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?


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]

  • $\begingroup$ Does it somehow follow easily from Theorem 4a in that paper? I don't quite see it. $\endgroup$
    – user5810
    May 9 '10 at 6:58
  • $\begingroup$ It actually follows from Theorem 3a, see Remark 3 which follows that theorem. $\endgroup$ May 18 '10 at 22:25

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