JAN KRUSCHEWSKI

Let n≥1 and assume that there is a Woodin cardinal. Fix x∈ℝ and let α be the least β such that
Lβ[x]⊨Σn-KP+∃κ(‘‘κ is inaccessible and κ+ exists").
We adapt the analysis of HODL[x,G] as a strategy mouse to Lαx[x,G] for a cone of reals x. That is, we identify a mouse n-ad and define a class H⊆Lαx[x,G] as a natural analogue of HODL[x,G]⊆L[x,G], and show that H=M∞[Σ0], where M∞ is an iterate of n-ad and Σ0 a fragment of its iteration strategy.

Let Σ1-KP be Kripke-Platek set theory where the full foundation scheme is replaced with ∈-Foundation and Th∞ be the theory which consists of Σ1-KP and the statement “Vλ exists, where λ is a limit of Woodin cardinals”. Let ThAD be the theory which consists of Σ1-KP, “V= L(R)”, “P(ω) exists”, the Axiom of Determinacy, and the statement “there is a Σ1-elementary substructure which contains the reals”. We will show in the metatheory ZFC that if Con(Th∞) holds, then Con(ThAD) holds. The proof involves a variant of the derived model theorem. The construction of the derived model proceeds by first identifying its Σ1 theory in real parameters via mouse witnesses and then constructing an appropriate term model from this theory. The appropriate version of the key reflection lemma which is used in the classical proof of the derived model theorem is then shown for this term model.

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if W and R are Ω+1-iterable, 1-small weasels, then W≤*R iff there is a club C ⊂ Ω such that for all α∈C, if α is regular, then the cardinal successor of α in W is less or equal than the cardinal successor of α in R . We will show that the conjecture fails, assuming that there is an iterable premouse which models KP and which has a Σ1-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds.
In the course of this we will also show that if M is an iterable admissible premouse with a largest, regular, uncountable cardinal δ, and ℙ is a forcing poset with the δ-c.c. in M, and g is M-generic, but not necessarily Σ1-generic, M[g] is a model of KP. Moreover, if M is such a mouse and T is maximal normal iteration tree on M such that T is non-dropping on its main branch, then its last model is again an iterable admissible premouse with a largest regular and uncountable cardinal. At last, we answer another open question from 'The Core Model Iterability Problem' regarding the S-hull property.