Is It Possible To Have An Actually Infinite Number of Things? (Part 2)

In part one of this series, we looked at why it's important to have a grasp on the concept of infinity as it relates to the real world.  We talked briefly about how people may ask questions like "Why can't the universe be infinitely old?" or "who designed the designer?" (in reference to the statement "God designed the universe").

As we showed with an example of infinite dominoes from Dallas Willard's book Knowing Christ Today, an actually infinite number of things is impossible because it's necessary to have a "first thing" or, to put it another way, a "necessary being" to start things off.

This time, we're going to look at some of the logical inconsistencies that emerge when we attempt to apply the concepts of infinity to a hotel.  This example, called "Hilbert's Hotel", was invented by David Hilbert who was a great German mathematician of the early 20th century.  In his article The Existence of God and the Beginning of the Universe, Dr. William Lane Craig discusses Hilbert's Hotel, showing the logical absurdities that would result if such a hotel existed:

"Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.

"But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things."

At this point we've seen that an actually infinite series of dominoes (or days or any infinite series) cannot exist, but needs a necessary being that is, itself, uncaused but which starts the series going.  With Hilbert's Hotel, we also see the logical absurdities that result from actual infinities.

So what do we say when people make claims such as "The universe is infinitely old" or "You can't say God designed the universe, because then we need to answer 'Who designed God?'"?  The answer is really a simple actually infinite number of things is impossible.

All things that exist have causes.  And at the beginning of every seemingly infinite chain of events, there must be a necessary being to start it all.  There is simply no other choice.

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