There is a general way to make this sequence of independent sentences over any effective, essentially incomplete theory. A theory $T$ is essentially incomplete if it is consistent and no consistent computable extension of $T$ is complete. Q, PA, ZF, and ZFC are examples of effective, essentially incomplete theories.
Assume that $T = T_0$ is effective and essentially incomplete. Then since $T$ is incomplete, there is a sentence $S_0$ such that $T + S_0$ and $T + \lnot S_0$ are both consistent. Let $R_0 = S_0$.
Now $T_1 = T + \lnot S_0$ is effective, and it is incomplete because $T$ is essentially incomplete. So there is a sentence $S_1$ such that both $T + \lnot S_0 + S_1$ and $T + \lnot S_0 + \lnot S_1$ are consistent. Let $R_1 = \lnot S_0 \land S_1$. Then $R_0$ cannot imply $R_1$ over $T$, and $R_1$ cannot imply $R_0$ over $T$, because both $T + R_0$ and $T + R_1$ are consistent.
Now $T_2 = T + \lnot S_0 + \lnot S_1$ is incomplete, because it is consistent and $T$ is essentially incomplete. So there is a sentence $S_2$ such that $T_2 + S_2$ and $T_2 + \lnot S_2$ are both consistent. Let $R_2 = \lnot S_0 \land \lnot S_1 \land S_2$ and let $T_3 = T_2 + \lnot S_2$. Then none of $R_0$, $R_1$, $R_2$ imply the others over $T$, and $T_3$ is again effective and incomplete. Continue inductively to generate an independent sequence of sentences $R_0, R_1, R_2, \ldots$.
The phenomenon in this proof can be stated more abstractly as: the Lindenbaum algebra of any effective, essentially incomplete theory is atomless.
It is interesting to analyze what the sentences $S_0$, $S_1$, etc. all say. The proof that the theories named above are essentially incomplete is constructive, and so we can actually construct examples of the sentences $S_i$. We have to use Rosser's variation of the incompleteness theorem, because theories such as $T_1$ will not be $\omega$ consistent. $S_i$ says, in informal terms, "if there is a proof of $S_i$ in $T_i$ then there is a shorter proof of $\lnot S_i$ in $T_i$". So $R_1$ says, informally, "There is a proof of $S_0$ in $T$ such that no proof of $\lnot S_0$ in $T$ is shorter, and if there is a proof of $S_1$ from $T + \lnot S_0$ then there is a shorter proof of $\lnot S_1$ from $T + \lnot S_0$."