Stable

• Definition and Variants
  • Variants
• Properties

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 ${\mathrm{\Sigma }}_1$-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 $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}L$; equivalently, $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{{\omega }_1}$. (Madore, 2017)

Variants

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

• A countable ordinal $\alpha$ is called $(+\beta )$-stable iff $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\alpha +\beta }$.
• A countable ordinal $\alpha$ is called $({}^{+})$-stable iff $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\beta }$ where $\beta$ is the least admissible ordinal larger than $\alpha$.
• A countable ordinal $\alpha$ is called $({}^{++})$-stable iff $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\beta }$ where $\beta$ is the least admissible ordinal larger than an admissible ordinal larger than $\alpha$.
• A countable ordinal $\alpha$ is called inaccessibly-stable iff $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\beta }$ where $\beta$ is the least computably inaccessible ordinal larger than $\alpha$.
• A countable ordinal $\alpha$ is called Mahlo-stable iff $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\beta }$ where $\beta$ is the least computably Mahlo ordinal larger than $\alpha$; that is, the least $\beta$ such that any $\beta$-recursive function $f:\beta \to \beta$ 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 $L_{\alpha }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\beta }$.
• A countable ordinal $\alpha$ is called nonprojectible iff the set of all $\beta <\alpha$ such that $L_{\beta }{\prec }_{{\mathrm{\Sigma }}_1}{L}_{\alpha }$ is unbounded in $\alpha$.

Properties

Any $L$-stable ordinal is stable. This is because $L_{\alpha }^L=L_{\alpha }$ and $L^L=L$. (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 $L_{\alpha }\models \text{KP}+{\mathrm{\Sigma }}_2^1\text{-reflection}$, which in turn is the smallest ordinal which is not the order-type of any ${\mathrm{\Delta }}_2^1$-ordering of the natural numbers. The smallest stable ordinal $\sigma$ has the property that any ${\mathrm{\Sigma }}_1(L_{\sigma })$ 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 ${\Sigma }_2$-admissible (the same for other weakened variants of stability defined above). (Madore, 2017)

References

1. Madore, D. (2017). A zoo of ordinals. http://www.madore.org/ david/math/ordinal-zoo.pdf
2. 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

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