Yes, one infinity can be bigger than another.  I cannot explain why precisely enough in order to try here but it is proven mathematically that this is true and it was explained to me by a philosopher of math.

Your question does have a definite mathematical answer. It comes down to the definition of magnitude, and the result was first obtained by Georg Cantor sometime around 1900 or so as I recall. (Until then, the orthodox answer was "infinity is infinity, and we shouldn't try to think about it".)

Cantor's approach was to ask "what does it mean for two numbers to be the same"? Of course, being a mathematician, he was concerned with sets, not with numbers per se. So he first asked "what does it mean for two sets to have the same cardinality?" Once he had an answer to that question — in brief, the two sets have equal cardinality if there exist a pair of one-to-one functions which map back and forth between the elements of the two sets — he asked the question "are there infinite sets with different cardinalities?"

He spent quite a long time thinking about this. The simple infinite sets, as it turns out, all have the same cardinality, which is usually known as ℵ0. (An aleph followed by a zero in subscript, which I was taught to read as "aleph sub naught".) That's the "number" of positive integers.

Obviously, you can double each integer in that set and get the set of even positive integers only, so there must be "the same number" of even positive integers as there are positive integers. And similarly the same number of odd positive integers, since you can subtract one from each even integer to get the set of all odd integers. And if you add or subtract some finite number of new entries, you can just bump everything over and recount, it means k + ℵ0 = ℵ0 for every integer k. (And, since the set of all positive integers is identical to the set of all even integers combined with the set of all odd integers, that means 2ℵ0 = ℵ0. And using some basic arithmetic and identities, kℵ0 = ℵ0 for any positive integer k. Nifty, eh? Unfortunately, for technical reasons subtracting ℵ0 is undefined — you aren't allowed to just subtract cardinalities in set theory, you have to say what you're removing from the set and what order you're doing it in. So ℵ0 - ℵ0 has no defined value.) The set of all integers (both positive and negative) also has cardinality ℵ0 because you can interleave the positives and negatives as though they were odd and even, so the case is analogous to the set of all positive integers.

Cantor was, by his own report, about to give up and decide that all infinite sets had the same cardinality after all, when he discovered a counterexample. Specifically, he found a proof by contradiction that there are "more" real numbers than there are integers. It's not too difficult to follow, either, in modern form:

Suppose that there are the same "number" of positive integers and real numbers between 0 and 1. This means that there is a function f such that for any positive integer X, f(X) is a real number between 0 and 1, and that if we start at X = 1 and work upward, sooner or later we will hit every real number between 0 and 1.

Now, suppose I define a number as follows: a real number whose decimal representation begins "0.", and then digit N after the decimal point is one more than digit N after the decimal point of f(N), unless that digit of f(N) is 9 in which case the digit of the new number is 0. (Strictly speaking, the real version of this proof is more rigorous, but you can think of it this way.) This number is definitely between 0 and 1, but it will be different from every single possible output of the function f by at least one decimal place. Therefore no such function can exist, and so there must be more real numbers between 0 and 1 than there are integers, because there is at least one real number in the range of 0 to 1 which the integers can never map. (And you can define an infinite number of other numbers which are likewise not in "the list" — it's not like there's only one of them!)

(It isn't hard to show that there are the same number of real numbers between 0 and 1 as there are real numbers, using a simple euclidean geometric construction. Draw a line. Now draw a circle — ideally of radius 1/π — whose center further away from the line than the length of the radius. The half of this circle which is between its center and the line has a finite arc length — 1, in fact, if the radius is 1/π — so the semicircle corresponds to the segment between 0 and 1. The line — as usual — corresponds to the real numbers, stretching off infinitely in both directions. But for every point on the line, there is one and only one line which can be drawn through that point and the center of the circle. For each point on the semicircle, there is also one and only one line. This provides a direct mapping between the infinitely long line and the finite semicircle. The real numbers are just weird.)

(Or, if you don't like geometry, the trigonometric tangent function provides a direct, invertable mapping between a limited range of real values and the full range of real numbers. I just like the visual.)

It all depends on what you mean by infinity. In the philosophy of mathematics this notion is generally taken to be of two kind: potential and actual. A potential infinity is sort of like the idea that we "could" have an infinity, but it never really exists an an entity itself. We see this in calculus with limiting processes. The sequence of numbers 1/n becomes zero as n goes to infinity. This is a limiting process, we take the sequence and see 1, 1/2, 1/3, ..., 1/10000, 1/10000000, etc. There is no actual number infinity involved. There is no entity that exists. This is just a process of "keep on going" and often denoted by that "..." we see in sequences that just "keep on going."

In contrast, the whole Cantorian set theory of transfinite numbers deals with how we construct actual infinities. The idea is rather genius, but stems from rather simple intuitions. Vicar already explained much of this, but I will restate it differently. As already said, we need a formal definition of size, and that requires us to develop a way to compare two entities (sets). What does it mean for {1, 2, 3} to be the same size as {a, b, c} or {bob, joe, frank}? A way that did not require anything beyond the existence of those sets themselves was to make a comparison function: a one-one correspondence.

Consider a herder wanting to count his flock but was not schooled in arithmetic. The man could not actually count his sheep, say. He could, however, compare the size of his flock to the size of his jar of stones. If the jar is filled with exactly one stone for each sheep, then all the man needs to do when counting his flock is move one stone from one jar into another as each sheep enters the pen. In this way, he can know precisely if all his sheep are in or not by whether or not any stones remain (or he comes up short).

This same simple intuition is precisely how we define sizes of sets. We need a way to compare the two entities. It simply means we need a way to define a relation between the two entities. In the case of the three element sets above, we can state it that {bob, joe, frank} is equal in size to {1, 2, 3} which is equal in size to {a, b, c}. They are the same size precisely because we can transform any of those sets into the other one by simply renaming the elements, e.g., sending bob to 1, or 1 to a, etc.

We can do this same thing for infinite sets, such as the natural numbers. What is the difference between {0, 1, 2, ...} and {5, 6, 7, ...} and {... -3, -2, -1, 0, 1, 2, ...} and {0, 2, 4, 6, ...} in terms of size? Nothing! We can find ways to rename these elements to look like any of the others, for instance, the first one is we simply add 5 to every element, i.e., we send 0 to 5, 1, to 6, and 2 to 7, etc. In the third one we can send 0 to 0, 1 to -1, 2 to 1, 3 to -2, etc. We simply make all even numbers the positive numbers and all odd numbers the negative numbers. The last one is also simple, we send 0 to 0, 1 to 2, 2, to 4, 3 to 6 and witness we're just doubling every number. We can define functions for all of these precisely and this demonstrates they are all the same size of the natural numbers N = {0, 1, 2, ...}.

This size is known as the countable size (aleph nought), whereas the uncountable is something like the size of the real numbers, as already pointed out. The proof for their difference is a beautiful proof known as diagonalization, and a proof by contradiction. Suppose we have a countable listing of all the real numbers between 0 and 1. Since it is countable, we can list all of these (uncountable sets cannot be listed because "listing" is a method of spelling out all their members, and then we could just assign natural numbers to each one). Say it is something like

0.abcdefg...
0.a'b'c'd'e'f'g'...
0.a*b*c*d*e*f*g*...
...
0.a^b^c^d^e^f^g^...

In this way, every sort of a, b, c, d, e, f, g, ... is a number between 0 and 9 for that decimal spot. This listing goes on forever downward and to the right, say. It may be infinite but we say that we have listed every real number and that each listing is unique. Even thought his is an infinite listing, it is not enough to capture all the real numbers since on the diagonal there exists another real number not in that listing. In particular, the number is

0.ab'c*...x^...

This number is different at some decimal spot from all the other numbers in that listing making it a unique real number between 0 and 1 that was not counted for. Therefore, the countable listing is not big enough. If it were not unique, then there would have to be a double entry of a number in our listing somewhere, violating our original assumption that the countable listing is complete with unique entries. It is quite an elegant proof that shows up in other areas of mathematical foundations. 

Therefore, when it comes to mathematical ideas we have a definite example of two different kinds of infinities being used, and we can say precisely that the countable infinity is less than the uncountable infinity. We can go even further, though, and show that there are an uncountable infinity of uncountable infinities! I allude to this in my [ordinal blog]. An ordinal is a number that represents, well, order! Not only can we transform {0, 1, 2} to {a, b, c}, but if we have an ordering such as the natural numbers so 0 is less than 1 is less than 2, then by sending 0 to a, 1 to b and 2 to c, we're saying a is less than b is less than c. If we send 0 to c, 1 to b and 2 to a, then c is less than b is less than a. Obvious, right? Well, it is the same with infinities. I give some examples. We say when we do this that we're ordering a given set by a given order type. So if we order the set of integers to the natural numbers, denoted by ω, then we've order the integers with the ω-th order type. We use the "-th" to indicate a type of order, just like we talk about being on the 5th page of a book, and that the 4th page is less than the 5th. This "-th" specifies order. In that, we can compare their size. This gives us a whole host of sets between sizes because, as I show in that ordinal blog, there are a LOT of ordinal infinities, and they're all countable! That is to say, when we take the set ω as infinite, we can also add one to it, say ω+1. This doesn't mean we're adding the number one to it. It means we're appending the entire set of natural numbers to the natural numbers (i.e., ω+1 is equal to the union of ω with the set containing ω, which in turn means that we have {0, 1, 2, ..., {0, 1, 2, ...}} ). So, ω+2 has "two copies" of ω in it, and so on.

This is called transfinite arithmetic because we're dealing with arithmetic just like with natural numbers, but with infinities! 1+ω is just ω, just like the cardinal arithmetic Vicar showed. We cannot subtract just like you cannot subtract in the natural numbers. The reason is because "+" is not an operation like we're thinking of it. As already said, it is a union of the whole entity back into itself. This comes from the very [construction of the natural numbers].

0=∅
1={∅}
2={∅,{∅}}
3={∅,{∅},{∅,{∅}}}
4={∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}

or more succinctly

0={}
1={0}
2={0,1}
3={0,1,2}
4={0,1,2,3}

Thus, with ω = {0, 1, 2, ...}, then ω+1 = {0, 1, 2, ..., ω} = {0, 1, 2, ... {0, 1, 2, ...}}, and ω+2 = {0, 1, 2, ..., ω, ω+1}, and so on.

We can continue this process to ω+ω = ω2, and continue all over again with ω2+1, ω2+2, ω2+2 = ω3. I show this graphically in the ordinal blog.

We can continue constructing all these infinities out to ω to the power of ω to the power of ω, ω-times! And guess what? ALL of these are still countable! We can, in theory, find a one-one correspondence with these back into ω. We can order ω^(ω^(ω^(...))) (i.e., ω exponentially to ω, ω-times!) to the natural numbers because there is always room in ω to match it to one of the elements in that insanely huge beast of a set. But all of these are still less than the first cardinal.

Now, to really blow your mind, consider the fact that aleph nought is only the first cardinal. There is aleph sub 1, aleph sub 2, as well. These are all uncountable. We can keep on going! There is aleph out to ω! There is ω2 alephs, there is ω to the ω to the ω, ω-times alephs! Guess what? There's aleph sub aleph-nought size! We can just keep on going on this process. Just imagine what kind of set would exist that has that size. You can't! It's not fathomable at this point, and we're still not at the largest sets because there are sets requiring new axioms just to recognize them as entities. These are known as "large cardinals". There are technical reasons why we need new axioms to specify what these entities are, but I wont get into that here. The point is that even on this process we cannot spell out all the infinities, and yet we can still order all of them such that one infinity is less than or greater than another.

This has been an exercise in mathematical thought, but it requires that we have a definite idea of infinity, and ordering. Ordering in this sense has been nothing more than inclusion. 0 is less than 4 because, as shown, 0 is IN the number 4, just like ω is less than ω+1 because we can see that ω is IN ω+1.

Interesting aside, even out to the aleph sub aleph-nought, there is a finite trek "back" to 0 because all the numbers before it are IN that set, and we can see that if we reversed the process, there is a finite end. There is a "cap" and it is 0. For instance, out to ω+2 = {0, 1, 2, ..., ω, ω+1} = {0, 1, 2, ..., {0, 1, 2, ...}, {0, 1, 2, ..., {0, 1, 2, ...}}} we can still "walk" back to zero because we can "jump" over the "..." from any of these infinities onto the "lower" level. If we use that spiral ordinal graphic I used in my ordinal blog, think of it as a staircase. The "jump" is going from one infinity step to the lower, and eventually we will get to the bottom, no matter how high we climb. Crazy.

Anyway, no one has to accept this notion of infinity or that actual infinities even exist in any substantial sense. The "existence" here is mathematical existence. It means that given our mathematical [structure] we can devise a way of constructing the entity. Now, there is a philosophical issue in foundations here, often in terms of the constructivists vs the realists (platonists). The realists want to think the entities exist independent of our knowing them and that we discover the truths with our mathematical "sight". The constructivists, on the other hand, say it just doesn't exist until we construct it, with the extension that all of mathematics is just a "game" of "symbol manipulation." While there is truth in both cases, the fact of the matter is that mathematical existence still requires that we can construct the entity in some important way. Even in set theory there are times when we refer to an entity that hasn't been specified specifically. We refer to it arbitrarily. The constructivist might have a problem with this because we don't know if the entity exists or that our reference is indistinct by not pointing to anything in particular, but still referring to definite properties. Important issues, but there is no "independent existence" of mathematical entities. They are simply logical consequences that we do, in fact, discover because we have limited knowledge of the breadth of any given mathematical system.

In any import sense, talking about ordering infinities is meaningless because it is always contingent on the system by which we construct those ideas. There are set theories that have no infinities because they refuse certain axioms that Cantor accepted. In reality, there may be no infinity. One professor asked "is the real world represented by the real numbers or by rationals?" It is interesting because we use the real number line, but who says there exists in reality an irrational number? Even if it does, is the world represented by the real numbers of hyperreal numbers? (where a hyperreal is an infinitesimal number smaller than any real number) The fact of the matter is that the world is none of these. How the world is represented in the abstract is just that, an abstraction! It is not the real world itself. When I want to represent the world by rational numbers, I do so justifiably to the extent what I am representing is corresponded by rationals. The same goes for reals or hyperreals. So does the world have infinities? Only if we can demonstrate it is justifiably represented by it, but that is an abstraction, not that it actually exists in the world. It is rather common that we take the world to not have ACTUAL infinities. Time is about the only thing we can think of that has an infinite structure, but it is still only a potential infinity in that time just "keeps on going." We thought at one time the universe might be infinite, but modern theories present it as limited, defined and having boundaries. So when we are concerned about whether one infinity can be bigger than another, it has to do with "what do you mean by infinity?" First, to compare them they need to be actual infinities. Then, you'd still have to make a definite idea of that infinity. In terms of potential infinities, there is no comparison. 1/n is equal to 1/2n because in the process of going out to infinity, we're not specifying numbers, we're still working within the domain of where n comes from, and in this case that is a natural number. Domain is everything, because it specifies the structure we are talking about.

@WAR_ON_ERROR - I'll have to give a more formal analysis when I find time, but right now I suspect that the probability of being in a simulated world would go nearly identically to 1. In short because the probability space needs to sum to one, by definition of being a probability measure. If we think about the cases successively, in each case we have one world to ten simulated worlds. Well, if we continue this over infinite worlds, then the ten simulated worlds to one world will continue to grow at a much larger rate than that of the one real world. In other words, the one real world will converge to zero and the simulated worlds will converge to 1, not half and half. It really depends on how we set up the scenario, what kind of functions we use to represent them, etc. No where in this does transfinite arithmetic or actual infinities matter because this is straight calculus (real-valued functions with a probability measure). It may be the case, if set up a certain way, that each converges to half. It might also converse back to a 1:10 ratio, or it can go to a 0 and 1. I think the most accurate representation would be the 0 and 1 because on infinite trials the addition of one more real world to infinitely more simulated worlds contributes less and less as the observations grow toward infinity. For that reason, the "growing less and less" is what a convergence to zero represents, like the series 1/n converging to zero. There is nothing that appears in the scenario that warrants a lower bound for it to stop at half, nor a property of the gross-more simulated worlds dropping to half. It also doesn't make any sense that we would maintain the same ratio in the limiting process because the growth of the simulated worlds simply dwarfs the real worlds. Thus, that kind of behavior elicits, to me, that the probability of the real worlds will go to 0 and the simulated worlds will grow to one. How to represent that formally will take a bit of work and getting the ideas ironed out precisely, but it has to be that it all sums to one. 

@WAR_ON_ERROR - I have to change course and say Carrier might be correct that the proportion of real worlds to simulated worlds won't change as we go to infinity because there's nothing to alter the behavior of these probabilities. Our chance of ending up in one world versus another remains the same because the proportion of real worlds to simulated worlds remains the same in terms of likelihood. Think of it this way, just because America has ten times the population of Canada doesn't mean our life expectancy should be ten times worse, as Bill O'Reilly tried to have us believe (link). The reason he was wrong is that he thinks the measure of life expectancy is impacted by population size. It isn't. It's a rate. That means it involves the relative size of the countries. Likewise, the probability measure involves the relative sizes of the samples, even as they go toward infinity. While we might say that if we do deal with actual infinities, we'd have a different issue, but that would mean we'd need to talk about a transfinite measure and a transfinite probability, both of which are beside the issue and rather different topics altogether. The potential infinity characterized by the limiting process of taking a probability over a metric space to infinity deals with real numbers, just infinitely many of them. Thus, the probability wont change.

I like numerical examples, and you could try to download R and get an idea about this stuff.

Consider the code

foo <- sample(c(0,1), 1000000, replace=TRUE, prob=c(0.1, 0.9))
length(foo[foo==0])/length(foo[foo==1])

What this does is grab a 1 million sized sample from our space of 0s and 1s with the given probabilities, and then looks at the proportion of 0s to 1s, which represent real and simulated worlds, respectively. Clearly it should be something similar to 0.1 as specified in the probability assignments (i.e., the concatenated vector c(0.1, 0.9) ). You can then run the code

size <- 10000
bob <- NULL
for(i in 1:size) {
  foo <- sample(c(0,1), i, replace=TRUE, prob=c(0.1, 0.9))
  bob[i] <- length(foo[foo==0])/length(foo[foo==1])
} ## end loop
plot(bob)

to see the convergence, though it might take some time to run with larger size values. What the extra baggage around the original code does is define a storage variable (named bob), the limit value you want to get to, and then it takes samples of the ith size and looks at the proportions as that size goes toward the given limit. It then plots those values out of the storage variable bob.

What I got was a graphic that shows everything clustering largely around 0.1 out toward 10,000 trials, but with some scatter, especially toward the beginning, which should be expected for small sample sizes at that point (i.e., larger error or confidence intervals). It also shows that it doesn't take that much to get that convergence since the "spread" quickly drops away (like, within the first couple hundred!). I recommend trying a size of 2,000 though. It really shows the variance in the beginning and how everything converges toward the probability. If we continue this process taking size to infinity, then we'd get the same result, but probably even less variance as we get out to the tip. In fact, the limiting process would have it converge exactly back to its probability.

We can think of it this way, if the probability of a real world is p and the probability of a simulated world is p', then with our space being of only real worlds (0) and simulated worlds (1), we have it necessary that P(real world OR simulated world) = 1, but we can reasonably assume that the probability of getting in one of these worlds is mutually exclusive and independent. Thus, P(real OR simulated) = P(real) + P(simulated) = p + p' = 1. In the code I used p = 0.1 and p' = 0.9.