Anthony Aguirre demonstrates:
Infinity can violate our human intuition, which is based on finite systems, and create perplexing philosophical problems. A classic example was invented by the mathematician David Hilbert. Rather than imagining an infinite universe, imagine an infinite hotel, with all rooms completely filled. Though the hotel is full, you can accommodate infinitely many more guests by moving each guest into the room of twice its current number and adding guests in all of the odd rooms. Yet although you double the number of guests, the hotel looks exactly the same. Applying this notion to an eternally inflating universe, suppose the whole bubbling mess is infinite at some time. Although infinitely many bubbles have formed during some time interval, it is rather unclear that after this time the universe is actually any bigger!
The Dish 