Is An Actual Infinite Coherent

Is An Actual Infinite Coherent?

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.

How does the concept of an actual infinite apply to the universe, time, creation, and Mormonism?

The Mormon Church denies creation ex nihilo, choosing instead to teach creation ex materia, the position that God organized the universe from pre-existing matter.  In fact, Mormonism takes this position even a step further, teaching that matter, the stuff everything is made of, has always existed.

Time and matter are relative, i.e., one cannot exist without the other (see Einstein’s theory of relativity).  Therefore, if matter has always existed, time has always existed.  Time is the succession of moments; in otherwords, one moment following another makes up time.  If time has always existed, prior to today there existed an actual infinite amount of time.  As a result, there were an actual infinite number of moments prior to today.

However, an actual infinite number of things is incoherent, and whatever is incoherent is impossible.  Therefore, an actual infinite number of moments prior to today, as well as the Mormon belief of the eternal existence of matter and the universe are all impossible.

Darrell

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