The Infinite Multiverse And Monkeys Typing Pi

The idea of a multiverse, once only a fantastic dream, has now found its way into everyday conversation. Part of the reason is that cosmic inflation seems to imply the possibility of other universes, string theory's 'extra dimensions' are another gateway to a multiplicity of worlds, and the very nature of quantum mechanics--at least in one of its interpretations--favors the possibility that our cosmos is not the only one. 
As physicists have come to accept the scenario of a multiverse, despite the lack of any convincing evidence for its existence both from a verifiable theoretical framework and from any kind of confirmed experimental or observational results, this concept has already been taken to its limit: infinity. But infinity has always been the scourge of physics: Any kind of 'infinite' answer to a calculation has always been taken as a sure sign that a theory had something very wrong with it. Lifetimes have been spent by theoretical physicists to correct such problems, that is, to 'renormalize' theories so that the nonsensical infinite answers would disappear: QED and QCD being the most well known renormalizable theories. And while many kinds of physics calculations often make use of 'potential infinity' through the infinitesimals of the calculus, no actual infinity has played a major role in the field. Not until now. The multiverse has almost invariably come to mean an actual infinite set of worlds. But what is its cardinality (or level of infinity)?When I asked this question of Brian Greene, a strong proponent of the multiverse, he told me: "It is the cardinality of the real line." When I posed the same question to Alan Guth, the inventor of the theory of cosmic inflation, he answered: "Aleph-zero" (This means that the number of universes in the multiverse is countable, meaning their level of infinity is lower than that of the real line and agrees with the cardinality of the integers and the rational numbers).
So who is right? I would favor Guth's answer, because it is more palatable to me. I personally do not believe in an infinite multiverse, so a lower order of infinity is easier for me to accept. If the universe is truly a multiverse, and it is really infinite in its number of member universes, then a more pedestrian order of infinity would suit me best. But it's important to understand why Greene would answer as he did. This has to do with the purpose you want your multiverse to fulfill.