Non-linearities in the analytical hierarchy

It is commonly known that there exist true $\Pi^0_1$ sentences which are mutually independent over PA. The corresponding fact for $\Pi^1_1$ fails: for every pair of true $\Pi^1_1$ sentences $\phi,\psi$, one of them implies the other over ACA_0 + all true $\Sigma^1_1$ sentences. What about other classes, such as $\Pi^1_n$? We prove in ZFC + “there are infinitely many Woodin cardinals” that if $\Gamma$ = any of the classes $\Pi^1_{2n}$ or $\Sigma^1_{2n+1}$, then there are true $\Gamma$ sentences $\phi,\psi$ which are mutually independent over the theory ACA_0 + all true negations of $\Gamma$ sentences. This is joint work with F. Pakhomov.