Can anything happen if there are an infinite number of universes each with an infinite number of possibilities in each? Can you be bald in one universe and fully haired in another? Can you have two eyeballs in this universe and three in another? The answer is no. In a nutshell, the reason is that some infinities are bigger than other infinities. (And this is not a claim like infinity plus one is bigger than infinity. Infinity plus one is still infinity.)

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.) And, no, true infinity is not the same as the symbol ∞. In mathematics, ∞ typically means “increasing without bound.” And no matter how high you count, you still have infinity to go.

The number of points on a line segment — the bigger infinity — can be referred to 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.)

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. It also begs the question of where our universe, counted by us as universe #1 in the multiverse, fits in this uncountable continuum.

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.

Like the multiverse, true infinity is a mathematical construct. Mathematician extraordinaire David Hilbert (1862–1943) said it succinctly: “… the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought…”

Infinity is conceptually weird. Assuming that it is a reality leads to absurdities as illustrated by Hilbert’s hotel. 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!

Such ridiculous situations commonly arise when considering the infinite. Isaac Newton recognized such an absurdity when he showed that an assumption of infinity can lead to the ridiculous conclusion that a foot was equal to inch in length. Newton wrote:

“…if an inch may be divided into an infinite number of parts the sum of those parts will be an inch; and if a foot may be divided into an infinite number of parts the sum of those parts must be a foot; and therefore, since all infinities are equal, those sums must be equal, that is, an inch equal to a foot.”

The infinite does not exist in reality, only in our minds.

So any claim of an infinite number of universes has no foundation in naturalistic reality. If there were an infinite number of universes each with a finite number of possibilities , all contingency combinations would be possible.

Are there implications here regarding the existence of God? Georg Cantor who pioneered the math that showed some infinites are larger than others, thought so. Cantor sought an audience with Pope Leo XIII to discuss the theological implications of his theory.

But we will leave the discussion of the theological implications of infinities to another time.

You may also wish to read: Yes, you CAN manipulate infinity — in math. The hyperreals are bigger (and smaller) than your average number — and better! Hyperreal numbers are a new type of number that was developed to simplify and rethink the way that we deal with very large and very small numbers. It reduces the complexity of the task and allows us to use our well-honed high-school algebra skills to solve complex problems easily. (Jonathan Bartlett)

and

The jump of Chaitin’s Omega number: Gregory Chaitin explains, “For any infinity, there’s a bigger infinity, which is the infinity of all subsets of the previous step.” Chaitin tells us, “In the mathematics that I would say is discovered not invented, you feel you’re touching a reality beyond normal reality.”