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Difference between revisions of "Uncountable"

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A set <math>S</math> is said to be '''uncountable''' if there is no [[injection]] <math>f:S\to\mathbb{N}</math>. A well-known example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>.
 
A set <math>S</math> is said to be '''uncountable''' if there is no [[injection]] <math>f:S\to\mathbb{N}</math>. A well-known example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>.
  
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=== Proof that <math>\mathbb{R}</math> is uncountable ===
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== Proof that <math>\mathbb{R}</math> is uncountable ==
  
 
We give an indirect proof here. This is one of the most famous indirect proofs and was given by George Cantor.
 
We give an indirect proof here. This is one of the most famous indirect proofs and was given by George Cantor.
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Revision as of 21:58, 7 December 2007

A set $S$ is said to be uncountable if there is no injection $f:S\to\mathbb{N}$. A well-known example of an uncountable set is the set of real numbers $\mathbb{R}$.

Proof that $\mathbb{R}$ is uncountable

We give an indirect proof here. This is one of the most famous indirect proofs and was given by George Cantor.

Suppose that the set $A=\{x\in\mathbb{R}:0<x< 1\}$ is countable. Let $\{\omega_1, \omega_2, \omega_3, ...\}$ be any enumeration of the elements of $A$. Consider the binary expansion of each $\omega_i$. For dyadic rationals, take only the expansion terminating in an infinite chain of zeros, so that we have a unique binary expansion for each element of $A$. Say $\omega_i=(.b_{i1}b_{i2}b_{i3}...)_2$ for all $i$. Now construct $\omega=(.b_1b_2b_3...)_2$ such that $b_i=1-b_{ii}$ for all $i$. So $\omega$ is different from $\omega_i$ for all $i$. So $\omega\not\in A$. But $\omega\in\mathbb{R},0<\omega<1$, a contradiction. So $\mathbb{R}$ is uncountable.

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