Sets in Prikry and Magidor generic extensions

Intermediate Models of Magidor-Radin Forcing- Part II

“…Kanovey, Koepke and the second author showed in [9] that the same is true for the standard Prikry forcing. 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].…”

“…For limit δ < ν + the sequences D (ρ) i | i < θ are defined for every ρ < δ. For each i < θ, the sequence D (ρ) i | ρ < δ is ⊆ * -increasing hence by corollary 4.17, there is a Mathias E (δ) i ⊆ θ i which is a ⊆ * -bound, this insures (2), (5) Hence the sequence D (ξ) j | j < θ is defined for every ξ < ν + . For every column j < θ, D (ξ) j | ξ < ν + is a ⊆ * -increasing sequence of subsets of θ j , thus there is ξ j < ν + from which this sequence stabilizes.…”

“…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…”

“…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 + .…”

“…The following lemma is a generalization of a combinatorical property that was proven in [5] for product of measures. It can be stated for more general trees, however, let us restrict the attention to our needs.…”