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Countability (Mathematics) (view source)
Revision as of 22:45, January 6, 2010
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==Examples of Uncountable Sets==
==Examples of Uncountable Sets==
The real numbers are not countable, nor is any set with positive [[Measure (Mathematics|Lebesgue measure]]. This is because any countable set of numbers can be completely contained within a set of arbitrarily small measure. A short proof follows:
The real numbers are not countable, nor is any set with positive [[Measure (Mathematics)|Lebesgue measure]]. This is because any countable set of numbers can be completely contained within a set of arbitrarily small measure. A short proof follows:
Let <math>\left\{{a_1, a_2, a_3, ... }\right\} \ </math> be a countable set, and let <math>w \ </math> be some arbitrarily small number. Construct the interval <math>[a_1-w/4,a_1+w/4] \ </math>.
Obviously, <math>a_1 \ </math> is contained in this interval, and this interval has length <math>w/4+w/4 = w/2 \ </math>.
But around <math>a_2 \ </math>, we can construct the interval <math>[a_2-w/8,a_2+w/8] \ </math>, around <math>a_3, [a_3-w/16,a_3+w/16] \ </math>, and in general, around <math>a_k \ </math>, we can construct <math>[a_k-w2^{k+1},a_k+w2^{k+1}] \ </math>.
The sum of all these lengths will be <math>w/2 + w/4 + w/8 + w/16 + ... = w \ </math>, but remember, <math>w \ </math> was chosen to be arbitrarily small, and so a countable set can be contained in a set of arbitrarily small measure.
Since the real numbers, or any set of positive measure, cannot, they must not be countable.