I think the infinity of the string is not to be seen in length, but is our possibility of performing infinitely many divisions of its length.

Nevertheless, I agree that there is some strange or inaccurate usage that we do of infinity. In this case the problem concerns also the use of limit. A limit is reached at infinity, because it is defined to be so. Numbers are defined as limiting procedures, and here is the matter..

If we consider 0.999... and 1 to be equivalent, then any nearby point is equivalent to the one we are considering. And "physically" this may arise from the fact that 1-0.9999...=h but h is zero (intended as the distance between two such points). If two points realize no distance between them, is mathematics an accurate model of the world? Can it represent distances?

However, apart from these philosphical questions, what astonishes me the most is that 0.999... is 1 even with simple mathematics such as 0.999... = x --->10x-x=9=9x. Hence x=1.

But maybe there is some problem of definition of operations with recurring decimals that we apply here without knowing if that is really correct on earth.

Show me one please.