An observer throws an object towards a black hole, and then an arbitrary amount of time later, throws a second object towards the black hole. Disregarding Hawking radiation and assuming the Black Hole will last forever, it will take an infinite amount of time from the point of the view of the observer for the first object to reach the event horizon. In other words, the first object crosses the event horizon right at the very end of time. But if this is true, then when does the second object reach the event horizon? After the first object reaches the event horizon, there is theoretically no more time that can be elapsed, yet we know that the second object must arrive at the event horizon after the first object.
Update: Perhaps it would be better to rephrase some of the above. Just for clarification, I am not thinking of infinity as a number, but more like indexes in infinite set theory. Infinity is not a number, but there is a concept denoting the last index in an infinite set, omega. In this case, the state of the object corresponds to an index of the time set, and the state where the object crosses the event horizon is defined to correspond to an index of omega. Two sets with last indices omega and omega + 2 have the same cardinality, but are just indexed differently. My question was that the state of the first object when it passes the horizon corresponds to index omega, and the index corresponding to the event when the second object crosses the event horizon must come after omega. However, time is defined with a last index of omega, so my question is: what is the index CORRESPONDING to (not equal to, as with numbers) the event when the second object crosses? Ie, if an index of omega is ascribed to the event when the first object reaches the event horizon, what index do we ascribe to the event when the second object reaches the event horizon?