I think you just nerd-sniped me but I’m not convinced it’s impossible to assign a consistent ordering to events with relativistic separations.
For starters, the spacetime interval between two events IS a Lorentz invariant quantity. That could probably be used to establish a universal order for timelike separations between events. I suspect that you could use a reference clock, like a pulsar or something to act as an event against which to measure the spacetime interval to other events, and use that for ordering. Any events separated by a light-like interval are essentially simultaneous to all observers under that measure.
The problem comes for events with a space like or light like separation. In that case, the spacetime interval is still conserved, but I’m not sure how you assign order to them. Perhaps the same system works without modification, but I’m not sure.
For any space-like event you can find reference frames where things happen in different order. For the time-like situation you described the order indeed exists within the cone, which is to say that causality exists.
You can still order them with the spacetime interval compared to a reference event, even for space like separated events.
It allows for differing elements of the set to share the same value but so does using time alone. It just also allows every observer to agree on the ordering.
Bc Assigning a distance function to elements of a set is a common way to do that in fact. It doesn’t work with just a time coordinate or space coordinate, because that’s effectively a Euclidean metric.
You just have to contend with a few nonintuitive aspects but it’s not so bad.
I think you meant compared to a reference observer? Events are not really independent of observers. Consider the case in baseball where a runner and the baseman tag the base at the "same" time from opposite sides of the base. Assume they move at equal speeds. If the umpire is closer to the baseman then the baseman has tagged it first, if he is closer to the runner, then the runner has tagged it first. The "event" of "touching the base" has two possible outcomes depending on where the observer stands, and there is no "view from nowhere" or observer-free view that we can reference.
No, I mean a reference event, though you bring up an interesting subtlety. (Essentially I just mean an event that definitely happened [A particle decay, a supernova, an omnidirectional radio signal, etc] which will serve essentially as an origin point on the spacetime manifold). You are right though that technically, we need at least one observer to define the coordinates of that event initially. Once that's done however, ALL observers can order events according to the spacetime interval between any event they observe and the reference point (transformed into their coordinates) and they will ALL agree on that ordering. A "good" event here would be something that observers can compare. I think using pulsar pulses counted from some epoch is a perfectly good reference here, assuming we could communicate that omnidirectionally. The difference, as measured by the spacetime interval, between any event in any observers reference frame, and a reference event in their past lightcones is something that ALL observers that can communicate will always agree on. Observers may disagree about how many pulses have occurred since that epoch at a particular time in their coordinate time, but it doesn't matter. As long as they're comparing in spacetime intervals to a particular count on the pulsar, no disagreement will occur. i.e. the spacetime interval between the 3rd pulse and some event will always be the same since it's a lorentz invariant scalar quantity (i.e. a rank zero tensor).
Your baseball analogy has flaws: No properly defined "event" in spacetime will have dual-outcomes. The events in that case are that "a baseman tagged the base", and "a runner tagged the base". "x tagged the base first" is NOT an event, that's a comparison between events, and it's one that was done in a particular observers time coordinate, which is not the correct procedure here. No Lorentz invariant transformation between observers within the light cone will disagree that those events happened, though observers may disagree which happened first within their coordinate time.
(Note the issue of observers needing to be in the same light-cone is a superficial one. I haven't defined that precisely, but I don't need to: If observers can communicate at all, they will agree, upon communication, that an event is within their past light cone. In the context of server synchronization, this will always be true.)
For starters, the spacetime interval between two events IS a Lorentz invariant quantity. That could probably be used to establish a universal order for timelike separations between events. I suspect that you could use a reference clock, like a pulsar or something to act as an event against which to measure the spacetime interval to other events, and use that for ordering. Any events separated by a light-like interval are essentially simultaneous to all observers under that measure.
The problem comes for events with a space like or light like separation. In that case, the spacetime interval is still conserved, but I’m not sure how you assign order to them. Perhaps the same system works without modification, but I’m not sure.