Trouble with a Statement in Arnold's "Mathematical Methods of Classical Mechanics" On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads

One can speak of two events occuring simultaneously in different places, but the expression 'two non-simultaneous events $a,b \in A^4$' occurring at one and the same place in three-dimensional space" has no meaning as long as we have not chosen a coordinate system.

(By $A^4$ we mean a four dimensional affine space over the vector space $\mathbf R^4$, coupled with a rank $1$ linear map $t:\mathbf R^4\to \mathbf R$ which measures the time interval between two events.)
I do not know what is the precise meaning of the two terms in bold in the above highlighted line. 
What is meant by "place"? Probably a point in the space of simultaneous points of an event.
Another thing which Arnold doesn't seem to define is a "coordinate system".
Can somebody help?
Thanks.
 A: I think he has a particular kind of coordinate system in mind: Choose a point $O$ as your origin, and then a basis $(e_0,e_1,e_2,e_3)$ for $\mathbb{R}^4$, where $e_0$ goes from $O$ to some point $P$ with $t(e_0)=t(\vec{OP})=1$, and where $(e_1,e_2,e_3)$ is an ON basis for the kernel of $t$.
Then every event $Q$ (= point in $A^4$) has unique coordinates $(t,x,y,z)$,
where $\vec{OQ}=t e_0 + x e_1 + y e_2 + z e_3$,
and we may say that $Q_1$ and $Q_2$ occur at the same place if their $(x,y,z)$ coordinates are the same. But this depends on the choice of the point $P$ (if $Q_1$ and $Q_2$ have different $t$ coordinates); if we choose another coordinate system with another point $P'$ giving the direction of the time axis, then $Q_1$ and $Q_2$ will have different $(x',y',z')$ coordinates.
The idea is that different coordinate systems of this type correspond to different inertial observers, and no inertial observer is "better" than any other, so there is no preferred coordinate system. The vector between the points $P$ and $P'$ for two inertial observers tells you their relative velocity. If one observer thinks that two non-simultaneous events occur at the same place, then another observer (moving relative to the first) will disagree. And both of them have the impression of "standing still"; they both think that the other observer is the one who's moving. So there is no way to objectively determine whether non-simultaneous events occur at the same place; it depends on the observer.
