Infinite shortest paths in graphs From Wikipedia:

"If there is no path connecting two
  vertices, i.e., if they belong to
  different connected components, then
  conventionally the distance is defined
  as infinite."

This seems to negate the possibility that there are graphs with vertices connected by an infinite shortest path (as opposed to being not connected). 
Why is it that for every (even infinite) path between two vertices there is a finite one?
Note that infinite paths between vertices do exist - e.g. in the infinite complete graph -, but they are not the shortest.
 A: The point I made in the comments is that standard graph-theoretic terminology does not allow you to make sense of the notion of two points being connected by an infinite path. However, for what it's worth, there is a formalism that allows you to make sense of this related to end compactification: see Diestel's survey Locally finite graphs with ends: a topological approach for details. Briefly, we have to add extra vertices "at infinity," then introduce a nontrivial topology, after which we can speak of two vertices connected by a continuous function from $[0, 1]$ that passes through infinitely many vertices. 
In any case, the excerpt from the Wikipedia article is about the standard graph-theoretic context. In the standard graph-theoretic context, we sometimes want to think of a graph as an extended metric space, and a natural way to do that is to define the distance to be infinite if two vertices aren't connected by an edge; certainly there's no sensible finite value, and if you choose infinity then the triangle inequality holds. 
A very general context in which to understand this assignment is to think of the distance $d(u, v)$ between two vertices as the infimum over the lengths of all paths between $u$ and $v$. If $u, v$ are not connected by a path, this infimum is empty, and the empty infimum in a poset is always the greatest element, if it exists. This is a special case of a general fact in category theory that the empty limit is the terminal object. (Here the category is the poset $0 < 1 < 2 < ... < \infty$, regarded as a category where there is a single arrow $a \to b$ if $a \le b$ and no arrows otherwise.)
A: Traditional graph theory focuses on finite graphs.  Two vertices are considered connected iff there is a finite walk between them (basically a sequence of vertices, each one adjacent to the last).
If we start considering infinite graphs, we have to consider some interesting consequences.  Suppose we want to consider vertices x,y as connected by an infinite walk.  Then we can order this walk linearly, and get a sequence $x=x_1,x_2,x_3,\ldots,x_\infty=y$.  If this is an infinite sequence, it must contain some limit point -- some point $x_k$ must have either no immediate predecessor or no immediate successor.  This idea doesn't match the old definition of a walk between vertices.
Although I don't know of any references, I do think this idea is interesting as a way to extend connected concepts to infinite graphs.
For example, consider the tree of surreal numbers.  Think of the surreals as sign sequences; each is a map from an ordinal to $\{-,+\}$.  Say that $s$ and $t$ are adjacent if the domain of $s$ is one more or one less than the domain of $t$.  So 0 is adjacent to -1 and 1, 1 is adjacent to 0, 2, and 1/2, etc.
This looks like a standard tree until we get surreals with infinite domains.  Consider some $s$ with domain $\omega$, and the sequence $s_i$ where $s_i$ has domain $i = \{j:j<i\}$, and $s_i(j)=s(j)$.  It feels like the $s_i$ walk should connect 0 with $s$.  A new definition to capture this idea would be interesting.
A: To expand on my comment: It's clear that if an infinite path is defined as a map from $\mathbb N$ to the edge set such that consecutive edges share a vertex, then any vertices connected by such an infinite path are in fact connected by a finite section of the path. To make sense of the question nevertheless, one might ask whether it is possible to use a different ordinal than $\omega$, say, $\omega\cdot2$, to define an infinite path. But that doesn't make sense either, since there's no way (at least I don't see one) to make the two parts of such a path have anything to do with each other -- at each limit ordinal, the path can start wherever it wants, since there's no predecessor for applying the condition that consecutive edges share a vertex.
Note that the situation is different in infinite trees, which can perfectly well contain infinite paths connecting the root to a node. This is because the definition of a path in an infinite tree is different; it explicitly attaches the nodes on levels corresponding to limit ordinals to entire sequences of nodes, not to individual nodes; such a concept doesn't exist in graphs.
