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EDIT: to my chagrin, the notion of "$n$-admissibility" is not what I thought it was! What I wanted was $\Sigma_n$-admissibility. You can find the definition of $n$-admissibles here; they are vastly smaller than their $\Sigma_n$ counterparts, and indeed for each $n$ the least $n$-admissible is less than the least $\Sigma_2$-admissible. Now $n$-admissibility is a rare notion these days and I've seen "$n$-admissible" used for "$\Sigma_n$-admissible before, but given the relevance of older papers to this topic it's probably a good idea for me to not butcher this distinction.

EDIT: to my chagrin, the notion of "$n$-admissibility" is not what I thought it was! What I wanted was $\Sigma_n$-admissibility. You can find the definition of $n$-admissibles here; they are vastly smaller than their $\Sigma_n$ counterparts, and indeed for each $n$ the least $n$-admissible is less than the least $\Sigma_2$-admissible. Now $n$-admissibility is a rare notion these days and I've seen "$n$-admissible" used for "$\Sigma_n$-admissible" before, but given the relevance of older papers to this topic it's probably a good idea for me to not butcher this distinction.

Embarrassingly, I think I was overthinking this: I believe that the Robinsonian admissibles are exactly the successor admissibles.

That last bit is what I was missing when I was worrying about non-Gandy-ness. I think it's worth elaborating on:

• First, note that it fails for $2$-admissibility, since by the Gandy Basis Theorem there is a model of $KP2$ with wellfounded part having height $\omega_1^{CK}$.

• The reason it works for ($1$-)admissibility is the upwards absoluteness of $\Sigma_1$ formulas. Let $M\models KP$ and $N$ be the wellfounded part of $M$. Let $a,\varphi$ be a $\Sigma_1$-Replacement instance in $N$: that is, $\varphi$ is $\Sigma_1$ and for each $b\in a$ there is exactly one $c\in N$ such that $N\models\varphi(b,c)$. Then in $M$ we can apply absoluteness to argue that $a,\hat{\varphi}$ is also a $\Sigma_1$-Replacement instance with the same solution class, where $\hat{\varphi}(x,y)$ is the formula "$\varphi(x,y)$ and no $z$ of rank $<rk(y)$ has $\varphi(x,z)$."

EDIT: to my chagrin, the notion of "$n$-admissibility" is not what I thought it was! What I wanted was $\Sigma_n$-admissibility. You can find the definition of $n$-admissibles here; they are vastly smaller than their $\Sigma_n$ counterparts, and indeed for each $n$ the least $n$-admissible is less than the least $\Sigma_2$-admissible. Now $n$-admissibility is a rare notion these days and I've seen "$n$-admissible" used for "$\Sigma_n$-admissible before, but given the relevance of older papers to this topic it's probably a good idea for me to not butcher this distinction.

Embarrassingly, I think I was overthinking this: I believe that the Robinsonian admissibles are exactly the successor admissibles.

That last bit is what I was missing when I was worrying about non-Gandy-ness. I think it's worth elaborating on:

• First, note that it fails for $\Sigma_2$-admissibility, since by the Gandy Basis Theorem there is a model of $KP2$ with wellfounded part having height $\omega_1^{CK}$.

• The reason it works for ($\Sigma_1$-)admissibility is the upwards absoluteness of $\Sigma_1$ formulas. Let $M\models KP$ and $N$ be the wellfounded part of $M$. Let $a,\varphi$ be a $\Sigma_1$-Replacement instance in $N$: that is, $\varphi$ is $\Sigma_1$ and for each $b\in a$ there is exactly one $c\in N$ such that $N\models\varphi(b,c)$. Then in $M$ we can apply absoluteness to argue that $a,\hat{\varphi}$ is also a $\Sigma_1$-Replacement instance with the same solution class, where $\hat{\varphi}(x,y)$ is the formula "$\varphi(x,y)$ and no $z$ of rank $<rk(y)$ has $\varphi(x,z)$."

That last bit is what I was missing when I was worrying about non-Gandy-ness. I think it's worth elaborating on:

 

• First, note that it fails for $2$-admissibility, since by the Gandy Basis Theorem there is a model of $KP2$ with wellfounded part having height $\omega_1^{CK}$.

 
• The reason it works for ($1$-)admissibility is the upwards absoluteness of $\Sigma_1$ formulas. Let $M\models KP$ and $N$ be the wellfounded part of $M$. Let $a,\varphi$ be a $\Sigma_1$-Replacement instance in $N$: that is, $\varphi$ is $\Sigma_1$ and for each $b\in a$ there is exactly one $c\in N$ such that $N\models\varphi(b,c)$. Then in $M$ we can apply absoluteness to argue that $a,\hat{\varphi}$ is also a $\Sigma_1$-Replacement instance with the same solution class, where $\hat{\varphi}(x,y)$ is the formula "$\varphi(x,y)$ and no $z$ of rank $<rk(y)$ has $\varphi(x,z)$."

That last bit is what I was missing when I was worrying about non-Gandy-ness. I think it's worth elaborating on:

• First, note that it fails for $2$-admissibility, since by the Gandy Basis Theorem there is a model of $KP2$ with wellfounded part having height $\omega_1^{CK}$.

• The reason it works for ($1$-)admissibility is the upwards absoluteness of $\Sigma_1$ formulas. Let $M\models KP$ and $N$ be the wellfounded part of $M$. Let $a,\varphi$ be a $\Sigma_1$-Replacement instance in $N$: that is, $\varphi$ is $\Sigma_1$ and for each $b\in a$ there is exactly one $c\in N$ such that $N\models\varphi(b,c)$. Then in $M$ we can apply absoluteness to argue that $a,\hat{\varphi}$ is also a $\Sigma_1$-Replacement instance with the same solution class, where $\hat{\varphi}(x,y)$ is the formula "$\varphi(x,y)$ and no $z$ of rank $<rk(y)$ has $\varphi(x,z)$."