On the consistency of the definable tree property on \aleph_1

Details On the consistency of the definable tree property on \aleph_1 On the consistency of the definable tree property on \aleph_1

2000-05-22

arxiv.org/abs/math/0005208v1

Mathematics

Logic

9 pages. To appear in Journal of symbolic Logic

Scientific paper

In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 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.

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