Stable

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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.

Definition and Variants

Stability is defined using a reflection principle. Let $\Sigma$ denote the "existential side" of the Levy hierarchy, and let $\prec_\Gamma$ denote the elementary substructure relation with respect to a set of formulae $\Gamma$. A countable ordinal $\alpha$ is called stable iff $L_\alpha\prec_{\Sigma_1}L$. [1]

Variants

An ordinal $\alpha$ is called $\beta$-stable iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ (for $\alpha<\beta$) (Definition 1.17).[2]

There are quite a few (weakened) variants of stability:[1]

  • A countable ordinal $\alpha$ is called $(+\beta)$-stable iff $L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}$ ($\alpha$ is $\alpha+\beta$-stable[2]).
  • A countable ordinal $\alpha$ is called $({}^+)$-stable iff $L_\alpha\prec_{\Sigma_1}L_{\beta}$ where $\beta$ is the least admissible ordinal larger than $\alpha$ ($\alpha$ is $\alpha^+$-stable[2]).
  • 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 $\alpha$.

Further variants have appeared in proof theory, for example in Arai's paper analyzing subsystems of the second-order arithmetic $Z_2$ "Introducing the hardline in proof theory".

Properties

Variants

A related property is nonprojectibility, which has many equivalent characterizations. An ordinal $\alpha$ is nonprojectible iff:

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]

$(+1)$-stable ordinals are exactly the $Π^1_0$-reflecting (i.e., $Π_n$-reflecting for every $n ∈ ω$[1]) ordinals (Theorem 1.18). $({}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals (Theorem 1.19).[2]

Properties of nonprojectible ordinals:

  • If $\alpha$ is nonprojectible, not only is $\alpha$ $\Pi_1$-reflecting on $\{\beta\in\alpha:L_\beta\prec_{\Sigma_1}L_\alpha\}$, but also $\alpha$ is $\Pi_2$-reflecting on $\{\beta\in\alpha:L_\beta\prec_{\Sigma_1}L_\alpha\}$. (E. Kranakis, Reflection and partition properties of admissible ordinals, theorem 2.2)

Stable

On the other hand, 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. In fact, the least stable ordinal is greater than the minimal heights of models of arbitrarily sufficiently satisfiable theories [1] (a more detailed characterization is given in proposition 0.7 of (W. Marek, K. Rasmussen, Spectrum of L)).

The smallest stable ordinal is also the smallest ordinal $\alpha$ that is $\Sigma_2^1$-reflecting [1] (where $\Sigma$ here denotes an extension of the Levy hierarchy) or 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]

Any $L$-stable ordinal is stable. This is because $L_\alpha^L=L_\alpha$ and $L^L=L$. [3] 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.

When $\sigma$ is stable, $L_\sigma$ is $\Sigma_1$-pointwise-definable. $L_{\sigma_{\alpha+1}}$ is pointwise definable for all $\alpha \in \omega_1^L$Theorem 4.8 where $\sigma_\alpha$ is consecutive enumeration of stable ordinals.before Theorem 4.4[4]. If $\alpha$ is stable and less then the first stable gap, then $L_\alpha$ is pointwise definable.Lemma 4.10

The intersection of the sets of countable stable ordinals and $\{\beta\in\omega_1:(L_{\beta+1}\setminus L_\beta)\cap P(\omega)=\varnothing\}$ is a very "sparse" set. For example, if we let $f$ enumerate the countable stable ordinals, and let $\alpha=\textrm{min}\{\sigma:L_\sigma\prec_{\Sigma_1}L\land(L_{\alpha+1}\setminus L_\alpha)\cap P(\omega)=\varnothing\}$ (i.e. $\alpha$ is stable gap), then $\alpha=f(\alpha)$.[4]

There are stronger properties then stability:[4]

  • (The first ordinal that is not $\Delta^1_n$ is called $\delta_n$.)in section 0
  • For $n ≥ 2$Theorem 4.16
    • $L_{\delta_n^L} \prec_{\Sigma_{n-1}} L_{\omega_1^L}$ and $\delta_n^L$ is the least ordinal with this property.
    • $L_{\delta_n^L}$ is $\Sigma_{n-1}$-pointwise definable and it consists exactly of $\Sigma_{n-1}$-definable elements of $\delta_n^L$.
  • $\delta_2$ is the least stable ordinal.Theorem 3.1

References

  1. Madore, David. A zoo of ordinals. , 2017. www   bibtex
  2. Richter, Wayne and Aczel, Peter. Inductive Definitions and Reflecting Properties of Admissible Ordinals. Generalized recursion theory : proceedings of the 1972 Oslo symposium, pp. 301-381, 1974. www   bibtex
  3. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  4. Marek, W. Stable sets, a characterization of $β_2$-models of full second order arithmetic and some related facts. Fundamenta Mathematicae 82(2):175-189, 1974. www   bibtex
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