A simplified ordinal analysis of first-order reflection

TL;DR

This paper presents a simplified ordinal analysis of first-order reflection using a new ordinal notation system based on ψ-functions, and demonstrates how provable Σ₁-sentences are bounded via cut-elimination.

Contribution

It introduces a simplified ordinal notation system for analyzing first-order reflection and applies cut-elimination to bound provable Σ₁-sentences.

Findings

Ordinal notation system based on ψ-functions

Bounded provable Σ₁-sentences on L_{ω₁^{CK}}

Simplified approach to ordinal analysis of reflection

Abstract

In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system $OT$ is introduced based on $\psi$-functions. Provable $\Sigma_{1}$-sentences on $L_{\omega_{1}^{CK}}$ are bounded through cut-elimination on operator controlled derivations.

Equations

$${\psi }_{\Omega }({\epsilon }_{\mathbb{K}+1})=|\mathsf{KP}{\Pi }_N{|}_{{\Sigma }_1^{\Omega }}:=\min \{\alpha \le {\omega }_1^{CK}:\forall \theta \in {\Sigma }_1(\mathsf{KP}{\Pi }_N⊢{\theta }^{L_{\Omega }}\Rightarrow L_{\alpha }\models \theta )\}.$$

$$\xi {=}_{NF}\sum _{i\le m}{\Lambda }^{{\xi }_i}{a}_i={\Lambda }^{{\xi }_m}{a}_m+\cdots +{\Lambda }^{{\xi }_0}{a}_0$$

$$\pi \in M{h}_k^a(\xi ):\iff \{\pi ,a\}\cup K(\xi )\subset {\mathcal{H}}_a(\pi )\&$$

$$\forall \vec{\nu }<\xi \left(K(\vec{\nu })\subset {\mathcal{H}}_a(\pi )\Rightarrow \pi \in M_k(M{h}_k^a(\vec{\nu }))\right)$$

$$\pi \in M{h}_k^a(\vec{\nu }):\iff \pi \in \bigcap _{k\le i\le n}M{h}_i^a({\nu }_i).$$

$${\psi }_{\pi }^{\vec{\xi }}(a):=\min (\{\pi \}\cup \{\kappa \in M{h}_2^a(\vec{\xi })\cap \pi :{\mathcal{H}}_a(\kappa )\cap \pi \subset \kappa ,K(\vec{\xi })\cup \{\pi ,a\}\subset {\mathcal{H}}_a(\kappa )\})$$

$$B(a):\iff \mathbb{K}\in M{h}_{N-1}^{\alpha }(a)\cap M_{N-1}(M{h}_{N-1}^{\alpha }(a))$$

$$M{h}_m^a(\vec{\nu }){\prec }_{k}M{h}_n^a(\vec{\xi }):\iff \forall \pi \in M{h}_n^a(\vec{\xi })(\{a,\pi \}\cup K(\vec{\nu })\subset {\mathcal{H}}_a(\pi )\Rightarrow \pi \in M_k(M{h}_m^a(\vec{\nu }))).$$

$$\forall k(K_{\alpha }({\nu }_k)<\max K({\nu }_k))$$

$$(b\varepsilon a):\equiv\left\{\begin{array}[]{ll}A(b)&\mbox{{\rm if }}a\equiv[x\in L_{\alpha}:A(x)]\\ b\not\in L_{0}&\mbox{{\rm if }}a\equiv L_{\alpha}\end{array}\right.$$

$$(b\in a):\simeq \bigvee (c\epsilon a\wedge c=b{)}_{c\in J}\text{mbox}\mathrm{and}(b\notin a):\simeq \bigwedge (c/\epsilon a\vee c\ne b{)}_{c\in J}$$

$$\exists x\in aA(x):\simeq \bigvee (b\epsilon a\wedge A(b){)}_{b\in J}\text{mbox}\mathrm{and}\forall x\in aA(x):\simeq \bigwedge (b/\epsilon a\vee A(b){)}_{b\in J}.$$

$$\mathsf{k}(\Gamma )\cup \{a\}\subset {\mathcal{H}}_{\gamma }[\Theta ]$$

$$|\iota |<a$$

$$\forall i<k({\nu }_i={\xi }_i)\&({\nu }_k,\dots ,{\nu }_{N-1}){<}_{sd}{\xi }_k\&K(\vec{\nu })\cup K(\vec{\xi })\subset {\mathcal{H}}_{\gamma }[\Theta ]$$

$$\forall \mu \in \vec{\nu }\cup \vec{\xi }\cup \vec{m}(\pi )(K(\mu )\subset {\mathcal{H}}_{\max K(\mu )}[\Theta ])$$

$$C(\pi ,\gamma ,\Theta )$$

$$\rho \in H(\vec{\nu },\pi ,\gamma ,\Theta )$$

$$\sup \{a_{\ell },a_r(\rho ):\rho \in H(\vec{\nu },\pi ,\gamma ,\Theta )\}\le a_0\in {\mathcal{H}}_{\gamma }[\Theta ]\cap a$$

$$({\mathcal{H}}_c,\Theta \cup \{\kappa \}){⊢}_{\kappa }^{\beta (a,\kappa )}{\Gamma }^{(\kappa ,\pi )}$$

$$\mathsf{k}(\Gamma )\cap \pi \subset {\mathcal{H}}_{\gamma }(\kappa )\cap \pi \subset \kappa$$

$$\{a_0,a_1\}\subset {\mathcal{H}}_{\gamma }[\Theta \cup {\Theta }_0]\&a_0<a_1\&{\Theta }_0\subset \kappa \Rightarrow \beta (a_0,\kappa )<\beta (a_1,\kappa ).$$

$$({\mathcal{H}}_{\gamma },\Theta ){⊢}_{\pi }^{a_{\ell }}\Gamma ,\neg \delta$$

$$({\mathcal{H}}_{\gamma },\Theta \cup \{\rho \}){⊢}_{\pi }^{a_r(\rho )}\Gamma ,{\Delta }^{(\rho ,\pi )}$$

$$({\mathcal{H}}_{\gamma },\Theta \cup \mathsf{k}(\vec{d})){⊢}_{\pi }^{a_{\ell }}\Gamma (\vec{d}),\neg \delta$$

$$\rho \in H(\vec{\nu },\kappa ,c,\Theta \cup \{\kappa \}\cup \mathsf{k}(\vec{d}))\Rightarrow \rho \in H(\vec{\nu },\pi ,\gamma ,\Theta )$$

$$({\mathcal{H}}_c,\Theta \cup \{\rho ,\kappa \}){⊢}_{\kappa }^{\beta (a_r(\rho ),\kappa )}{\Gamma }^{(\kappa ,\pi )},{\Delta }^{(\rho ,\pi )}$$

$$({\mathcal{H}}_c,\Theta \cup \mathsf{k}(\vec{d})\cup \{\rho \}){⊢}_{\rho }^{\beta (a_{\ell },\rho )}\Gamma (\vec{d}{)}^{(\rho ,\pi )},\neg {\delta }^{(\rho ,\pi )}$$

$$\{({\mathcal{H}}_c,\Theta \cup \mathsf{k}(\vec{d})\cup \{\kappa ,\rho \}){⊢}_{\kappa }^{\beta (a(\rho ),\kappa )+p}\Gamma (\vec{d}{)}^{(\rho ,\pi )},{\Gamma }^{(\kappa ,\pi )}:\rho \in M_{\ell }\}$$

$$({\mathcal{H}}_{\gamma },\Theta \cup \mathsf{k}(\vec{d})\cup \{\kappa \}){⊢}_0^{2\text{mbox}\mathrm{rk}(\theta (\vec{d}{)}^{(\kappa ,\pi )})}\Gamma (\vec{d}{)}^{(\kappa ,\pi )},\neg \theta (\vec{d}{)}^{(\kappa ,\pi )}$$

$$({\mathcal{H}}_{\gamma },\Theta ){⊢}_{\pi }^a\Gamma$$

Full text

1 Introduction

Let $ORD$ denote the class of all ordinals, $A\subset ORD$ and $\alpha$ a limit ordinal. $\alpha$ is said to be $\Pi_{n}$-reflecting on $A$ iff for any $\Pi_{n}$-formula $\phi(x)$ and any $b\in L_{\alpha}$, if $\langle L_{\alpha},\in\rangle\models\phi(b)$, then there exists a $\beta\in A\cap\alpha$ such that $b\in L_{\beta}$ and $\langle L_{\beta},\in\rangle\models\phi(b)$. Let us write $\alpha\in rM_{n}(A):\Leftrightarrow\alpha\mbox{ is }\Pi_{n}\mbox{-reflecting on }A$. Also $\alpha$ is said to be $\Pi_{n}$-reflecting iff $\alpha$ is $\Pi_{n}$-reflecting on $ORD$.

It is not hard for us to show that the assumption that the universe is $\Pi_{n}$-reflecting is proof-theoretically reducible to iterabilities of the lower operation $rM_{n-1}$ (and Mostowski collapsings), cf.[3].

In this paper we aim an ordinal analysis of $\Pi_{n}$-reflection. Such an analysis was done by Pohlers and Stegert [7] using reflection configurations introduced in M. Rathjen[9], and an alternative analysis in [1, 2, 4] with the complicated combinatorial arguments of ordinal diagrams and finite proof figures. Our approach is simpler in view of combinatorial arguments. In [1], a $\Pi_{n}$-reflecting universe is resolved in ramified hierarchies of lower Mahlo operations, and ultimately in iterations of recursively Mahlo operations. Our ramification process is akin to a tower, i.e., has an exponential structure. It is natural that an exponential structure emerges in lowering and eliminating first-order formulas (in reflections), cf. ordinal analysis for the fragments ${\rm I}\Sigma_{n-3}$ of the first-order arithmetic. Mahlo classes $Mh_{k}(\xi)$ defined in Definition 2.5 to resolve or approximate $\Pi_{n}$-reflection are based on similar structure. As in Rathjen’s analysis for $\Pi_{3}$-reflection in [8], thinning operations are applied on the Mahlo classes $Mh_{k}(\xi)$, and this yields an exponential structure similar to the one in [1] as follows.

Let us consider the simplest case $N=4$. Let $\Lambda:=\varepsilon_{\mathbb{K}+1}$, the next epsilon number above the lease $\Pi_{4}$-reflecting ordinal $\mathbb{K}$. Roughly $\pi\in Mh_{3}(\xi)$ designates the fact that an ordinal $\pi$ is $\Pi_{3}$-reflecting on $Mh_{3}(\nu)$ for any $\nu<\xi<\Lambda$. Suppose a $\Pi_{3}$-sentence $\theta$ on $L_{\pi}$ is derived from the assumption $\pi\in Mh_{3}(\xi)$. We need to find an ordinal $\kappa<\pi$ for which $L_{\kappa}\models\theta$ holds. It turns out that $\kappa\in Mh_{2}(\Lambda^{\xi}a)$ suffices for an ordinal $a<\Lambda$, where the ordinal $\kappa$ in the class $Mh_{2}(\Lambda^{\xi}a)$ is $\Pi_{2}$-reflecting on classes $Mh_{2}(\Lambda^{\xi}b)\cap Mh_{3}(\nu)$ for any $b<a$ and any $\nu<\xi$. Note that the class $Mh_{2}(\Lambda^{\xi}a)$ is not obtained through iterations of recursively Mahlo operations since it involves $\Pi_{4}$-definable classes $Mh_{3}(\nu)$. The classes $Mh_{3}(\nu)\,(\nu<\xi)$ for the assumption $\pi\in Mh_{3}(\xi)$ are thinned out with the new classes $Mh_{2}(\Lambda^{\xi}b)\,(b<\Lambda)$, cf. Lemma 5.1.

Our theorem runs as follows. Let $\mbox{{\sf KP}}\Pi_{N}$ denote the set theory for $\Pi_{N}$-reflecting universes, and $\mbox{{\sf KP}}\omega$ the Kripke-Platek set theory with the axiom of infinity. $OT$ is a computable notation system of ordinals defined in section 3, $\Omega=\omega_{1}^{CK}$ and $\psi_{\Omega}$ is a collapsing function such that $\psi_{\Omega}(\alpha)<\Omega$. $\mathbb{K}$ is an ordinal term denoting the least $\Pi_{N}$-reflecting ordinal in the theorems.

Theorem 1.1

Suppose ${\sf KP}\Pi_{N}\vdash\theta$ for a $\Sigma_{1}(\Omega)$-sentence $\theta$. Then we can find an $n<\omega$ such that for $\alpha=\psi_{\Omega}(\omega_{n}(\mathbb{K}+1))$, $L_{\alpha}\models\theta$.

Actually the bound is seen to be tight, cf. [5].

Theorem 1.2

${\sf KP}\Pi_{N}$* proves that each initial segment

$\{\alpha\in OT:\alpha<\psi_{\Omega}(\omega_{n}(\mathbb{K}+1))\}\,(n=1,2,\ldots)$ is well-founded.*

Thus the ordinal $\psi_{\Omega}(\varepsilon_{\mathbb{K}+1})$ is seen to be the proof-theoretic ordinal of ${\sf KP}\Pi_{N}$.

Theorem 1.3

[TABLE]

$A\subset ORD$ is $\Pi^{1}_{n}$-indescribable in $\alpha$ iff for any $\Pi^{1}_{n}$-formula $\phi(X)$ and any $B\subset ORD$, if $\langle L_{\alpha},\in;B\cap\alpha\rangle\models\phi(B\cap\alpha)$, then there exists a $\beta\in A\cap\alpha$ such that $\langle L_{\beta},\in;B\cap\beta\rangle\models\phi(B\cap\beta)$. A regular cardinal $\pi$ is $\Pi^{1}_{n}$-indescribable ifff $ORD$ is $\Pi^{1}_{n}$-indescribable in $\pi$.

Let us mention the contents of this paper. In the next section 2 we define simultaneously iterated Skolem hulls $\mathcal{H}_{\alpha}(X)$ of sets $X$ of ordinals, ordinals $\psi^{\vec{\xi}}_{\kappa}(\alpha)$ for regular cardinals $\kappa$, $\alpha<\varepsilon_{\mathbb{K}+1}$ and sequences $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ of ordinals $\xi_{i}<\varepsilon_{\mathbb{K}+2}$, and classes $Mh^{\alpha}_{k}(\xi)$ under the assumption that a $\Pi^{1}_{N-2}$-indescribable cardinal $\mathbb{K}$ exists. It is shown that for $2\leq k<N$, $\alpha<\varepsilon_{\mathbb{K}+1}$ and each $\xi<\varepsilon_{\mathbb{K}+2}$, $(\mathbb{K}\mbox{ {\rm is a }}\Pi^{1}_{N-2}\mbox{{-indescribable cardinal}})\to\mathbb{K}\in Mh^{\alpha}_{k}(\xi)$ in ${\sf ZF}+(V=L)$.

In section 3 a computable notation system $OT$ of ordinals is extracted. Following W. Buchholz[6], operator controlled derivations for $\mbox{{\sf KP}}\Pi_{N}$ is introduced in section 4, and inference rules for $\Pi_{N}$-reflection are eliminated from derivations in section 5. This completes a proof of Theorem 1.1 for an upper bound.

IH denotes the Induction Hypothesis, MIH the Main IH and SIH the Subsidiary IH. We are assuming tacitly the axiom of constructibility $V=L$. Throughout of this paper $N\geq 3$ is a fixed integer.

2 Ordinals for $\Pi_{N}$-reflection

In this section we work in the set theory ${\sf ZFLK}_{N}$ obtained from ${\sf ZFL}={\sf ZF}+(V=L)$ by adding the axiom $\exists\mathbb{K}(\mathbb{K}\mbox{ is }\Pi^{1}_{N-2}\mbox{-indescribable})$ for a fixed integer $N\geq 3$. For ordinals $\alpha$, $\varepsilon(\alpha)$ denotes the least epsilon number above $\alpha$.

Let $ORD\subset V$ denote the class of ordinals, $\mathbb{K}$ the least $\Pi^{1}_{N-2}$-indescribable cardinal, and $Reg$ the set of regular ordinals below $\mathbb{K}$. $\Theta$ denotes finite sets of ordinals$\leq\mathbb{K}$. $u,v,w,x,y,z,\ldots$ range over sets in the universe, $a,b,c,\alpha,\beta,\gamma,\ldots$ range over ordinals$<\Lambda$, $\xi,\zeta,\nu,\mu,\iota,\ldots$ range over ordinals$<\varepsilon(\Lambda)=\varepsilon_{\mathbb{K}+2}$, $\vec{\xi},\vec{\zeta},\vec{\nu},\vec{\mu},\vec{\iota},\ldots$ range over finite sequences over ordinals$<\varepsilon(\Lambda)$, and $\pi,\kappa,\rho,\sigma,\tau,\lambda,\ldots$ range over regular ordinals. $\theta$ denotes formulas.

Let $\vec{\xi}=(\xi_{0},\ldots,\xi_{m-1})$ be a sequence of ordinals. The length $lh(\vec{\xi}):=m$. Sequences consisting of a single element $(\xi)$ is identified with the ordinal $\xi$, and $\emptyset$ denotes the empty sequence. $\vec{0}$ denotes ambiguously a zero-sequence $(0,\ldots,0)$ with its length $0\leq lh(\vec{0})\leq N-1$. $\vec{\xi}*\vec{\mu}=(\xi_{0},\ldots,\xi_{m-1})*(\mu_{0},\ldots,\nu_{n-1})=(\xi_{0},\ldots,\xi_{m-1},\mu_{0},\ldots,\mu_{n-1})$ denotes the concatenated sequence of $\vec{\xi}$ and $\vec{\mu}$.

$\Lambda=\varepsilon(\mathbb{K})=\varepsilon_{\mathbb{K}+1}$ denotes the next epsilon number above the least $\Pi_{N-2}$-

indescribable cardinal $\mathbb{K}$, and $\varepsilon(\Lambda)=\varepsilon_{\mathbb{K}+2}$ the next epsilon number above $\Lambda$.

Definition 2.1

For a non-zero ordinal $\xi<\varepsilon(\Lambda)$, its Cantor normal form with base $\Lambda$ is uniquely determined as

[TABLE]

where $\xi_{m}>\cdots>\xi_{0},\,0<a_{i}<\Lambda$.

$K(\xi)=\{a_{i}:i\leq m\}\cup\bigcup\{K(\xi_{i}):i\leq m\}$ is the set of components of $\xi$ with $K(0)=\emptyset$. For a sequence $\vec{\xi}=(\xi_{0},\ldots,\xi_{n-1})$ of ordinals $\xi_{i}<\varepsilon(\Lambda)$, $K(\vec{\xi}):=\bigcup\{K(\xi_{i}):i<n\}$. 2. 2.

For $\xi>1$, $te(\xi)=\xi_{0}$ in (1) is the tail exponent, and $he(\xi)=\xi_{m}$ is the head exponent of $\xi$, resp. The head $Hd(\xi):=\Lambda^{\xi_{m}}a_{m}$, and the tail $Tl(\xi):=\Lambda^{\xi_{0}}a_{0}$ of $\xi$. 3. 3.

$he^{(i)}(\xi)$ is the $i$-th head exponent of $\xi$, defined recursively by

$he^{(0)}(\xi)=\xi$, $he^{(i+1)}(\xi)=he(he^{(i)}(\xi))$.

The $i$-th tail exponent $te^{(i)}(\xi)$ is defined similarly. 4. 4.

$\zeta$ is a part of $\xi$, denoted by $\zeta\leq_{pt}\xi$ iff

$\zeta=_{NF}\sum_{i\geq n}\Lambda^{\xi_{i}}a_{i}=\Lambda^{\xi_{m}}a_{m}+\cdots+\Lambda^{\xi_{n}}a_{n}$ for an $n\,(0\leq n\leq m+1)$.

$\zeta<_{pt}\xi:\Leftrightarrow\zeta\leq_{pt}\xi\,\&\,\zeta\neq\xi$. 5. 5.

A sequence $\vec{\mu}=(\mu_{0},\ldots,\mu_{n})$ is an iterated tail parts of $\xi$, denoted by $\vec{\mu}\subset_{pt}\xi$ iff $\mu_{0}\leq_{pt}\xi\,\&\,\forall i<n(\mu_{i+1}\leq_{pt}te(\mu_{i}))$. 6. 6.

$\vec{\nu}=(\nu_{0},\ldots,\nu_{n})*\vec{0}<\xi$ iff there exists a sequence $\vec{\mu}=(\mu_{0},\ldots,\mu_{n})$ such that $\vec{\mu}\subset_{pt}\xi$ and $\nu_{i}<\mu_{i}$ for every $i\leq n$. 7. 7.

Let $\vec{\nu}=(\nu_{0},\ldots,\nu_{n})$ and $\vec{\xi}=(\xi_{0},\ldots,\xi_{n})$ be sequences of ordinals in the same length, and $0\leq k\leq n$.

$\vec{\nu}<_{k}\vec{\xi}:\Leftrightarrow\forall i<k(\nu_{i}\leq\xi_{i})\land(\nu_{k},\ldots,\nu_{n})<\xi_{k}$. 8. 8.

$\zeta$ is a step-down of $\xi$, denoted by $\zeta<_{sd}\xi$ iff

$\zeta=\Lambda^{\xi_{m}}a_{m}+\cdots+\Lambda^{\xi_{1}}a_{1}+\Lambda^{\xi_{0}}b+\nu$ for some ordinals $b<a_{0}$ and $\nu<\Lambda^{\xi_{0}}$. 9. 9.

$\vec{\nu}=(\nu_{0},\ldots,\nu_{n})*\vec{0}<_{sd}\xi$ iff $\nu_{i}<_{sd}te^{(i)}(\xi)$ for every $i\leq n$. 10. 10.

$\zeta\leq_{sp}\xi:\Leftrightarrow\exists\mu\leq_{pt}\xi(\zeta\leq_{sd}\mu)$, and $\zeta<_{sp}\xi:\Leftrightarrow\exists\mu\leq_{pt}\xi(\zeta<_{sd}\mu)$. 11. 11.

$\vec{\nu}<_{sp}\xi$ iff $\vec{\nu}<_{sd}\mu$ for a $\mu\leq_{pt}\xi$.

Let $p(\vec{\nu},\xi)$ denote the number $p\,(0\leq p<m)$ such that $\xi=_{NF}\mu+\sum_{i<p}\Lambda^{\xi_{i}}a_{i}$ for $\mu=\Lambda^{\xi_{m}}a_{m}+\cdots+\Lambda^{\xi_{p}}a_{p}$ and $\vec{\nu}<_{sd}\mu$.

Note that $(\nu)*\vec{0}<\xi\Leftrightarrow\nu<\xi$, and $(\xi,te(\xi),te^{(2)}(\xi),\ldots)\subset_{pt}\xi$. Also $\zeta<_{sd}\xi\Leftrightarrow\zeta<\xi$ if $\xi<\Lambda$.

Proposition 2.2

$\xi<\mu<\varepsilon(\Lambda)\Rightarrow te(\xi)\leq he(\xi)\leq he(\mu)$.

Proposition 2.3

$\vec{\nu}<\xi\leq\zeta\Rightarrow\vec{\nu}<\zeta$.

Proof by induction on the lengths $n=lh(\vec{\nu})$. Let $\vec{\mu}=(\mu_{0},\ldots,\mu_{n-1})$ be a sequence for $\vec{\nu}=(\nu_{0},\ldots,\nu_{n-1})$ such that $\vec{\mu}\subset_{pt}\xi$ and $\forall i\leq n-1(\nu_{i}<\mu_{i})$, cf. Definition 2.1.6.

If $n=1$, then $\nu_{0}<\mu_{0}\leq_{pt}\xi\leq\zeta$. $\nu_{0}<\zeta\leq_{pt}\zeta$ yields $\vec{\nu}=(\nu_{0})<\zeta$.

Let $n>1$. We have $(\nu_{1},\ldots,\nu_{n-1})<te(\mu_{0})$ with $(\mu_{1},\ldots,\mu_{n-1})\subset_{pt}te(\mu_{0})$. We show the existence of a $\lambda$ such that $\mu_{0}\leq\lambda\leq_{pt}\zeta$ and $te(\mu_{0})\leq te(\lambda)$. Then IH yields $(\nu_{1},\ldots,\nu_{n-1})<te(\lambda)$, and $\vec{\nu}<\zeta$ follows.

If $\mu_{0}\leq_{pt}\zeta$, then $\lambda=\mu_{0}$ works. Suppose $\mu_{0}\not\leq_{pt}\zeta$. On the other hand we have $\mu_{0}\leq_{pt}\xi\leq\zeta$. This means that $\xi<\zeta$ and there exists a $\lambda\leq_{pt}\zeta$ such that $\mu_{0}<\lambda$ and $te(\mu_{0})\leq te(\lambda)$. $\Box$

2.1 Ordinals

Definition 2.4

For $i<\omega$ and $\xi<\varepsilon(\Lambda)$, $\Lambda_{i}(\xi)$ is defined recursively by $\Lambda_{0}(\xi)=\xi$ and $\Lambda_{i+1}(\xi)=\Lambda^{\Lambda_{i}(\xi)}$. 2. 2.

For $A\subset ORD$, limit ordinals $\alpha$ and $i\geq 0$, let $\alpha\in M_{2+i}(A)$ iff $A\cap\alpha\mbox{ {\rm is} }\Pi^{1}_{i}\mbox{{\rm-indescribable in} }\alpha$. 3. 3.

$\kappa^{+}$ denotes the next regular ordinal above $\kappa$. 4. 4.

$\Omega_{\alpha}:=\omega_{\alpha}$ for $\alpha>0$, $\Omega_{0}:=0$, and $\Omega=\Omega_{1}$.

Define simultaneously classes $\mathcal{H}_{\alpha}(X)$, $Mh^{\alpha}_{k}(\xi)$, and ordinals $\psi_{\kappa}^{\vec{\xi}}(\alpha)$ as follows. We see that these are $\Sigma_{1}$-definable as a fixed point in ${\sf ZFL}$, cf. Proposition 2.7.

Let $a<\Lambda$, and $\varphi$ denote the binary Veblen function. Let us define a Skolem hull $\mathcal{H}_{a}(X)$ of $\{0,\mathbb{K}\}\cup X$ under the functions $+,\alpha\mapsto\omega^{\alpha},(\alpha,\beta)\mapsto\varphi\alpha\beta\,(\alpha,\beta<\mathbb{K}),\alpha\mapsto\Omega_{\alpha}\,(\alpha<\mathbb{K})$ and $\psi$-functions. $Reg$ denotes the set of regular ordinals$\leq\mathbb{K}$.

Definition 2.5

${\cal H}_{a}[Y](X):={\cal H}_{a}(Y\cup X)$ for sets $Y\subset\mathbb{K}$.

(Inductive definition of $\mathcal{H}_{a}(X)$).

1. (a)

$\{0,\mathbb{K}\}\cup X\subset\mathcal{H}_{a}(X)$. 2. (b)

$x,y\in\mathcal{H}_{a}(X)\Rightarrow x+y\in\mathcal{H}_{a}(X)$, $x\in{\cal H}_{a}(X)\Rightarrow\omega^{x}\in\mathcal{H}_{a}(X)$, and $x,y\in\mathcal{H}_{a}(X)\cap\mathbb{K}\Rightarrow\varphi xy\in\mathcal{H}_{a}(X)$. 3. (c)

$\mathbb{K}>\alpha\in\mathcal{H}_{a}(X)\Rightarrow\Omega_{\alpha}\in\mathcal{H}_{a}(X)$. 4. (d)

If $\pi\in\mathcal{H}_{a}(X)\cap Reg$ and $b\in\mathcal{H}_{a}(X)\cap a$, then $\psi_{\pi}(b)\in\mathcal{H}_{a}(X)$. 5. (e)

If $\{b,\xi\}\subset\mathcal{H}_{a}(X)$ with $\xi\leq b<a$, then $\kappa=\psi_{\mathbb{K}}^{\vec{0}*(\xi)}(b)\in\mathcal{H}_{a}(X)$, where $lh(\vec{0})=N-3$. 6. (f)

Let $\{\pi,b,c\}\subset{\cal H}_{a}(X)$ with $\pi<\mathbb{K}$, $2\leq k<N-1$ an integer, and $\vec{\xi}=(\xi_{2},\ldots,\xi_{k},\xi_{k+1})*\vec{0}$ a sequence of ordinals $\xi_{i}<\varepsilon(\Lambda)$ with $lh(\vec{0})=N-2-k$ such that $\xi_{k+1}\neq 0$ and $K(\vec{\xi})\subset{\cal H}_{a}(X)$. Assume $\max(K(\vec{\xi})\cup\{c\})\leq b<a$, and $\pi\in Mh_{2}^{b}(\vec{\xi})$.

Then $\kappa=\psi_{\pi}^{\vec{\nu}}(b)\in{\cal H}_{a}(X)$ for the sequence $\vec{\nu}=(\xi_{2},\ldots,\xi_{k}+\Lambda^{\xi_{k+1}}c)*\vec{0}$ with $lh(\vec{0})=N-1-k$. 7. (g)

Let $\{\pi,b\}\subset{\cal H}_{a}(X)$ with $\pi<\mathbb{K}$, and $0\neq\xi<\varepsilon(\Lambda)$ an ordinal with $K(\xi)\subset{\cal H}_{a}(X)$. Let $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})$ be a sequence of ordinals$<\varepsilon(\Lambda)$ such that $K(\vec{\nu})\subset{\cal H}_{a}(X)$. Assume $\max K(\vec{\nu})\leq b<a$, $K(\vec{\nu})\subset{\cal H}_{b}(\pi)$, $\pi\in Mh_{2}^{b}(\xi)$, and $\vec{\nu}<\xi$, cf. Definition 2.1.6.

Then $\kappa=\psi_{\pi}^{\vec{\nu}}(b)\in{\cal H}_{a}(X)$. 2. 2.

(Definitions of $Mh^{a}_{k}(\xi)$ and $Mh^{a}_{k}(\vec{\xi})$)

First let $\mathbb{K}\in Mh^{a}_{N}(0):\Leftrightarrow\mathbb{K}\in M_{N}\Leftrightarrow\mathbb{K}\mbox{ {\rm is }}\Pi^{1}_{N-2}\mbox{{\rm-indescribable}}$.

The classes $Mh^{a}_{k}(\xi)$ are defined for $2\leq k<N$, and ordinals $a<\Lambda$, $\xi<\varepsilon(\Lambda)$. Let $\pi$ be a regular ordinal$\leq\mathbb{K}$. Then for $\xi>0$

[TABLE]

where $\vec{\nu}=(\nu_{k},\ldots,\nu_{n})\,(2\leq k\leq n\leq N-1)$ varies through non-empty sequences of ordinals$<\varepsilon(\Lambda)$ and

[TABLE]

By convention, let for $2\leq k<N$, $\pi\in Mh^{a}_{k}(0):\Leftrightarrow\pi\in Mh^{a}_{2}(\emptyset):\Leftrightarrow\pi\mbox{ {\rm is a limit ordinal}}$. Note that by letting $\vec{\nu}=(0)$, $\pi\in Mh^{a}_{k}(\xi)\Rightarrow\pi\in M_{k}$ for $\xi>0$. Also $\vec{0}<1$, and $Mh^{a}_{k}(1)=M_{k}$. 3. 3.

(Definition of $\psi_{\pi}^{\vec{\xi}}(a)$)

Let $a<\Lambda$ be an ordinal, $\pi\leq\mathbb{K}$ a regular ordinal and $\vec{\xi}$ a sequence of ordinals$<\varepsilon(\Lambda)$ such that $lh(\vec{\xi})=N-2$. Then let

[TABLE]

Let $\psi_{\pi}a:=\psi_{\pi}^{\vec{0}}a$, where $lh(\vec{0})=N-2$, $Mh^{a}_{2}(\vec{0})=Lim$, and $\pi\in M_{2}$, i.e., $\pi$ is a regular ordinal.

Note that $\pi\in Mh^{a}_{k}(\xi)\Rightarrow\forall\nu<\xi\left(\pi\in M_{k}(Mh^{a}_{k}(\nu))\right)$, since $(\nu)<\xi$ holds with $(\xi)\subset_{pt}\xi$ for $\nu<\xi$.

Proposition 2.6

$b+c\in{\cal H}_{a}[\Theta](d)\Rightarrow c\in{\cal H}_{a}[\Theta](d)$, and $\omega^{c}\in{\cal H}_{a}[\Theta](d)\Rightarrow c\in{\cal H}_{a}[\Theta](d)$.

The following Proposition 2.7 is easy to see.

Proposition 2.7

Each of $x=\mathcal{H}_{a}(y)\,(a<\Lambda,y<\mathbb{K})$, $x=\psi_{\kappa}a$, $x\in Mh^{a}_{k}(\xi)$ and $x=\psi^{\vec{\xi}}_{\kappa}(a)$, is a $\Sigma_{1}$-predicate as fixed points in ZFL.

Proof.  This is seen from the facts that there exists a universal $\Pi^{1}_{n}$-formula, and by using it, $\alpha\in M_{n}(x)$ iff $\langle L_{\alpha},\in\rangle\models m_{n}(x\cap L_{\alpha})$ for some $\Pi^{1}_{n+1}$-formula $m_{n}(R)$ with a unary predicate $R$. $\Box$

Let $A(a)$ denote the conjunction of $\forall u<\mathbb{K}\exists!x[x=\mathcal{H}_{a}(u)]$, and

$\forall\vec{\xi}\forall x(\max K(\vec{\xi})\leq a\,\&\,K(\vec{\xi})\cup\{\kappa,a\}\subset x=\mathcal{H}_{a}(\kappa)\to\exists!b\leq\kappa(b=\psi^{\vec{\xi}}_{\kappa}(a)))$, where $lh(\vec{\xi})=N-2$.

Since the cardinality of the set $\mathcal{H}_{\varepsilon_{\mathbb{K}+1}}(\pi)$ is $\pi$ for any infinite cardinal $\pi\leq\mathbb{K}$, pick an injection $f:\mathcal{H}_{\Lambda}(\mathbb{K})\to\mathbb{K}$ so that $f"\mathcal{H}_{\Lambda}(\pi)\subset\pi$ for any weakly inaccessibles $\pi\leq\mathbb{K}$.

Lemma 2.8

$\forall a<\Lambda\,A(a)$. 2. 2.

$\pi\in Mh^{a}_{k}(\xi)$* is a $\Pi^{1}_{k-1}$-class on $L_{\pi}$ uniformly for weakly inaccessible cardinals $\pi\leq\mathbb{K}$ and $a,\xi$. This means that for each $k$ there exists a $\Pi^{1}_{k-1}$-formula $mh^{a}_{k}(x)$ such that $\pi\in Mh^{a}_{k}(\xi)$ iff $L_{\pi}\models mh^{a}_{k}(\xi)$ for any weakly inaccessible cardinals $\pi\leq\mathbb{K}$ with $f"(\{a\}\cup K(\xi))\subset L_{\pi}$.* 3. 3.

$\mathbb{K}\in Mh^{\alpha}_{N-1}(\Lambda)\cap M_{N-1}(Mh^{\alpha}_{N-1}(\Lambda))$.

Proof.  

2.8.1. We show that $A(a)$ is progressive, i.e., $\forall a<\Lambda[\forall c<a\,A(c)\to A(a)]$.

Assume $\forall c<a\,A(c)$ and $a<\Lambda$. $\forall b<\mathbb{K}\exists!x[x=\mathcal{H}_{a}(b)]$ follows from IH in ${\sf ZFL}$. $\exists!b\leq\kappa(b=\psi^{\vec{\xi}}_{\kappa}a)$ follows from this.

2.8.2. Let $\pi$ be a weakly inaccessible cardinal with $f"(\{a\}\cup K(\xi))\subset L_{\pi}$. Let $f$ be an injection such that $f"{\cal H}_{\Lambda}(\pi)\subset L_{\pi}$. Then for $\forall\alpha\in K(\xi)(f(\alpha)\in f"{\cal H}_{\alpha}(\pi))$, $\pi\in Mh^{a}_{k}(\xi)$ iff for any $f(\vec{\nu})=(f(\nu_{k}),\ldots,f(\nu_{N-1}))$, each of $f(\nu_{i})\in L_{\pi}$, if $\forall\alpha\in K(\vec{\nu})(f(\alpha)\in f"{\cal H}_{a}(\pi))$ and $\vec{\nu}<\xi$, then $\pi\in M_{k}(Mh^{a}_{k}(\vec{\nu}))$, where $f"{\cal H}_{a}(\pi)\subset L_{\pi}$ is a class in $L_{\pi}$.

2.8.3. We show the following $B(a)$ is progressive in $a<\Lambda$:

[TABLE]

Note that $a\in\mathcal{H}_{a}(\mathbb{K})$ holds for any $a<\Lambda$.

Suppose $\forall b<a\,B(b)$. We have to show that $Mh^{\alpha}_{N-1}(a)$ is $\Pi^{1}_{N-3}$-indescribable in $\mathbb{K}$. It is easy to see that if $\pi\in M_{N-1}(Mh^{\alpha}_{N-1}(a))$, then $\pi\in Mh^{\alpha}_{N-1}(a)$ by induction on $\pi$. Let $\theta(u)$ be a $\Pi^{1}_{N-3}$-formula such that $L_{\mathbb{K}}\models\theta(u)$.

By IH we have $\forall b<a[\mathbb{K}\in M_{N-1}(Mh^{\alpha}_{N-1}(b))]$. In other words, $\mathbb{K}\in Mh^{\alpha}_{N-1}(a)$, i.e., $L_{\mathbb{K}}\models mh^{\alpha}_{N-1}(a)$, where $mh^{\alpha}_{N-1}(a)$ is a $\Pi^{1}_{N-2}$-sentence in Proposition 2.8.2. Since the universe $L_{\mathbb{K}}$ is $\Pi^{1}_{N-2}$-indescribable, pick a $\pi<\mathbb{K}$ such that $L_{\pi}$ enjoys the $\Pi^{1}_{N-2}$-sentence $\theta(u)\land mh^{\alpha}_{N-1}(a)$, and $\{f(\alpha),f(a)\}\subset L_{\pi}$. Therefore $\pi\in Mh^{\alpha}_{N-1}(a)$ and $L_{\pi}\models\theta(u)$. Thus $\mathbb{K}\in M_{N-1}(Mh^{\alpha}_{N-1}(a))$. $\Box$

2.2 Normal forms in ordinal notations

In this subsection we introduce an irreducibility of sequences, which is needed to define a normal form in ordinal notations.

Proposition 2.9

$\pi\in Mh^{a}_{k}(\zeta)\,\&\,\xi\leq\zeta\Rightarrow\pi\in Mh^{a}_{k}(\xi)$.

Proof.  (2) for $\pi\in Mh^{a}_{k}(\xi)$ in Definition 2.5.2 follows from $\pi\in Mh^{a}_{k}(\zeta)$ and Proposition 2.3. $\Box$

Lemma 2.10

(Cf. Lemma 3 in [1].)* Assume $\mathbb{K}\geq\pi\in Mh^{a}_{k}(\xi)\cap Mh^{a}_{k+1}(\xi_{0})$ with $2\leq k\leq N-1$, $he(\mu)\leq\xi_{0}$ and $\{a\}\cup K(\mu)\subset{\cal H}_{a}(\pi)$. Then $\pi\in Mh^{a}_{k}(\xi+\mu)$ holds. Moreover if $\pi\in M_{k+1}$, then $\pi\in M_{k+1}(Mh^{a}_{k}(\xi+\mu))$ holds.*

Proof.  Suppose $\pi\in Mh^{a}_{k}(\xi)\cap Mh^{a}_{k+1}(\xi_{0})$ and $K(\mu)\subset{\cal H}_{a}(\pi)$ with $he(\mu)\leq\xi_{0}$. We show $\pi\in Mh^{a}_{k}(\xi+\mu)$ by induction on ordinals $\mu$. First note that if $b\in{\cal H}_{a}(\pi)$, then $f(b)\in f"{\cal H}_{\Lambda}(\pi)\subset L_{\pi}$. We have $K(\xi+\mu)\subset{\cal H}_{a}(\pi)$. $\pi\in M_{k+1}(Mh^{a}_{k}(\xi+\mu))$ follows from $\pi\in Mh^{a}_{k}(\xi+\mu)$ and $\pi\in M_{k+1}$.

Let $(\zeta)*\vec{\nu}<\xi+\mu$ and $K(\zeta)\cup K(\vec{\nu})\subset{\cal H}_{a}(\pi)$ for $\vec{\nu}=(\nu_{0},\ldots,\nu_{n-1})$. We need to show that $\pi\in M_{k}(Mh^{a}_{k}((\zeta)*\vec{\nu}))$. By Definition 2.1.6, let $(\zeta_{0})*(\mu_{0},\ldots,\mu_{n-1})$ be a sequence such that $\zeta<\zeta_{0}\leq_{pt}\xi+\mu$, $\mu_{0}\leq_{pt}te(\zeta_{0})$, $\forall i\leq n-1(\nu_{i}<\mu_{i})$, and $\forall i<n-1(\mu_{i+1}\leq_{pt}te(\mu_{i}))$.

If $\zeta_{0}\leq_{pt}\xi$, then $(\zeta)*\vec{\nu}<\xi$, and $\pi\in M_{k}(Mh^{a}_{k}((\zeta)*\vec{\nu}))$ by $\pi\in Mh^{a}_{k}(\xi)$.

Let $\zeta_{0}=\xi+\zeta_{1}$ with $0<\zeta_{1}\leq_{pt}\mu$. If $\zeta_{1}<_{pt}\mu$, then by IH with $he(\zeta_{1})=he(\mu)$ we have $\pi\in Mh^{a}_{k}(\zeta_{0})$. On the other hand we have $(\zeta)*\vec{\nu}<\zeta_{0}$. Hence $\pi\in M_{k}(Mh^{a}_{k}((\zeta)*\vec{\nu}))$.

Finally consider the case when $0<\zeta_{1}=\mu$. Then we obtain $\vec{\nu}<te(\xi+\mu)=te(\mu)\leq he(\mu)\leq\xi_{0}$. $\pi\in Mh^{a}_{k+1}(\xi_{0})$ with Proposition 2.9 yields $\pi\in M_{k+1}(Mh^{a}_{k+1}(\vec{\nu}))$.

On the other side we see $\pi\in Mh^{a}_{k}(\zeta)$ as follows. We have $\zeta<\xi+\mu$. If $\zeta\leq\xi$, then this follows from $\pi\in Mh^{a}_{k}(\xi)$ and Proposition 2.9, and if $\zeta=\xi+\lambda<\xi+\mu$, then IH yields $\pi\in Mh^{a}_{k}(\zeta)$.

Since $\pi\in Mh^{a}_{k}(\zeta)$ is a $\Pi^{1}_{k-1}$-sentence holding on $L_{\pi}$ by Lemma 2.8.2 and $\{a\}\cup K(\zeta)\subset{\cal H}_{a}(\pi)$, we obtain $\pi\in M_{k+1}(Mh^{a}_{k}((\zeta)*\vec{\nu}))$, a fortiori $\pi\in M_{k}(Mh^{a}_{k}((\zeta)*\vec{\nu}))$. $\Box$

Definition 2.11

For sequences of ordinals $\vec{\xi}=(\xi_{k},\ldots,\xi_{N-1})$ and $\vec{\nu}=(\nu_{k},\ldots,\nu_{N-1})$ and $2\leq k,m,n\leq N-1$,

[TABLE]

Corollary 2.12

Let $\vec{\nu}$ be a sequence defined from a sequence $\vec{\xi}$ as follows. $\forall i<k(\nu_{i}=\xi_{i})$, $\forall i>k(\nu_{i}=0)$, and $\nu_{k}=\xi_{k}+\Lambda^{\xi_{k+1}}b$, where $2\leq k<N$, $b<\Lambda$ and $\xi_{k+1}\neq 0$. Then $Mh^{a}_{2}(\vec{\nu})\prec_{k+1}Mh^{a}_{2}(\vec{\xi})$ holds. In particular if $\pi\in Mh^{a}_{2}(\vec{\xi})$ and $K(\vec{\nu})\cup\{\pi,a\}\subset{\cal H}_{a}(\pi)$, then $\psi_{\pi}^{\vec{\nu}}(a)<\pi$.

Proof.  This is seen from Lemma 2.10. $\Box$

Proposition 2.13

Let $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})$, $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ be sequences of ordinals$<\varepsilon(\Lambda)$ such that $\vec{\nu}<_{k}\vec{\xi}$ for an integer $k$ with $2\leq k\leq N-1$. Then $Mh^{a}_{2}(\vec{\nu})\prec_{k}Mh^{a}_{2}(\vec{\xi})$. In particular if $\pi\in Mh^{a}_{2}(\vec{\xi})$ and $K(\vec{\nu})\cup\{\pi,a\}\subset{\cal H}_{a}(\pi)$, then $\psi_{\pi}^{\vec{\nu}}(a)<\pi$.

Proof.  Assume $\pi\in Mh^{a}_{2}(\vec{\xi})$ and $K(\vec{\nu})\subset{\cal H}_{a}(\pi)$. We have $\pi\in Mh^{a}_{k}(\xi_{k})$. By the definition (2) and $(\nu_{k},\ldots,\nu_{N-1})<\xi_{k}$, we obtain $\pi\in M_{k}(\bigcap_{k\leq i\leq N-1}Mh^{a}_{i}(\nu_{i}))$.

On the other hand we have $\pi\in\bigcap_{i<k}Mh^{a}_{i}(\xi_{i})$, and hence $\pi\in\bigcap_{i<k}Mh^{a}_{i}(\nu_{i})$ by $\forall i<k(\nu_{i}\leq\xi_{i})$ and Proposition 2.9. Since $\pi\in\bigcap_{i<k}Mh^{a}_{i}(\nu_{i})$ is a $\Pi^{1}_{k-2}$-sentence holding in $L_{\pi}$, we obtain $\pi\in M_{k}(\bigcap_{i\leq N-1}Mh^{a}_{i}(\nu_{i}))=M_{k}(Mh^{a}_{2}(\vec{\nu}))$, a fortiori $\pi\in M_{2}(Mh^{a}_{2}(\vec{\nu}))$.

Suppose $\{\pi,a\}\subset{\cal H}_{a}(\pi)$. The set $C=\{\kappa<\pi:{\cal H}_{a}(\kappa)\cap\pi\subset\kappa,K(\vec{\nu})\cup\{\pi,a\}\subset{\cal H}_{a}(\kappa)\}$ is a club subset of the regular cardinal $\pi$. This shows the existence of a $\kappa\in Mh_{2}^{a}(\vec{\nu})\cap C\cap\pi$, and hence $\psi_{\pi}^{\vec{\nu}}(a)<\pi$ by the definition (3). $\Box$

Proposition 2.14

Let $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ be a sequence of ordinals$<\varepsilon(\Lambda)$ such that $\{\pi,a\}\cup K(\vec{\xi})\subset{\cal H}_{a}(\pi)$. Assume $Tl(\xi_{i})<\Lambda_{k}(\xi_{i+k}+1)$ for some $i<N-1$ and $k>0$. Then $\pi\in Mh^{a}_{2}(\vec{\xi})\Leftrightarrow\pi\in Mh^{a}_{2}(\vec{\mu})$, where $\vec{\mu}=(\mu_{2},\ldots,\mu_{N-1})$ with $\mu_{i}=\xi_{i}-Tl(\xi_{i})$ and $\mu_{j}=\xi_{j}$ for $j\neq i$.

Proof.  When $0<\xi_{i}=\Lambda^{\gamma_{m}}a_{m}+\cdots+\Lambda^{\gamma_{1}}a_{1}+\Lambda^{\gamma_{0}}a_{0}$ with $\gamma_{m}>\cdots>\gamma_{1}>\gamma_{0},\,0<a_{i}<\Lambda$, $\mu_{i}=\Lambda^{\gamma_{m}}a_{m}+\cdots+\Lambda^{\gamma_{1}}a_{1}$ for $Tl(\xi_{i})=\Lambda^{\gamma_{0}}a_{0}$. If $\xi_{i}=0$, then so is $\mu_{i}=0$.

Let $\pi\in Mh^{a}_{2}(\vec{\mu})$ and $Tl(\xi_{i})<\Lambda_{k}(\xi_{i+k}+1)$. We obtain $\forall j\leq k(he^{(j)}(Tl(\xi_{i}))<\Lambda_{k-j}(\xi_{i+k}+1))$, and $he^{(k)}(Tl(\xi_{i}))\leq\xi_{i+k}$. On the other hand we have $\pi\in Mh^{a}_{i+k}(\xi_{i+k})$. From Lemma 2.10 we see inductively that for any $j<k$, $\pi\in Mh^{a}_{i+j}(he^{(j)}(Tl(\xi_{i})))$. In particular $\pi\in Mh^{a}_{i+1}(he(Tl(\xi_{i})))$, and once again by Lemma 2.10 and $\pi\in Mh^{a}_{i}(\mu_{i})$ we obtain $\pi\in Mh^{a}_{i}(\xi_{i})$. Hence $\pi\in Mh^{a}_{2}(\vec{\xi})$. $\Box$

Definition 2.15

A sequence of ordinals $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ is said to be irreducible iff $\forall i<N-1\forall k>0(\xi_{i}>0\Rightarrow Tl(\xi_{i})\geq\Lambda_{k}(\xi_{i+k}+1))$. **

Proposition 2.16

Let $\vec{\nu}=(\nu_{k},\ldots,\nu_{N-1})\neq\vec{0}$ be an irreducible sequence, and $k_{0}\geq k$ be the least number such that $\nu_{k_{0}}\neq 0$. Assume $\nu_{k_{0}}<he^{(k_{0}-k)}(\xi)$. Then $\vec{\nu}<\xi$ holds in the sense of Definition 2.1.6.

Proof.  Let $\ell<N-k$ be the largest number such that $\nu_{k+\ell}\neq 0$. We show $(\nu_{k},\ldots,\nu_{k+\ell})<\xi$. Since $\vec{\nu}$ is irreducible, we have $\Lambda_{i}(\nu_{k_{0}+i}+1)\leq Tl(\nu_{k_{0}})$. From $\nu_{k_{0}}<he^{(k_{0}-k)}(\xi)$ and $te(\mu)\leq he(\mu)$ we obtain $\nu_{k_{0}+i}<\nu_{k_{0}+i}+1\leq he^{(i)}(\nu_{k_{0}})\leq he^{(k_{0}-k+i)}(\xi)$. Let $(\mu_{k},\ldots,\mu_{N-1})\subset_{pt}\xi$ such that $\mu_{k}=Hd(\xi)$ and $\mu_{i+1}=he(\mu_{i})=te(Hd(\mu_{i}))$. Then $te(\mu_{k+i})=he(\mu_{k+i})$ and $\mu_{k_{0}+i}=he(\mu_{k_{0}+i-1})=he^{(k_{0}-k+i)}(\xi)$ for $k_{0}-k+i>0$. Therefore $(\mu_{k},\ldots,\mu_{k+\ell})\subset_{pt}\xi$ witnesses $(\nu_{k},\ldots,\nu_{k+\ell})<\xi$. $\Box$

Definition 2.17

Let* $\vec{\xi}=(\xi_{k},\ldots,\xi_{N-1})$, $\vec{\nu}=(\nu_{k},\ldots,\nu_{N-1})$ and $\vec{\nu}\neq\vec{\xi}$. Let $i\geq k$ be the minimal number such that $\nu_{i}\neq\xi_{i}$. Suppose $(\xi_{i},\ldots,\xi_{N-1})\neq\vec{0}$, and let $k_{1}\geq i$ be the minimal number such that $\xi_{k_{1}}\neq 0$. Then $\vec{\nu}<_{lx,k}\vec{\xi}$ iff one of the followings holds:*

$(\nu_{i},\ldots,\nu_{N-1})=\vec{0}$. 2. 2.

In what follows assume* $(\nu_{i},\ldots,\nu_{N-1})\neq\vec{0}$, and let $k_{0}\geq i$ be the minimal number such that $\nu_{k_{0}}\neq 0\,(i=\min\{k_{0},k_{1}\})$. Then $\vec{\nu}<_{lx,k}\vec{\xi}$ iff one of the followings holds:*

1. (a)

$i=k_{0}<k_{1}$* and $he^{(k_{1}-k_{0})}(\nu_{k_{0}})\leq\xi_{k_{1}}$.* 2. (b)

$k_{0}\geq k_{1}=i$* and $\nu_{k_{0}}<he^{(k_{0}-k_{1})}(\xi_{k_{1}})$.*

Proposition 2.18

Suppose that both of $\vec{\nu}$ and $\vec{\xi}$ are irreducible. Then $\vec{\nu}<_{lx,k}\vec{\xi}\Rightarrow Mh^{a}_{k}(\vec{\nu})\prec_{k}Mh^{a}_{k}(\vec{\xi})$.

Proof.  Let $\pi\in Mh^{a}_{k}(\vec{\xi})$, $K(\vec{\nu})\subset{\cal H}_{a}(\pi)$, and $i\geq k$ be the minimal number such that $\nu_{i}\neq\xi_{i}$. We have $\pi\in\bigcap_{k\leq j<i}Mh^{a}_{j}(\nu_{j})$, which is a $\Pi^{1}_{i-2}$-sentence holding on $L_{\pi}$. In the case $\xi_{i}\neq 0$, it suffices to show that $\pi\in M_{i}(\bigcap_{j\geq i}Mh^{a}_{j}(\nu_{j}))$, since then we obtain $\pi\in M_{i}(Mh^{a}_{k}(\vec{\nu}))$ by $\pi\in Mh^{a}_{i}(\xi_{i})\subset M_{i}$, a fortiori $\pi\in M_{k}(Mh^{a}_{k}(\vec{\nu}))$.

If $(\nu_{i},\ldots,\nu_{N-1})=\vec{0}$, then $\xi_{i}\neq 0$ and $\bigcap_{j\geq i}Mh^{a}_{j}(\nu_{j})$ denotes the class of limit ordinals. Obviously $\pi\in M_{i}(\bigcap_{j\geq i}Mh^{a}_{j}(\nu_{j}))$.

In what follows assume $(\nu_{i},\ldots,\nu_{N-1})\neq\vec{0}$, and let $k_{0}\geq i$ be the minimal number such that $\nu_{k_{0}}\neq 0$, and $k_{1}\geq i$ be the minimal number such that $\xi_{k_{1}}\neq 0$.

Case 1. $k_{0}\geq k_{1}=i$: Then we have $\nu_{k_{0}}<he^{(k_{0}-k_{1})}(\xi_{k_{1}})$. Proposition 2.16 yields $(\nu_{k_{0}},\ldots,\nu_{N-1})<\xi_{k_{1}}=\xi_{i}$, which in turn yields $\pi\in M_{i}(\bigcap_{j\geq i}Mh^{a}_{j}(\nu_{j}))$ by the definition (2) of $\pi\in Mh^{a}_{i}(\xi_{i})$.

Case 2. $i=k_{0}<k_{1}$: Then we have $he^{(k_{1}-i)}(\nu_{i})\leq\xi_{k_{1}}$. Also $\nu_{i+p}<he^{(p)}(\nu_{i})$ for any $p>0$ since $\vec{\nu}$ is irreducible and $\nu_{i}\neq 0$. Let $j\geq k_{1}$. Then $\nu_{j}<he^{(j-i)}(\nu_{i})\leq he^{(j-k_{1})}(\xi_{k_{1}})$. In particular $\nu_{k_{1}}<\xi_{k_{1}}$. Proposition 2.16 yields $(\nu_{k_{1}},\ldots,\nu_{N-1})<\xi_{k_{1}}$. $\pi\in Mh^{a}_{k_{1}}(\xi_{k_{1}})$ yields $\pi\in M_{k_{1}}(\bigcap_{j\geq k_{1}}Mh^{a}_{j}(\nu_{j}))$. Moreover for any $p<k_{1}-i$, $he^{(k_{1}-i-p)}(\nu_{i+p})\leq\xi_{k_{1}}$ by Proposition 2.2. Lemma 2.10 yields $\pi\in\bigcap_{k_{1}>j\geq i}Mh^{a}_{j}(\nu_{j})$. Therefore $\pi\in M_{k_{1}}(Mh^{a}_{k}(\vec{\nu}))$, a fortiori $\pi\in M_{k}(Mh^{a}_{k}(\vec{\nu}))$. $\Box$

Proposition 2.19

(Cf. Proposition 4.20 in [8])

Let $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})$, $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ be irreducible sequences of ordinals$<\varepsilon(\Lambda)$, and assume that $\psi_{\pi}^{\vec{\nu}}(b)<\pi$ and $\psi_{\kappa}^{\vec{\xi}}(a)<\kappa$.

Then $\beta_{1}=\psi_{\pi}^{\vec{\nu}}(b)<\psi_{\kappa}^{\vec{\xi}}(a)=\alpha_{1}$ iff one of the following cases holds:

$\pi\leq\psi_{\kappa}^{\vec{\xi}}(a)$. 2. 2.

$b<a$, $\psi_{\pi}^{\vec{\nu}}(b)<\kappa$ and $K(\vec{\nu})\cup\{\pi,b\}\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$. 3. 3.

$b>a$* and $K(\vec{\xi})\cup\{\kappa,a\}\not\subset{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$.* 4. 4.

$b=a$, $\kappa<\pi$ and $\kappa\not\in{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$. 5. 5.

$b=a$, $\pi=\kappa$, $K(\vec{\nu})\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$, and $\vec{\nu}<_{lx,2}\vec{\xi}$. 6. 6.

$b=a$, $\pi=\kappa$, $K(\vec{\xi})\not\subset{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$.

Proof.  If the case (2) holds, then $\psi_{\pi}^{\vec{\nu}}(b)\in{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))\cap\kappa\subset\psi_{\kappa}^{\vec{\xi}}(a)$.

If one of the cases (3) and (4) holds, then $K(\vec{\xi})\cup\{\kappa,a\}\not\subset{\cal H}_{a}(\psi_{\pi}^{\vec{\nu}}(b))$. On the other hand we have $K(\vec{\xi})\cup\{\kappa,a\}\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$. Hence $\psi_{\pi}^{\vec{\nu}}(b)<\psi_{\kappa}^{\vec{\xi}}(a)$.

If the case (5) holds, then Proposition 2.18 yields $Mh^{a}_{2}(\vec{\nu})\prec_{2}Mh^{a}_{2}(\vec{\xi})\ni\psi_{\kappa}^{\vec{\xi}}(a)$. Hence $\psi_{\kappa}^{\vec{\xi}}(a)\in M_{2}(Mh^{a}_{2}(\vec{\nu}))$. Since $K(\vec{\nu})\cup\{\kappa,a\}\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$, the set $\{\rho<\psi_{\kappa}^{\vec{\xi}}(a):{\cal H}_{a}(\rho)\cap\kappa\subset\rho,K(\vec{\nu})\cup\{\kappa,a\}\subset{\cal H}_{a}(\rho)\}$ is club in $\psi_{\kappa}^{\vec{\xi}}(a)$. Therefore $\psi_{\pi}^{\vec{\nu}}(b)=\psi_{\kappa}^{\vec{\nu}}(a)<\psi_{\kappa}^{\vec{\xi}}(a)$ by (3) in Definition 2.5.3.

Finally assume that the case (6) holds. Since $K(\vec{\xi})\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$, $\psi_{\pi}^{\vec{\nu}}(b)<\psi_{\kappa}^{\vec{\xi}}(a)$ holds.

Conversely assume that $\psi_{\pi}^{\vec{\nu}}(b)<\psi_{\kappa}^{\vec{\xi}}(a)$ and $\psi_{\kappa}^{\vec{\xi}}(a)<\pi$.

First consider the case $b<a$. Then we have $K(\vec{\nu})\cup\{\pi,b\}\subset{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$. Hence (2) holds.

Next consider the case $b>a$. $K(\vec{\xi})\cup\{\kappa,a\}\subset{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$ would yield $\psi_{\kappa}^{\vec{\xi}}(a)\in{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))\cap\pi\subset\psi_{\pi}^{\vec{\nu}}(b)$, a contradiction $\psi_{\kappa}^{\vec{\xi}}(a)<\psi_{\pi}^{\vec{\nu}}(b)$. Hence (3) holds.

Finally assume $b=a$. Consider the case $\kappa<\pi$. $\kappa\in{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))\cap\pi$ would yield $\psi_{\kappa}^{\vec{\xi}}(a)<\kappa<\psi_{\pi}^{\vec{\nu}}(b)$, a contradiction. Hence $\kappa\not\in{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$, and (4) holds. If $\pi<\kappa$, then $\pi\in{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))\cap\kappa\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))\cap\kappa$, and $\pi<\psi_{\kappa}^{\vec{\xi}}(a)$, a contradiction, or we should say that (1) holds. Finally let $\pi=\kappa$. We can assume that $K(\vec{\xi})\subset{\cal H}_{b}(\psi_{\pi}^{\vec{\nu}}(b))$, otherwise (6) holds. If $\vec{\xi}<_{lx,2}\vec{\nu}$, then by (5) $\psi_{\kappa}^{\vec{\xi}}(a)<\psi_{\pi}^{\vec{\nu}}(b)$ would follow. If $K(\vec{\nu})\not\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$, then by (6) again $\psi_{\kappa}^{\vec{\xi}}(a)<\psi_{\pi}^{\vec{\nu}}(b)$ would follow. Hence $K(\vec{\nu})\subset{\cal H}_{a}(\psi_{\kappa}^{\vec{\xi}}(a))$ and $\vec{\nu}\leq_{lx}\vec{\xi}$. If $\vec{\nu}=\vec{\xi}$, then $\psi_{\kappa}^{\vec{\xi}}(a)=\psi_{\pi}^{\vec{\nu}}(b)$. Therefore (5) must be the case. $\Box$

Definition 2.20 is utilized to define a computable notation system in the next section 3.

Definition 2.20

A set $SD$ of sequences $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})$ of ordinals $\xi_{i}<\varepsilon(\Lambda)$ is defined recursively as follows.

$\vec{0}*(a)\in SD$ for each $a<\Lambda$. 2. 2.

(Cf. Definition 2.1.9.) Let $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})\in SD$, $1\leq k<N-1$, $\zeta<\varepsilon(\Lambda)$ be an ordinal such that $(\xi_{k+1},\ldots,\xi_{N-1})<_{sd}\zeta$, and $(\xi_{2},\ldots,\xi_{k-1},\xi_{k},\zeta)*\vec{0}\in SD$. Then for $\zeta_{k}=\xi_{k}+\Lambda^{\zeta}a$ with an ordinal $a<\Lambda$, $(\xi_{2},\ldots,\xi_{k-1})*(\zeta_{k})*(\xi_{k+1},\ldots,\xi_{N-1})\in SD$ and $(\xi_{2},\ldots,\xi_{k-1})*(\zeta_{k})*\vec{0}\in SD$.

Proposition 2.21

Let $\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})\in SD$.

$(\xi_{2},\ldots,\xi_{i})*\vec{0}\in SD$* for each $i$ with $1\leq i<N$.* 2. 2.

For $2\leq i<j<k<N$, if $\xi_{i}\neq 0$ and $\xi_{k}\neq 0$, then $\xi_{j}\neq 0$. 3. 3.

Let $\xi_{i}\neq 0$. Then $(\xi_{i+1},\ldots,\xi_{N-1})<_{sd}te(\xi_{i})$. 4. 4.

$\vec{\xi}$* is irreducible.*

Proof.  Let $1\leq k<N-1$, $\zeta<\varepsilon(\Lambda)$ be an ordinal such that $(\xi_{k+1},\ldots,\xi_{N-1})<_{sd}\zeta$, and $(\xi_{2},\ldots,\xi_{k-1},\xi_{k},\zeta)*\vec{0}\in SD$. Also let $\zeta_{k}=\xi_{k}+\Lambda^{\zeta}a$ with an ordinal $a<\Lambda$.

2.21.1 is seen by induction on the recursive definition of $\vec{\xi}\in SD$.

2.21.2 is seen by induction on the recursive definition of $\vec{\xi}\in SD$. Suppose $\xi_{i}\neq 0$ for an $i<k$. From $(\xi_{2},\ldots,\xi_{k-1},\xi_{k},\zeta)*\vec{0}\in SD$ and $\zeta\neq 0$, IH yields $\xi_{k}\neq 0$.

2.21.3 and 2.21.4. We show these by simultaneous induction on the recursive definition of $\vec{\xi}\in SD$.

2.21.3. We show Proposition 2.21.3 for the sequence $(\xi_{2},\ldots,\xi_{k-1})*(\zeta_{k})*(\xi_{k+1},\ldots,\xi_{N-1})\in SD$. The proposition holds for the sequence $\vec{\xi}$, and we can assume $a\neq 0$. We obtain $(\xi_{i+1},\ldots,\xi_{N-1})<_{sd}te(\xi_{i})$ for $i>k$ if $\xi_{i}\neq 0$, and $(\xi_{k+1},\ldots,\xi_{N-1})<_{sd}te(\zeta_{k})=\zeta$ by the assumption. Let $2\leq i<k$ and $\xi_{i}\neq 0$. We show $(\xi_{i+1},\ldots,\xi_{k-1})*(\zeta_{k})*(\xi_{k+1},\ldots,\xi_{N-1})<_{sd}te(\xi_{i})$. It suffices to show that $\zeta_{k}<_{sd}te^{(k-i)}(\xi_{i})$. By IH we have $\xi_{k}<_{sd}te^{(k-i)}(\xi_{i})$. On the other hand we have $\xi_{k}\neq 0$ by $(\xi_{2},\ldots,\xi_{k-1},\xi_{k},\zeta)*\vec{0}\in SD$, $\zeta\neq 0$, and Proposition 2.21.2. Moreover $(\xi_{2},\ldots,\xi_{k-1},\xi_{k},\zeta)*\vec{0}$ is irreducible by Proposition 2.21.4, and hence $Tl(\xi_{k})\geq\Lambda^{\zeta+1}$. Therefore $te(\xi_{k})>\zeta$. This means that $\zeta_{k}=_{NF}\xi_{k}+\Lambda^{\zeta}a$, and $\xi_{k}<_{sd}te^{(k-i)}(\xi_{i})$ yields $\zeta_{k}<_{sd}te^{(k-i)}(\xi_{i})$ by Definition 2.1.8.

2.21.4. If $(\xi_{i+1},\ldots,\xi_{N-1})<_{sd}te(\xi_{i})$ for $\xi_{i}\neq 0$, then $\xi_{i+k}<_{sd}te^{(k)}(\xi_{i})$ for $k>0$, and $\xi_{i+k}+1\leq te^{(k)}(\xi_{i})$. Hence $\Lambda_{k}(\xi_{i+k}+1)\leq\Lambda^{te(\xi_{i})}\leq Tl(\xi_{i})$, and $\vec{\xi}$ is irreducible. $\Box$

3 Computable notation system $OT$

In this section (except Propositions 3.3) we work in a weak fragment of arithmetic, e.g., in the fragment $I\Sigma_{1}$ or even in the bounded arithmetic $S^{1}_{2}$. Referring Proposition 2.19 the sets of ordinal terms $OT\subset\Lambda=\varepsilon_{\mathbb{K}+1}$ and $E\subset\varepsilon(\Lambda)=\varepsilon_{\mathbb{K}+2}$ over symbols $\{0,\mathbb{K},\Lambda,+,\omega,\varphi,\Omega,\psi\}$ are defined recursively. $OT$ is isomorphic to a subset of ${\cal H}_{\Lambda}(0)$. Simultaneously we define finite sets $K_{\delta}(\alpha)\subset OT$ for $\delta,\alpha\in OT$, and sequences $(m_{k}(\alpha))_{2\leq k\leq N-1}$ for $\alpha\in OT\cap\mathbb{K}$, where in $\alpha=\psi_{\pi}^{\vec{\nu}}(a)$, $m_{k}(\alpha)=\nu_{k}$, i.e., $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})=(m_{2}(\alpha),\ldots,m_{N-1}(\alpha))=(m_{k}(\alpha))_{k}=\vec{m}(\alpha)$. For $\{\alpha_{0},\ldots,\alpha_{m},\beta\}\subset OT$, $K_{\delta}(\alpha_{0},\ldots,\alpha_{m}):=\bigcup_{i\leq m}K_{\delta}(\alpha_{i})$, $K_{\delta}(\alpha_{0},\ldots,\alpha_{m})<\beta:\Leftrightarrow\forall\gamma\in K_{\delta}(\alpha_{0},\ldots,\alpha_{m})(\gamma<\beta)$, and $\beta\leq K_{\delta}(\alpha_{0},\ldots,\alpha_{m}):\Leftrightarrow\exists\gamma\in K_{\delta}(\alpha_{0},\ldots,\alpha_{m})(\beta\leq\gamma)$.

An ordinal term in $OT$ is said to be a regular term if it is one of the form $\mathbb{K}$, $\Omega_{\beta+1}$ or $\psi_{\pi}^{\vec{\nu}}(a)$ with the non-zero sequences $\vec{\nu}\neq\vec{0}$. $\mathbb{K}$ and the latter terms $\psi_{\pi}^{\vec{\nu}}(a)$ are Mahlo terms.

$\alpha=_{NF}\alpha_{m}+\cdots+\alpha_{0}$ means that $\alpha=\alpha_{m}+\cdots+\alpha_{0}$ and $\alpha_{m}\geq\cdots\geq\alpha_{0}$ and each $\alpha_{i}$ is a non-zero additive principal number. $\alpha=_{NF}\varphi\beta\gamma$ means that $\alpha=\varphi\beta\gamma$ and $\beta,\gamma<\alpha$. $\alpha=_{NF}\omega^{\beta}$ means that $\alpha=\omega^{\beta}>\beta$. $\alpha=_{NF}\Omega_{\beta}$ means that $\alpha=\Omega_{\beta}>\beta$.

Let $pd(\psi_{\pi}^{\vec{\nu}}(a))=\pi$ (even if $\vec{\nu}=\vec{0}$). Moreover for $n$, $pd^{(n)}(\alpha)$ is defined recursively by $pd^{(0)}(\alpha)=\alpha$ and $pd^{(n+1)}(\alpha)\simeq pd(pd^{(n)}(\alpha))$.

For terms $\pi,\kappa\in OT$, $\pi\prec\kappa$ denotes the transitive closure of the relation $\{(\pi,\kappa):\exists\vec{\xi}\exists b[\pi=\psi_{\kappa}^{\vec{\xi}}(b)]\}$, and its reflexive closure $\pi\preceq\kappa:\Leftrightarrow\pi\prec\kappa\lor\pi=\kappa\Leftrightarrow\exists n(\kappa=pd^{(n)}(\pi))$.

For each ordinal term $\alpha=\psi_{\pi}^{\vec{\nu}}(a)$, a series $(\pi_{i})_{i\leq L}$ of ordinal terms is uniquely determined as follows: $\pi_{L}=\alpha$, $\pi_{i}=pd(\pi_{i+1})$ and $\pi_{0}=\mathbb{K}$. Let us call the series $(\pi_{i})_{i\leq L}$ the collapsing series of $\alpha=\pi_{L}$.

Then we see that an ordinal term $\alpha=\psi_{\pi}^{\vec{\nu}}(a)$ with $\vec{\nu}\neq\vec{0}$ is constructed by Definition 3.1.2g below iff $L=1$. $\alpha$ is constructed by Definition 3.1.2i iff $L\equiv 1\!\!\!\!\pmod{(N-2)}$. Otherwise $\alpha$ is constructed by Definition 3.1.2h.

Definition 3.1

$\ell\alpha$ denotes the number of occurrences of symbols

$\{0,\mathbb{K},\Lambda,+,\omega,\varphi,\Omega,\psi\}$ in terms $\alpha\in OT\cup E$.

2. (a)

$0\in E$. 2. (b)

If $0<a\in OT$, then $a\in E$. $K(a)=\{a\}$. 3. (c)

If $\{\xi_{i}:i\leq m\}\subset E$, $\xi_{m}>\cdots>\xi_{0}>0$ and $0<b_{i}\in OT$, then $\sum_{i\leq m}\Lambda^{\xi_{i}}b_{i}=\Lambda^{\xi_{m}}b_{m}+\cdots+\Lambda^{\xi_{0}}b_{0}\in E$. $K(\sum_{i\leq m}\Lambda^{\xi_{i}}b_{i})=\{b_{i}:i\leq m\}\cup\bigcup\{K(\xi_{i}):i\leq m\}$. 4. (d)

For sequences $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})$, let $K(\vec{\nu})=\bigcup_{2\leq i\leq N-1}K(\nu_{i})$. 2. 2.

1. (a)

$0,\mathbb{K}\in OT$. $m_{k}(0)=0$ for any $k$, and $K_{\delta}(0)=K_{\delta}(\mathbb{K})=\emptyset$. 2. (b)

If $\alpha=_{NF}\alpha_{m}+\cdots+\alpha_{0}\,(m>0)$ with $\{\alpha_{i}:i\leq m\}\subset OT$, then $\alpha\in OT$, and $m_{k}(\alpha)=0$ for any $k$. $K_{\delta}(\alpha)=K_{\delta}(\alpha_{0},\ldots,\alpha_{m})$. 3. (c)

If $\alpha=_{NF}\varphi\beta\gamma$ with $\{\beta,\gamma\}\subset OT\cap\mathbb{K}$, then $\alpha\in OT$, and $m_{k}(\alpha)=0$ for any $k$. $K_{\delta}(\alpha)=K_{\delta}(\beta,\gamma)$. 4. (d)

If $\alpha=_{NF}\omega^{\beta}$ with $\mathbb{K}<\beta\in OT$, then $\alpha\in OT$, and $m_{k}(\alpha)=0$ for any $k$. $K_{\delta}(\alpha)=K_{\delta}(\beta)$. 5. (e)

If $\alpha=_{NF}\Omega_{\beta}$ with $\beta\in OT\cap\mathbb{K}$, then $\alpha\in OT$. $m_{2}(\alpha)=1,m_{k}(\alpha)=0$ for any $k>2$ if $\beta$ is a successor ordinal. Otherwise $m_{k}(\alpha)=0$ for any $k$. In each case $K_{\delta}(\alpha)=K_{\delta}(\beta)$. 6. (f)

Let $\alpha=\psi_{\pi}(a):=\psi_{\pi}^{\vec{0}}(a)$ where $\pi$ is a regular term , i.e., either $\pi=\mathbb{K}$ or $\vec{m}(\pi)\neq\vec{0}$, and $K_{\alpha}(\pi,a)<a$.

Then $\alpha=\psi_{\pi}(a)\in OT$. Let $m_{k}(\alpha)=0$ for any $k$. $K_{\delta}(\psi_{\pi}(a))=\emptyset$ if $\alpha<\delta$. $K_{\delta}(\psi_{\pi}(a))=\{a\}\cup K_{\delta}(a,\pi)$ otherwise. 7. (g)

Let $\alpha=\psi_{\mathbb{K}}^{\vec{\nu}}(a)$ with $\vec{\nu}=\vec{0}*(b)\,(lh(\vec{\nu})=N-2)$ and $b,a\in OT$ such that $0<b\leq a$ and $K_{\alpha}(b,a)<a$.

Then $\alpha=\psi_{\mathbb{K}}^{\vec{\nu}}(a)\in OT$. Let $m_{N-1}(\alpha)=b$, $m_{k}(\alpha)=0$ for $k<N-1$. $K_{\delta}(\psi_{\mathbb{K}}^{\vec{\nu}}(a))=\emptyset$ if $\alpha<\delta$. $K_{\delta}(\psi_{\mathbb{K}}^{\vec{\nu}}(a))=\{a\}\cup\bigcup\{K_{\delta}(\gamma):\gamma\in K(\nu)\}$ otherwise. 8. (h)

Let $\pi\in OT\cap\mathbb{K}$ be such that $m_{k+1}(\pi)\neq 0$ and $\forall i>k+1(m_{i}(\pi)=0)$ for a $k\,(2\leq k\leq N-2)$, and $b,a\in OT$ such that $0<b\leq a$. Let $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})$ be a sequence defined by $\forall i<k(\nu_{i}=m_{i}(\pi))$, $\nu_{k}=m_{k}(\pi)+\Lambda^{m_{k+1}(\pi)}b$, and $\forall i>k(\nu_{i}=0)$.

Then $\alpha=\psi_{\pi}^{\vec{\nu}}(a)\in OT$ if $K_{\alpha}(\pi,a,b)\cup K_{\alpha}(K(\vec{m}(\pi)))<a$. Let $m_{i}(\alpha)=\nu_{i}$ for each $i$. $K_{\delta}(\psi_{\pi}^{\vec{\nu}}(a))=\emptyset$ if $\alpha<\delta$. Otherwise $K_{\delta}(\psi_{\pi}^{\vec{\nu}}(a))=\{a\}\cup K_{\delta}(a,\pi)\cup\bigcup\{K_{\delta}(b):b\in K(\vec{\nu})\}$. 9. (i)

Let $\pi\in OT\cap\mathbb{K}$ be such that $m_{2}(\pi)\neq 0$ and $\forall i>2(m_{i}(\pi)=0)$, and $a\in OT$. Let $\vec{0}\neq\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})\in SD$ be a sequence of ordinal terms $\nu_{i}\in E$ such that $\vec{\nu}<_{sp}m_{2}(\pi)$.

Then $\alpha=\psi_{\pi}^{\vec{\nu}}(a)$ if $K_{\alpha}(\pi,a)<a$, and

[TABLE]

Let $m_{i}(\alpha)=\nu_{i}$ for each $i$.

$K_{\delta}(\psi_{\pi}^{\vec{\nu}}(a))=\emptyset$ if $\alpha<\delta$. Otherwise $K_{\delta}(\psi_{\pi}^{\vec{\nu}}(a))=\{a\}\cup K_{\delta}(a,\pi)\cup\bigcup\{K_{\delta}(b):b\in K(\vec{\nu})\}$.

Let $\{\pi,a,\xi\}\subset{\cal H}_{a}(\pi)$. Then $\xi=m_{k}(\pi)$ is intended to be equivalent to $\pi\in Mh^{a}_{k}(\xi)$. For Definition 3.1.2h, see Corollary 2.12, and for Definition 3.1.2i, see Proposition 2.13.

Proposition 3.2

For each Mahlo term $\alpha=\psi_{\pi}^{\vec{\nu}}(a)\in OT$, $\vec{m}(\alpha)=\vec{\nu}\in SD$ for the class $SD$ in Definition 2.20.

Proposition 3.3

For any $\alpha\in OT$ and any $\delta$ such that $\delta=0,\mathbb{K}$ or $\delta=\psi_{\pi}^{\vec{\nu}}(b)$ for some $\pi,b,\vec{\nu}$, $\alpha\in{\cal H}_{\gamma}(\delta)\Leftrightarrow K_{\delta}(\alpha)<\gamma$.

Proof.  By induction on $\ell\alpha$. $\Box$

Lemma 3.4

$(OT,<)$* is a computable notation system of ordinals. In particular the order type of the initial segment $\{\alpha\in OT:\alpha<\Omega_{1}\}$ is less than $\omega_{1}^{CK}$.*

Specifically each of $\alpha<\beta$ and $\alpha=\beta$ is decidable for $\alpha,\beta\in OT$, and $\alpha\in OT$ is decidable for terms $\alpha$ over symbols $\{0,\mathbb{K},\Lambda,+,\omega,\varphi,\Omega,\psi\}$.

Proof.  These are shown simultaneously referring Propositions 2.19 and 3.3. Let us give recursive definitions only for terms $\Omega_{\alpha},\psi_{\kappa}^{\vec{\nu}}(a)\in OT$.

First $\Omega_{\psi_{\kappa}^{\vec{\nu}}(a)}=\psi_{\kappa}^{\vec{\nu}}(a)$, i.e., $\Omega_{\alpha}<\psi_{\kappa}^{\vec{\nu}}(a)\Leftrightarrow\alpha<\psi_{\kappa}^{\vec{\nu}}(a)$, $\psi_{\kappa}^{\vec{\nu}}(a)<\Omega_{\alpha}\Leftrightarrow\psi_{\kappa}^{\vec{\nu}}(a)<\alpha$. Next $\Omega_{\alpha}<\psi_{\Omega_{\alpha+1}}(a)<\Omega_{\alpha+1}$.

Finally for $\psi_{\pi}^{\vec{\nu}}(b),\psi_{\kappa}^{\vec{\xi}}(a)\in OT$, $\psi_{\pi}^{\vec{\nu}}(b)<\psi_{\kappa}^{\vec{\xi}}(a)$ iff one of the following cases holds:

$\pi\leq\psi_{\kappa}^{\vec{\xi}}(a)$. 2. 2.

$b<a$, $\psi_{\pi}^{\vec{\nu}}(b)<\kappa$, and $K_{\psi_{\kappa}^{\vec{\xi}}(a)}(\{\pi,b\}\cup K(\vec{\nu}))<a$. 3. 3.

$b\geq a$, and $b\leq K_{\psi_{\pi}^{\vec{\nu}}(b)}(\{\kappa,a\}\cup K(\vec{\xi}))$. 4. 4.

$b=a$, $\pi=\kappa$, $K_{\psi_{\kappa}^{\vec{\xi}}(a)}(K(\vec{\nu}))<a$, and $\vec{\nu}<_{lx,2}\vec{\xi}$.

$\Box$

Proposition 3.5

Let $\beta=\psi_{\pi}^{\vec{\nu}}(b)$ with $\pi=\psi_{\kappa}^{\vec{\xi}}(a)$. Then $a<b$. 2. 2.

For $\alpha=\psi_{\pi}^{\vec{\nu}}(a)\in OT$, $\max K(\vec{\nu})\leq a$ holds.

Proof.  3.5.1. Let $\beta=\psi_{\pi}^{\vec{\nu}}(b)$ with $\pi=\psi_{\kappa}^{\vec{\xi}}(a)$. Then $K_{\beta}(\{\pi,b\}\cup K(\vec{\nu}))<b$. On the other hand we have $\beta<\pi$. Hence $a\in K_{\beta}(\pi)<b$.

3.5.2. This is seen by induction on $\ell\alpha$. Ww have $c<a$ by Proposition 3.5.1 when $\pi=\psi_{\sigma}^{\vec{\mu}}(c)$

When $\alpha$ is constructed by Definition 3.1.2h, $\nu_{k}=m_{k}(\pi)+\Lambda^{m_{k+1}(\pi)}b$ holds for $b\leq a$. By IH we have $\max K(\vec{m}(\pi))\leq c<a$ when $\pi=\psi_{\sigma}^{\vec{\mu}}(c)$.

Suppose $\alpha$ is constructed by Definition 3.1.2i. We obtain $\vec{\nu}<_{sp}m_{2}(\pi)$, and hence $\max K(\vec{\nu})\leq\max K(m_{2}(\pi))\leq c<a$ by IH. $\Box$

4 Operator controlled derivations

In this section, operator controlled derivations are defined, which are introduced by W. Buchholz[6].

In this and the next sections except otherwise stated $\alpha,\beta,\gamma,\ldots,a,b,c,d,\ldots$ range over ordinal terms in $OT\subset{\cal H}_{\Lambda}(0)$, $\xi,\zeta,\nu,\mu,\iota,\ldots$ range over ordinal terms in $E$, $\vec{\xi},\vec{\zeta},\vec{\nu},\vec{\mu},\vec{\iota},\ldots$ range over finite sequences over ordinal terms in $E$, and $\pi,\kappa,\rho,\sigma,\tau,\lambda,\ldots$ range over regular ordinal terms $\mathbb{K}$, $\Omega_{\beta+1}$, $\psi_{\pi}^{\vec{\nu}}(a)$ with $\vec{\nu}\neq\vec{0}$. $Reg$ denotes the set of regular ordinal terms. We write $\alpha\in{\cal H}_{a}(\beta)$ for $K_{\beta}(\alpha)<a$.

4.1 Classes of sentences

Following Buchholz[6] let us introduce a language for ramified set theory $RS$.

Definition 4.1

$RS$-terms and their levels are inductively defined.

For each $\alpha\in OT\cap\mathbb{K}$, $L_{\alpha}$ is an $RS$-term of level $\alpha$.** 2. 2.

If $\phi(x,y_{1},\ldots,y_{n})$ is a set-theoretic formula in the language $\{\in\}$, and $a_{1},\ldots,a_{n}$ are $RS$-terms of levels$<\alpha$, then $[x\in L_{\alpha}:\phi^{L_{\alpha}}(x,a_{1},\ldots,a_{n})]$ is an $RS$-term of level $\alpha$.**

Each ordinal term $\alpha$ is denoted by the ordinal term $[x\in L_{\alpha}:x\mbox{ is an ordinal}]$, whose level is $\alpha$.

Definition 4.2

$|a|$* denotes the level of $RS$-terms $a$, and $Tm(\alpha)$ the set of $RS$-terms of level$<\alpha$. $Tm=Tm(\mathbb{K})$ is then the set of $RS$-terms, which are denoted by $a,b,c,d,\ldots$* 2. 2.

$RS$-formulas are constructed from literals $a\in b,a\not\in b$ by propositional connectives $\lor,\land$, bounded quantifiers $\exists x\in a,\forall x\in a$ and unbounded quantifiers $\exists x,\forall x$. Unbounded quantifiers $\exists x,\forall x$ are denoted by $\exists x\in L_{\mathbb{K}},\forall x\in L_{\mathbb{K}}$, resp.** 3. 3.

For $RS$-terms and $RS$-formulas $\iota$, ${\sf k}(\iota)$ denotes the set of ordinal terms $\alpha$ such that the constant $L_{\alpha}$ occurs in $\iota$.** 4. 4.

For a set-theoretic $\Sigma_{n}$-formula $\psi(x_{1},\ldots,x_{m})$ in $\{\in\}$ and $a_{1},\ldots,a_{m}\in Tm(\kappa)$, $\psi^{L_{\kappa}}(a_{1},\ldots,a_{m})$ is a $\Sigma_{n}(\kappa)$-formula, where $n=0,1,2,\ldots$ and $\kappa\leq\mathbb{K}$. $\Pi_{n}(\kappa)$-formulas are defined dually.** 5. 5.

For* $\theta\equiv\psi^{L_{\kappa}}(a_{1},\ldots,a_{m})\in\Sigma_{n}(\kappa)$ and $\lambda<\kappa$, $\theta^{(\lambda,\kappa)}:\equiv\psi^{L_{\lambda}}(a_{1},\ldots,a_{m})$.*

Note that the level $|t|=\max(\{0\}\cup{\sf k}(t))$ for $RS$-terms $t$. In what follows we need to consider sentences. Sentences are denoted $A,C$ possibly with indices.

The assignment of disjunctions and conjunctions to sentences is defined as in [6].

Definition 4.3

For* $b,a\in Tm(\mathbb{K})$ with $|b|<|a|$,*

[TABLE]

and* $(a=b):\equiv(\forall x\in a(x\in b)\land\forall x\in b(x\in a))$.* 2. 2.

For* $b,a\in Tm(\mathbb{K})$ and $J:=Tm(|a|)$*

[TABLE] 3. 3.

$(A_{0}\lor A_{1}):\simeq\bigvee(A_{\iota})_{\iota\in J}$* and $(A_{0}\land A_{1}):\simeq\bigwedge(A_{\iota})_{\iota\in J}$ for $J:=2$.* 4. 4.

For* $a\in Tm(\mathbb{K})\cup\{L_{\mathbb{K}}\}$ and $J:=Tm(|a|)$*

[TABLE]

The rank $\mbox{{\rm rk}}(\iota)$ of sentences or terms $\iota$ is defined as in [6].

Definition 4.4

$\mbox{{\rm rk}}(\lnot A):=\mbox{{\rm rk}}(A)$. 2. 2.

$\mbox{{\rm rk}}(L_{\alpha})=\omega\alpha$. 3. 3.

$\mbox{{\rm rk}}([x\in L_{\alpha}:A(x)])=\max\{\omega\alpha+1,\mbox{{\rm rk}}(A(L_{0}))+2\}$. 4. 4.

$\mbox{{\rm rk}}(a\in b)=\max\{\mbox{{\rm rk}}(a)+6,\mbox{{\rm rk}}(b)+1\}$. 5. 5.

$\mbox{{\rm rk}}(A_{0}\lor A_{1}):=\max\{\mbox{{\rm rk}}(A_{0}),\mbox{{\rm rk}}(A_{1})\}+1$. 6. 6.

$\mbox{{\rm rk}}(\exists x\in a\,A(x)):=\max\{\omega\mbox{{\rm rk}}(a),\mbox{{\rm rk}}(A(L_{0}))+2\}$* for * $a\in Tm(\mathbb{K})\cup\{L_{\mathbb{K}}\}$.

Proposition 4.5

Let $A$ be a sentence with $A\simeq\bigvee(A_{\iota})_{\iota\in J}$ or $A\simeq\bigwedge(A_{\iota})_{\iota\in J}$.

$\mbox{{\rm rk}}(A)<\mathbb{K}+\omega$. 2. 2.

$|A|\leq\mbox{{\rm rk}}(A)\in\{\omega|A|+i:i\in\omega\}$. 3. 3.

$\forall\iota\in J(\mbox{{\rm rk}}(A_{\iota})<\mbox{{\rm rk}}(A))$. 4. 4.

$\mbox{{\rm rk}}(A)<\lambda\Rightarrow A\in\Sigma_{0}(\lambda)$**

4.2 Operator controlled derivations

By an operator we mean a map $\mathcal{H}$, $\mathcal{H}:\mathcal{P}(OT)\to\mathcal{P}(OT)$, such that

$\forall X\subset OT[X\subset\mathcal{H}(X)]$. 2. 2.

$\forall X,Y\subset OT[Y\subset\mathcal{H}(X)\Rightarrow\mathcal{H}(Y)\subset\mathcal{H}(X)]$.

For an operator $\mathcal{H}$ and $\Theta,\Theta_{1}\subset OT$, $\mathcal{H}[\Theta](X):=\mathcal{H}(X\cup\Theta)$, and $\mathcal{H}[\Theta][\Theta_{1}]:=(\mathcal{H}[\Theta])[\Theta_{1}]$, i.e., $\mathcal{H}[\Theta][\Theta_{1}](X)=\mathcal{H}(X\cup\Theta\cup\Theta_{1})$.

Obviously $\mathcal{H}_{\alpha}$ is an operator for any $\alpha$, and if ${\cal H}$ is an operator, then so is $\mathcal{H}[\Theta]$.

Sequents are finite sets of sentences, and inference rules are formulated in one-sided sequent calculus. Let $\mathcal{H}={\cal H}_{\gamma}\,(\gamma\in OT)$ be an operator, $\Theta$ a finite set of $\mathbb{K}$, $\Gamma$ a sequent, $a\in OT$ and $b\in OT\cap(\mathbb{K}+\omega)$.

We define a relation $(\mathcal{H}_{\gamma},\Theta)\vdash^{a}_{b}\Gamma$, which is read ‘there exists an infinitary derivation of $\Gamma$ which is $\Theta$-controlled by $\mathcal{H}_{\gamma}$, and whose height is at most $a$ and its cut rank is less than $b$’.

$\kappa,\lambda,\sigma,\tau,\pi$ ranges over regular ordinal terms.

Definition 4.6

$(\mathcal{H}_{\gamma},\Theta)\vdash^{a}_{b}\Gamma$ holds if

[TABLE]

and one of the following cases holds:

$(\bigvee)$

$A\simeq\bigvee\{A_{\iota}:\iota\in J\}$, $A\in\Gamma$ and there exist $\iota\in J$ and $a(\iota)<a$ such that

[TABLE]

and $(\mathcal{H}_{\gamma},\Theta)\vdash^{a(\iota)}_{b}\Gamma,A_{\iota}$.

$(\bigwedge)$

$A\simeq\bigwedge\{A_{\iota}:\iota\in J\}$, $A\in\Gamma$ and for every $\iota\in J$ there exists an $a(\iota)<a$ such that $(\mathcal{H}_{\gamma},\Theta\cup\{{\sf k}(\iota)\})\vdash^{a(\iota)}_{b}\Gamma,A_{\iota}$.

$(cut)$

There exist $a_{0}<a$ and $C$ such that $\mbox{{\rm rk}}(C)<b$ and $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{0}}_{b}\Gamma,\lnot C$ and $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{0}}_{b}C,\Gamma$.

$(\Omega\in M_{2})$

There exist ordinals $a_{\ell}$, $a_{r}(\alpha)$ and a sentence $C\in\Pi_{2}(\Omega)$ such that $\sup\{a_{\ell}+1,a_{r}(\alpha)+1:\alpha<\Omega\}\leq a$, $b\geq\Omega$, $({\cal H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{b}\Gamma,C$ and $({\cal H}_{\gamma},\Theta\cup\{\omega\alpha\})\vdash^{a_{r}(\alpha)}_{b}\lnot C^{(\alpha,\Omega)},\Gamma$ for any $\alpha<\Omega$.

$({\rm rfl}(\pi,k,\vec{\xi},\vec{\nu}))$

There exist a Mahlo ordinal $\mathbb{K}\geq\pi\in{\cal H}_{\gamma}[\Theta]\cap(b+1)$, an integer $2\leq k\leq N$ and sequences $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1}),\vec{\xi}=(\xi_{2},\ldots,\xi_{N-1})\in SD$ of ordinals $\nu_{i},\xi_{i}\in E$, ordinals $a_{\ell},a_{r}(\rho),a_{0}$, and a finite set $\Delta$ of $\Sigma_{k}(\pi)$-sentences enjoying the following conditions: When $\pi=\mathbb{K}$, $k=N$ and $\vec{\nu}=\vec{0}$ with $lh(\vec{\nu})=N-1$ hold. Also let $\vec{\xi}=\vec{0}$ in this case. When $\pi<\mathbb{K}$, $\xi_{k}\neq 0$ with $k<N$, $\vec{0}\neq\vec{\xi}$, and $\forall i(\xi_{i}\leq_{sp}m_{i}(\pi))$.

1. 1.

When $\pi<\mathbb{K}$, cf. Definitions 2.1.9,

[TABLE]

and

[TABLE]

cf. (4). 2. 2.

For each $\delta\in\Delta$, $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{b}\Gamma,\lnot\delta$. 3. 3.

$H(\vec{\nu},\pi,\gamma,\Theta)$ denotes the resolvent class for $\vec{\nu}$, $\pi$, $\gamma$ and $\Theta$ defined as follows:

[TABLE]

for $\rho\in Reg\cap C(\pi,\gamma,\Theta)$.

Then for each $\rho\in H(\vec{\nu},\pi,\gamma,\Theta)$, $(\mathcal{H}_{\gamma},\Theta\cup\{\rho\})\vdash^{a_{r}(\rho)}_{b}\Gamma,\Delta^{(\rho,\pi)}$. 4. 4.

[TABLE]

In the inference rule $({\rm rfl}(\pi,k,\vec{\xi},\vec{\nu}))$ for $\pi=\psi_{\sigma}^{\vec{\xi}}(c)<\mathbb{K}$, we have $\pi\in Mh^{c}_{2}(\vec{\xi})$. In particular, $\pi\in\bigcap_{i<k}Mh^{c}_{i}(\xi_{i})\cap Mh^{c}_{k}(\xi_{k})$. Also we are assuming $(\nu_{k},\ldots,\nu_{N-1})<_{sd}\xi_{k}$, a fortiori $(\nu_{k},\ldots,\nu_{N-1})<\xi_{k}$. Since $\pi\in\bigcap_{i<k}Mh^{c}_{i}(\nu_{i})$ is a $\Pi_{k}$-sentence holding on $L_{\pi}$, we obtain $\pi\in M_{k}(Mh^{c}_{2}(\vec{\nu}))$. Thus the reflection rule $({\rm rfl}(\pi,k,\vec{\nu}))$ says that $\pi$ is $\Pi_{k}$-reflecting on the class $H(\vec{\nu},\pi,\gamma,\gamma_{0},\Theta)$ for the club subset $C(\pi,\gamma,\Theta)$ of $\pi$, cf. Proposition 2.13. On the other side we see $\rho\in Mh_{2}^{a}(\vec{\nu})$ from Proposition 2.9 if $\forall i(\nu_{i}\leq m_{i}(\rho))$ for $\rho\in Mh_{2}^{a}(\vec{m}(\rho))$.

We will state some lemmas for the operator controlled derivations. These can be shown as in [6]. In what follows by an operator $\mathcal{H}$ we mean an $\mathcal{H}_{\gamma}$ for an ordinal $\gamma$.

Lemma 4.7

Let $({\cal H}_{\gamma},\Theta)\vdash^{a}_{b}\Gamma$.

$({\cal H}_{\gamma^{\prime}},\Theta\cup\Theta_{0})\vdash^{a^{\prime}}_{b^{\prime}}\Gamma,\Delta$* for any $\gamma^{\prime}\geq\gamma$, any $\Theta_{0}$, and any $a^{\prime}\geq a$, $b^{\prime}\geq b$ such that ${\sf k}(\Delta)\cup\{a^{\prime}\}\subset{\cal H}_{\gamma^{\prime}}[\Theta\cup\Theta_{0}]$.* 2. 2.

Assume $\Theta_{1}\cup\{c\}=\Theta$, $c\in{\cal H}_{\gamma}[\Theta_{1}]$. Then $({\cal H}_{\gamma},\Theta_{1})\vdash^{a}_{b}\Gamma$.

Lemma 4.8

(Tautology)* $(\mathcal{H},{\sf k}(\Gamma\cup\{A\}))\vdash^{2\footnotesize{\mbox{{\rm rk}}}(A)}_{0}\Gamma,\lnot A,A$.*

Lemma 4.9

(Inversion)* Let $A\simeq\bigwedge(A_{\iota})_{\iota\in J}$, and $({\cal H},\Theta)\vdash^{a}_{b}\Gamma$ with $A\in\Gamma$. Then for any $\iota\in J$, $({\cal H},\Theta\cup{\sf k}(\iota))\vdash^{a}_{b}\Gamma,A_{\iota}$ holds.*

Lemma 4.10

(Boundedness)* Suppose $(\mathcal{H},\Theta)\vdash^{a}_{c}\Gamma,C$ for a $C\in\Sigma_{1}(\lambda)$, and $a\leq b\in\mathcal{H}\cap\lambda$. Then $(\mathcal{H},\Theta)\vdash^{a}_{c}\Gamma,C^{(b,\lambda)}$.*

Lemma 4.11

(Persistency)* Suppose $(\mathcal{H},\Theta)\vdash^{a}_{c}\Gamma,C^{(b,\lambda)}$ for a $C\in\Sigma_{1}(\lambda)$ and a $b<\lambda\in{\cal H}[\Theta]$. Then $(\mathcal{H},\Theta)\vdash^{a}_{c}\Gamma,C$.*

Lemma 4.12

(Predicative Cut-elimination)* Suppose $(\mathcal{H},\Theta)\vdash^{b}_{c+\omega^{a}}\Gamma$, $a\in\mathcal{H}[\Theta]$ and $]c,c+\omega^{a}]\cap Reg=\emptyset$. Then $(\mathcal{H},\Theta)\vdash^{\varphi ab}_{c}\Gamma$.*

Lemma 4.13

(Embedding of Axioms)

For each axiom $A$ in $\mbox{{\sf KP}}\Pi_{N}$, there is an $m<\omega$ such that for any operator $\mathcal{H}={\cal H}_{\gamma}$, $(\mathcal{H},\emptyset)\vdash^{\mathbb{K}\cdot 2}_{\mathbb{K}+m}A$ holds.

Proof.  The axiom $\lnot A,\exists z\,A^{(z)}$ for $\Pi_{N}$-reflection follows from $A,\lnot A$ and

$\exists z\,A^{(z)},\lnot A^{(\rho)}$ for regular ordinals $\rho<\mathbb{K}$ by an inference $({\rm rfl}(\mathbb{K},N,\vec{0},\vec{0}))$. $\Box$

Lemma 4.14

(Embedding)* If $\mbox{{\sf KP}}\Pi_{N}\vdash\Gamma$ for sets $\Gamma$ of sentences, there are $m,k<\omega$ such that for any operator ${\cal H}={\cal H}_{\gamma}$, $({\cal H},\emptyset)\vdash_{\mathbb{K}+m}^{\mathbb{K}\cdot 2+k}\Gamma$ holds*

5 Lowering and eliminating higher Mahlo operations

In the section inferences $({\rm rfl}(\mathbb{K},N,\vec{0},\vec{0}))$ for $\Pi_{N}$-reflecting ordinals $\mathbb{K}$ are eliminated from operator controlled derivations of $\Sigma_{1}$-sentences $\varphi^{L_{\Omega}}$ over $\Omega$.

$\alpha\#\beta$ denotes the natural (commutative) sum of ordinal terms $\alpha,\beta$.

Lemma 5.1

For a Mahlo term $\pi\in OT$, $\vec{\xi}\in SD$ denotes a sequence with $lh(\vec{\xi})=N-2$, and $2\leq k\leq N-1$ an integer for which the following hold: When $\pi=\mathbb{K}$, let $\vec{\xi}=\vec{0}$ and $k=N-1$. Otherwise $\vec{\xi}=(\xi_{2},\ldots,\xi_{k+1})*\vec{0}$ with $\xi_{k+1}\neq 0$ such that $\forall i\leq k+1(\xi_{i}\leq_{sp}m_{i}(\pi))$.

For ordinal terms $\gamma,a\in OT$ let us define a sequence $\vec{\zeta}(a):=(\zeta_{2}(a),\ldots,\zeta_{k}(a))*\vec{0}$ with $lh(\vec{\zeta}(a))=N-2$ as follows. $\vec{\zeta}(a)=\vec{0}*(\gamma+a)$ when $\pi=\mathbb{K}$. Otherwise $\zeta_{k}(a)=\xi_{k}+\Lambda^{\xi_{k+1}}(\gamma+a)$ and $\zeta_{i}(a)=\xi_{i}$ for $i<k$.

Let $\kappa\in H(\vec{\zeta}(a),\pi,\gamma,\Theta)$ for a finite set $\Theta\subset OT$.

Now suppose $(\mathcal{H}_{\gamma},\Theta)\vdash^{a}_{\pi}\Gamma$ where $\{\gamma,\pi\}\cup K(\vec{\xi})\subset{\cal H}_{\gamma}[\Theta]$, $\Theta\subset\pi$, $\forall i(K(\xi_{i})\subset\mathcal{H}_{\max K(\xi_{i})}[\Theta])$, and $\Gamma\subset\Pi_{k+1}(\pi)$.

Let $\gamma(a,b)=\gamma\#a\#b$, $\beta(a,b)=\psi_{\pi}(\gamma(a,b))$, and $c>\gamma(a,\kappa)$. Then the following holds:

[TABLE]

Proof by induction on $a$. Let $\kappa\in H(\vec{\zeta}(a),\pi,\gamma,\Theta)$. We see $\vec{\zeta}(a)\in SD$, and from (5) and $\Theta\subset\kappa$ that

[TABLE]

For any $a\in{\cal H}_{\gamma}[\Theta]$, we obtain $\{\gamma,\pi,a,\kappa\}\subset{\cal H}_{\gamma}(\pi)$ by $\Theta\cup\{\kappa\}\subset\pi$. Hence for $\gamma(a,\kappa)=\gamma\#a\#\kappa$, $\{\gamma(a,\kappa),\pi\}\subset{\cal H}_{\gamma}(\pi)$, and $\{\gamma(a,\kappa),\pi\}\subset{\cal H}_{\gamma(a,\kappa)}(\beta(a,\kappa))$ by the definition (3). Therefore $\kappa\in{\cal H}_{\gamma(a,\kappa)}(\beta(a,\kappa))\cap\pi\subset\beta(a,\kappa)$ by Proposition 2.6, and $\Theta\subset\beta(a,\kappa)<\pi$. Thus we obtain

[TABLE]

Case 1. First consider the case when the last inference is a $({\rm rfl}(\pi,k+1,\vec{\xi},\vec{\nu}))$.

We have $a_{\ell}\in\mathcal{H}_{\gamma}[\Theta]\cap a$, $a_{r}(\rho)\in\mathcal{H}_{\gamma}[\Theta\cup\{\rho\}]\cap a$, and a finite set $\Delta$ of $\Sigma_{k+1}(\pi)$-sentences. We have for each $\delta\in\Delta$

[TABLE]

and for each $\rho\in H(\vec{\nu},\pi,\gamma,\Theta)$

[TABLE]

When $\pi<\mathbb{K}$, $\vec{\nu}=(\nu_{2},\ldots,\nu_{N-1})\in SD$ is a sequence such that $\forall i<k+1(\nu_{i}=\xi_{i})$, $(\nu_{k+1},\ldots,\nu_{N-1})<_{sd}\xi_{k+1}$, $K(\vec{\nu})\cup K(\vec{\xi})\subset{\cal H}_{\gamma}[\Theta]$, and $\forall i(K(\nu_{i})\subset\mathcal{H}_{\max K(\nu_{i})}[\Theta])$, cf. (7) and (8).

Let $\Gamma_{0}=\Gamma\cap\Sigma_{k}(\pi)$ and $\{\forall x\in L_{\pi}\,\theta_{i}(x):i=1,\ldots,n\}\,(n\geq 0)=\Gamma\setminus\Gamma_{0}$ for $\Sigma_{k}(\pi)$-formulas $\theta_{i}(x)$. Let us fix $\vec{d}=\{d_{1},\ldots,d_{n}\}\subset Tm(\kappa)$ arbitrarily. Put ${\sf k}(\vec{d})=\bigcup\{{\sf k}(d_{i}):i=1,\ldots,n\}$ and $\Gamma(\vec{d})=\Gamma_{0}\cup\{\theta_{i}(d_{i}):i=1,\ldots,n\}$.

By Inversion lemma 4.9 from (13) we obtain for each $\delta\in\Delta$

[TABLE]

Let $\rho\in C(\kappa,c,\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))$. We see $\rho<\kappa$, and ${\sf k}(\vec{d})<\rho$ from ${\sf k}(\vec{d})<\kappa$. By $\Theta\cap\pi\subset{\cal H}_{\gamma}(\kappa)\cap\pi\subset\kappa$ and $\gamma\leq c$ we obtain $C(\kappa,c,\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))\subset C(\pi,\gamma,\Theta)$. Namely, cf. (9)

[TABLE]

For each $\rho\in H(\vec{\nu},\kappa,c,\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))$, IH with (14) and (16) yields for $c>\gamma(a_{r}(\rho),\kappa)$ and $\kappa\in H(\vec{\zeta}(a_{r}(\rho)),\pi,\gamma,\Theta\cup\{\rho\})$

[TABLE]

Let $\rho\in M_{\ell}:=\{\rho\in Reg:\forall i(\zeta_{i}(a_{\ell})\leq_{sp}m_{i}(\rho))\}\cap H(\vec{\nu},\kappa,c,\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))$. Then $M_{\ell}\subset H(\vec{\zeta}(a_{\ell}),\pi,\gamma,\Theta\cup{\sf k}(\vec{d}))$ and $\Theta\cup{\sf k}(\vec{d})\subset\rho$. For each $\delta\in\Delta$, IH with (15) yields for $c>\gamma(a_{\ell},\rho)$

[TABLE]

From (17) and (18) by several $(cut)$’s of $\delta^{(\rho,\pi)}$ with $\mbox{rk}(\delta^{(\rho,\pi)})<\kappa$ we obtain for $a(\rho)=\max\{a_{\ell},a_{r}(\rho)\}$ and some $p<\omega$

[TABLE]

On the other hand we have by Tautology lemma 4.8 for each $\theta(\vec{d})^{(\kappa,\pi)}\in\Gamma(\vec{d})^{(\kappa,\pi)}$

[TABLE]

where $2\mbox{{\rm rk}}(\theta(\vec{d})^{(\kappa,\pi)})\leq\kappa+p$ for some $p<\omega$.

Moreover we have $\sup\{2\mbox{{\rm rk}}(\theta(\vec{d})^{(\kappa,\pi)}),\beta(a(\rho),\kappa)+p:\rho\in M_{\ell}\}\leq\beta(a_{0},\kappa)+p\in{\cal H}_{\gamma}[\Theta\cup\{\kappa\}]$, where $\sup\{a_{\ell},a_{r}(\rho):\rho\in H(\vec{\nu},\pi,\gamma,\Theta)\}\leq a_{0}<a$ by (10).

Now let $\vec{\mu}=(\mu_{2},\ldots,\mu_{N-1})=\max\{\vec{\zeta}(a_{\ell}),\vec{\nu}\}$ with $\mu_{i}=\max\{\zeta_{i}(a_{\ell}),\nu_{i}\}$. Since $\nu_{i}=\xi_{i}\leq_{pt}\zeta_{i}(a_{\ell})$ for $i<k+1$, we obtain \mu_{i}=\left\{\begin{array}[]{ll}\zeta_{i}(a_{\ell})&i\leq k\\ \nu_{i}&i>k\end{array}\right.. We see that $M_{\ell}=H(\vec{\mu},\kappa,c,\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))$. Moreover we have $\forall i<k(\mu_{i}=\xi_{i}=\zeta_{i}(a))$ and $(\mu_{k},\ldots,\mu_{N-1})=(\zeta_{k}(a_{\ell}))*(\nu_{k+1},\ldots,\nu_{N-1})<_{sd}\zeta_{k}(a)$. Also $\forall i(K(\zeta_{i}(a))\subset\mathcal{H}_{\max K(\zeta_{i}(a))}[\Theta])$ and $\forall i(K(\mu_{i})\subset\mathcal{H}_{\max K(\mu_{i})}[\Theta])$. For $\lnot\Gamma(\vec{d})^{(\kappa,\pi)}\subset\Pi_{k}(\kappa)$, by an inference rule $({\rm rfl}(\kappa,k,\vec{\zeta}(a),\vec{\mu}))$ with its resolvent class $M_{\ell}$, we conclude from (20) and (19) that $({\cal H}_{c},\Theta\cup\{\kappa\}\cup{\sf k}(\vec{d}))\vdash^{\beta(a_{0},\kappa)+p+1}_{\kappa}\Gamma(\vec{d})^{(\kappa,\pi)},\Gamma^{(\kappa,\pi)}$. Since $\vec{d}\subset Tm(\kappa)$ is arbitrary, several $(\bigwedge)$’s yield (11).

Case 2. Second consider the case when the last inference is a $({\rm rfl}(\pi,j,\vec{\xi},\vec{\nu}))$ for a $j<k+1$. We have $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,\lnot\delta$ for each $\delta\in\Delta\subset\Sigma_{j}(\pi)$ with $a_{\ell}\in\mathcal{H}_{\gamma}[\Theta]\cap a$, and $(\mathcal{H}_{\gamma},\Theta\cup\{\rho\})\vdash^{a_{r}(\rho)}_{\pi}\Gamma,\Delta^{(\rho,\pi)}$ for each $\rho\in H(\vec{\nu},\pi,\gamma,\Theta)$ with $a_{r}(\rho)\in\mathcal{H}_{\gamma}[\Theta\cup\{\rho\}]\cap a$. $\vec{\nu}\in SD$ is a sequence such that $\forall i<j(\nu_{i}=\xi_{i})$ and $(\nu_{j},\ldots,\nu_{N-1})<_{sd}\xi_{j}$.

We see that the resolvent class $H(\vec{\nu},\kappa,c_{1},\Theta\cup\{\kappa\})$ is a subclass of $H(\vec{\nu},\pi,\gamma,\Theta)$. By IH we have $(\mathcal{H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{\ell},\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\lnot\delta^{(\kappa,\pi)}$ for each $\delta\in\Delta$, and $(\mathcal{H}_{c},\Theta\cup\{\kappa,\rho\})\vdash^{\beta(a_{r}(\rho),\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\Delta^{(\rho,\pi)}$ for each $\rho\in H(\vec{\nu},\kappa,c,\Theta\cup\{\kappa\})$ with $\Delta^{(\rho,\pi)}=(\Delta^{(\kappa,\pi)})^{(\rho,\kappa)}$. We claim that $\forall i\leq j(\xi_{j}\leq_{sp}m_{i}(\kappa))$. Consider the case when $i=j=k$. Then we have $\xi_{k}\leq_{sp}m_{k}(\pi)$ and $\zeta_{k}(a)\leq_{sp}m_{k}(\kappa)$ with $\xi_{k}<_{pt}\zeta_{k}(a)$. We obtain $\xi_{k}\leq_{sp}m_{k}(\kappa)$. Hence by an inference rule $({\rm rfl}(\kappa,j,\vec{\xi}(j),\vec{\nu}))$ for the sequence $\vec{\xi}(j)=(\xi_{2},\ldots,\xi_{j})*\vec{0}\in SD$, cf. Proposition 2.21.1, we obtain (11).

Case 3. Third consider the case when the last inference is a $({\rm rfl}(\sigma,j,\vec{\mu},\vec{\nu}))$ for a $\sigma<\pi$. We have $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,\lnot\delta$ for each $\delta\in\Delta\subset\Sigma_{j}(\sigma)$, and $(\mathcal{H}_{\gamma},\Theta\cup\{\rho\})\vdash^{a_{r}(\rho)}_{\pi}\Gamma,\Delta^{(\rho,\sigma)}$ for each $\rho\in H(\vec{\nu},\sigma,\gamma,\Theta)$. We obtain $\sigma<\kappa$ by (12) for $\sigma\in{\cal H}_{\gamma}[\Theta]$. Hence $\Delta\subset\Sigma^{1}_{0}(\sigma)\subset\Sigma_{0}(\kappa)$ and $\delta^{(\kappa,\pi)}\equiv\delta$ for any $\delta\in\Delta$. Let $H(\vec{\nu},\sigma,c,\Theta\cup\{\kappa\})$ be the resolvent class for $\sigma$, $\vec{\nu}$, $c$ and $\Theta\cup\{\kappa\}$. Then $H(\vec{\nu},\sigma,c,\Theta\cup\{\kappa\})\subset H(\vec{\nu},\sigma,\gamma,\Theta)$.

From IH we have $(\mathcal{H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{\ell},\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\lnot\delta$ for each $\delta\in\Delta$, and $(\mathcal{H}_{c},\Theta\cup\{\kappa,\rho\})\vdash^{\beta(a_{r}(\rho),\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\Delta^{(\rho,\sigma)}$ for each $\rho\in H(\vec{\nu},\sigma,c,\Theta\cup\{\kappa\})$. We obtain (11) by an inference rule $({\rm rfl}(\sigma,j,\vec{\mu},\vec{\nu}))$ with the resolvent class $H(\vec{\nu},\sigma,c,\Theta\cup\{\kappa\})$.

Case 4. Fourth consider the case when the last inference $(\bigwedge)$ introduces a $\Pi_{k+1}(\pi)$-sentence $(\forall x\in L_{\pi}\,\theta(x))\in\Gamma$. We have $(\mathcal{H}_{\gamma},\Theta\cup{\sf k}(d))\vdash^{a(d)}_{\pi}\Gamma,\theta(d)$ for each $d\in Tm(\pi)$. For each $d\in Tm(\kappa)$, IH with ${\sf k}(d)<\kappa$ yields $(\mathcal{H}_{c},\Theta\cup\{\kappa\}\cup{\sf k}(d))\vdash^{\beta(a(d),\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\theta(d)^{(\kappa,\pi)}$. $(\bigwedge)$ yields (11) for $\forall x\in L_{\kappa}\,\theta(x)^{(\kappa,\pi)}\equiv(\forall x\in L_{\pi}\,\theta(x))^{(\kappa,\pi)}\in\Gamma^{(\kappa,\pi)}$.

Case 5. Fifth consider the case when the last inference $(\bigwedge)$ introduces a $\Sigma_{0}(\pi)$-sentence $(\forall x\in c\,\theta(x))\in\Gamma$ for a $c\in Tm(\pi)$. We have $(\mathcal{H}_{\gamma},\Theta\cup{\sf k}(d))\vdash^{a(d)}_{\pi}\Gamma,\theta(d)$ for each $d\in Tm(|c|)$. Then we have $|d|<|c|<\kappa$ by (12). IH yields $(\mathcal{H}_{c},\Theta\cup\{\kappa\}\cup{\sf k}(d)\vdash^{\beta(a(d),\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\theta(d)$, and we obtain (11) by an inference $(\bigwedge)$.

Case 6. Sixth consider the case when the last inference $(\bigvee)$ introduces a $\Sigma_{k}(\pi)$-sentence $(\exists x\in L_{\pi}\,\theta(x))\in\Gamma$. We have $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}\Gamma,\theta(d)$ for a $d\in Tm(\pi)$. Without loss of generality we can assume that ${\sf k}(d)\subset{\sf k}(\theta(d))$. Then we see that $|d|<\kappa$ from (12), and $d\in Tm(\kappa)$. Also $|d|<\kappa<\beta(a,\kappa)$ for (6). IH yields with $(\exists x\in L_{\pi}\,\theta(x))^{(\kappa,\pi)}\equiv(\exists x\in L_{\kappa}\,\theta(x)^{(\kappa,\pi)})\in\Gamma^{(\kappa,\pi)}$, $({\cal H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{0},\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\theta(d)^{(\kappa,\pi)}$, and we obtain (11) by an inference $(\bigvee)$.

Case 7. Seventh consider the case when the last inference is a $(cut)$. We have $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}\Gamma,\lnot C$ and $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}C,\Gamma$ for $a_{0}<a$ with $\mbox{rk}(C)<\pi$. Then $C\in\Sigma_{0}(\pi)$ by Proposition 4.5.4. On the other side ${\sf k}(C)\subset\pi$ holds by Proposition 4.5.2. Then ${\sf k}(C)\subset\kappa$ by (12). Hence $C^{(\kappa,\pi)}\equiv C$ and $\mbox{rk}(C^{(\kappa,\pi)})<\kappa$ again by Proposition 4.5.2. IH yields $({\cal H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{0},\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},\lnot C^{(\kappa,\pi)}$ and $({\cal H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{0},\kappa)}_{\kappa}C^{(\kappa,\pi)},\Gamma^{(\kappa,\pi)}$. Hence by a $(cut)$ we obtain (11).

Case 8. Eighth consider the case when the last inference is an $(\Omega\in M_{2})$. We have $({\cal H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,C$ and $({\cal H}_{\gamma},\Theta\cup\{\omega\alpha\})\vdash^{a_{r}(\alpha)}_{\pi}\lnot C^{(\alpha,\Omega)},\Gamma$ for each $\alpha<\Omega$ with $\sup\{a_{\ell}+1,a_{r}(\alpha)+1:\alpha<\Omega\}\leq a$ and $C\in\Pi_{2}(\Omega)$.

We obtain $\omega\alpha<\kappa$ for $\alpha<\Omega$. IH with $C^{(\kappa,\pi)}\equiv C$ yields for each $\alpha<\Omega$, $({\cal H}_{c},\Theta\cup\{\kappa,\omega\alpha\})\vdash^{\beta(a_{r}(\alpha),\kappa)}_{\kappa}\lnot C^{(\alpha,\Omega)},\Gamma^{(\kappa,\pi)}$, and $({\cal H}_{c},\Theta\cup\{\kappa\})\vdash^{\beta(a_{\ell},\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)},C$. An $(\Omega\in M_{2})$ yields (11)

All other cases are seen easily from IH. $\Box$

Lemma 5.2

Let $\lambda\leq\pi$ be a regular ordinal term such that $\forall i(K(m_{i}(\pi))\subset\mathcal{H}_{\max K(m_{i}(\pi))}[\Theta])$, and $\Gamma\subset\Sigma_{1}(\lambda)$.

Suppose for an ordinal term $a\in OT$

[TABLE]

where $\{\gamma,\lambda,\pi\}\subset{\cal H}_{\gamma}[\Theta]$.

*Assume *

[TABLE]

Let $\hat{a}=\gamma\#\omega^{\pi+a+1}$ and $\beta=\psi_{\lambda}(\hat{a})$. Then the following holds

[TABLE]

Proof by main induction on $\pi$ with subsidiary induction on $a$. We can assume $a>0$.

We see that $\Theta\subset\beta=\psi_{\lambda}(\hat{a})$ from (21). Hence

[TABLE]

Let $\vec{\xi}\in SD$ be a sequence of ordinals and $k$ a number for which the following hold: If $\pi=\mathbb{K}$, then let $\vec{\xi}=\vec{0}$ with $lh(\vec{\xi})=N-1$ and $k=N-1$. Let $\pi<\mathbb{K}$. If $\vec{m}(\pi)\neq\vec{0}$, then $K(\vec{\xi})\subset{\cal H}_{\gamma}[\Theta]$, $\vec{\xi}\leq\vec{m}(\pi)$ and $k=\max\{k\leq N-2:\xi_{k+1}>0\}$. Otherwise let $\vec{\xi}=\vec{0}$ and $k=1$. By the assumption (21), and (5) we obtain

[TABLE]

Case 1. First consider the case when $k\geq 2$.

Let $\vec{\xi}=\vec{m}(\pi)$, and $\vec{\zeta}(a):=(\zeta_{2}(a),\ldots,\zeta_{k}(a))*\vec{0}$ be the sequence defined as in Lemma 5.1 from $\gamma,a$: $\vec{\zeta}(a)=\vec{0}*(\gamma+a)$ when $\pi=\mathbb{K}$, otherwise $\zeta_{k}(a)=\xi_{k}+\Lambda^{\xi_{k+1}}(\gamma+a)$ and $\zeta_{i}(a)=\xi_{i}$ for $i<k$. Also let $\gamma(a,b)=\gamma\#a\#b$ and $\beta(a,b)=\psi_{\pi}\gamma(a,b)$.

Let $\kappa:=\psi_{\pi}^{\vec{\zeta}(a)}(\gamma(a,0))$. By the assumption (21) we have $\Theta\subset\psi_{\pi}(\gamma\#a)$. On the other hand we have $\psi_{\pi}(\gamma\#a)=\psi_{\pi}(\gamma(a,0))\leq\kappa$, and $\Theta\subset\kappa$. $\pi\in{\cal H}_{\gamma}[\Theta]$ with $\Theta\subset\pi$ yields $K(\vec{\xi})=K(\vec{m}(\pi))\subset{\cal H}_{\gamma}[\Theta]\subset{\cal H}_{\gamma(a,0)}(\kappa)$. Hence $K(\vec{\xi})\cup\{\pi,\gamma(a,0)\}\subset{\cal H}_{\gamma(a,0)}(\kappa)$, and $\kappa\in OT$ by $\gamma(a,0)=\gamma\#a>0$ and Definition 3.1.2h such that $\kappa<\pi$ and ${\cal H}_{\gamma}(\kappa)\cap\pi\subset\kappa$. Moreover we have $\forall i(K(\zeta_{i}(a))\subset\mathcal{H}_{\max K(\zeta_{i}(a))}[\Theta])$ by $\forall i(K(m_{i}(\pi))\subset\mathcal{H}_{\max K(m_{i}(\pi))}[\Theta])$ and $\{\gamma,a\}\subset\mathcal{H}_{\gamma}[\Theta]$ with $\Theta\subset\kappa$. In other words, $\kappa\in H(\vec{\zeta}(a),\pi,\gamma,\Theta)$.

By Lemma 5.1 we obtain $(\mathcal{H}_{\gamma(a,\kappa)+1},\Theta\cup\{\kappa\})\vdash^{\beta(a,\kappa)}_{\kappa}\Gamma^{(\kappa,\pi)}$, and Lemma 4.7.2 with $\kappa\in\mathcal{H}_{\gamma(a,0)+1}[\Theta]$

[TABLE]

If $\lambda=\pi$, then $\Gamma^{(\kappa,\pi)}\subset\Sigma_{1}(\kappa)\subset\Sigma_{0}(\lambda)$. We have $\Theta\subset\psi_{\pi}(\hat{a})=\beta$, and $\kappa\in{\cal H}_{\hat{a}}(\beta)$. Hence $\{\gamma,\pi,a,\kappa\}\subset{\cal H}_{\hat{a}}(\beta)$, and $\gamma(a,\kappa)=\gamma\#a\#\kappa<\gamma\#\omega^{\pi+a+1}=\hat{a}$. Therefore $\kappa<\beta(a,\kappa)\leq\psi_{\pi}(\hat{a})=\beta$. We obtain (22) by Persistency lemma 4.11.

Next consider the case when $\lambda<\pi$. Then $\lambda<\kappa$ and $\Gamma^{(\kappa,\pi)}=\Gamma$. We have for (21), $\forall d\forall\rho\in[\lambda,\kappa](\Theta\subset\psi_{\rho}(\gamma(a,\kappa)+1\#d))$. By MIH on (24) we obtain $(\mathcal{H}_{b_{0}+1},\Theta)\vdash^{\beta_{0}}_{\beta_{0}}\Gamma$ for $\beta_{0}=\psi_{\lambda}(b_{0})$ with $b_{0}=(\gamma(a,\kappa)+1)\#\omega^{\kappa+\beta(a,\kappa)+1}$. We have $b_{0}=\gamma\#a\#\kappa\#1\#\omega^{\beta(a,\kappa)+1}<\gamma\#\omega^{\pi+a+1}=\hat{a}$ by $\beta(a,\kappa)<\pi$. This yields $\psi_{\lambda}(b_{0})=\beta_{0}<\beta=\psi_{\lambda}(\hat{a})$ by $\Theta\subset\beta$ and $\{\gamma,\kappa,\pi,a\}\subset{\cal H}_{\hat{a}}(\beta)$. Hence (22) follows.

In what follows suppose $k=1$.

Case 2. Consider the case when the last inference rule is a $({\rm rfl}(\pi,2,\vec{\xi},\vec{\nu}))$.

We have an ordinal term $a_{\ell}\in\mathcal{H}_{\gamma}[\Theta]\cap a$, and a finite set $\Delta$ of $\Sigma_{2}(\pi)$-sentences for which $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,\lnot\delta$ holds for each $\delta\in\Delta$. On the other hand we have sequences $\vec{\nu},(\xi_{2})*\vec{0}\in SD$ such that $\vec{\nu}<_{sd}\xi_{2}$ and $K(\vec{\nu})\cup K(\vec{\xi})\subset{\cal H}_{\gamma}[\Theta]$ by (7), and an ordinal term $a_{r}(\rho)\in\mathcal{H}_{\gamma}[\Theta\cup\{\rho\}]\cap a$ for which $(\mathcal{H}_{\gamma},\Theta\cup\{\rho\})\vdash^{a_{r}(\rho)}_{\pi}\Gamma,\Delta^{(\rho,\pi)}$ holds for each $\rho\in H(\vec{\nu},\pi,\gamma,\Theta)$, where $\xi_{2}\leq_{sp}m_{2}(\pi)$.

Let $\rho:=\psi_{\pi}^{\vec{\nu}}(\widehat{a_{\ell}}\#\pi)$ for $\widehat{a_{\ell}}=\gamma\#\omega^{\pi+a_{\ell}+1}$. By the assumption (21) we have $\Theta\subset\psi_{\pi}(\widehat{a_{\ell}})\subset\rho$. $K(\vec{\nu})\cup\{\pi,\gamma,a\}\subset{\cal H}_{\gamma}[\Theta]$ yields $K(\vec{\nu})\cup\{\pi,\widehat{a_{\ell}}\}\subset{\cal H}_{\widehat{a_{\ell}}\#\pi}(\rho)$. Next consider the condition (4). We have $\forall i(K(\nu_{i})\subset\mathcal{H}_{\max K(\nu_{i})}[\Theta])$ by (8), and hence $\forall i(K(\nu_{i})\subset\mathcal{H}_{\max K(\nu_{i})}(\rho))$ by $\Theta\subset\rho$. Therefore $\rho\in OT$ by Definition 3.1.2i. Moreover $\rho\in C(\pi,\gamma,\Theta)$, i.e., ${\cal H}_{\gamma}(\rho)\cap\pi\subset\rho\,\&\,\Theta\cap\pi\subset\rho$. Hence $\rho\in H(\vec{\nu},\pi,\gamma,\Theta)$.

By Inversion lemma 4.9 we obtain for each $\delta\equiv(\exists x\in L_{\pi}\delta_{1}(x))\in\Delta$ and each $d\in Tm(\rho)$ with $|d|=\max(\{0\}\cup{\sf k}(d))$, $(\mathcal{H}_{\gamma\#|d|},\Theta\cup{\sf k}(d))\vdash^{a_{\ell}}_{\pi}\Gamma,\lnot\delta_{1}(d)$.

We have $\{\pi,\gamma,|d|\}\subset{\cal H}_{\gamma\#|d|}(\pi)$ by $|d|<\rho<\pi$, and this yields $|d|\in{\cal H}_{\gamma\#|d|}(\psi_{\pi}(\gamma\#|d|))\cap\pi\subset\psi_{\pi}(\gamma\#|d|)$. Hence $|d|<\psi_{\pi}(\gamma\#|d|)$, and $\forall e(\Theta\cup{\sf k}(d)\subset\psi_{\pi}(\gamma\#|d|\#e))$, i.e., (21) holds for $\lambda=\pi$ and $\gamma\#|d|$. Let $\beta_{d}=\psi_{\pi}(\widehat{a_{d}})$ for $\widehat{a_{d}}=\gamma\#|d|\#\omega^{\pi+a_{\ell}+1}=\widehat{a_{\ell}}\#|d|$. SIH yields $({\cal H}_{\widehat{a_{d}}+1},\Theta\cup{\sf k}(d))\vdash^{\beta_{d}}_{\beta_{d}}\Gamma,\lnot\delta_{1}(d)$, which in turn Boundedness lemma 4.10 yields $({\cal H}_{\widehat{a_{\pi}}+1},\Theta\cup{\sf k}(d))\vdash^{\beta_{d}}_{\beta_{d}}\Gamma,\lnot\delta_{1}^{(\beta_{d},\pi)}(d)$ for $\widehat{a_{\pi}}=\gamma\#\pi\#\omega^{\pi+a_{\ell}+1}=\widehat{a_{\ell}}\#\pi$. By persistency we obtain $({\cal H}_{\widehat{a_{\pi}}+1},\Theta\cup{\sf k}(d))\vdash^{\beta_{d}}_{\rho}\Gamma,\lnot\delta_{1}^{(\rho,\pi)}(d)$ for $\beta_{d}<\psi_{\pi}(\widehat{a_{\pi}})=\rho\in{\cal H}_{\gamma}[\Theta]$. Since $d\in Tm(\rho)$ is arbitrary, $(\bigwedge)$ yields

[TABLE]

Now pick the $\rho$-th branch from the right upper sequents

[TABLE]

By $\rho\in\mathcal{H}_{\widehat{a_{\pi}}+1}[\Theta]$ and Lemma 4.7.2 we obtain

[TABLE]

Case 2.1. First consider the case $\lambda=\pi$. Then $\Delta^{(\rho,\pi)}\subset\Sigma_{0}(\lambda)$. Let $\beta_{\rho}=\psi_{\pi}(b_{\rho})$ with $b_{\rho}=\widehat{a_{\pi}}\#1\#\omega^{\pi+a_{r}(\rho)+1}=\gamma\#\omega^{\pi+a_{\ell}+1}\#\omega^{\pi+a_{r}(\rho)+1}\#\pi\#1$. Then $\beta_{\rho}>\rho$ and $\forall d[\Theta\cup\{\rho\}\subset\psi_{\pi}(\widehat{a_{\pi}}+1\#d)]$. SIH yields for (26)

[TABLE]

Several $(cut)$’s with (27), (25) yield $({\cal H}_{\hat{a}+1},\Theta)\vdash^{\beta_{\rho}+p}_{\beta_{\rho}}\Gamma$ for $\beta_{\rho}\geq\rho$, $\widehat{a_{\pi}}<b_{\rho}<\hat{a}$ and some $p<\omega$, where $\beta_{\rho}<\beta=\psi_{\pi}(\hat{a})$ by $b_{\rho}<\hat{a}$. (22) follows.

Case 2.2. Next consider the case when $\lambda<\pi$. Then $\lambda<\rho$ and $\Delta^{(\rho,\pi)}\subset\Sigma_{1}(\rho^{+})$ with $\rho^{+}=\Omega_{\rho+1}$. SIH with (26) yields $(\mathcal{H}_{b_{\rho}+1},\Theta\cup\{\rho\})\vdash^{\beta_{\rho^{+}}}_{\beta_{\rho^{+}}}\Gamma,\Delta^{(\rho,\pi)}$ for $\beta_{\rho^{+}}=\psi_{\rho^{+}}(b_{\rho})>\rho$, and by Lemma 4.7.2 we obtain

[TABLE]

Several $(cut)$’s with (25), (28) yield $({\cal H}_{b_{0}+1},\Theta)\vdash^{\beta_{\rho^{+}}+p}_{\beta_{\rho^{+}}}\Gamma$ for $\beta_{\rho^{+}}>\rho$ and $b_{0}=\gamma\#(\omega^{\pi+a_{\ell}+1}\cdot 2)\#\omega^{\pi+a_{r}(\rho)+1}\#1\geq\max\{b_{\ell},b_{\rho}\}$. Predicative cut-elimination lemma 4.12 yields for $\beta_{1}=\varphi(\beta_{\rho^{+}})(\beta_{\rho^{+}}+p)<\rho^{+}$

[TABLE]

We obtain $\lambda<\rho\in{\cal H}_{b_{0}+1}[\Theta]$ by $\gamma<\widehat{a_{\ell}}<b_{0}$. MIH with (29) yields $({\cal H}_{c+1},\Theta)\vdash^{\psi_{\lambda}c}_{\psi_{\lambda}c}\Gamma$ for $c=b_{0}\#1\#\omega^{\rho+\beta_{1}+1}$. We obtain $c=b_{0}\#\omega^{\rho+\beta_{1}+1}\#1=\gamma\#(\omega^{\pi+a_{\ell}+1}\cdot 2)\#\omega^{\pi+a_{r}(\rho)+1}\#\omega^{\rho+\beta_{1}+1}\#2<\gamma\#\omega^{\pi+a+1}=\hat{a}$ since $a_{\ell},a_{r}(\rho)<a$ and $\rho,\beta_{1}<\rho^{+}<\pi$. Hence $\psi_{\lambda}c<\psi_{\lambda}(\hat{a})=\beta$, and (22) follows.

Case 3. Third consider the case when the last inference introduces a $\Sigma_{1}(\lambda)$-sentence $(\forall x\in c\,\theta(x))\in\Gamma$ for $c\in Tm(\lambda)$. We have $(\mathcal{H}_{\gamma},\Theta\cup{\sf k}(d))\vdash^{a(d)}_{\pi}\Gamma,\theta(d)$ for each $d\in Tm(|c|)$. Then we see from (23) that $|d|<|c|\in{\cal H}_{\gamma}(\psi_{\rho}(\gamma\#e))\cap\rho\subset\psi_{\rho}(\gamma\#e)$ for any $\rho\in[\lambda,\pi]$ and any $e$. Hence $|d|\in\psi_{\rho}(\gamma\#e)$. (21) is enjoyed for $\Theta\cup{\sf k}(d)$. SIH yields $(\mathcal{H}_{\hat{a}+1},\Theta\cup{\sf k}(d))\vdash^{\beta_{d}}_{\beta_{d}}\Gamma,\theta(d)$ for $\beta_{d}=\psi_{\lambda}(\widehat{a(d)})$. $(\bigwedge)$ yields (22) for $\beta=\psi_{\lambda}(\hat{a})>\beta_{d}$.

Case 4. Fourth consider the case when the last inference introduces a $\Sigma_{1}(\lambda)$-sentence $(\exists x\in L_{\lambda}\,\theta(x))\in\Gamma$. We have $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}\Gamma,\theta(d)$ for a $d\in Tm(\lambda)$. SIH yields $({\cal H}_{\hat{a}+1},\Theta)\vdash^{\beta_{0}}_{\beta_{0}}\Gamma,\theta(d)$ for $\beta=\psi_{\lambda}(\hat{a})>\psi_{\lambda}(\widehat{a_{0}})=\beta_{0}$. Without loss of generality we can assume that ${\sf k}(d)\subset{\sf k}(\theta(d))$. Then we see from (23) that $|d|\in{\cal H}_{\gamma}(\psi_{\lambda}(\gamma+1))\cap\lambda\subset\psi_{\lambda}(\gamma+1)<\beta$. Thus is enjoyed in the following inference rule $(\bigvee)$. We obtain $({\cal H}_{\hat{a}+1},\Theta)\vdash^{\beta}_{\beta}\Gamma$ by a $(\bigvee)$, which enjoys (6).

Case 5. Fifth consider the case when the last inference is a $({\rm rfl}(\tau,j,\vec{\mu},\vec{\nu}))$ for a $\tau\in{\cal H}_{\gamma}[\Theta]\cap\pi$. We have an $a_{\ell}<a$ and a finite set $\Delta$ of $\Sigma_{j}(\tau)$-sentences such that $(\mathcal{H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,\lnot\delta$ for each $\delta\in\Delta$. On the other hand we have a sequence $\vec{\nu}$ and an ordinal term $a_{r}(\rho)<a$ for each $\rho\in H(\vec{\nu},\tau,\gamma,\Theta)$ such that $(\mathcal{H}_{\gamma},\Theta\cup\{\rho\})\vdash^{a_{r}(\rho)}_{\pi}\Gamma,\Delta^{(\rho,\tau)}$. By (23), for any $\rho\in H(\vec{\nu},\tau,\gamma,\Theta)$ we obtain

[TABLE]

Case 5.1. First consider the case when $\tau<\lambda$. Then $\rho<\psi_{\kappa}(\gamma\#e)$ for any $\kappa\in[\lambda,\pi]$ and $e$. From SIH with (30) we obtain $(\mathcal{H}_{\hat{a}+1},\Theta)\vdash^{\beta_{\ell}}_{\beta_{\ell}}\Gamma,\lnot\delta$ for each $\delta\in\Delta$ with $\beta_{\ell}=\psi_{\lambda}(\widehat{a_{\ell}})$, and $(\mathcal{H}_{\hat{a}+1},\Theta\cup\{\rho\})\vdash^{\beta_{r}(\rho)}_{\beta_{r}(\rho)}\Gamma,\Delta^{(\rho,\tau)}$ for each $\rho\in H(\vec{\nu},\tau,\gamma,\Theta)$ with $\beta_{r}(\rho)=\psi_{\lambda}(\widehat{a_{r}(\rho)})$. We see $\max\{\beta_{\ell},\beta_{r}(\rho),\tau\}<\beta=\psi_{\lambda}(\hat{a})$, and an inference rule $({\rm rfl}(\tau,j,\vec{\mu},\vec{\nu}))$ yields $(\mathcal{H}_{\hat{a}+1},\Theta)\vdash^{\beta}_{\beta}\Gamma$.

Case 5.2. Second consider the case when $\lambda\leq\tau$. Then $\Delta\cup\Delta^{(\rho,\tau)}\subset\Sigma_{1}(\tau^{+})$, and $\rho<\psi_{\kappa}(\gamma\#e)$ for $\tau<\kappa\leq\pi$ and $e$ by (30). SIH yields $(\mathcal{H}_{\widehat{a_{\ell}}+1},\Theta)\vdash^{\beta_{2}}_{\beta_{2}}\Gamma,\lnot\delta$ for each $\delta\in\Delta$, where $\beta_{2}=\psi_{\tau^{+}}\left(\widehat{a_{\ell}}\right)$. On the other side SIH yields $(\mathcal{H}_{\widehat{a_{r}(\rho)}+1},\Theta\cup\{\rho\})\vdash^{\beta_{\rho}}_{\beta_{\rho}}\Gamma,\Delta^{(\rho,\tau)}$ for each $\rho\in H(\vec{\nu},\tau,\gamma,\Theta)$, where $\beta_{\rho}=\psi_{\tau^{+}}\left(\widehat{a_{r}(\rho)}\right)$. Predicative cut-elimination lemma 4.12 yields $(\mathcal{H}_{\widehat{a_{\ell}}+1},\Theta)\vdash^{\delta_{2}}_{\tau}\Gamma,\lnot\delta$ and $(\mathcal{H}_{\widehat{a_{r}(\rho)}+1},\Theta\cup\{\rho\})\vdash^{\delta_{\rho}}_{\tau}\Gamma,\Delta^{(\rho,\tau)}$ for $\delta_{2}=\varphi(\beta_{2})(\beta_{2})$ and $\delta_{\rho}=\varphi(\beta_{\rho})(\beta_{\rho})$. From these with the inference rule $({\rm rfl}(\tau,j,\vec{\mu},\vec{\nu}))$ we obtain

[TABLE]

where $\sup\{\delta_{2},\delta_{\rho}:\rho\in H(\vec{\nu},\tau,\widehat{a_{0}}+1,\Theta)\}\leq\delta_{0}:=\varphi(\beta_{0})(\beta_{0})\in{\cal H}_{\widehat{a_{0}}+1}[\Theta]$ with $\sup\{\beta_{2},\beta_{\rho}:\rho\in H(\vec{\nu},\tau,\gamma,\Theta)\}\leq\beta_{0}:=\psi_{\tau^{+}}\left(\widehat{a_{0}}\right)$, and $\sup\{a_{\ell},a_{r}(\rho):\rho\in H(\vec{\nu},\tau,\gamma,\Theta)\}\leq a_{0}\in{\cal H}_{\gamma}[\Theta]\cap a$, cf. (10).

MIH with (31) yields $({\cal H}_{\hat{a}+1},\Theta)\vdash_{\delta}^{\delta}\Gamma$ for $\delta=\psi_{\lambda}((\widehat{a_{0}}+1)\#\omega^{\tau+\delta_{0}+2})$ and $(\widehat{a_{0}}+1)\#\omega^{\tau+\delta_{0}+2}<\hat{a}$. We have $\delta=\psi_{\lambda}(\widehat{a_{0}}\#1\#\omega^{\tau+\delta_{0}+2})<\psi_{\lambda}(\hat{a})=\beta$ by $\widehat{a_{0}}<\hat{a}$ and $\tau,\delta_{0}<\tau^{+}<\pi$ and $\tau\in{\cal H}_{\gamma}[\Theta]$. (22) follows.

Case 6. Sixth consider the case when the last inference is a $(cut)$. For an $a_{0}<a$ and a $C$ with $\mbox{rk}(C)<\pi$, we have $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}\Gamma,\lnot C$ and $({\cal H}_{\gamma},\Theta)\vdash^{a_{0}}_{\pi}C,\Gamma$.

Case 6.1. First consider the case when $\mbox{{\rm rk}}(C)<\lambda$. Then $C\in\Sigma_{0}(\lambda)$. SIH yields the lemma.

Case 6.2. Second consider the case when $\lambda\leq\mbox{{\rm rk}}(C)<\pi$. Let $\rho^{+}=(\mbox{{\rm rk}}(C))^{+}=\min\{\kappa\in Reg:\mbox{{\rm rk}}(C)<\kappa\}$. Then $C\in\Sigma_{0}(\rho^{+})$ and $\lambda\leq\rho\in{\cal H}_{\gamma}[\Theta]\cap\pi$. SIH yields $({\cal H}_{\widehat{a_{0}}+1},\Theta)\vdash^{\beta_{0}}_{\beta_{0}}\Gamma,\lnot C$ and $({\cal H}_{\widehat{a_{0}}+1},\Theta)\vdash^{\beta_{0}}_{\beta_{0}}C,\Gamma$ for $\beta_{0}=\psi_{\rho^{+}}\left(\widehat{a_{0}}\right)\in{\cal H}_{\widehat{a_{0}}+1}[\Theta]$. By a $(cut)$ we obtain $({\cal H}_{\widehat{a_{0}}+1},\Theta)\vdash^{\beta_{1}}_{\beta_{1}}\Gamma$ for $\beta_{1}=\max\{\beta_{0},\mbox{{\rm rk}}(C)\}+1$ with $\rho<\beta_{1}<\rho^{+}$. Predicative cut-elimination lemma 4.12 yields $({\cal H}_{\widehat{a_{0}}+1},\Theta)\vdash^{\delta_{1}}_{\rho}\Gamma$ for $\delta_{1}=\varphi(\beta_{1})(\beta_{1})$, where $\widehat{a_{0}}\in{\cal H}_{\widehat{a_{0}}+1}[\Theta]$, and $\forall e\forall\tau\in[\lambda,\rho][\Theta\subset\psi_{\tau}(\widehat{a_{0}}\#e)]$ hold. Hence MIH with $\rho\in{\cal H}_{\widehat{a_{0}}+1}[\Theta]$ yields $({\cal H}_{b+1},\Theta)\vdash^{\psi_{\lambda}(b)}_{\psi_{\lambda}(b)}\Gamma$ for $b=\widehat{a_{0}}\#1\#\omega^{\rho+\delta_{1}+1}$. We see $b<\hat{a}$ and $\psi_{\lambda}(b)<\psi_{\lambda}(\hat{a})=\beta$, and (22) follows.

Case 7. Seventh consider the case when the last inference is an $(\Omega\in M_{2})$. We have $({\cal H}_{\gamma},\Theta)\vdash^{a_{\ell}}_{\pi}\Gamma,C$ for an $a_{\ell}<a$, and $({\cal H}_{\gamma},\Theta\cup\{\alpha\})\vdash^{a_{r}(\alpha)}_{\pi}\lnot C^{(\alpha,\Omega)},\Gamma$ for an $a_{r}(\alpha)<a$ for each $\alpha<\Omega$, where $C\in\Pi_{2}(\Omega)$.

The case $\lambda>\Omega$ is seen as in Case 5.1. The case $\lambda=\Omega$ is seen as in Case 5.2. $\Box$

Let us conclude Theorem 1.1. Let $\Omega=\Omega_{1}$.

Proof of Theorem 1.1. Let ${\sf KP}\Pi_{N}\vdash\theta$. By Embedding lemma 4.14 pick an $m$ so that $({\cal H}_{0},\emptyset)\vdash_{\mathbb{K}+m}^{\mathbb{K}\cdot 2+m}\theta$. Predicative cut-elimination lemma 4.12 yields $({\cal H}_{0},\emptyset)\vdash_{\mathbb{K}}^{\omega_{m+1}(\mathbb{K}+1)}\theta$ for $\omega_{m}(\mathbb{K}\cdot 2+m)<\omega_{m+1}(\mathbb{K}+1)$. Lemma 5.2 yields $({\cal H}_{a+1},\emptyset)\vdash^{\beta}_{\beta}\theta$ for $a=\omega^{\mathbb{K}+\omega_{m+1}(\mathbb{K}+1)+1}$ and $\beta=\psi_{\Omega}(a)$. Predicative cut-elimination lemma 4.12 yields $({\cal H}_{a+1},\emptyset)\vdash^{\varphi(\beta)(\beta)}_{0}\theta$. We obtain $\varphi(\beta)(\beta)<\alpha:=\psi_{\Omega}(\omega_{n}(\mathbb{K}+1))$ for $n=m+3$, and hence $({\cal H}_{\omega_{n}(\mathbb{K}+1)},\emptyset)\vdash^{\alpha}_{0}\theta$. Boundedness lemma 4.10 yields $({\cal H}_{\omega_{n}(\mathbb{K}+1)},\emptyset)\vdash^{\alpha}_{0}\theta^{(\alpha,\Omega)}$. Since each inference rule other than reflection rules $({\rm rfl}(\pi,k,\vec{\xi},\vec{\nu}))$ and $(\Omega\in M_{2})$ is sound, we see by induction up to $\alpha=\psi_{\Omega}(\omega_{n}(\mathbb{K}+1))$ that $L_{\alpha}\models\theta$.

This completes a proof of Theorem 1.1.