TITLE: Shoenfield's Absoluteness Theorem
QUESTION [7 upvotes]: From wikipedia: http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem
"Shoenfield's theorem shows that if there is a model ZF in which a given $\Pi^1_3$  statement $\phi$ is false, then $\phi$ is also false in the constructible universe of that model."
The problem is that the only proofs I can find seem to use "a model of ZF+DC" or "a model of ZFC".
Is wikipedia right?
If so, is there a proof from just "a model of ZF" online or short enough to sketch here?

REPLY [13 votes]: 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]