[Enginerrrd]

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.

[tobias2014]

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.

[Enginerrrd]

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.