Your statement would imply that a point has a direction, which it does not. Also, a line does not have a length; both you and Posner meant line segment. So a correct statement would be:
A zero-length line segment is a point.
Edit: not even this is true if your definition of a line segment explicitly states that the endpoints must be distinct, as does PlanetMath's definition, as Aethaeryn points out below.
The Wikipedia definition of a line segment says that it is bound by two endpoints.[1] It provides a reference to Planet Math that goes into specifics.[2] In this page, it is made clear that the two endpoints cannot be equal. Planet Math provides an equation for a closed[3] line segment:
L = {a + tb | t in [0, 1]}
This means that a line segment can be expressed as all of the points a + tb, where t is the range [0, 1] (which contains 0, 1 and all the points between them). It also limits a, b as real or complex numbers with b != 0. In other words, any line segment is just "morphing" the basic 0 to 1 range, with a shifting it and b scaling it.
Now, since b can't be 0, and 0 != 1, you're not going to get the endpoints of the range 0 and 1 to equal each other no matter how you scale them with b or shift them with a.[4] In other words, line segments will always have length.
Because you cannot get the endpoints to equal each other, a point cannot be thought of as a line segment under what appears to be the common definition.
No, the basis for the modern treatment of Euclidean geometry is the explicit construction of the plane as R^2. Axiomatization is reserved for things like set theory and elementary number theory. Although Hilbert's axioms aren't exactly even a proper axiomatization anyway, seeing as it's a second-order axiomatization, and thus requires some ambient set theory. But if you've got set theory, you may as well construct it. You could think of the axioms as just conditions, of course, but then you need an existence proof, which is provided by the explicit construction as R^2, so...
Good point, my statement was poorly formulated. While I agree that analytic geometry and other developments in the field could be more fundamental, Hilbert's formalization can still be useful to express more precisely what one means by "points" and "lines", being, I think, applicable to the current discussion.
On the contrary, it's an axiomatization, and thus only describes the relation between them. If by "what one means by them" you mean a definition, then you'll need a construction for that. Of course, for the most part, the relation between them is what we mean by them; but Hilbert's formalism doesn't seem to be a good way to address such statements as the one that started this, "a point is a line segment of zero length".
I'm actually surprised by the lack of rigor in many of the sources I checked. It's obvious that b != 0 is necessary because otherwise you can't get results like line segments having an infinite number of points, and hence equal to the amount of points in a line.[1] The thing is, a lot of common references seem to leave out that the endpoints can't be equal.[2] It's so obvious that it's just implied, but it's dangerous to treat mathematics like that!
I think there's a difference between what's referred to as a "line" in mathematical terms and in conventional English. I believe the judge and mikek were both using the term in the latter sense.
A zero-length line segment is a point.
Edit: not even this is true if your definition of a line segment explicitly states that the endpoints must be distinct, as does PlanetMath's definition, as Aethaeryn points out below.