There’s a 2009 article by Arai on interated recursive Mahloness. The main idea of the paper is that stronger systems of reflection can be analysed in terms of iterated $\Pi_2$-reflection.

In section 5, Arai writes the following:

A $\Pi_3$-reflecting ordinal $K$ is understood to be $<\varepsilon_{K+1}$[-ply]-recursively Mahlo.The implication here is that in terms of strength, the existence of a $\Pi_3$-reflecting is

*equivalent*to the existence of a $K$ that is $<\varepsilon_{K+1}$-ply-recursively Mahlo.

So a $\Pi_3$-reflecting is equivalent, in some sense, to a $K$ that is $F(\varepsilon_{K+1})$-ply-recursively Mahlo, were $F(\alpha)=\alpha$. Now the question is: what are the ‘Arai functions’ $F$ for higher reflecting ordinals?

For a $\Pi_2$-reflecting, the Arai function is (obviously) $1$. For a $\Pi_{n+1}$-reflecting, the arai function is $F(x)=x^{G(x)}$ where $G$ is the Arai function for $\Pi_n$-reflection.

The final cases are $X$-reflecting ordinals that are also $\Pi_n$-reflecting onto $Y$-reflecting ordinals. Suppose the Arai function for $X$-reflection is $x^{x^{\ldots^{G(x)}}}$ with $n-2$ $x$s and that for $Y$-reflection is $x^{x^{\ldots^{H(x)}}}$ with $n-2$ $x$s, then the Arai function for $X$-reflecting ordinals that are also $\Pi_n$-reflecting onto $Y$-reflecting ordinals is $x^{x^{\ldots^{H(x)+G(x)}}}$.

## Collapsing functions

Fortunately, Arai has a solution for that. First, we agree to notate Arai functions by the value of $F(K)$, where $K$ is some suitable large ordinal. Essentially, given an iterated Mahlo hierarchy $M$ and Skolem hulls $C$, define $\psi^\alpha_\kappa(\beta)$ to be the least ordinal $\pi$ that is in $M^\alpha$ and such that $C(\beta,\pi)\cap\kappa=\pi$.

Since defining $C$ is trivial, we focus on defining $M$. We see that the definition must somehow unpack the Arai function $F$ knowing $F(K)$. However, there are ordinals not described by Arai functions, such as $\kappa$ which are $\kappa$-ply-recursively Mahlo, and are important to reach the full strength of reflection. So we instead notate Arai functions by the value of $F(\varepsilon_{K+1})$. To obtain the Arai function from the value, the solution is the standard diagonalization trick:

$M^\alpha$ is the set of $\kappa$ such that:

- $\kappa< K$
- $C(\alpha,\kappa)\cap K=\kappa$
- For all $\beta\in C(\alpha,\kappa)\cap\alpha$, $\kappa$ is $\Pi_2$-reflecting on $M^\beta$

For completeness, we also define the Skolem hulls $C(\alpha,\kappa)$ as the colsure of $\kappa\cup\{0,K,\varepsilon_{K+1}\}$ under:

- $\beta+\gamma$
- $\omega^\beta$
- $\psi^\beta_\kappa(\gamma)$, provided $\beta,\gamma<\alpha$

If we restrict our ordinals to be below $\varepsilon_{K+1}$, then *voilà*, we have rediscovered Rathjen’s $\Psi$, modulo “technical convenience” (Rathjen’s wording).

## Degrees of Reflection

Those of you familiar with Taranovsky may have noticed a resemblance between the Arai function assignments above and Taranovsky’s Degrees of Reflection. Indeed, if the Arai function is $F$, then the Taranovsky degree is $\Omega^{F(\Omega)}$.

Thus, we can use this to analyse DoR. Note that both DoR and our $\Pi_n$-ref OCF both have additional structure beyond the Arai functions.

In our OCF, an Arai function is coded as $F(\varepsilon_{\Omega+1})$, and in Taranovsky’s, it is coded as $\Omega^{F(\Omega)}$. For the Arai functions corresponding to basic reflection, the former is much larger than the latter, so it seems plausible that DoR misses too much structure to reach $\Pi_n$-ref. Consider the original degrees of reflection. Then the degrees range over precisely the same range as Arai and Rathjen use to analyse $\Pi_3$-ref. Thus, the strength of raw DoR is probably $\Pi_3$-ref. However, using Taranovsky’s suggestion to use Veblen’s function with DoR makes $F(\varepsilon_{\Omega+1})\approx\Omega^{F(\Omega)}$ for most functions $F$ under consideration, so Taranovsky’s claims in this case are most likely correct, though the assignments may need to be lowered.

## Degrees of Reflection ordinal assignments (conjectural)

Based on the previous section, we construct a function $C$ which, conjecturally, models the DoR notation system. We use Taranovskian notation.

Skolem hulls $H(a,b)$ are defined as the closure of $b\cup\{0,\Omega\}$ under $C$ with first argument below $a$ and $O$ provided the output is above $\Omega$.

Define hirarchies $M(a)$ as the set of $b<\Omega$ such that:

- $H(a,b)\cap\Omega=b$
- For all $c\in H(a,b)\cap a$, $b$ is $\Pi_2$-reflecting on $M(c)$

$C(a,b)$ is the least element of $M(a)$ that is greater than $b$.

Of course this assignment is suboptimal–it does not use limits, so the outputs of $C$ are above $\omega_1^{CK}$. However, modifying it to use limits and yield ordinals below $\omega_1^{CK}$ decreases the simplicity.