# Shoenfield's 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?

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## 1 Answer

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]

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Does it somehow follow easily from Theorem 4a in that paper? I don't quite see it. –  Ricky Demer May 9 '10 at 6:58
It actually follows from Theorem 3a, see Remark 3 which follows that theorem. –  François G. Dorais May 18 '10 at 22:25