Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.

The self-iterability of L[E]

Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ.As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω1 are cardinals, then holds true, and if in addition λ is regular, then holds true.

Local Kc constructions

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , thenΉ is Woodin in V ⇒ Ή is Woodin in ,for all Ή. It is natural to ask whetherκ is strong in V ⇒ κ is λ-strong in ,for all κ, or even better,κ is λ-strong in V ⇒ κ is λ-strong in .As one might suspect, the more useful answer would be “yes”.For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in .We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that(1) iteration trees on can be lifted to iteration trees on V,(2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and(3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).

Core models with more Woodin cardinals

In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of Kc with respect to uncountable iteration trees.Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2].Theorem. Let Ω be measurable. Then the following are equivalent:(a) for all posets,(b) for every poset,(c) for every poset ℙ ∈ VΩ, Vℙ ⊨ there is no uncountable sequence of distinct reals in L(ℝ)(d) there is an Ω-iterable premouse of height Ω which satisfies “there are infinitely many Woodin cardinals”.It is an immediate corollary that if every set of reals in L(ℝ) is weakly homogeneous, then ADL(ℝ) holds. We shall also indicate some extensions of the theorem to pointclasses beyond L(ℝ), and mice with more than ω Woodin cardinals.