Stability was developed as a large countable ordinal property in order
to try to generalize the different strengthened variants of
[admissibility](Admissible%20ordinal.md "Admissible").
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](Reflecting.md "Reflecting")
(which is trivial) to a nontrivial form.
## Definition and Variants
Stability is defined using a reflection principle. A countable ordinal
$\alpha$ is called **stable** iff $L_\alpha\prec_{\Sigma_1}L$;
equivalently, $L_\alpha\prec_{\Sigma_1}L_{\omega_1}$.
{% cite Madore2017 %}
### Variants
There are quite a few (weakened) variants of
stability:{% cite Madore2017 %}
- 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](Admissible%20ordinal.md "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 [](Admissible%20ordinal.md)
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 [](Admissible%20ordinal.md)
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 $\alpha$.
## Properties
Any $L$-stable ordinal is stable. This is because
$L_\alpha^L=L_\alpha$ and $L^L=L$.
{% cite Jech2003 %} 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.
{% cite Madore2017 %}
If there is an ordinal $\eta$ such that $L_\eta\models\text{ZFC}$
(i.e. the minimal height of a
<a href="Transitive_ZFC_model" class="mw-redirect" title="Transitive ZFC model">transitive model of $\text{ZFC}
lt;/a>)
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). {% cite Madore2017 %}