Is an infinite set of random real numbers equivalent to the Reals?

Question

Edit for clarity: I'm going to create a set of Real numbers. To do so I start with a copy of the Reals, then for each element I replace it with a number randomly selected from the Reals (equal weighting). Is every element of the reals represented in my randomized set? The stuff about the multiverse was added for context and has no bearing on my actual question. I hope this helps clarify the question, though maybe it is still poorly defined due to some concept I'm missing. Thanks for the comments so far!

Original post: If I randomly generated an (uncountably) infinite set of real numbers, would the Reals be a surjection of this set? Put another way, if I randomized every element of the Reals, would I get back the Reals?

I can easily think of an infinite set that isn't the Reals; the Reals but without the number 7 is still uncountably infinite, but 7 is in the Reals and not my set.

I don't really have any intuition here- on the one hand I have an example of an infinite set that doesn't contain all the Reals, but I also think this might just be an example of really not being able to comprehend what infinite means.

Some context for those interested (what follows is not relevant to understand the question): I'm a physicist and the other day I was hanging out with a bunch of other physicists and we were talking about multiverse theory. The general question arose- if there are an infinite set of universes, is every (physically possible) alternative represented? Statistically speaking it's possible that all the air molecules might just leave NYC, so there would be a universe where that happened. In fact there'd be a universe where that happened every Thursday. If you don't like that example, think of anything else unlikely, it's not important. But I started thinking about the multiverse as a set of elements and thought it need not necessarily map 1 to 1 and onto the Reals (or Natural numbers or whatever else depending on what size of infinity the multiverse might turn out to be...). I figured the multiverse would have been a statistical process as well, right? Not a curated one? After all, in the Reals, there's one of every element- why all? why not multiples? Wouldn't it be possible, if not statistically reeeeeally improbable, of generating a set with no 7s? With only 7s?

I guess what I'm trying to ask... is there a universe where I'm president? :)

Answer 1

As mentioned in comments, there is no such thing as a uniform probability distribution on $\mathbb{R}$. But we could use $[0,1]$ instead, in which case the uniform distribution is Lebesgue measure.

So here is one way to precisely formulate the question: consider the set $\Omega = [0,1]^{[0,1]}$ of all functions from $[0,1]$ into itself, equipped with the uncountable product $\sigma$-algebra $\mathcal{F}$. Then there exists a probability measure $P$ on $\mathcal{F}$ which is the uncountable product of Lebesgue measure with itself. If we let $X_t(\omega) = \omega(t)$ be the random variable on $\Omega$ which evaluates a function $\omega$ at $t$, then under the measure $P$, the random variables $X_t$ are iid uniform on $[0,1]$. So far so good.

Now let $A \subset \Omega$ be the set of all functions which are surjective (onto). Thus an outcome in which "every number is represented" would correspond to an $\omega \in A$, and so the question becomes: does $P(A) = 1$?

Unfortunately, the answer is that $A$ is not a measurable set (it is not in the $\sigma$-algebra $\mathcal{F}$), and therefore $P(A)$ is undefined. The basic issue is that an uncountable product $\sigma$-algebra is too small; we can only talk about the probabilities of sets that only "look at" countably many values of a function $\omega$. But clearly we cannot tell whether a given $\omega$ is surjective by only looking at $\omega(t)$ for countably many values of $t$.

Basically, probability theory has no really good way to deal with uncountably many independent random variables. But on the other hand, it's not really clear that this would be a good model for anything in the real world anyhow. Most actual random processes can be reduced somehow to countably many independent trials.

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A much better-behaved model, which is also of great interest in physics, is something like Brownian motion. Asking about the position of a Brownian particle at time $t$ again gives you an uncountable family of random variables, but this time they are not independent; indeed, because a particle moves continuously, its position $B_{t+\epsilon}$ at time $t+\epsilon$ is much more likely to be close to $B_t$ than far away. But it is meaningful to ask whether a Brownian particle will visit every point in space, given enough time; this is the question of whether it is transient or recurrent.

The answer is that Brownian motion on a line is recurrent (it visits every point uncountably many times); in a plane it is point-transient but neighborhood recurrent (it will not visit every point, but it will come arbitrarily close to every point infinitely many times); and in three-dimensional space it is transient (for each point $x$ and each sufficiently small $\epsilon$, there is a nonzero probability that it will never get within distance $\epsilon$ of $x$). This is a famous result and is proved in nearly every textbook that discusses Brownian motion.

Answer 2

This question

If I randomly generated an (uncountably) infinite set of real numbers, would the Reals be a surjection of this set? Put another way, if I randomized every element of the Reals, would I get back the Reals?

is not precise enough to have a precise answer.

What is true is that any uncountable subset of the real numbers (however you specify it) has the same cardinality as the set of real numbers, so there is a bijection between them. In that sense, yes, you do "get back the reals". But that bijection is likely to be wildly discontinuous, and won't respect any arithmetic or physics you want to transfer from one "universe" to the other.

Thinking about random real numbers or random sets of real numbers is hard to get right.

The multiverse question is part mathematics, part physics and part philosophy. Mathematically, you need a structure with infinitely many universe objects (themselves mathematical abstractions). That number of universes might be uncountable, or even larger. Physics would inform the mathematical rules governing that structure.

This question

I guess what I'm trying to ask... is there a universe where I'm president? :)

belongs on some other stackexchange site. Maybe there's one in area 51.

Edit in response to this new information in the question.

I'm going to create a set of Real numbers. To do so I start with a copy of the Reals, then for each element I replace it with a number randomly selected from the Reals (equal weighting).

There is no good way to "choose a random real number with equal weighting", let alone to do that once for each real number. But suppose you could, somehow. That would create a "random" function $f$ from $\mathbb{R}$ to itself. You seem to be asking whether $f$ would be surjective. I would expect that to happen with "probability $0$". Getting rid of those quotation marks and making a rigorous argument would depend on solving the difficult problem involved in constructing $f$ in the first place.

If you choose a random function from a finite $n$-element set to itself the probability that the function is surjective tends to $0$ as $n$ increases. That's because there are $n^n$ functions but only $n!$ bijections.