\section{Really Big Sets}

\definition{}<infiniteset>
%We say a set $S$ is \textit{finite} if there exists a bijection from $S$ to $\{1, 2, 3, ..., n\}$ for some integer $n$. \par
%In other words
We say set is \textit{finite} if its elements can be consecutively numbered from 1 to some maximum index $n$. \par
Informally, we could say that a set is finite if it \say{ends.} \par
For example, the set $\{\star, \diamond, \heartsuit\}$ is (obviously) finite. We can number its elements 1, 2, and 3.

\vspace{2mm}

If a set is not finite, we say it is \textit{infinite}.

\vspace{2mm}



\problem{}
Which of the following sets are finite?
\begin{itemize}
	\item $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$
	\item $\{ \text{all rats in Europe} \}$
	\item $\{ \text{all positive numbers} \}$
	\item $\{ \mathbb{ \text{all rational numbers} } \}$
\end{itemize}


\vfill

\generic{Remark:}
Note that our definition of \say{infinite-ness} is based on a property of the set. Saying \say{a set is infinite} is much like saying \say{a cat is black} or \say{a number is even}. There are many different kinds of black cats, and there are many different even numbers --- some large, some small. \par

\vspace{2mm}

In general, \textbf{$\infty$ is not a well-defined mathematical object\footnotemark{}}. Infinity is not a number. There isn't a single \say{infinity.} Infinity is the the general concept of endlessness, used in many different contexts.

\vspace{2mm}

%The Russian language (as well as many others, no doubt) captures this well: \say{infinity} in Russian is \say{бес-конеч-ность}, which can be literally translated as \say{without-end-ness}.


\footnotetext{
	In most cases. There are exceptions, but you need not worry about them for now. If you're curious, you may ask an instructor to explain. There's also a chance we'll see a well-defined \say{infinity} in a handout later this quarter.
}


\vfill
\pagebreak

%Say we have two finite sets $A$ and $B$. Comparing the sizes of these is fairly easy: all we need to do is count the elements %in each. It is not difficult to see that $\{1, 2, 3\}$ is bigger than $\{1, 2\}$.
%
%\vspace{2mm}
%
%We could extend this notion of \say{size} to infinite sets. \par
%For example, consider $\mathbb{R}$ and $\mathbb{Z}$. Intuitively, we'd expect $\mathbb{R}$ to be larger, \par
%since there are many elements in $\mathbb{R}$ between every two elements in $\mathbb{Z}$.
%
%\vspace{1mm}
%
%We could also try to compare the sizes of $\mathbb{Q}$ and $\mathbb{Z}$. There are bIntuitively, we'd expect $\mathbb{R}$ to %be larger, \par
%since there are many elements in $\mathbb{R}$ between every two elements in $\mathbb{Z}$.
%
%
%\vfill
%\pagebreak