Am I missing something about the OP’s question? Because the answer looks to me to be trivially simple (as I’ve already posted), but this thread has gotten pretty complex.
Well, first of all quantum mechanics isn’t about random shit poping in and out of the universe.
What you described is mostly possible. There could be a planet where the native sentient population by some quirk of genetics strongly resemembled Earthly Sqarepants. They could have a moon where they manufacture some kind of cheese-like products and be advanced to the point where travel to and from that moon was commonplace. They wouldn’t fly in swan-powered boats because by defination a boat is a water vehicle and an animal that resembled an Earth swan is not capable of space travel. Some things are impossible, not just highly improbable. Others are just a matter of semantics.
Yes. Things that defy the laws of physics. For example a creature biologically the same as human who flys like superman under his own power. Doesn’t matter what color sun you have, you could have an infinite number of humans and none will fly.
Popular Science did an artical about this very subject a few years ago. Basically, given an infinite universe and an even distribution of matter, theoretically you could have a planet identical to Earth somewhere out there. Problem is it could be a trillion trillion light years away. Given that the universe is estimated to be only about 11 billion years old, you can see the problems with that.
(Note - this post is merely in reference to a potential universe, not one which must occur; in other words it has no bearing on the OP’s question.)
Nothing wrong with the humans flying in a universe that allows for probabilistic variations in the position of particles. In this hypothetical universe the person simply takes a random walk up, up, and away, defying probability* but not possibility.
True, you can’t say it’s ‘under his own power’, but that’s more of a logical (or semantic) problem, not a physical one.
Or maybe even philosophical - posit a hypothetical universe in which the human thinks that he wills the flight, and that he came from another planet, and the more likely phenomena happen every time he is near a certain substance, etc.
Your point about ‘laws of physics’ may still stand, of course.
*in a colloquial sense of ‘going against that which is more likely to happen’.
If every possible universe existed, then what about “the only universe in existence”? If it exists, then by definition, there are no other universes. And if it doesn’t exist, then not every possible universe exists.
Or it’s not possibe.
If there is more than one possible universe, then there is no such universe as “the only universe in existence.”
-FrL-
If the letters are chosen randomly, then there will eventually be a C. It’s possible that it could take a trillion digits before it’s reached, but it’s impossible that it would never show up unless something is influencing it.
I don’t see why. Can you explain it?
It seems to me that if each individual one is chosen at random, then each individual one can fail to be C. So, given any string of letters which fails to include a C in it, it is possibe for the next letter to fail to be a C as well, producing thereby another string with no Cs in it. This can be repeated ad infinitum. So it seems possible for it to be the case that no matter how far out in the infinite string you go, you never get a case of the letter C.
(Note also, however, that I did not specify, nor intend to imply, that the sequence I was talking about was chosen at random. I don’t think the OP required this either.)
-FrL-
I guess we’re having a semantic issue over the meaning of “possible”. If we define possible as “a universe which can exist”, then it doesn’t matter how many universes there are - one, forty two, infinite - than all the universes that exist are the only possible ones that could exist and all the ones that don’t are impossible.
If, on the other hand, we define a possible universe as “a universe that can imagined or described” then we have a genuine question of whether or not all possible universes must exist in an infinite number of non-identical universes. Which is, I believe, the point of the OP. And by that definition, “the only universe in existence” is as valid a universe as “a version of Earth where everyone looks like Spongebob Squarepants and flies to the moon on weekends in boats pulled by zombie swans to eat moon cheese”.
No, it’s possible for C never to appear in the sequence. As I’ve said before, the probability of this happening is exactly 0 (assuming that the letters are chosen uniformly at random), but that doesn’t mean that it cannot happen.
As a nonmathematician, I ask: Is the probability zero, or is it “undefined?”
-FrL-
The second law of thermodynamics would be violated. I don’t know how to mathematically explain that though. The same would be true for a functioning volkswagon bug to randomly assemble itself from the raw elements. Even in an infinite universe, such a thing is impossible.
To my understanding, the second law of thermodynamics is probabilistic, not deterministic. Is that not true, though?
-FrL-
This is way out of my mathematical abilities, but I would say that understanding this is fundamental to the OP. Entropy is definitely an indirectly measurable quantity. My intuition says there are some probabilities so unlikely that even in an infinite universe, they do not occur.
The probability of C not appearing in the (infinite) sequence is defined, and it is exactly 0. Consider this: if you take a sequence of n letters, the probability of C not appearing is (2/3)^n (given that when we choose a letter in {A, B, C}, we have a probability of 2/3 of not choosing C). When n goes to infinity, this probability goes to 0. However, we do know that there is one (actually, infinitely many) sequences where C does not appear. Therefore, an event having probability 0 does not imply that it may not occur.
When dealing with infinite spaces, a lot of what people usually consider intuitive doesn’t hold anymore. Only when you develop a mathematical intuition does this not surprise you anymore. 
Well, as we’ve seen, even some things with probability 0 may occur. This said, I don’t know enough about physics to say if there are some things that simply may not happen, or if we live in a probabilistic universe and everything may happen, but is unlikely to.
That the limit goes to zero as n approaches infinity does not mean that the value is zero when n is infinity, does it? Can’t it be that the limit goes to x as n is approached, but the value at n is not x?
-FrL-
Indeed, what I did wasn’t a formal proof. I believe you can formalize it this way: call X_n the event that “no C appears in a given sequence of length n” and call X the event that “no C appears in a given infinite sequence”. We know that P(X) <= P(X_n) for all n >= 1; after all, if no C appears in an infinite sequence S, then no C appears in the subsequence S_n that we obtain by taking the first n elements of S. What this means is that P(X) is smaller than (2/3)^n for all n, but we also know that P(X) >= 0. Take any e > 0, we know that there exists a certain n such that P(X_n) = (2/3)^n < e. So P(X) is smaller than any positive number e, and so P(X) = 0.
As for your second question, you are correct if you have a non-continuous function. Consider the function f that is identically 0, except at 0, where f(0) = 1. Now take the sequence x_n where x_n = 1/n. This sequence goes to 0 as n goes to infinity, and f(x_n) goes to 0 (since f(x_n) = 0 for all n), but f(0) = 1. Now, are probabilities continuous? If X_n is a sequence of events that converges to an event X, can we say that lim(P(X_n)) = P(X)? I’m afraid I don’t know enough about probability theory to answer this, and I don’t even know if there exists a meaningful definition of the “limit of a sequence of events”. It seems compelling to say that, in my first paragraph, the event X is the limit of the sequence of events X_n, but I have no idea if it means anything.
Does this help you?
I haven’t seen this. I will admit, the mathmatics may be beyond my understanding. My intuition still tells me, that in an infinite universe, there are still some things that are too improbable to happen. Call it faith if you will. I just think some things are too unlikely to happen. I can’t prove it.
Mine too. That’s why I asked this question. I have a hard time believing that somewhere in the universe bowling balls levitate. However from what little I know of probability it seems like even something ridiculously improbable must happen somewhere if the universe is infinite.
If by that you mean something may be ‘too improbable to ever happen’ then it is almost by definition impossible, and can’t have a very low probability attached to it. What may be true, however, is that our scientific view of probabilities (e.g. with regards to quantum mechanics) is flawed. Because some events are so improbable, according to our theories, that we should never expect to see them in any reasonable time frame, we can’t be absolutely certain that there actually should be a probability attached to them. This would, however, be a violation of the mathematical elegance that the vast majority of our scientific observations align with (either that or we will discover some new branch of mathematics to apply if it ever becomes an issue).
This issue is separate from the idea that with a continuous (i.e. uncountably infinite) range of probabilities, some individual event has probability 0, even if it might happen.