Abstract Let M be a tame mouse modelling $\mathrm {ZFC}$ . We show that M satisfies “ $V=\mathrm {HOD}_x$ for some real x ”, and that the restriction $\mathbb {E}^M\!\upharpoonright \![\omega _1^M,\mathrm {OR}^M)$ of the extender sequence $\mathbb {E}^M$ of M to indices above $\omega _1^M$ is definable without parameters over the universe of M . We show that M has universe $\mathrm {HOD}^M[X]$ , where $X=M|\omega _1^M$ is the initial segment of M of height $\omega _1^M$ (including $\mathbb {E}^M\!\upharpoonright \!\omega _1^M$ ), and that $\mathrm {HOD}^M$ is the universe of a premouse over some $t\subseteq \omega _2^M$ . We also show that M has no proper grounds via strategically $\sigma $ -closed forcings. We then extend some of these results partially to non-tame mice, including a proof that many natural $\varphi $ -minimal mice model “ $V=\mathrm {HOD}$ ”, assuming a certain fine structural hypothesis whose proof was almost established in Closson [1], and has since been completed in the preprint [14].

Abstract We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$ -iterable, $1$ -small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$ , if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$ . We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a -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 a premouse which models KP with a largest, regular, uncountable cardinal $\delta $ , and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$ , and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$ . Additionally, we study the preservation of admissibility under iteration maps. At last, we will prove a fact about the closure of the set of ordinals at which a weasel has the S-hull property. This answers another question implicit in remarks in [10].

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

Abstract 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 $u_n$ .

Abstract Assume ZF + AD +V=L(ℝ) and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof (κ) = ω thenκis Rowbottom. We also establish some other partition properties.

Abstract 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 sets. InMnevery hom set is correctly , and (δ+ 1)-universally Baire whereδis the least Woodin. InMω, every hom set is <λ-hom, whereλis the supremum of the Woodins.