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Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) not sharp. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KP$\omega$), I'll write "$Ad(M)$" for the smallest transitive model of KP$\omega$ with $M$ as an element.

• I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest $E$-closed set containing $M$ as an element is a possible candidate, being in general (even always, I think) smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.

Bumping an old question, Andreas' bound of the next admissible ordinal is in general not sharp (although Dmytro Taranovsky showed that it sometimes is). The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KP$\omega$), I'll write "$Ad(M)$" for the smallest transitive model of KP$\omega$ with $M$ as an element.

• I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest $E$-closed set containing $M$ as an element is a possible candidate, being in general (even always, I think) smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.

Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) not sharp. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KPi), I'll write "$Ad(M)$" for the smallest transitive model of KPi with $M$ as an element.

• I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest $E$-closed set containing $M$ as an element is a possible candidate, being in general (even always, I think) smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.

Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) not sharp. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KP$\omega$), I'll write "$Ad(M)$" for the smallest transitive model of KP$\omega$ with $M$ as an element.

• I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest $E$-closed set containing $M$ as an element is a possible candidate, being in general (even always, I think) smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.

Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) not sharp. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KPi), I'll write "$\omega_1^{CK}(M)$" for the height of the smallest transitive model of KPi with $M$ as an element, and "$Ad(M)$" for that model.

• I'll "$\omega_1^{CK}(\gamma)$" for $\omega_1^{CK}(L_\gamma)$ when $\gamma$ is admissible.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present.

Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) not sharp. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. (It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)

Part of the point of this answer is that the notation in the OP winds up being highly misleading. So let me notationally start from scratch:

• I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$.

• For $M$ a transitive model of KP + Infinity (= KPi), I'll write "$Ad(M)$" for the smallest transitive model of KPi with $M$ as an element.

• I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

(This MSE answer of mine contains the argument below in more detail, and this old MO answer of mine is also relevant:)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is computable in $Ad(M)$: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ (in fact, all we're really using here is $M^\omega\subseteq Ad(M)$). This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is never sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest $E$-closed set containing $M$ as an element is a possible candidate, being in general (even always, I think) smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.