[ANewFormation]

Relativistic mass increases are an observed effect. So if you're on the relativistic starship (as these are usually called) your mass does not change as your relative velocity does.

But for an at rest observer, your ship's mass would approach infinity as its speed approaches the speed of light. This is the reason the ship would never be observed as hitting the speed of light.

In practical terms this is also why particle accelerators can't just infinitely accelerate the particles - their apparent mass exponentially increases and so too does the amount of energy required to continue to accelerate them.

[Rhapso]

I really really wish they would call it "apparent mass". It is entirely an observer effect. Its entirely "that ship throws propellant out the back but they don't end up going as fast as Newton thinks it should, so it must have more mass than we think"

[whatshisface]

If there are many objects moving in many directions, you won't be able to find a reference frame in which the mass of the system is equal to the sum of the rest masses of the objects.

[Rhapso]

correct. This is not a contradiction.

A reference frame is definitionally a point from which you would measure an "apartment mass". The rest mass isn't relevant to a reference frame where that mass isn't at rest. This is true at all scales.