 1998.
Penrose, R. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 84-85, 1989.
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