Making Your Own Fun

I thought I'd take some time to work through this one.

The question asks, essentially, whether "If anything is possible, then something is possibly impossible" is satisfiable. The antecedent establishes the existence of a world - at this world, which we'll call "world-0", everything is possible. This means that this world accesses a world where A and a world where not-A, for all A. I'll interpret "something is impossible" to mean that some p is not possible, i.e., that for some p, it is not possible that p. Converting modal operators, this means that necessarily not-p.

If, at world-0, everything is possible, then world-0 accesses a world where necessarily not-p. That world accesses either no other worlds, or only worlds where not-p.

Is "If anything is possible, then something is possibly impossible" satisfiable? Sure. Say world-0 accesses, inter alia, world-1. It also accesses any worlds sufficient to make "anything is possible" true at world-0. World-1 accesses no worlds. Then world-0 accesses a world at which not-p is necessary. In other words, from world-0, it is possibly impossible for p to be true.

QED