# Difference between revisions of "Admissible"

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===Properties=== | ===Properties=== | ||

*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ [https://igi-web.tugraz.at/PDF/4.pdf#page=3]. | *$\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ [https://igi-web.tugraz.at/PDF/4.pdf#page=3]. | ||

− | **Note that although admissibility is considered to be "recursive regularity"{{citation needed}}, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}(\omega_1 | + | **Note that although admissibility is considered to be "recursive regularity"{{citation needed}}, $\Sigma_1\textrm{-cof}$ 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<!--Barwise, "Part C: α-Recursion (p.157)-->, and $f$ is also $\omega_1^{CK}\times 2$-recursive) |

*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ [https://core.ac.uk/download/pdf/30905237.pdf#page=39]. | *$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ [https://core.ac.uk/download/pdf/30905237.pdf#page=39]. | ||

**For the alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18]. | **For the alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18]. | ||

*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. [https://igi-web.tugraz.at/PDF/4.pdf#page=3] | *When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. [https://igi-web.tugraz.at/PDF/4.pdf#page=3] | ||

*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_{\Sigma_1\textrm{-cof}(\beta)}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_{\Sigma_1\textrm{-proj}(\beta)}\prec_{\Sigma_1}L_\beta$. | *Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_{\Sigma_1\textrm{-cof}(\beta)}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_{\Sigma_1\textrm{-proj}(\beta)}\prec_{\Sigma_1}L_\beta$. | ||

− | <!--Σ_n-projecta of | + | <!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--> |

{{references}} | {{references}} |

## Revision as of 19:02, 13 September 2021

This article is a stub. Please help us to improve Cantor's Attic by adding information.

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.

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:

## 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 and further

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]

There are also *recursively weakly compact* i.e. *$Π_3$-reflecting* or *2-admissible* ordinals.[1]

## Higher admissibility

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), 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 $\Sigma_n$-definable function mapping $\xi$ cofinally into $\beta$. [1]
- 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 $L_\beta$ [2]. (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. [3][Barwise, p.157]

$\Delta_n$-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}: [4], Δ_{n}: [5])

### Properties

- $\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ [6].
- Note that although admissibility is considered to be "recursive regularity"[
*citation needed*], $\Sigma_1\textrm{-cof}$ 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, and $f$ is also $\omega_1^{CK}\times 2$-recursive)

- Note that although admissibility is considered to be "recursive regularity"[
- $\beta$ is nonprojectible iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ [7].
- For the alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals [8].

- When $\beta$ is admissible[
*citation needed*], $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. [9] - Applying a result from here, $L_{\Sigma_1\textrm{-cof}(\beta)}$$\prec_{\Sigma_1}$$L_\beta$ and $L_{\Sigma_1\textrm{-proj}(\beta)}\prec_{\Sigma_1}L_\beta$.

## References

Main library