What is a good way to measure the distance between finite subsets of the reals? I have some sets of numbers, and I'd like to have a way to talk about how close these sets are to each other.  I'm not sure what properties it should have (e.g. does it need to be a metric?).  But I hope there are examples of better defined problems like this out there with some good solutions.
Some considerations.  {1} should be closer to {1.1} than it is to {2}.  And {.9,1.1} should be closer to {1} than {0,2} is.  But is {1,1.1} closer to {1} than {2}?  I suspect any distance function will have at least one parameter to control the tradeoff between nearness of points and cardinality of the sets.
 A: Let $A$ be a finite subset of $\mathbb{R}$. For $x\in\mathbb{R}$ define $d(x,A)$, the distance from $x$ to $A$, to be $$d(x,A) = \min\{\vert x-a\vert:a\in A\},$$ the distance from $x$ to the closest point of $A$. If $B$ is another finite subset of $\mathbb{R}$, define $d^*(A,B)$, the asymmetric distance from $A$ to $B$, to be $$d^*(A,B) = \max\{d(a,B):a\in A\}.$$ Finally, define $d(A,B)$, the distance between $A$ and $B$, to be $$d(A,B) = \max\{d^*(A,B),d^*(B,A)\}.$$ This is the Hausdorff distance mentioned by deinst.
A few examples: 
$$\begin{align*}d(\{0.9,1.1\},\{1\}) &= \max\{0.1,0.1\} = 0.1\\
d(\{0.9,1.1\},\{2\}) &= \max\{1.1,0.9\} = 1.1\\
d(\{1,1.1\},\{1\}) &= \max\{0.1,0\} = 0.1\\
d(\{1,1.1\},\{2\}) &= \max\{1,0.9\} = 1\\
d(\{0.9,1.1\},\{1.9,2.1\}) &= \max\{1,1\} = 1\\
d(\{0.9,1.1\},\{1.9,2.0,2.1\}) &= \max\{1,1\} = 1\\
d(\{1.9,2.0,2.1\},\{1.9,2.1\}) &= \max\{0.1,0\} = 0.1
\end{align*}$$
A: There are multiple distances which you might consider in principle. Given that you're considering finite sets, it makes sense to reduce to metrics popular on real vector spaces.
Let $S$ and $T$ be finite sets of real numbers. In the case that $S$ and $T$ have the same cardinality $n$, let $\mathbf s, \mathbf t \in \mathbb R^n$ be vectors whose coefficients are elements of $S$ and $T$ respectively (in any order). Then, for any positive integer $p$ (and for $p = \infty$), you may consider a $p$-distance defined by
$$ \min_\Pi\; \| \Pi \mathbf s - \mathbf t \|_p$$
where $\Pi$ ranges over all permutation matrices on $\mathbb R^n$, and where $$\| \mathbf v \|_p := \sqrt[\Big.^{\scriptstyle p}]{\sum_{j=1}^n |v_j|^p\;}$$ (taking the limit as $p \to \infty$ for the norm $\| \mathbf v \|_\infty$). If $S$ and  $T$ have different cardinalities, you may vary this (in the case that $S$ is larger) by replacing $\Pi$ with an arbitrary matrix whose columns are distinct standard basis vectors (selecting some subset of $S$ to perform the distance computation with).
Regardless of the dimensions, you could also generalize this to consider a distance of the form 
$$ \min_{\Pi_S, \Pi_T}\; \bigl\| \Pi_{\!S} \;\mathbf s \;-\; \Pi_{\!T} \;\mathbf t \bigr\|_p$$
where $\Pi_S$ and $\Pi_T$ range over matrices whose columns are standard basis vectors, possibly with repetition; this allows you to compute vector metrics using arbitrary pairs of elements of $S$ and $T$.
Note that as the cardinalities of the sets grow, these quantities are likely to be prohibitive to compute; but the sets of vectors which I describe using these permutation/selection matrices will describe nice convex sets, for which the usual vector metrics make a certain amount of sense, and which may prove amenable to analysis. And if you consider instead the limit $p \to -\infty$, you should encounter the minimum-pair-distance norm described by Brian in his earlier post.
