Companion of the pointclass of inductive sets

Question

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, which are just a type of boldface pointclass. The smallest one is IND, the class of (boldface) inductive sets, which I will consider as a typical example.

The companion of a Spector class $\bf \Gamma$ on $\mathbb{R}$ is a structure $(M,\in,R)$ with certain properties (listed in the book) such that $\bf \Gamma$ is the class of pointsets that are $\Sigma_1$-definable in $M$ with real parameters from the relation $R$. The companion of $\bf \Gamma$ is not unique, but its underlying set $M$ is unique and also the class of relations on $M$ that are $\Sigma_1$-definable from $R$ in $M$ with real parameters is unique.

The pointclass IND can also be described as the class of pointsets that are $\Sigma_1$-definable over $M$ from parameters in $\mathbb{R} \cup \lbrace\mathbb{R} \rbrace$ where $M = L_\kappa(\mathbb{R})$ is the least admissible level of $L(\mathbb{R})$. We must allow $\lbrace\mathbb{R} \rbrace$ itself as a parameter here or we would just get the ${\bf \Sigma}^1_2$ sets. For any companion $(M,\in,R)$ of IND we must have $M = L_\kappa(\mathbb{R})$.

Question: Is there a companion $(M,\in,R)$ of IND where the relation $R$ has a simple definition over $M = L_\kappa(\mathbb{R})$ (simpler than in Moschovakis's general construction of a companion?) Maybe something that is already studied in the fine structure of $L(\mathbb{R})$?

Answer 1

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$.

The existence of a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. Moreover, from this we can easily get a $\Delta_1^{J_\kappa(\mathbb{R})}$ partial surjection as follows.

(I have edited my answer below to replace the more convoluted argument that I wrote before.)

Let $F: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ be a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection and let $\theta$ be a $\Sigma_1$ formula defining it over $J_\kappa(\mathbb{R})$. Then define $G: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by letting $(x,a) \in G$ if and only if there is an $\alpha < \kappa$ such that $$J_\alpha(\mathbb{R}) \models \theta(x,a) \quad\And\quad \forall \xi < \alpha\, J_\xi(\mathbb{R}) \models \forall y \in \mathbb{R}\, \neg\theta(y,a).$$ Then $F \subset G$ and it is easy to check that $G$ is $\Delta_1^{J_\kappa(\mathbb{R})}$ and that the range of $G$ is equal to the range of $F$, which is all of $J_\kappa(\mathbb{R})$.