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In this paper we prove the equiconsistency of “Every ω1 –tree which is first order definable over (, ε) has a cofinal branch” with the existence of a reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
In this paper we prove the independence of for n ≥ 3. We show that can be forced to be above any ordinal of L using set forcing. For we prove that it can be forced, using set forcing, to be above any L cardinal κ such that κ is Π1 definable without parameters in L. We then show that cannot be forced by a set forcing to be above every cardinal of L Finally we present a class forcing construction to make greater than any given L cardinal.
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