Admissible
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An ordinal $\gamma$ is admissible if the $L_\gamma$ level of the constructible universe satisfies the Kripke-Platek axioms of set theory. The term was coined by Richard Platek in 1966. (G. Sacks: Higher Recursion Theory, (part C))
The smallest admissible ordinal is often considered to be $\omega$, the least infinite ordinal. However, some authors include Infinity in the KP axioms, in which case $\omega_1^{CK}$,[1] the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$.
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.[1]
Contents
Equivalent definitions
The following properties are also equivalent to admissibility:
$Π_2$-reflecting ordinals are precisely the admissible ordinals $>\omega$. (theorem 1.8)[2]
Higher admissibility
Computably inaccessible ordinal
An ordinal $\alpha$ is computably inaccessible, also known as recursively inaccessible, if it is admissible and a limit of admissible ordinals.[1] If \(f\) enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals \(\alpha\) where \(\alpha=f(\alpha)\).
Recursively Mahlo
An ordinal $α$ is recursively Mahlo iff for any $α$-recursive function $f : α → α$ there is an admissible $β < α$ closed under $f$.[1]
2-admissible
We call $\kappa\in\mathrm{Ad}$ 2-admissible iff every $\xi<\kappa$ such that $\{\xi\}_\kappa$ maps $\kappa$-recursive functions to $\kappa$-recursive functions has a witness ($\alpha\in\mathrm{Ad}$ such that $\xi<\alpha<\kappa$ and $\{\xi\}_\kappa$ maps $\alpha$-recursive functions to $\alpha$-recursive functions). ($\mathrm{Ad}$ is the class of admissible ordinals greater than $\omega$.) TODO: complete definition (definition 1.15)[2]
2-admissible ordinals are precisely the $Π_3$-reflecting ordinals. (theorem 1.16)[2]
2-admissibility is a recursive analogue of 2-regularity, which is equivalent to weak compactness. (theorem 1.14)[2] 2-admissible ordinals can be called recursively weakly compact.[1] More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. (after definition 1.12)[2]
$\Sigma_n$-admissible
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$-admissible if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom.
$\Sigma_n$-admissible ordinals need not necessarily satisfy the $\Sigma_n$-separation schema. For example, the least $\Sigma_2$-admissible ordinal doesn't satisfy $\Sigma_2$-separation.
Here are some properties of $\Sigma_n$-admissibility:
- $\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.
- For $n>1$, $\Sigma_n$-admissibility can be couched in terms of reflection onto sets of stable ordinals (Kranakis, Reflection and partition properties of admissible ordinals), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest nonprojectible ordinal and weaker variants of stable ordinals but smaller than the height of the minimal model of ZFC (if it exists).[1]
Cofinality and projectum
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.
- The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. (W. Maass, Inadmissibility, tame R.E. sets and the admissible collapse, 1976)
- The $\Sigma_n$-projectum of $\beta$ is equal to the least $\delta$ such that some $\Sigma_n(L_\alpha)$-definable function maps a subset of $\delta$ onto $L_\beta$ (K. Devlin, An introduction to the fine structure of the constructible hierarchy, 1972). (However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)
- Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. (W. Maass, Inadmissibility, tame R.E. sets and the admissible collapse, 1976) (Sacks, Higher Recursion Theory, p.157) This is claimed to extend to $n>1$ in [1]
- Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n(L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.[3]^{p.549}
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $\delta$, employing just the ordinal $\delta$ instead. (Compare Σ_{n}: [2], Δ_{n}: [3])
Properties
- $\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ (W. Maass, Inadmissibility, tame R.E. sets and the admissible collapse, 1976).
- Note that although admissibility is considered to be "recursive regularity"[4]^{p.4}, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}(\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}(\omega_1^{CK}\times 2)=\omega$. (This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive (Sacks, Higher Recursion Theory (p.157)),therefore also $\omega_1^{CK}\times 2$-recursive)
- $\beta$ is nonprojectible iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ (K. Devlin, An introduction to the fine structure of the constructible hierarchy, 1972).
- For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals (M. Rathjen, The Art of Ordinal Analysis).
- A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_n\textrm{-proj}(\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. (K. Devlin, An introduction to the fine structure of the constructible hierarchy, 1972). Similarly, if $\Delta_n\textrm{-proj}(\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.
- When $\beta$ is admissible[ citation needed ], $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. (W. Maass, Inadmissibility, tame R.E. sets and the admissible collapse, 1976)
- Applying a result from here, $L_{\Sigma_1\textrm{-cof}(\beta)}$$\prec_{\Sigma_1}$$L_\beta$ and $L_{\Sigma_1\textrm{-proj}(\beta)}\prec_{\Sigma_1}L_\beta$.
Patterns
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}(\alpha)=\Sigma_2\textrm{-proj}(\alpha)>\Sigma_3\textrm{-proj}(\alpha)$. In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $(\Sigma_k\textrm{-proj}(\alpha))_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. [4]
References
- Madore, David. A zoo of ordinals. , 2017. www bibtex
- 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
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex
- Arai, Toshiyasu. A sneak preview of proof theory of ordinals. , 1997. www bibtex