(I'm reposting this here from an old Dreamwidth post of mine, since I've seen people reference it occasionally and figure it would be easier to find here.)

So people throw around the word "impossible" a lot, but oftentimes they actually mean different things by it. (I'm assuming here we're talking about real-world discussions rather than mathematical discussions, where things are clearer.) I thought I'd create a list of different things that people mean by "impossible", in the hopes that it might clarify things. Note -- because we're talking about real-world things, time is going to play a role. (Yes, there's not really any universal clock. Whatever.)

I'm listing these as "levels of impossibility", going roughly from "most impossible" to "least impossible", even though they're not necessarily actually linearly ordered. Also, some of the distinctions between some of these may be fuzzy at times.

Level 0. Instantaneously inconsistent. The given description contains or logically implies a contradiction. It rules out all possible states at some point in time, in any universe. People often claim this one when they really mean level 2 or level 3.

Level 1. Instantaneously impossible (contingently). In the actual universe we live in, the given description is instantaneously impossible; it rules out all possible states at some point in time. I think in most discussion that isn't about physics this isn't actually strongly distinguished from level 0.

Level 2. Non-equilibrium. The described system fails to propagate itself forward in time; or, if a system extended in time is described, it contains an inconsistency. This is one that people often actually mean when they say something is "impossible".

Level 3. Unstable equilibrium or possible non-equilibrium. The described system is not resilient to noise; it will not propagate itself forward in time unless exceptional conditions hold continually. This is another one that people often really mean when they say something is "impossible".

Level 4. Unachievable. The described system is unreachable from our present state -- it may make sense on its own, it may not be inconsistent with the way the world evolves in time, but it's inconsistent with the initial conditions that hold in the real world. Yet another one that people often mean when they say "impossible".

Level 5. Not "stably achievable". The only path from the current state to the described state is not resilient to noise and requires exceptional conditions to hold, possibly for an extended period of time. We might also want to require that in addition that the failure modes of such a path leave us worse off than we were before or somehow prevent the same path from being used again (so that you can't just try over and over for free).

I'm not sure that this is really complete; but, overall, the point is that when you say something is "impossible", you should think about whether you're actually talking about instantaneous impossibility, non-equilibrium / instability, or unachievability (and then yeah I've introduced some finer distinctions here).