Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}

In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T2n], which are very natural canonical inner models of ZFC. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us in particular to reprove a strong partition property result on the ordinal of a Steel pointclass and derive a new boundedness principle which could be useful in the study of the cardinal structure of L(R). We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson’s proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Π12n+1 pointclasses using Jackson’s theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T2n]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T2n] are unique. This generalizes previous work of Hjorth. We then characterize the L[T2n] in term of inner models theory, showing that they actually are constructible models over direct limit of mice with Woodin cardinals, a counterpart to Steel’s result that the L[T2n+1] are extender models, and finally show that the generalized contiuum hypothesis holds in these models, solving a conjecture of Woodin.

In dieser Arbeit werden lambda-Strukturen und s-Strukturen eingeführt, und Funktionen S und Lambda entwickelt, die lambda-Strukturen auf s-Strukturen abbilden und umgekehrt. lambda-Strukturen sind eng verwandt mit den in von Jensen untersuchten Prämäusen (iterierbare Prämäuse dieser Art sind lambda-Strukturen), und s-Strukturen wurden in Anlehnung an die von Mitchell und Steel betrachteten Prämäuse definiert. Wieder sind iterierbare Prämäuse dieser Art auch s-Strukturen. Für die Definition dieser Strukturen wurde eine neue, schwache Form der initial segment condition entwickelt (die s'-ISC), die stark genug für die Anwendungen ist. Um zu zeigen, dass die hier entwickelten Funktionen die gewünschte Korrespondenz realisieren, wurden Methoden zur Übersetzung von Formeln entwickelt, die teilweise sehr allgemein gehalten sind. So ist die Übersetzung von Sigma-1-Formeln, die in einer Nachfolgerstufe der Jensen-Hierarchie gelten, in entsprechende Sigma-omega-Formeln in der Vorgängerstufe, anwendbar auf beliebige J-Strukturen. Es werden normale s-Iterationen eingeführt, die den normalen Iterationen von Prämäusen im Sinne von Mitchell-Steel nachgebildet sind, aber auf lambda-Strukturen angewandt werden, und es wird gezeigt, dass die entwickelten Funktionen komponentenweise auf Iterationen angewandt werden können, um normale s-Iterationen von lambda-Strukturen in normale Iterationen von s-Strukturen zu übersetzen, und umgekehrt. Mit diesen Methoden lassen sich auch Iterationsstrategien übersetzen, und man erhält, dass die entwickelten Funktionen normal s-iterierbare lambda-Strukturen auf normal iterierbare s-Strukturen abbilden, und umgekehrt. Auch bleiben die wesentlichen feinstrukturellen Größen, wie bspw. Projekta, und unter gewissen Voraussetzungen (soundness und 1-solidity) auch die Standard-Parameter, erhalten. / In this work we introduce lambda-structures and s-structures, and develop functions S and Lambda, which map lambda-structures to s-structures and vice versa. lambda-structures are closely related to the premice studied in recent work of Jensen (iterable premice of this kind are lambda-structures), and s-structures were defined with the premice developed by Mitchell and Steel in mind. Again, iterable premice of this kind are s-structures. For the definition of these structures, a new form of the initial segment condition condition, called s'-ISC, was developed, which is a common weakening of the versions used in by Steel and Jensen. It still suffices for the applications. In order to show that the functions introduced establish the desired correspondence, we developed methods for translating formulae, which in part are very generally applicable. For instance, the translation of Sigma-1-formulae which hold in a successor level of the Jensen-hierarchy into corresponding Sigma-omega-formulae in the predecessor level, can be applied to arbitrary J-structures. We introduce normal s-iterations, which have been designed so as to rebuild the iterations of premice in the sense of Mitchell-Steel but are applied to lambda-structures. It is shown that the translation functions can be applied component-wise to normal iterations, in order to translate normal s-iterations of lambda-structures into normal iterations of s-structures, and vice versa. Using these methods, we can also translate iteration strategies and the result is that the functions introduced in this work map normally s-iterable lambda-structures to normally iterable s-structures, and vice versa. Also,the fundamental fine structural notions, such as projecta, and under additional hypotheses (soundness and 1-solidity) standard-parameters, are preserved.