We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let Ω be a regular uncountable cardinal. Let m < ω and M be an m-sound premouse and Σ be...

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that (...) [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe [Formula: see text] (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming ZF [Formula: see text] [Formula: see text] [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model [Formula: see text] containing [Formula: see text] and satisfying ZF [Formula: see text] [Formula: see text] [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text] ⊆[Formula: see text], where [Formula: see text] witnesses [Formula: see text] in [Formula: see text]. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text]-DC holds in [Formula: see text], then the model [Formula: see text] will also satisfy [Formula: see text]-DC. We show that ZFC [Formula: see text] “[Formula: see text] is even” [Formula: see text] [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists. (shrink)

This paper studies homogeneously Suslin (hom) sets of reals in tame mice. The following results are established: In 0 ¶ the hom sets are precisely the [Symbol] sets. In M n every hom set is correctly [Symbol] and (δ + 1)-universally Baire where ä is the least Woodin. In M u every hom set is <λ-hom, where λ is the supremum of the Woodins.

A significant open problem in inner model theory is the analysis of HODL[x] as a strategy premouse, for a Turing cone of reals x. We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals x there are proper class 1-small premice M,N, with Woodin cardinals δ,ε, respectively, such that M|δ and N|ε are in L[x], M and N are countable in L[x], and the pseudo-comparison of M with N succeeds, is in (...) L[x], and lasts exactly ω1L[x] stages. Moreover, we can take M=M1, the minimal iterable proper class inner model with a Woodin cardinal, and take N to be M1-like and short-tree-iterable. (shrink)

Assume. Let κ be a cardinal. A ‐ground is a transitive proper class W modelling such that V is a generic extension of W via a forcing of cardinality. The κ‐mantle is the intersection of all ‐grounds. We prove that certain partial choice principles in are the consequence of κ being inaccessible/weakly compact, and some other related facts.

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $ “E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$ ”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the (...) Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$ ”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “ $V=\mathrm {HOD}$ ”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$. (shrink)