Yes, by Post's theorem, for every $\Sigma^0_{m+1}$ formula $\phi(x)$ there is an oracle Turing machine $e$ such that, for every $i$, $\phi(i)$ holds if and only if machine $e$ halts on input $i$ when run with oracle $\emptyset^{(m)}$. Here $\emptyset^{(m)}$ is the $m$th Turing jump of the empty set, and is a $\Sigma^0_m$ set.
In general, Post's theorem shows that there is an extremely close connection between the arithmetical hierarchy, computability, and the sets $\emptyset^{(m)}$. The claim in the Wikipedia article is just one part of this connection.
Start with the negation of the twin primes conjecture. The negation is $\Sigma^0_2$ and thus by Post's theorem is equivalent to the claim that some particular oracle machine halts when it is run with $\emptyset'$ an oracle. Since there are no free variables, that machine does the same thing regardless of input - it ignores the input.
In this particular case, what the machine does is to search for the least $k$ such that there is no pair of twin primes greater than $k$, and then halts if it finds such a $k$. The oracle $\emptyset'$ is able to decide the set $\{ k : \text{ there is a pair of twin primes greater than } k\}$ because that set is $\Sigma^0_1$ and $\emptyset'$ is $\Sigma^0_1$ complete. So the oracle machine is able to use the oracle $\emptyset'$ to repeatedly test whether each $k$ is in the set and then halt if it finds a $k$ that is not in the set.
The the original twin primes conjecture is then equivalent to the claim that that machine does not halt when run with an oracle for $\emptyset'$. If it never halts, that means it never finds a $k$ such that there is no pair of twin primes greater than $k$, so there are arbitrarily large pairs of twin primes, and vice versa.