I've been reading Sacks' Countable admissible ordinals and hyperdegrees as I'm interested in Theorem 5.3 of the paper:
Let $M$ be a countable standard model of $\mathsf{ZF}$ and $V=L$. Suppose $\kappa$ is a regular uncountable cardinal in $M$, then there exists a $T\subseteq \omega$ such that
- $\kappa = \omega_1^{M[T]}$
- if $S \in M[T]$ and $\kappa = \omega_1^{M[S]}$ then $T \in M[S]$.
Hence the theorem says that we collapse any uncountable regular cardinal to $\omega_1$ in a "minimal" way. Sacks doesn't prove directly the theorem, he just says that the proof is analogous to the one of the main result of the paper (Theorem 4.6) whose statement is: for every countable admissible ordinal $\alpha$ there is an $X\subseteq \omega$ such that $\alpha = \omega_1^X$ and for all $Y <_h X, \ \omega_1^Y < \alpha$, where $\le_h$ is the relation of "being hyperarithmetic in".
I'm having a hard time trying to understand some (most) of the things he writes. These are some of the (numerous) questions I have. Here $\alpha$ is a countable admissible ordinal:
In Section 2.14 he introduces the Unbounded Levy Forcing, which is defined in $L_\alpha$ and adds generic bijections between $\omega$ and all ordinals below $\alpha$. Obviously this is not a set forcing, from the perspective of $L_\alpha$, is it some sort of class forcing?
Lemma 2.18 says, among other things, that $\alpha$ is the least non-countable ordinal in $L_\alpha[G]$, where $G$ is the generc unbounded levy collapse. But $\alpha$ is not an element of $L_\alpha[G]$, so why does it says that it is the least non-countable ordinal in $L_\alpha[G]$?
In section 4.1 he introduces the Perfect unbounded Levy forcing, whose conditions now are perfect trees (in $L_\alpha$) whose nodes are conditions of the unbounded Levy forcing. Hence, the ranges of the nodes of such a perfect tree should be uniformly bounded in $\alpha$, as the perfect tree belongs to $L_\alpha$. But then, why should this forcing collapse every ordinal below $\alpha$? If the nodes of a tree only talk about ordinals below a $\beta < \alpha$, then the generic branch is not able to collapse ordinals above $\beta$. What am I missing?
Reading the properties he imposes on the trees that are condtions of the perfect levy forcing, I, again, don't see why the generic should collapse all ordinals below $\alpha$. In particular, the diversification property (4.1.6) seems too weak to me to guarantee that the generic branch is a Levy collapse map, since I can think of a tree that satisfies the diversification property and, still, does not collapse any ordinal above some $\beta < \alpha$. Still Sacks says that this property guarantees that any generic branch through these trees collapses every ordinal below $\alpha$. Again, what am I missing?
By the way: is Theorem 5.3 (stated at the beginning of this post) proved somewhere else, ideally with a more modern approach?
Many thanks to anyone who can help me with some of the questions I wrote.
