Is infinity real?

1. Can a number that is finite but very large substitute for infinity?

This first question is just a warm-up to show how we can replace infinitistic thinking with finitistic thinking. It concerns the famous Hilbert’s Hotel, an idea introduced by David Hilbert in 1924.

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. Can it accommodate 1,000 new guests without increasing the number of guests in any of the occupied rooms? If you had a finite number of rooms, the pigeonhole principle would apply. In this context, this common sense principle says that you cannot have n+1 pigeons in n holes if there is only room for one pigeon in each hole. But in an infinite hotel, it’s easy! We just move every resident from his or her room n to room n + 1,000. Voilà! Rooms one to 1,000 are now empty!

Notice the sleight of hand involved in using infinity in this way. This solution cannot work with a finite number of rooms, no matter how large. Let us restrict ourselves to the notion that the number of rooms can be as large as the size of the universe allows, but must be finite. Can the question still be answered positively? Well, it turns out that you can easily accommodate 1,000 new guests in a finite physical hotel that is currently full. And the arrangement will take less time than moving a single person from one room to another. All it takes is the reasonable assumption that there is a tiny, nonzero probability of a person checking out within a given time. Let’s assume, conservatively, that the probability of a guest checking out on a given day is one in a hundred. Can you see how the hotel can put up its additional guests?