The most straightforward way of saying that the miner will eventually fall through is to say he will if no events occur. This is much too strong a condition. Here's an idea. Introduce a concept of causal space . In many of the approximate theories we will want to use, causal space will not correspond to real space.
Causal space has points.
Events occur at points. Points persist from situation to situation. Thus we can talk about the same point in related situations. Distance in causal space is defined by the time of propagation of effects of events. Events cannot affect fluents associated with far away points in a short time.