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Stability was developed as a large countable ordinal property in order to try to generalize the different strengthened variants of admissibility. More specifically, they capture the various assertions that $L_\alpha\models\text{KP}+A$ for different axioms $A$ by saying that $L_\alpha\models\text{KP}+A$ for many axioms $A$. One could also argue that stability is a weakening of $\Sigma_1$-correctness (which is trivial) to a nontrivial form.

Stability is defined using a reflection principle. A countable ordinal
$\alpha$ is called **stable** iff $L_\alpha\prec_{\Sigma_1}L$;
equivalently, $L_\alpha\prec_{\Sigma_1}L_{\omega_1}$.
(Madore, 2017)

There are quite a few (weakened) variants of stability:(Madore, 2017)

- A countable ordinal $\alpha$ is called
**$(+\beta)$-stable**iff $L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}$. - A countable ordinal $\alpha$ is called
**$({}^+)$-stable**iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ where $\beta$ is the least admissible ordinal larger than $\alpha$. - A countable ordinal $\alpha$ is called
**$({}^{++})$-stable**iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ where $\beta$ is the least admissible ordinal larger than an admissible ordinal larger than $\alpha$. - A countable ordinal $\alpha$ is called
**inaccessibly-stable**iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ where $\beta$ is the least computably inaccessible ordinal larger than $\alpha$. - A countable ordinal $\alpha$ is called
**Mahlo-stable**iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ where $\beta$ is the least computably Mahlo ordinal larger than $\alpha$; that is, the least $\beta$ such that any $\beta$-recursive function $f:\beta\rightarrow\beta$ has an admissible $\gamma<\beta$ which is closed under $f$. - A countable ordinal $\alpha$ is called
**doubly $(+1)$-stable**iff there is a $(+1)$-stable ordinal $\beta>\alpha$ such that $L_\alpha\prec_{\Sigma_1}L_\beta$. - A countable ordinal $\alpha$ is called
**nonprojectible**iff the set of all $\beta<\alpha$ such that $L_\beta\prec_{\Sigma_1}L_\alpha$ is unbounded in $\alpha$.

Any $L$-stable ordinal is stable. This is because $L_\alpha^L=L_\alpha$ and $L^L=L$. (Jech, 2003) Any $L$-countable stable ordinal is $L$-stable for the same reason. Therefore, an ordinal is $L$-stable iff it is $L$-countable and stable. This property is the same for all variants of stability.

The smallest stable ordinal is also the smallest ordinal $\alpha$ such that $L_\alpha\models\text{KP}+\Sigma_2^1\text{-reflection}$, which in turn is the smallest ordinal which is not the order-type of any $\Delta_2^1$-ordering of the natural numbers. The smallest stable ordinal $\sigma$ has the property that any $\Sigma_1(L_\sigma)$ subset of $\omega$ is $\omega$-finite. (Madore, 2017)

If there is an ordinal $\eta$ such that $L_\eta\models\text{ZFC}$ (i.e. the minimal height of a transitive model of $\text{ZFC}$) then it is smaller than the least stable ordinal. On the other hand, the sizes of the least $(+1)$-stable ordinal and the least nonprojectible ordinal lie between the least recursively weakly compact and the least $Σ_2$-admissible (the same for other weakened variants of stability defined above). (Madore, 2017)

- Madore, D. (2017).
*A zoo of ordinals*. http://www.madore.org/ david/math/ordinal-zoo.pdf - Jech, T. J. (2003).
*Set Theory*(Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf