Intermediate Models of Magidor-Radin Forcing- Part II

We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(\kappa )<{\kappa }^{+}$, there is $C^{\mathrm{\prime }}\subseteq C_G$, such that $V[A]=V[C^{\mathrm{\prime }}]$. Also we prove that the supremum of a fresh set in a Prikry, tree Prikry, Magidor, Radin-Magidor and Radin forcing, changes cofinality to $\omega$.