A model of geometry with the negation of Pasch’s axiom? [duplicate]

How would a geometry with all the usual axioms of Euclidean geometry, except that instead of Pasch’s axiom, we take it negation, look like?

marked as duplicate by colormegone, hardmath, Claude Leibovici, user91500, user228113 Jul 10 '16 at 10:30

• What do you take as "the usual axioms of Euclidean geometry"? Hilbert's? – Blue Feb 5 '14 at 23:05

For elementary geometry without the Pasch Axiom, every model is isomorphic to a Cartesian plane over a formally real Pythagorean semi-ordered field $\mathcal{F}$.
More can be said if our geometry has the full second-order continuity axiom. For then $\mathcal{F}$ is (as a field) isomorphic to the reals.
Now let $f:\mathbb{R}\to \mathbb{R}$ be a non-linear solution of the Cauchy functional equation $f(x+y)=f(x)+f(y)$, where $f$ is onto and $0\lt f(1)$. Define the order relation $\lt^\ast$ by $x\lt^\ast y$ if $f(x)\lt f(y)$. Then under $\lt^\ast$ and ordinary addition, the reals form an ordered group. The Cartesian plane over the reals, with order relation given by $\lt^\ast$, is a non-Paschian model of geometry with second-order continuity.