Intermediate models of Magidor-Radin forcing-Part II

Intermediate Models of Magidor-Radin Forcing- Part II

“…We continue the work done in [9],[5], [4]. We prove that for every set A in a Magidor-Radin generic extension using a coherent sequence such that o U (κ) < κ + , there is…”

“…Remark 2.3 In [5], [4] we had another requirement in definition 2.2, that given a condition p, if we would like to add an ordinal α to the sequence in the interval (κ i−1 (p), κ i (p)) then we needed to make sure that o U (α) < o U (κ i (p)). This condition is not essential as any condition p can be directly extended to a condition in the set…”

“…The result was generalized to the Magidor forcing in [5]. This was pushed further to versions of the Magidor-Radin forcing with o U (κ) < κ, in [4]. The result for o U (κ) < κ, splits into two parts.…”

“…The major difference between the case o U (κ) < κ and o U (κ) ≥ κ, is that we cannot split M[ U] to the part below o U (κ) and above it. As proven in [4], this decomposition provided the ability to run over all possible extension types. In terms of C G this means that we cannot split C G below κ in a way that will determine what are the measure used in the construction of C G .…”

“…In context of [4] and [5], we are working by induction on κ. Formally we prove the following inductive step: Theorem 1.1 Let U be a coherent sequence with maximal measurable κ, such that o U (κ) < κ + .…”