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\begin{document}





A set $S$ is \emph{infinite} if it is not \htmladdnormallink{finite}{http://planetmath.org/encyclopedia/Finite.html}; that is, there is no $n \in \mathbb{N}$ for which there is a \htmladdnormallink{bijection}{http://planetmath.org/encyclopedia/BijectiveFunction.html} between $n$ and $S$.

Assuming the \htmladdnormallink{Axiom of Choice}{http://planetmath.org/encyclopedia/AxiomOfChoice.html} (or the \htmladdnormallink{Axiom of Countable Choice}{http://planetmath.org/encyclopedia/CountableChoice.html}), this definition of infinite sets is equivalent to that of \htmladdnormallink{Dedekind-infinite sets}{http://planetmath.org/encyclopedia/DedekindInfinite.html}.

Some examples of \htmladdnormallink{finite sets}{http://planetmath.org/encyclopedia/Finite.html}:

\begin{itemize}
\item The \htmladdnormallink{empty set}{http://planetmath.org/encyclopedia/EmptySet.html}: $\{\}$.
\item $\{0, 1\}$
\item $\{1, 2, 3, 4 , 5\}$
\item $\{1,1.5, e, \pi\}$
\end{itemize}

Some examples of infinite sets:

\begin{itemize}
\item $\{1, 2, 3, 4, \ldots\}$.
\item The \htmladdnormallink{primes}{http://planetmath.org/encyclopedia/RationalPrime.html}: $\{2, 3, 5, 7, 11, \ldots\}$.
\item The \htmladdnormallink{rational numbers}{http://planetmath.org/encyclopedia/RationalNumbers.html}: $\mathbb{Q}$.
\item An \htmladdnormallink{interval}{http://planetmath.org/encyclopedia/Segment.html} of the \htmladdnormallink{reals}{http://planetmath.org/encyclopedia/MathbbR.html}: $(0, 1)$.
\end{itemize}

The first three examples are \htmladdnormallink{countable}{http://planetmath.org/encyclopedia/Countable.html}, but the last is \htmladdnormallink{uncountable}{http://planetmath.org/encyclopedia/UncountableSet.html}.

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