On Monday, February 17, 2014 6:20:09 PM UTC-5, David Hartley wrote: > In message <firstname.lastname@example.org>, Ken > > Quirici <email@example.com> 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. > >
Righto. They don't. The Heath translation (and I think most of the others) divide the axioms into Postulates and Common Notions and none of them relate to the plane.
> > 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. > > > > -- > > David Hartley
OK. I'll just have to check it out (literally) when it arrives from ILL.