Stable
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.
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, $L_\alpha\prec_{\Sigma_1}L_{\omega_1}$. [1]
Variants
There are quite a few (weakened) variants of stability:[1]
- 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 $\alpha$.
Properties
Any $L$-stable ordinal is stable. This is because $L_\alpha^L=L_\alpha$ and $L^L=L$. [2] 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. [1]
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). [1]
References
- Madore, David. A zoo of ordinals. , 2017. www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex