In message <email@example.com>, Ken Quirici <firstname.lastname@example.org> writes >It would be an interesting exercise (at least it seems to me) to >imagine this Euclidean axiomatic system INDEPENDENT of ANY pictures. >How many 'realities' would this system model? I suspect there are many >that have absolutely nothing to do with geometry. That is, take the >axioms as simply words that relate to each other in logical ways and >see what they can represent.
I was going to suggest this to you, but you've got there on your own. Imagine trying to explain Euclid's proofs to a blind man. If it can't be done solely from the axioms and postulates, with no diagrams, then it's not a valid proof.
It is indeed true that the explicit axioms and postulates in the Elements are not sufficient for even the simplest of his proofs.
Most trivially, there's nothing to ensure that there even exist any points or lines.
More importantly, as was raised at the start of this thread, without an axiom that ensures that certain circles intersect, most of the constructions fail.
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.
So even if you take Euclid's postulates and add the existence of point and line, Pasch's axiom and any necessary order axioms, you still don't have enough to prove Proposition 1.
Add an axiom that ensures there is a point not on a given line and it may be possible, I don't know.