I'm studying proof of stake. I haven't found a clear definition of what "liveness" is. It is a main thing in PoS content, but I don't think I understand it clearly.
Distributed systems - be they blockchains or otherwise - have a concept of correctness. Correctness can be broken down into two parts: safety and liveness.
Liveness, in very general terms, could be considered the guarantee that something good will happen, eventually. The "eventually" is important because it means there's no bound on the time it takes for the good thing to happen.
Making things a little more specific, in a distributed system, an example of liveness is the guarantee that a distributed computation will terminate.
In a consensus mechanism, liveness is the guarantee that all processes/validators/actor/whatever, eventually agree on a value.
To complete the picture, safety is the guarantee that nothing bad will ever happen in a given system. In terms of consensus, this would mean that no two processes/validators/actors/whatever ever decide on different values.