We have introduced the notions of \(\kappa\)-Borel class, \(\kappa\)-analytic class, \(\kappa\)-analytic-coanalytic class, \(\kappa\)-Borel* class in the previous talk. In descriptive set theory the Borel class, the analytic-coanalytic class, and the Borel* class are the same class, we showed that this doesn't hold in the generalized descriptive set theory.
In this talk, we will show the consistency of "\(\kappa\)-Borel* class is equal to the \(\kappa\)-analytic class". This was initially proved by Hyttinen and Weinstein (former Kulikov), under the assumption V=L. We will show a different proof that shows that this holds in L but also can be forced by a cofinality-preserving GCH-preserving forcing from a model of GCH, but also by a \(<\!\kappa\)-closed \(\kappa^+\)‑cc forcing.