The Multiverse and Hilbert's Hotel

Cosmic fine-tuning is so astonishing that numerous philosophers and scientists have remarked that it provides extraordinary evidence that the universe has been intelligently designed. Others have sought to evade that conclusion by suggesting that there are an infinite variety of universes comprising what's been called the multiverse.

Watch this video for a quick explanation of both fine-tuning and the multiverse hypothesis:

The reasoning behind the multiverse hypothesis is that if there are an infinity of universes all with different laws and parameters then a universe that's just right to support life, as astronomically improbable as it may be, just has to exist.

In other words, in an infinite array of disparate worlds every possible world must exist and since a world like ours is obviously possible, it's existence isn't so remarkable after all.

There are lots of problems with the multiverse idea, not the least of which is that there's no evidence for its existence, but Robert Marks explains in an article at Mind Matters that even if there were an infinite ensemble of worlds, it doesn't follow that every possible world would exist.

Can anything happen if there are an infinite number of universes each with an infinite number of possibilities in each? ....The answer is no. In a nutshell, the reason is that some infinities are bigger than other infinities.

The number of points on a line segment from, say zero to one, is a bigger infinity than the number of counting numbers {1,2,3,…}. We can label the infinite number of universes in the multiverse as universe #1, #2, #3, etc. Because they can be counted, this infinity is said to be countably infinite.

This looks to be the smallest infinity. (“Smallest infinity” sounds like an oxymoron but isn’t.)

The number of points on a line segment — the bigger infinity — can be referred to as a “continuous infinity.” The points on a line are too many to count. They can’t be ordered as points 1,2,3, etc. Given any point on a line, for example, there is no closest point.

No matter how close a point is chosen to a given point, there will be a closer third point midway between the first two points.

This situation is not true for the countably infinite. Given any number, say 112, the numbers 111 and 113 are the closest numbers. Not so with the set of numbers on the line segment from zero to one. Consider the midpoint ½ =0.5. Is 0.501 the closest number to 0.5? No. 0.5001 is closer and 0.50001 even closer.

This can go on forever, getting closer and closer. But there is no closest number to 0.5.

How does this apply to claims that there is an infinite number of universes where — as a result — anything can happen?

If there is a countably infinite number of possibilities (e.g. we have three eyes in one universe, two in another), then the infinity of universes must be continuous in order to include all possibility combinations. (The proof is here.)

In order to have every possible set of forces and parameters represented in the multiverse there would have to be not merely a countably infinite ensemble of worlds but rather a continuous infinity of worlds. It would far exceed 10^1000 which is usually stated as the upper bound for countable universes in the multiverse:

The universes in the multiverse cannot therefore be counted but would correspond rather to a smear on the number line. Such a multiverse is inconceivable.

Such observations are fun, but stories about a multiverse look more and more to be fairy tales. There is no experimental proof of parallel universes and many, including me, feel the infinite multiverse hypothesis is a fantasy built on soft sand by imaginative minds and speculative mathematics.

No physical proof exists.

Moreover, the notion of infinity creates all manner of absurdities. Mathematician David Hilbert (1862-1943) imagines a hotel, Hilbert's hotel, in which there are an infinite number of rooms:

Hilbert’s hotel has an infinite number of rooms labeled 1,2,3, etc. There is no vacancy in the hotel. All the rooms are occupied. Nevertheless, a room can be made available by moving the lodger in room 1 to room 2, the lodger in room 2 to room 3, 3 to 4, etc.

Doing so leaves room 1 unoccupied for a new guest. In Hilbert’s hotel, there is literally always room for one more.

In fact, a hundred rooms can be vacated in Hilbert’s fully occupied hotel. Move the occupant of room 1 to room 101, the occupant of room 2 to 102, 3 to 103, etc. Doing so vacates the first 100 rooms.

But here’s the real surprise. A countably infinite number of rooms can be vacated in Hilbert’s fully occupied hotel.

Every occupant looks at their room number, doubles it, and moves to that room. So room 1’s occupant is moved to room 2, room 2’s occupant is moved to room 4, room 3 to room 6, 4 to 8, etc. This leaves all of the odd numbered rooms, (1,3,5,…), empty so Hilbert’s hotel, despite being totally full, can still accept a countably infinite number of new guests!

Hilbert wrote that, “… the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought…”

Nevertheless, as the British mathematician and astronomer Bernard Carr once said, “If you don’t want God, you’d better have a multiverse.” There will always be those so determined to avoid the God hypothesis that they'll embrace any alternative no matter how bizarre, even the belief that the cosmological equivalent of Hilbert's hotel actually exists.