**Notations:** Let $L_\alpha$ stand for the Gödel constructible hierarchy ($L_0=\varnothing$ and $L_{\alpha+1} = \mathrm{def}(L_\alpha)$ is the set of definable subsets of $L_\alpha$ and $L_\delta = \bigcup_{\beta<\delta} L_\beta$ for limit $\delta$), and $J_\alpha$ for the Jensen hierarchy ($J_0=\varnothing$ and $J_{\alpha+1} = \mathrm{rud}(J_\alpha)$ is the rud-closure of $J_\alpha \cup \{J_\alpha\}$ and $J_\delta = \bigcup_{\beta<\delta} J_\beta$ for limit $\delta$). Let $M \mathrel{\preceq_1} N$ (if $M$ and $N$ are transitive sets mean “$(M,{\in})$ is a $\Sigma_1$-elementary submodel of $(N,{\in})$”.

Consider the following two relations between two ordinals $\sigma<\gamma$:

say that “$\sigma$ is $\gamma$-stable” when $L_\sigma \mathrel{\preceq_1} L_\gamma$,

say that “$\sigma$ is $\gamma$-J-stable” when $J_\sigma \mathrel{\preceq_1} J_\gamma$.

Question:Are the above relations equivalent? If not, is there still a way to define “$\sigma$ is $\gamma$-stable” in terms of the Jensen hierarchy and/or “$\sigma$ is $\gamma$-J-stable” in terms of the Gödel hierarchy?

Clearly, any one of the above relations implies that $L_\sigma = J_\sigma$ (because $\sigma$ is, at the very least, admissible $>\omega$), so the problem concerns the right-hand side, as $\gamma$ is not required to be admissible, or even a limit ordinal. Certainly $J_\sigma \mathrel{\preceq_1} J_\gamma$ implies $L_\sigma \mathrel{\preceq_1} L_\gamma$, so the question is whether the converse holds, or, if not, whether we can still find a way to express stability for one hierarchy in terms of the other.

I know that $\sigma$ is $(\sigma+1)$-stable iff it is $\Pi_m$-reflecting for each $m$, for example, but I don't see whether this also applies to being $(\sigma+1)$-J-stable (or, if not, what the corresponding criterion would be).

**Bonus question:** What if we replace $1$ by $n$ (i.e., $\Sigma_1$ by $\Sigma_n$) throughout?

stability? Frankly speaking I opened this question in the hope of finding an unlikely relationship between stability in the model theoretic sense and fine structure theory! ;-) $\endgroup$ – Morteza Azad Aug 26 '17 at 12:46stablein the sense ofremaning truewhile transfering between two levels of these hierarchies. Am I missing anything more important and less transparent at first glance? $\endgroup$ – Morteza Azad Aug 26 '17 at 13:30stable in$V$ thus $L_{\sigma} \prec_{\Sigma_1} V$. The notation $\sigma_1$ is sometimes used for the least such (so countable) $\Sigma_1$ stable in $V$. $\Sigma_n $-stability is no different as a concept. (Except there may be no $\Sigma_2$-stable in $V$ level if $V\neq L$ .) $\endgroup$ – Philip Welch Aug 26 '17 at 18:181more comment