Originally Posted by **topsquark**

I think I know what this is about, but some statements are poorly made. I hope I can explain clearly what I am seeing here.

I.1 Every line contains at least two distinct points.

I take this as a given: in fact a line is made from a continuum of distinct points, so a line contains an infinite number of them.

Now look at I.3. I don't agree with the statement as it is written. I think what it is *trying* to say is "Given two distinct points there exists at most one line containing these points." This is the only statement I can write that doesn't either repeat or contradict I.1.

However, I don't understand why I.2 is there at all. I can't find any reason for it. If I am correct in my guess about I.3 we can simply rewrite I.3 (or I.2) to explicitly state "Given two distinct points there exists exactly one line containing these points." This seems to me to be a much more explicit axiom, it replaces two of the given axioms by one, and we would want the fewest axioms to work with in building such a logical system. Otherwise all I.2 is stating is that a line exists that contains two points; points which may not even be distinct (distinctness is not mentioned in I.2). This is already taken as a given in I.1. I am open to another interpretation but, as written, this one just looks bad to me.

-Dan