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Stable ordinals are large ordinals defined using the notion of $$\Sigma_1$$-elementary substructure on levels of constructible universe.[1] (invalid link)[2][footnote 1]

For a natural number $$n$$, let $$M \prec_{\Sigma_n} N$$ denote the relation "$$M$$ is a $$\Sigma_n$$-elementary substructure of $$N$$. Then an ordinal $$\alpha$$ is stable if $$L_\alpha \prec_{\Sigma_1} L$$.[2] The following are various weakenings of the notion of stability:

• An ordinal $$\alpha$$ is $$(+\beta)$$-stable if $$L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}$$.[citation needed]
• An ordinal $$\alpha$$ is $$(^+)$$-stable if $$L_\alpha\prec_{\Sigma_1}L_\beta$$ where $$\beta$$ is next admissible ordinal after $$\alpha$$.[2]
• An ordinal $$\alpha$$ is $$(^{++})$$-stable if $$L_\alpha\prec_{\Sigma_1}L_\beta$$ where $$\beta$$ is next admissible ordinal after the next admissible ordinal after $$\alpha$$.[2]
• An ordinal $$\alpha$$ is inaccessibly-stable if $$L_\alpha\prec_{\Sigma_1}L_\beta$$ where $$\beta$$ is next recursively inaccessible ordinal after $$\alpha$$.[2]
• An ordinal $$\alpha$$ is Mahlo-stable if $$L_\alpha\prec_{\Sigma_1}L_\beta$$ where $$\beta$$ is next recursively Mahlo ordinal after $$\alpha$$.[2]
• An ordinal $$\alpha$$ is doubly $$(+\beta)$$-stable if there exists ordinal $$\gamma$$ such that $$L_\alpha\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}$$.[citation needed]
• An ordinal $$\alpha$$ is nonprojectible if $$\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha$$.[2] Nonprojectible $$\alpha$$ are also $$\Pi_2$$-reflecting on $$\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}$$. [3]

The general definition by Kripke is that for ordinals $$\alpha$$ and $$\beta$$ with $$\beta>\alpha$$, $$\alpha$$ is $$\beta$$-stable iff $$L_\alpha\prec_{\Sigma_1}L_\beta$$.[​3]​​ The enumerating function of stable ordinals is continuous.[4][5]

Suppose $$n\ge 1$$. An ordinal $$\alpha$$ is $$n$$-stable if $$L_\alpha\prec_{\Sigma_n}L$$. For $$\rho>\alpha$$, an ordinal $$\alpha$$ is $$n$$-$$\rho$$-stable (also called "$$(\rho,n)$$-stable"[6]) if $$L_\alpha\prec_{\Sigma_n}L_\rho$$[7].

1-$$\rho$$-stability has the following properties:[8] invalid link[citation needed]

1. If $$\alpha<\beta<\gamma$$ and $$L_\alpha\prec_{\Sigma_1}L_\gamma$$, then $$L_\alpha\prec_{\Sigma_1}L_\beta$$.
2. If $$L_\alpha\prec_{\Sigma_1}L_\beta$$ and $$L_\beta\prec_{\Sigma_1}L_\gamma$$, then $$L_\alpha\prec_{\Sigma_1}L_\gamma$$.
3. If $$\alpha<\beta$$ and $$\beta$$ is stable, then $$\alpha$$ is stable iff $$L_\alpha\prec_{\Sigma_1}L_\beta$$.
4. If a set $$A$$ is nonempty and $$\forall\alpha\in A(L_\alpha\preceq_{\Sigma_1}L_\beta)$$, then $$L_{\sup A}\prec_{\Sigma_1}L_\beta$$.

For an ordinal $$\alpha$$, $$\alpha$$ is nonprojectible iff $$L_\alpha\models\text{KP}\omega+\Sigma_1\text{-Sep}$$[9], and "$$\alpha$$ is nonprojectible" implies "$$\alpha$$ is recursively Mahlo"[10]. For a limit ordinal $$\alpha$$, $$L_\alpha\models\Sigma_n\text{-Sep}+\Sigma_n\text{-Coll}$$ iff $$\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)$$.[11]

Properties of stability:

• (+1)-stable ordinals are exactly $$\Pi_0^1$$-reflecting ordinals.[12]
• $$(^+)$$-stable ordinals are exactly $$\Pi_1^1$$-reflecting ordinals.[13]
• The least $$\Sigma_1^1$$-reflecting ordinal is greater than the least $$\Pi_1^1$$-reflecting ordinal, but every $$(^{++})$$-stable ordinal is $$\Sigma_1^1$$-reflecting.
• Stable ordinals are exactly $$\Sigma_2^1$$-reflecting ordinals, while the least of them is less than the least $$\Pi_2^1$$-reflecting ordinal.[14]
• $$\zeta$$, the supremum of all eventually writable ordinals, is the least ordinal that is 2-$$\rho$$-stable for some $$\rho$$; and $$\Sigma$$, the supremum of all accidentally writable ordinals, is the least ordinal $$\rho$$ such that $$\zeta$$ is 2-$$\rho$$-stable.[15]

## Footnotes

1. Here, $$\Sigma_1$$ denotes Levy's hierarchy. (reference 4) On the other hand, Madore's zoo refers to $$\Pi$$, $$\Sigma$$, and $$\Delta$$ as the hierarchy for arithmetic in other parts. If it is actually intended, then the statement on the reflection property of a $$(+1)$$-stable ordinal conflicts the definition based on $$\Sigma_1$$-elementary substructure. Therefore it might be a mistake, or simply an abuse of notation.

## Sources

1. Cantor's attic
2. David A. Madore, A zoo of ordinals.
3. E. Kranakis, Reflection and partition properties of admissible ordinals (1982, p.218). The proof of interest ((ii)→(iii)) does not require $$\Sigma_2\textrm{-cof}(\alpha)>\omega$$ or $$J_\alpha\vDash\lnot\textrm{Powerset axiom}$$. Accessed 2022-03-25.
4. W. Marek and M. Srebrny, Gaps in the Constructible Universe (1973) (p.382)
5. J. Barwise, Admissible Sets and Structures (p.178)
6. T. Arai, A sneak preview of proof theory of ordinals (1997) (p.13)
7. M. Rathjen, The Higher Infinite in Proof Theory (p.19)
8. Jon Barwise, "Admissible Sets and Structures", page 179
9. M. Rathjen, The Higher Infinite in Proof Theory (p.19)
10. J. Barwise, Admissible Sets and Structures (p.188)
11. M. Rathjen, The Higher Infinite in Proof Theory (p.19)
12. Richter & Aczel (1973) Inductive Definitions and Reflecting Properties of Admissible Ordinals (p.44)
13. Richter & Aczel (1973) Inductive Definitions and Reflecting Properties of Admissible Ordinals (p.16)
14. Wayne Richter, "The Least $$\Sigma^1_2$$ and $$\Pi^1_2$$ Reflecting Ordinals", ISILC Logic Conference, ISBN 3-540-07534-8, 568--578
15. Ansten Morch Klev, "Extending Kleene’s O Using Infinite Time Turing Machines"