# The Core Model Induction and Other Inner Model Theoretic Tools

### Program

The workshop will consist of 4 tutorials on topics related to the core model induction and an additional tutorial by Omer Ben-Neria. The tutorials include lectures as well as discussion sessions.

The dependencies between the different tutorials are roughly as follows.

#### Fine Structure and the Core Model (Martin Zeman)

This series of lectures will be a brief course in fine structure theory of extender models, with focus on models up to one Woodin cardinal. The material here basically constitutes the successor step in the core model induction. The main topics presented will be
1. Abstract fine structure theory, where I attempt to provide some insight that is independent of which presentation (of many in the literature) you choose to work with, as well as of indexing of extenders.
2. Iterability, and fine structure of iterable extender models. Here it will be explained how iterability is used to develop basic fine structural properties of extender models. A proof of iterability in a simple case of tame mice will be presented.
3. Background constructions. Here the two basic constructions will be discussed in quite a detail: The fully backgrounded construction, and the $K^c$ construction, which only uses extender fragments in the background universe.
4. Core model theory. This will include the proof of iterability of the model K^c, introducing the true core model K, and developing its basic properties.
• (Mitchell-Steel) Fine structure and iteration trees.
• (Schindler-Zeman) Fine structure theory, Handbook of Set Theory.
• (Steel) An outline of inner model theory, Handbook of Set Theory.
• (Steel) The core model iterability problem.
• (Welch) $\Sigma^*$ fine structure, Handbook of Set Theory.
• (Zeman) Inner models and large cardinals.

#### Determinacy and Scales (Trevor Wilson)

We present some results and techniques involving scales and Suslin representations that are relevant to the core model induction, including
1. Steel's results on the pattern of scales in $L(\mathbb{R})$,
2. The Kechris-Woodin theorem that Suslin determinacy implies determinacy in $L(\mathbb{R})$, and
3. Woodin's construction of Suslin representations of $\Pi^2_1$ sets assuming $\mathsf{AD}^+ + \theta_0 < \Theta$.
We discuss how this construction can be generalized to produce Suslin representations of the $\Pi^2_1$ sets of a model of $\mathsf{AD}^+$, assuming various hypotheses outside the model instead of assuming $\theta_0 < \Theta$ in the model.

• (Steel) Scales in $L(\mathbb{R})$.
• (Kechris) Determinacy and the structure of $L(\mathbb{R})$.
• (Martin-Woodin) Weakly homogeneous trees.
• (Wilson) The envelope of a pointclass under a local determinacy hypothesis.
Necessary requirements:
Familiarity with the definition of "scale" and basic results about scales as in Kanamori, The Higher Infinite, 2nd ed., Section 30 up to Corollary 30.9.

#### Prikry-type Forcings and Inner Model Theory (Omer Ben-Neria)

The purpose of the series is to introduce the theory of Prikry-type forcing notions and explore its connections with inner-model theory, and especially with the Mitchell covering Lemma, and consistency results concerning HOD. Topics that are expected to be covered include
1. Prikry and Magidor forcing notions, and extender-based forcings;
2. the Mitchell Covering Lemma;
3. consistency results at the level of hyper-measurability; and
4. (if time permits) iterations of Prikry-type forcings with applications to HOD.
• (Gitik) Prikry-Type Forcings, Handbook of Set Theory.
• (Mitchell) The Covering Lemma, Handbook of Set Theory.
Necessary requirements:
Basic theory of forcing and large cardinals

#### HOD Computations (Sandra Müller)

An essential question regarding the theory of inner models is the analysis of the class of all hereditarily ordinal definable sets HOD inside various inner models $M$ of the set theoretic universe $V$ under appropriate determinacy hypotheses. Examples for such inner models $M$ are $L(\mathbb{R})$ or $L[x]$ on a cone of reals $x$. We will outline Steel's and Woodin's analysis of $HOD^{L(\mathbb{R})}$. Moreover, we will discuss their analysis of $HOD^{L[x,G]}$ on a cone of reals $x$, where $G$ is $Col(\omega,\kappa)$-generic and $\kappa$ is the least inaccessible cardinal in $L[x]$. We will point out were the problems are when trying to adapt this to analyze $HOD^{L[x]}$.

• (Steel) An outline of inner model theory, Handbook of Set Theory, Section 8.
• (Steel, Woodin) HOD as a core model, Cabal III.
Necessary requirements:
A good understanding of mice, the comparison process and genericity iterations, e.g. the fine structure tutorial given in the first week or the relevant parts of Steel's handbook chapter (Sections 1-3 and 7).

#### The Core Model Induction (Grigor Sargsyan and Nam Trang)

We present a proof of Steel's theorem: PFA implies AD holds in $L(\mathbb{R})$. The proof is done via a core model induction argument. The key ingredients that make up the proof are: core model theory at the level of Woodin cardinals (the $K^c$-dichotomy theorem) and the optimal scales analysis in $L(\mathbb{R})$. We will outline the general ideas (in particular, how the key ingredients are combined in the proof) before going into the details.

• (Steel) PFA implies $AD^{L(\mathbb{R})}$. J. Symbolic Logic, 70(4):1255–1296, 2005.