Shoenfield's absoluteless lemma asserts that if $\sigma$ is a $\Sigma_2^1$ statement (in arithmetic), then it is absolute to all inner models of $\mathsf{ZF} + \mathsf{DC}$. I am aware that this result applies to all models of $\mathsf{ZF}$.
However, all proofs that I see uses the absoluteness of well-foundedness, which requires $\mathsf{DC}$. According to this answer on MathOverflow, the trees in the proof can be "canonically well-ordered". I do not see why this is the case – following the proof of Theorem 25.20 in Jech's book, all trees involved are subsets of $\omega^{<\omega}$. Is he saying that the absoluteness of well-foundedness of subtrees of $\omega^{<\omega}$ does not require $\mathsf{DC}$?
Best Answer
Subtrees of $\omega^{<\omega}$ are well orderable, so they are well founded if and only if they do not have infinite branches.