The following questions (in no particular order) which I had submitted have been "Deleted by Community":
3. For flat space-time as axiomatized by John W. Schutz can the notion of "betweenness" be defined through "paths" and "events"?
The monograph "Independent axioms for Minkowski space-time" by John W. Schutz specifies as primitive undefined basis (Sec. 2.1):
a set $\mathcal E$ whose elements are called events,
a set $\mathcal P$ whose elements are called paths${}^{\text{(1)}}$, together with
a ternary relation on the set of events of $\mathcal E$ called a betweenness relation.
Schutz also notes (App. 4) that
It is possible to consider other alternative primitive undefined bases
laying out ([...]) the possibility of
"local betweenness" relations defined such that each path has its own betweenness relation
and (rather more elborately) the possibility
to dispense with $\mathcal P$ as a primitive concept [...] where the set of paths is then defined in much the same way as the set of lines [...] in the absolute geometries [can be defined through the] ternary betweenness relation on [the set of points.]
However, apparently Schutz doesn't give any consideration at all to a possibility of dispensing with the notion of betweenness as primitive concept, and to defining this relation instead through the given sets of events and of paths, $\mathcal E$ and $\mathcal P$.
[ Omission (char. limit). ]
My question:
Can the axiomatic system for Minkowski (flat) space-time presented by Schutz be expressed using only events and paths as undefined primitive basis?
Can a suitable notion of "betweenness" be defined, at least for this purpose, as a relation involving only events and paths?
This could involve the existence of certain pairs of events for which there is no path containing${}^{\text{(2)}}$ both of them;
as used in the definition of "unreachable sets" whose existence is affirmed through Axiom I5 (Non-Galileian).
Specificly:
Considering any three distinct events $\varepsilon_{AP}, \varepsilon_{BP}, \varepsilon_{JP} \in \mathcal E$ which are all contained in the same path $P \in \mathcal P$, is it a theorem of the axiomatic system presented by Schutz that
- if for each event $\varepsilon_{KQ}$ which is unreachable from $\varepsilon_{AP}$ as well as unreachable from $\varepsilon_{JP}$ it is true that event $\varepsilon_{KQ}$ is unreachable from $\varepsilon_{BP}$, too,
then event $\varepsilon_{BP}$ is between events $\varepsilon_{AP}$ and $\varepsilon_{JP}$
?
And first of all:
Considering any two distinct events $\varepsilon_{AP}, \varepsilon_{JP} \in \mathcal E$ which are both contained in the same path $P \in \mathcal P$, is it a theorem that
- there exists at least one event $\varepsilon_{KQ}$ which is unreachable from $\varepsilon_{AP}$ as well as from $\varepsilon_{JP}$
?
