I'm only in high school (junior) so goes easy on me. But, how do physicist refute these paradoxes? Considering there are a number of theories regarding the origin of the universe and some postulate that the universe is infinite, I assume they've come across this. Just looking for a direct answer to what should be a relatively simple question. Here are the two paradoxes (for those unaware):
"A similar paradox arises if the past is infinite. If there exists an infinite past, we would never have the present day. If there was an infinite set of past events and each event requires the previous event to occur, would we ever have the present? Of course not. This is because if today is dependent upon the fact that yesterday happened, and there is an infinite set of these dependencies (i.e. forever) - today will have not occurred. This is similar to the library book example mentioned earlier."
"If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning.
Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems.
The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupied by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accommodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible."