Are the implications of an infinite universe necessarily so unsettling

Question

I have often heard it said (by professional cosmologists) that if the universe is infinite, then there necessarily exist infinitely many copies of me repeated throughout.

The reasoning seems to be that any finite volume of space can contain at most finitely many fundamental particles existing in finitely many configurations. From here it is implied that all possible (permissible) configurations must both exist and occur infinitely often.

This does not sound like a sound argument to me, so I have to believe that I have misunderstood the argument.

Is this unsettling implication true in an infinite universe, and if so how is it justified.

Answer 1

The argument is sound given a few oft-omitted (but not too unreasonable) assumptions. Here is one way it can be formulated.

Consider a volume $V$. Suppose it has a (possibly infinite) set of possible configurations; call this set of states $S$. Suppose we are interested in a particular configuration, $c \in S$, to within a certain tolerance. Let $C \subseteq S$ be the set of all states close enough to $c$ to count for our purposes.

Assumption 1: There is a probability measure $\mu$ on $S$ corresponding to the probability of $V$ manifesting in a particular state in some selection process I'm intentionally being vague about.

Assumption 2: $\mu(C) > 0$. That is, there is a strictly positive chance of a "randomly" (again, being vague) selected state matching our desired configuration.

Assumption 3: There exists a "horizon distance" $d$ such that if two volumes are more than $d$ apart, their states are entirely independent.

Assumption 4: The universe is infinite.

Assumption 5: The universe is homogeneous. In particular, $S$ and $\mu$ are the same for any $V$ chosen.

(1) means we can meaningfully talk about the chances a state $s_i$ manifests in a volume $V_i$. (2) and (5) tell us that for any $V_i$, $P(s_i \in C)$ is a constant, positive number. If we define the indicator variable $$ \chi_i = \begin{cases} 1, & s_i \in C \\ 0, & s_i \not\in C, \end{cases}$$ then the expectation of indicator, $\langle \chi_i \rangle$, is this same positive number. (3) and (4) together mean there are infinitely many uncorrelated volumes $V_i$ in the universe (in addition to possibly many other volumes correlated with these) to choose from. If $I$ indexes finitely many mutually uncorrelated $V_i$, then we can see $\langle \sum_{i\in I} \chi_i \rangle$ can be made arbitrarily large by augmenting $I$.

Of course, no one is asserting the claim "there are infinitely many copies of you in existence" as fact, because these assumptions can always be questioned, some in important ways.

(2) seems justified based on your existence, and (5) is a common assumption about the universe.¹ But (1) really calls for more than a little bit of philosophy, and (3) and (5) together worried enough cosmologists in another context that they came up with inflation to essentially rid themselves of (3). And of course (4) is certainly not known, and strict scientific positivism would say that sentence isn't even deserving of being called science, for it is fundamentally untestable.

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¹ I find the history of 20th century cosmology interesting, in that homogeneity was assumed/hoped for before it was validated observationally. After all, most of the universe doesn't look like Earth, or the Solar System, or the Milky Way, or the Local Group, etc. Only with really big galaxy surveys did we see an end to the hierarchical structure.

Answer 2

The argument rests on the assumed validity of Ergodic theory (see http://en.wikipedia.org/wiki/Ergodic_theory). Quoting it "A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set". Thus, unless you can assume that a physical mechanisms is violating the ergodic assumption, then if the universe is infinite, there necessarily exist infinitely many copies of it repeated throughout. Actually, there is an extra assumption (by symmetry?) to get this last assertion: that the ergodic property also holds in a collection of separated systems at a single time, not only in a single system across time. The reason is that if the recurring states across time are random, then you get the same result if instead you have an infinite random collection of "disconnected" regions in the universe (the distribution of states across time should be no different than the distribution of many samples at a given time).