Difference between revisions of "Stable"

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(Definition and Variants)
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== Definition and Variants ==
 
== Definition and Variants ==
  
Stability is defined using a reflection principle, and although the uncountable analogue is guaranteed to exist by the [[reflecting|reflection theorem]], the existence of countable stable ordinals cannot be proven by ZFC. In fact, the least countable stable ordinal $\beta$ will have an $\alpha<\beta$ such that $L_\alpha\models\text{ZFC}$. <cite>Madore2017:OrdinalZoo</cite>
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Stability is defined using a reflection principle. The least countable stable ordinal $\beta$ will have an $\alpha<\beta$ such that $L_\alpha\models\text{ZFC}$, if such an $\alpha$ exists. <cite>Madore2017:OrdinalZoo</cite>
  
 
A countable ordinal $\alpha$ is called '''stable''' iff $L_\alpha\prec_{\Sigma_1}L$; equivalently, $L_\alpha\prec_{\Sigma_1}L_{\omega_1}$. <cite>Madore2017:OrdinalZoo</cite>  
 
A countable ordinal $\alpha$ is called '''stable''' iff $L_\alpha\prec_{\Sigma_1}L$; equivalently, $L_\alpha\prec_{\Sigma_1}L_{\omega_1}$. <cite>Madore2017:OrdinalZoo</cite>  

Revision as of 10:36, 22 November 2017


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. The least countable stable ordinal $\beta$ will have an $\alpha<\beta$ such that $L_\alpha\models\text{ZFC}$, if such an $\alpha$ exists. [1]

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:

  • 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]

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
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