I'm interested in (categorical) axiomatization of the real projective line in terms of a quaternary separation relation. First I'll give some axioms which describe the separation relation i.e. the order of a non-oriented circle.
We consider a set $L$ (projective line) and a quaternary relation on points, which will be denoted $ab//cd$.
$ab//cd$ is interpreted as "$a$ and $b$ separate $c$ from $d$". Axioms are the following:
- If $ab//cd$, then $a,b,c,d$ are pairwise distinct.
- If $ab//cd$, then $ba//cd$ and $cd//ab$.
- If $a,b,c,d$ are pairwise distinct, then $ab//cd$ or $ac//bd$ or $ad//bc$.
- If $ab//cd$, then $\neg ac//bd$ and $\neg ad//bc$.
- If $ab//cd$ and $ab//ce$, then $\neg ab//de$.
- If $ab//cd$ and $\neg ab//ce$, then $ab//de$.
These axioms are not fully independent, but it's not my concern here. Secondly, there are no existence axioms here yet and I did it on purpose. Two obvious existence axioms which should be added here are:
- There exist three distinct points.
- If $a,b,c$ are pairwise distinct, there exists $d$ such that $ab//cd$.
My question is what are the usual continuity axioms which should be added here (to make this axiomatization categorical, if possible)? Any reference appreciated.
My idea is as follows:
Suggested axiom 1. If the line $L$ is divided into two nonempty, disjoint sets $X,Y$ such that $X\cup Y=L$ and there are no two points $x_1,x_2\in X$ and two points $y_1,y_2\in Y$ such that $x_1x_2//y_1y_2$, then there exist two distinct points $p,q\in L$ such that $pq//xy$ for any $x\in X\setminus\{p,q\},y\in Y\setminus\{p,q\}$.
Suggested axiom 2. There exists a countable set $F\subseteq L$ such that if $a,b,c$ are pairwise distinct, there exists $d\in F$ such that $ab//cd$.