set theory - Shoenfield's Absoluteness Theorem

Wikipedia is correct in that the Shoenfield Absoluteness Theorem holds for plain ZF.

Since the proof of the theorem relies heavily on the absoluteness of well-foundedness, it is tempting to assume DC. However, since the trees that occur in the usual proof of the theorem are canonically well-ordered, DC is not necessary to prove that the well-foundedness of these trees is absolute. For a different approach, see the proof given by Barwise and Fisher in The Shoenfield Absoluteness Lemma. [Israel J. Math. 8 1970, 329-339, MR278934]