Cosmologists’ problems with aleph-null and the multiverse

This is another insight from Brian Greene’s book The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos – well worth reading.

Aleph-null ($\aleph_0$) is the order (size) of the set of countably infinite objects. The counting numbers are the obvious example: one can start from one and keep on going. But any infinite set where one can number the members has the order of $\aleph_0$. (There are other infinities – eg that of the continuum, which have a different size of infinity.)

It is the nature of $\aleph_0$ that proportions of it are also infinite with the same order. So 1% of a set with the order $\aleph_0$ is also of order $\aleph_0$. To understand why, think of the counting numbers. If we took a set of 1%, then the first member would be 1, the second 101, the third 201 and so on. It would seem this set is $\frac{1}{100}^{th}$ of the size of the counting numbers, but it is also the case that because the counting number set is infinite with order $\aleph_0$, the 1% set must also be infinite and have the same order. In other words, if paradoxically, the sets are in fact of the same order (size) – $\aleph_0$.

The problem for cosmologists comes when considering the whether we can use observations of our “universe” to point to the experimental provability of theories of an infinite number of universes – the multiverse.

The argument runs like this: we have a theory that predicts a multiverse. Such a theory also predicts that certain types of universe are more typical, perhaps much more typical than others. Applying the Copernican argument we would expect that we, bog-standard observers of the universe – nothing special in other words – are likely to be in one of those typical universes. If we were in a universe that was atypical it would weaken the case for the theory of the multiverse.

But what if there were an infinite number of universes in the multiverse? Then, no matter how atypical any particular universe was (as measured by the value of various physical constants) then there would be an infinite number of such a typical universes. It would hardly weaken the case of the multiverse theory if it turned out we were stuck inside one of these highly atypical universes: because there were an infinite number of them.

This “measure problem” is a big difficulty for cosmologists who, assuming we cannot build particle accelerators much bigger than the Large Hadron Collider, are stuck with only one other “experiment” to observe – the universe. If all results of that experiment are as likely as any other, it is quite difficult to draw conclusions.

Greene seems quite confident that the measure problem can be overcome. I am not qualified to pass judgement on that, though it is not going to stop me from saying it seems quite difficult to imagine how.