Brun's theorem and the twin prime conjecture

Question

According to the following extract taken from Wikipedia, almost all prime numbers are isolated given Brun's theorem. Doesn't that mean that there is only a finite number of twin prime numbers (they form a set of density 0 across the natural numbers)? As this hasn't been proven yet, I can only assume the extract is wrong. Or have I misunderstood something?

I normally understand almost never as meaning finite but this has led me to thinking about the following: $\lim_{x \to \infty}{\frac{x}{x^2}}=0$ and according to the extended continuum hypothesis $(\aleph_0)^2=\aleph_0$ (Schröder–Bernstein theorem). So $x$ can be taken to be the cardinality of a set of density 0 and still be infinite. Unless we consider $x=x^2$ at the limit in which case the output gives 1. I believe there is a mistake in my understanding somewhere but I can't quite point it out.

Isolated prime

An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite.

The first few isolated primes are

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, ... OEIS: A007510.

It follows from Brun's theorem that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold n and the number of all primes less than n tends to 1 as n tends to infinity.

Answer 1

The quoted Wikipedia passage gives the exact sense of "almost all" which is intended. To rephrase: for any $n$, we can look at the number of primes less than $n$ (call it $p_n$) and the number of isolated primes less than $n$ (call it $q_n$). Then the statement is that $\lim_{n \to \infty} q_n/p_n = 1$.

As you suggest, this essentially says that the twin primes have density zero in the primes. But this is a much weaker statement than saying that there are only finitely many twin primes. As an example, the density of square numbers in the positive integers is zero: the ratio of square numbers to all numbers up to $n$ is about $\sqrt{n}/n = 1 / \sqrt{n}$, which tends to zero as $n$ tends to infinity. But that doesn't mean there are only finitely many square numbers.