Analysis of HOD for Admissible Structures

Jan Kruschewski (TU Vienna)

Jun 27. 2024, 10:20 — 11:05

Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \R$ let $\alpha_x$ be the least $\beta$ such that

\[

L_\beta [x] \models \Sigma_n \text{-} \kp + \exists \kappa (``\kappa \text{ is inaccessible and }\kappa^+ \text{ exists}").

\]

Then there is a cone of reals $x$ such that letting $\kappa$ be the inaccessible of $L_{\alpha_x}[x]$, $G \subset \col(\omega, <\kappa)$ be $(L_{\alpha_x}[x],\col(\omega, <\kappa))$-generic, $\Sigma_n$-$\hod$ be the class of elements $a$ of $L_{\alpha_x}[x,G]$ such that $\tc(\{a\}) \subset \Sigma_n\text{-}\od$, where $\Sigma_n\text{-}\od$ is the class of all elements which are ordinal definable over $L_{\alpha_x}[x,G]$ via a $\Sigma_n$-formula, we have that $\Sigma_n$-$\hod$ is an iterate of a mouse adjoined with a fragment of its iteration strategy. Moreover, $\Sigma_n$-$\hod \models \Sigma_n \text{-} \kp$, $\omega_2^{L_{\alpha_x}[x,G]}$ is Woodin in $\Sigma_n$-$\hod$ and $\Sigma_n$-$\hod$ is a forcing ground of $L_{\alpha_x}[x,G]$. This is joint work with Farmer Schlutzenberg.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Determinacy, Inner Models and Forcing Axioms (Workshop)
Organizer(s):
Sandra Müller (TU Vienna)
Grigor Sargsyan (Polish Academy of Science, Warsaw)
Ralf Schindler (WWU Münster)
John Steel (UC, Berkeley)