As a part of the workshop on “The Core Model Induction and Other Inner Model Theoretic Tools” in Rutgers I gave a tutorial on HOD Computations.

Abstract: 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 $HOD^{L[x]}$.

Reading List:

• (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).

See here for more information about the meeting and here for lecture notes typed by James Holland.