In message <email@example.com>, Ken Quirici <firstname.lastname@example.org> writes >> Consider a system consisting of a single straight line and the points >> >> thereon. It satisfies all Euclid's explicit postulates (and Pasch's >> >> axiom) but circles have only two points in their circumferences and >> >> circles with a common radius do not intersect. There are no equilateral >> >> triangles. >> >> > >there's problem in that his definitions include (at least in Heath's >translation) surface - that which has length and breadth only. > >Does this vitiate your statement? > No, the axioms don't imply that a surface exists.
Of course the problem of non-standard models like the one I've suggested is trivial to fix. Just add axioms to say that there is at least one line and a point not on that line, (and a plane which does not contain the line if you want 3-D). I assume, from the discussions I've seen on the web, there are models which satisfy this but still have circles which don't intersect even though they "should". Further Googling would probably turn up an example but for now it's more fun trying to find one on my own.
.... >Thanks. I'm hoping Moise's book on Elementary Geometry from an Advanced >Standpoint will provide a sufficiently rigorous approach. Have you read >it? If so, is it sufficiently rigorous?
No I haven't read it. I've never studied Euclid's axioms before.