Difference between revisions of "Stable"

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== Properties ==
 
== Properties ==
 
===Variants===
 
===Variants===
Nonprojectibility has many equivalent properties. An ordinal $\alpha$ is nonprojectible iff:
+
A related property is nonprojectibility, which has many equivalent characterizations. An ordinal $\alpha$ is nonprojectible iff:
 
*$L_\alpha\vDash\Sigma_1\textrm{-separation}$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
 
*$L_\alpha\vDash\Sigma_1\textrm{-separation}$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
 
*There is no [https://en.wikipedia.org/wiki/Alpha_recursion_theory $\alpha$-recursive] injection $f:\alpha\rightarrow\alpha'$ for some $\alpha'\in\alpha$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
 
*There is no [https://en.wikipedia.org/wiki/Alpha_recursion_theory $\alpha$-recursive] injection $f:\alpha\rightarrow\alpha'$ for some $\alpha'\in\alpha$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
*Alternatively, there is no $\alpha$-recursive injection<!--Devlin says its domain is γ∈L_α?--> $f:A\rightarrow\alpha$ mapping a bounded subset of $\alpha$ to $\alpha$. <!--Devlin, "An introduction to the fine structure of the constructible hierarchy", 1974?-->
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*Alternatively, there is no $\alpha$-recursive injection<!--Devlin says its domain is γ∈L_α?--> $f:A\rightarrow\alpha$ mapping a bounded subset of $\alpha$ to $\alpha$. (Devlin, "An introduction to the fine structure of the constructible hierarchy", 1974)
  
 
The sizes of the least $(+1)$-stable ordinal and the least nonprojectible ordinal lie between the least recursively weakly compact and the least [[Admissible#Higher_admissibility|$Σ_2$-admissible]] (the same for other weakened variants of stability defined above). <cite>Madore2017:OrdinalZoo</cite>
 
The sizes of the least $(+1)$-stable ordinal and the least nonprojectible ordinal lie between the least recursively weakly compact and the least [[Admissible#Higher_admissibility|$Σ_2$-admissible]] (the same for other weakened variants of stability defined above). <cite>Madore2017:OrdinalZoo</cite>
 +
 
===Stable===
 
===Stable===
On the other hand, if there is an ordinal $\eta$ such that $L_\eta\models\text{ZFC}$ (i.e. the minimal [[heights of models|height]] of a [[Transitive ZFC model|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.<!--See Taranovsky's "Ordinal Notation" page, section 6.2, find quote "arbitrary sufficiently satisfiable axioms"-->
+
On the other hand, if there is an ordinal $\eta$ such that $L_\eta\models\text{ZFC}$ (i.e. the minimal [[heights of models|height]] of a [[Transitive ZFC model|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 [http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm#A6.2].<!--See Taranovsky's "Ordinal Notation" page, section 6.2, find quote "arbitrary sufficiently satisfiable axioms"-->
  
 
The smallest stable ordinal is also the smallest ordinal $\alpha$ that is $\Sigma_2^1$-[[reflecting]] <cite>Madore2017:OrdinalZoo</cite> (where $\Sigma$ here denotes an extension of the Levy hierarchy<!--RichterAczel73?? Different source-->) 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. <cite>Madore2017:OrdinalZoo</cite>
 
The smallest stable ordinal is also the smallest ordinal $\alpha$ that is $\Sigma_2^1$-[[reflecting]] <cite>Madore2017:OrdinalZoo</cite> (where $\Sigma$ here denotes an extension of the Levy hierarchy<!--RichterAczel73?? Different source-->) 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. <cite>Madore2017:OrdinalZoo</cite>

Revision as of 21:58, 6 January 2022

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

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 $\alpha$.

Further variants have appeared in proof theory, for example this paper by Arai analyzing subsystems of the second-order arithmetic $Z_2$.

Properties

Variants

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

  • $L_\alpha\vDash\Sigma_1\textrm{-separation}$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
  • There is no $\alpha$-recursive injection $f:\alpha\rightarrow\alpha'$ for some $\alpha'\in\alpha$ (Arai, "A sneak preview of proof theory of ordinals, 1997)
  • Alternatively, there is no $\alpha$-recursive injection $f:A\rightarrow\alpha$ mapping a bounded subset of $\alpha$ to $\alpha$. (Devlin, "An introduction to the fine structure of the constructible hierarchy", 1974)

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]

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].

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 $L^L=L$. [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.

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[3]. 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 $\alpha=f(\alpha)$.[3]

There are stronger properties then stability:[3]

  • (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. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  3. 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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