I remember from my geometry class as a child the lesson regarding these three mathematical constructs:
A line stretches both ways to infinity.
A line segment is finite…a definite beginning and a definite end.
A ray begins at a point and stretches out to infinity.
Is there some kind of anti-ray? Comes in from infinity and ends at a definite point?
A ray doesn’t really have a direction. It is just a portion of a line.
All roads lead from Rome. Some people just walk them in the wrong direction.
Also note that a ray (or line segment) doesn’t have to include its end point(s). For example, the interval (0;1) on the x-axis, while of finite length, doesn’t technically have any “beginning” or “end”!
Can’t you define the ends as a limit? I guess that’s not really classic geometry tho.
To the best of my knowledge, that’s precisely how it’s done. But my point (heh!) is, there is no point in the (open) interval that could by any standards count as an “end point”.
Are you sure about that? Wouldn’t it be axiomatic that the endpoints are indeed its endpoints?
It wouldn’t be axiomatic as much as a tautology. 
(And you would still have to presuppose that they actually existed!)
The supremum and infimum of an open interval could be considered its “endpoints”. But as Ignotus notes, the important point (no pun intended) is that these points do not belong to the interval. They are outside the interval. There is no point which is a member of the interval and is larger than any other point in the interval. This is fundamentally different than a closed interval which does contain its endpoints.
I’ve always been irked by rays vs vectors and how they are depicted. I’ve tutored a bunch, and it is quite common to have problems getting the idea across that vectors are of finite length when they use the same symbol as the infinite length ray that the students have had conditioned into them.
Solution? Dunno.
You could start by drawing the vector of (say) gravitational pull increasingly thinner (or, alternatively, increasingly less dense) but still, in theory, stretching to infinity with your piece of chalk on the blackboard; and then somehow convince your students that what really matters is how much chalk you have wasted on this; so from now on we stick to a uniform standard of x mg of chalk per centimeter…
No guarantees it would work.
Are a ray and a line therefore always both the same length? (infinitely long?)
Yes, they are of the same (infinite) length. You can easily (I’ll leave it to you as an exercise!) map the one onto the other.
Just like mapping the set of odd integers to the set of all integers. You’ll never run out of numbers to map from one to the other, even though it seems like the odd set should be half the size.
Ooooh, be careful with this one. “Mapping” implies set cardinality, which has some very non-intuitive results if you’re not comfortable with the behavior of infinite sets.
Basically, any open interval on the reals is just as big as the whole real number line. That means you can construct a mapping from (a,b) (with a < b, of course) to the real line as a whole without leaving any portions of the real line unmapped; I personally like using the tangent function for this purpose, because it’s familiar, but I’m sure a number of other functions could fill the role just as well. You can do the same trick with the open interval (a,b) and the whole complex plane, too; you can see this if you’re familiar with space-filling curves.
My point is, if you have mappings in that set-theoretic sense, where a mapping is a bijective function, and your lines, line segments, and rays are all equivalent to or fractions of the real line, all of the elements, lines, line segments, and rays, are the same size in a very precise sense. You need a different way to define “size” if you want to avoid that; the vector algebra notion of magnitude springs to mind, but I can’t see how that definition would make lines bigger than rays.
See, this is what I really love with maths, as much as with the Straight Dope! At whatever level of knowledge you’re at, there’s always someone at a higher level to point out your mistakes! 
I wouldn’t even call it a mistake, more a matter of taste, or a caveat most people wouldn’t know about.
Math is made by humans, for human purposes. Every mathematical concept has three parts to it:
[ul]
[li]The intuition, which is central, and is the somewhat hand-wavey intuitive notion of what a concept means.[/li][li]The definitions, or axioms, which attempt to capture the intuition in a rigorous way which squeezes out self-contradiction and allows us to be confident we’re not descending into nonsense when we use the concepts we’ve invented.[/li][li]The implementation, which includes notation on paper and blackboards and also software, which allows us to talk about the ideas in a way which is, ideally, compact and unambiguous.[/li][/ul]
My point is, humans make the definitions to capture intuitions, and, sometimes, the definitions don’t quite do it. This was proven quite explosively at the turn of the previous century, when Russell proved that the previous definitions of set theory lead to logical self-contradiction; the fact mathematicians were then trying to formalize all of mathematics on a foundation of set theory meant this was deeply troubling, and lead to a new, more complicated set of axioms for set theory to be proposed. The most widely used of these new axioms is called ZF, for Zermelo-Fraenkel, the names of the people who worked on it.
This didn’t erase the previous intuition, at least not entirely, but it did greatly modify certain parts of it, particularly the intuition around infinite sets. Which, as you’ve seen, can lead to bizarre results, which is the other way definitions can fail to capture intuition: You can have an idea, come up with a set of definitions you think capture it, and, lo and behold, your definitions lead to a result you never imagined. This doesn’t mean your intuition is wrong, just that you didn’t capture it in the definitions you laid down; maybe a new set would do a better job.
Sometimes, however, it’s impossible to capture an intuitive idea in consistent axioms. This was proven by the First Incompleteness Theorem: It’s possible to prove that if you have a consistent axiom system (that is, an axiom system which does not allow self-contradiction) which is powerful enough to express basic arithmetic on the natural numbers which is possible to program into a computer such that the computer would list all statements which are true according to that axiom system and no statements which are not true, that computer would never be able to prove all truths about the natural numbers. The Second Incompleteness Theorem says that it’s impossible for that axiom system to prove its own consistency.
ZF is subject to the First Incompleteness Theorem, so there are statements which are widely believed to be true which it cannot prove. An example of such a statement is the Axiom of Choice, which simply states that, given any collection of sets, each containing at least one object, it’s possible to pick an object from each set. This doesn’t require any of the sets be infinite, or that you have an infinite number of sets, but it applies to infinite sets/infinite numbers of sets just fine. This is so intuitively obvious that ZF is commonly extended into ZFC, which is ZF plus Choice, with Choice taken as an axiom added to the usual axioms of ZF. And ZFC is fun: It allows us to prove the Banach-Tarski Paradox*, which allows us to cut a sphere into two spheres of the same volume.
*(Paradox is a paradoxically ill-defined term in math: It can be used to mean either a logical contradiction, such as the one Russell used to show that the old-fashioned set theory of his time was fatally flawed, or it can be used to mean a really unintuitive result, like Banach-Tarski Paradox, which doesn’t involve any logical contradictions at all. I like Quine’s practice of distinguishing falsidical paradoxes, which involve logical contradiction, from viridical paradoxes, which do not. Sadly, I appear to be in a small minority in this.)
So you had a notion of the “size” of lines, line segments, and rays, and you pulled a definition of size from set theory. It didn’t quite do what you expected, that’s all. It’s happened to much better mathematicians than any of us, I assure you.
Interestingly in quantum mechanics interestingly the term ‘ray’ is used to describe the equivalence class of vectors v~w iff v=aw for some non-zero scalar a.
Two roads, even if both infinitely long, don’t necessarily pass through all the same places?
Paradoxes necessarily being “true statements that would appear to be mutually exclusive but in fact are not”, and optionally also fitting certain sub-categories, makes sense to me - or have I missed the point?