Post Author: Darrell
In short, no. The story of Hilbert’s Hotel helps to demonstrate this fact. It goes like this… Let’s say we have a hotel that has an infinite number of rooms and an infinite number of guests; as a result, the hotel is full. If a prospective guest walks in and asks for a room, can he check in? Since there are an infinite number of rooms, the answer must be “Yes.” How about if an infinite number of guests arrive wanting to check in. Can they? Again, despite the fact that the hotel already has an infinite number of guests, since there are an infinite number of rooms, guests can always check in – even an infinite number more.
Now, let’s say that the guests in all of the odd-numbered rooms check out, how many guests are left? There are an infinite number of total rooms. However, there are also an infinite number of odd-numbered rooms, the guests of which checked out, and there are also an infinite number of even-numbered rooms, the ones still left occupied. So in reality, there are still an infinite number of guests left in the hotel even though an infinite number of guests just checked out. This means when you take an infinite away from an infinite, you still get an infinite.
Where does this leave us? Even though Hilbert’s Hotel has an infinite number of guests and rooms, more rooms and guests can always be added. In addition, no matter how many guests check out there will always be an infinite number of guests left. As a result, the hotel could have a sign which reads, “Hilbert’s Hotel: Always full, Yet Rooms Are Always Available.”
This illustration points out how an actual infinite is incoherent. In an actual infinite the whole and the parts are always equal. You can take half away and still have an infinite, or you can add more and still have the same amount – an infinite. However, in reality, a part can never equal a whole. For example, two is part of four (half of it to be exact). Thus, two can never equal four.
Is there any infinite that is coherent? Yes, a potential infinite. A potential infinite is always finite and the whole is always greater than the parts. In a potential infinite you can always add more, but it will never become actually infinite. For example, let’s say you have 100 Jelly Beans in a pile. You can always add more Jelly Beans to the pile. In fact, you can continue to add Jelly Beans and never reach a maximum. As a result, you could say that the pile you are creating as you add more is potentially infinite. It is not an actual infinite because no matter how many you add, there are always a finite number of Jelly Beans in the pile. However, it is potentially infinite because more can always be added.
In the next post, we will look at how the concept of an actual infinite applies to the universe, time, creation, and Mormonism. Stick around.