The Independence of $delta^1_n$

Journal of Symbolic Logic 64 (1):350-362 (1999)
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In this paper we prove the independence of $\delta^1_n$ for n $\geq$ 3. We show that $\delta^1_4$ can be forced to be above any ordinal of L using set forcing. For $\delta^1_3$ we prove that it can be forced, using set forcing, to be above any L cardinal $\kappa$ such that $\kappa$ is $\Pi_1$ definable without parameters in L. We then show that $\delta^1_3$ cannot be forced by a set forcing to be above every cardinal of L. Finally we present a class forcing construction to make $\delta^1_3$ greater than any given L cardinal



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