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First off, it's not certain that space is infinite, but I'd say it's probable.
But even if it's infinite, infinity has some unusual properties that make this make sense. For this, I'm going to borrow from Hilbert's paradox of the Grand Hotel.
Imagine a hotel with an infinite number of rooms numbered 0, 1, 2, 3... with no end. The room numbers (and rooms) just keep going on forever. And imagine that hotel has no vacancy. That is, an infinite number of guests are already staying there and there are no hotel rooms that are vacant. But then, someone shows up and asks the hotel clerk for a room. The clerk, being a clever fellow, has all the current patrons change rooms to the room numbered one higher than the one they were previously in. (The person in room 0 moves to room 1. The person in room 1 moves to room 2. etc.) That operation is one that can go on forever. (It couldn't go on forever in a hotel with a finite number of rooms, but in an infinite hotel, an infinite number of patrons can move to the next room up and not a single one of the infinite number of patrons will be unable to do so for lack of a room numbered one greater than their previous room.) Then, the clerk books room 0 for the new arrival. But also notice that the number of patrons before the new arrival is the same as the total number of patrons including the new arrival.
Said another way,
∞+1=∞. (Not only that, but∞+∞=∞. Thinking about the previous thought experiment, if an infinite number of people arrived to a fully occupied infinite hotel, the clerk could have all the existing patrons move to the room that was double their previous room number and then book all of the infinite number of new arrivals in all the odd-numbered rooms.)Final thoughts:
That makes sense, but while I suppose we don’t have proof of space being infinite, is it not pretty intuitively certain that’s the case? Otherwise that would imply there’s some sort of border, which I guess is possible, but doesn’t make a lot of sense.
No, finite doesn't necessarily mean it has a border. The surface of the earth has finite area, but one can theoretically travel along the surface of the earth forver in any one direction without ever hitting any border. (You'd of course eventually return to where you started, but not hit a border.) The universe may well be the same way. A "hypersphere" if you will. That is, maybe theoretically if you traveled in "a straight line" forever, you'd eventually find yourself where you started rather than ever hitting an edge or boundary or border.
Ah, makes sense