The heap paradox is a false paradox known since the times of ancient Greece. The paradox states that if you have a heap of sand, that taking a single particle of sand away will not change the fact that the heap is a heap. Likewise to taking a second or third piece of sand away. Eventually, by the process of taking single pieces away, you eventually have a single piece of sand in the entire heap.

So the paradox lies in the fact that by going through a series of infintismal changes, you eventually get a noticable result. The reason that this is not a true paradox is that it is based merely on the human inability to count accuratly past a certain point, and the inability to notice the effects of small changes. There is also the fact that "heap" is defined rather ambiguosly. If "heap" is defined as being an actual integer, then there would be a point where we could say that the heap was no longer a heap. However, if we define heap as an uncountably large number, then we do have a paradox of sorts. However, in order for heap to be uncountable, it would also have to be infinitly large enough we could never succeed in removing parts down to the countable level.

Although this may not be a paradox on a mathematical level, there does seem to be some application of this to human affairs. I have noticed that in (for example) keeping my room clean, I can do it when there is a certain discrete, countable things that need to be picked up. However, if my room gets messy beyond a certain point, it turns into a "heap" for me...that is, it seems so messy that I wouldn't know where to start cleaning, and it seems that no combinations of discrete cleaning actions on my part will change the fact that the whole thing is covered with old recepits and dirty socks.