Some descriptive set theory and core models (Q1110505)

We show how the assumption of a \({\underset \tilde{} \Sigma}^ 1_ 3\)-absolute core model can be used to lift directly results about \({\underset \tilde{} \Delta}^ 1_ 2\) and \({\underset \tilde{} \Sigma}^ 1_ 2\) sets of reals to the third level of the projective hierarchy. We sketch the idea of how decomposing the Martin-Solovay tree on \(\omega \times u_{\omega}\) for a complete \(\Pi^ 1_ 2\) set can do this; and remark how many applications of this give the same effect as assuming the hypothesis of \({\underset \tilde{} \Delta}^ 1_ 2\)-determinacy.