In this chapter, we show that under AD^+, the derived model of certain hod pairs satisfies the LSA. We also prove results that are important elsewhere. In particular, we show the derived model of an active \omega.2 lsa Woodin mouse satisfies LSA. This result will be important in Chapter 12, where we obtain the consistency of LSA from PFA.

This chapter presents a proof $\square_{\kappa,2}$ holds in a lsa-small hod mouse $\mathcal{P}$ for all cardinals $\kappa$ of $\mathcal{P}$. The proof adapts a well-known construction of $\square$ in extender models by Schimmerling-Zeman. The main challenge to overcome in this situation is that the full condensation lemma, which holds for extender models, does not hold in hod mice. The main application of this result is in the proof of consistency of LSA in Chapter 12.

This chapter develops the theory of condensing sets. Condensing sets give rise to iteration strategies with nice condensation properties. We show the existence of condensing sets under various hypotheses: AD^+ and PFA. We will use the existence of condensing sets in AD^+ in the proof of generation of pointclasses in Chapter 10. We will use the existence of condensing sets in Chapter 12 to construct a model of LSA under PFA.

This chapter is devoted to proving a comparison theorem for hod pairs. We will have two comparison theorems: one is useful in determinacy context while the other is useful in Core Model Induction applications.

This chapter gives various applications of the theory developed in the previous chapters. The first application is a proof of generation of mouse full pointclasses assuming Strong Mouse Capturing. The second application is a proof that Strong Mouse Capturing holds in the minimal model of LSA; so the Mouse Set Conjecture is true in all models of AD^+ up to the minimal model of LSA. The third application is a proof of consistency of LSA from the existence of a Woodin limit of Woodin cardinals.

This chapter analyzes HOD up to $\Theta$ in the minimal model of LSA under the assumption that Strong Mouse Capturing (SMC) holds.

The main purpose of this chapter is to isolate the definition of short tree strategy mice. The main problem with defining this concept is the fact that it is possible that maximal iteration trees (which should not have branches indexed in the strategy predicate) may core down to short iteration trees (which must have branches indexed in the strategy predicate), thus causing indexing issues. To resolve this issue we will design an authentication procedure which will carefully choose iteration trees and index their branches. Thus, if some iteration tree doesn’t have a branch indexed in the strategy predicate then it is because the authentication procedure hasn’t yet found an authenticated branch, and therefore, such iteration trees cannot core down to an iteration tree whose branch is authenticated.

This chapter introduces the main concepts and the problems to be investigated by the book. In particular, the chapter defines the Largest Suslin Axiom (LSA) and the minimal model of LSA. The chapter summarizes the main theorems to be proved in the book: HOD of the minimal model of LSA satisfies the Generalized Continuum Hypothesis, the Mouse Set Conjecture holds in the minimal model of LSA, the consistency of LSA from large cardinals, the consistency of LSA from strong forcing axioms like PFA.

The main goal of this chapter is to prepare some terminology to be used in the rest of the book. One important notion introduced in this chapter is that of the undropping game. We will use it to prove a comparison theorem for hod mice in Chapter 4.

This chapter studies internal theory of lsa hod mice. Suppose $(\mathcal{P},\Sigma)$ is a hod pair of an sts hod pair, $X$ is a self-wellordered set such that $\mathcal{P}\in X$, and $\mathcal{N}$ is a $\Sigma$ or $\Sigma$-sts mouse over $X$. The main theorem of this chapter shows that N is $\Sigma$-closed and has fullness preserving iteration strategy, then $\Sigma \restriction \mathcal{N}[g]$ is definable in $\mathcal{N}[g]$ for any generic $g$ over $\mathcal{N}$ . The main idea behind the proof is that the branch of an iteration tree $\mathcal{T}$ on $\mathcal{P}$ can be identified by the authentication process introduced in Chapter 3.

This chapter gives a proof of generic interpretability for (pre)hod pairs, studies derived models of hod mice, and proves branch condensation holds on a tail for anomalous hod pairs of type II and III.

This chapter presents a construction of the minimal model of LSA from a hypothesis implied by strong forcing axioms such as PFA and by large cardinal hypotheses such as the existence of a strongly compact cardinal. Consequently, LSA is consistent relative to PFA and LSA is consistent relative to the existence of a strongly compact cardinal. This chapter is an application of the theory developed in the previous chapters and the core model induction technique, which is a general method for calibrating consistency strength of strong theories.