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 it 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.

Comparing Infinities

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. 

Read the rest at Mind Matters News, published by Discovery Institute’s Bradley Center for Natural and Artificial Intelligence.