Arithmetical completeness for some extensions of the pure logic of necessitation

Haruka Kogure¹¹1Email:[email protected] ²²2Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan.

Abstract

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation $\mathbf{N}$ . For $m,n\in\omega$ , let $\mathbf{N}^{+}\mathbf{A}_{m,n}$ , which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^{n}A\to\Box^{m}A$ and the rule $\dfrac{\neg\Box A}{\neg\Box\Box A}$ . In this paper, among other things, we prove that for each $m,n\geq 1$ , the logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$ becomes a provability logic.

1 Introduction

Let $T$ denote a primitive recursively axiomatized consistent extension of Peano Arithmetic $\mathsf{PA}$ . We say that a formula $\mathrm{Pr}_{T}(x)$ is a provability predicate of $T$ if $\mathrm{Pr}_{T}(x)$ weakly represents the set of all theorems of $T$ in $\mathsf{PA}$ , that is, for any formula $\varphi$ , $T\vdash\varphi$ if and only if $\mathsf{PA}\vdash\mathrm{Pr}_{T}(\ulcorner\varphi\urcorner)$ . Here, $\ulcorner\varphi\urcorner$ is the numeral of the Gödel number of $\varphi$ . In a proof of the second incompleteness theorem, it is essential to construct a canonical $\Sigma_{1}$ provability predicate $\mathrm{Prov}_{T}(x)$ of $T$ satisfying the following conditions:

$\mathbf{D2}$ :

$T\vdash\mathrm{Prov}_{T}(\ulcorner\varphi\to\psi\urcorner)\to(\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)\to\mathrm{Prov}_{T}(\ulcorner\psi\urcorner))$ .
$\mathbf{D3}$ :

$T\vdash\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)\to\mathrm{Prov}_{T}(\ulcorner\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)\urcorner)$ .

A provability predicate $\mathrm{Pr}_{T}(x)$ can be considered as a modality. For example, the schemes $\Box(A\to B)\to(\Box A\to\Box B)$ and $\Box A\to\Box\Box A$ are modal counterparts of $\mathbf{D2}$ and $\mathbf{D3}$ respectively. An arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ is a mapping from modal formulas to sentences of arithmetic, which maps $\Box$ to $\mathrm{Pr}_{T}(x)$ . For each provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ , let $\mathsf{PL}(\mathrm{Pr}_{T})$ denote the set of all modal formulas $A$ satisfying $T\vdash f(A)$ for all arithmetical interpretations $f$ based on $\mathrm{Pr}_{T}(x)$ . The set $\mathsf{PL}(\mathrm{Pr}_{T})$ is called the provability logic of $\mathrm{Pr}_{T}(x)$ . A well-known result for the study of provability logics is Solovay’s arithmetical completeness theorem [13] saying that if $T$ is $\Sigma_{1}$ -sound, then $\mathsf{PL}(\mathrm{Prov}_{T})$ is exactly the modal logic $\mathbf{GL}$ . For each non-standard provability predicate, which is not canonical provability predicate $\mathrm{Prov}_{T}(x)$ , its provability logic may not be $\mathbf{GL}$ , and there has been recent research on provability logics in this direction. One goal of studies in this direction is to obtain an in-depth understanding of the following problem.

Problem 1.1 ([6, Problem 2.2]).

For which modal logic $L$ is there a provability predicate $\mathrm{Pr}_{T}(x)$ such that $L=\mathsf{PL}(\mathrm{Pr}_{T})$ ?

Kurahashi [7, 6, 9] proved that modal logics such as $\mathbf{K}$ and $\mathbf{KD}$ are provability logics, and Kogure and Kurahashi [5] proved that so are non-normal modal logics such as $\mathbf{MN}$ and $\mathbf{MN4}$ . Also, not all modal logics can be a provability logic of some provability predicate. It is known that each of modal logics $\mathbf{KT}$ , $\mathbf{K4}$ , $\mathbf{B}$ , and $\mathbf{K5}$ is not a provability logic of any provability predicate (cf. [12, 8]).

Fitting, Marek, and Truszczyński [2] introduced the pure logic of necessitation $\mathbf{N}$ , which is obtained by removing the distribution axiom $\Box(A\to B)\to(\Box A\to\Box B)$ from the basic normal modal logic $\mathbf{K}$ , and Kurahashi proved that $\mathbf{N}$ is a provability logic of some provability predicate. Kurahashi [6] also introduced an extension $\mathbf{N4}$ of $\mathbf{N}$ and proved that $\mathbf{N4}$ has the finite frame property and $\mathbf{N4}$ is a provability logic of some provability predicate by using its finite frame property. The finite frame property of $\mathbf{N4}$ was generalized by Kurahashi and Sato [10]. For each natural numbers $m$ and $n$ , Kurahashi and Sato introduced the extension $\mathbf{N}^{+}\mathbf{A}_{m,n}$ obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^{n}A\to\Box^{m}A$ and the inference rule $(\mathrm{Ros}^{\Box})$ $\dfrac{\neg\Box A}{\neg\Box\Box A}$ , and proved that $\mathbf{N}^{+}\mathbf{A}_{m,n}$ has the finite frame property.

In the viewpoint of Problem 1.1, it is natural to ask the following question: “For each $m,n\in\omega$ , can the logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$ be a provability logic?” In this paper, we clarify the situation on this problem. If $m=n$ , then $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is exactly $\mathbf{N}$ , and Kurahashi proved that $\mathbf{N}$ is a provability logic of some $\Sigma_{1}$ provability predicate, so it suffices to consider the case $m\neq n$ . The following table summarizes the situation when $m\neq n$ .

Table 1: Is there a

\Sigma_{1}

\mathrm{Pr}_{T}(x)

T

such that

\mathsf{PL}(\mathrm{Pr}_{T})=\mathbf{N}^{+}\mathbf{A}_{m,n}

	$m,n\geq 1$	$m=0$ & $n\geq 1$	$m\geq 1$ & $n=0$
$T$ is $\Sigma_{1}$ -sound	Yes (Theorems 4.2, 4.3)	No (Proposition 3.4)	No (Proposition 3.5)
$T$ is $\Sigma_{1}$ -ill	Yes (Theorems 4.3, 5.2)	No (Proposition 3.4)	Yes (Theorem 5.3)

The following shows the details of the theorems and propositions in the table.

•

If $n>m\geq 1$ and $T$ is $\Sigma_{1}$ -sound, then $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is a provability logic of some $\Sigma_{1}$ provability predicate (Theorem 4.2).
•

If $m>n\geq 1$ , then $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is a provability logic of some $\Sigma_{1}$ provability predicate (Theorem 4.3).
•

If $n>m\geq 1$ and $T$ is $\Sigma_{1}$ -ill, then $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is a provability logic of some $\Sigma_{1}$ provability predicate (Theorem 5.2).
•

If $m\geq 1$ and $T$ is $\Sigma_{1}$ -ill, then $\mathbf{N}^{+}\mathbf{A}_{m,0}$ is a provability logic of some $\Sigma_{1}$ provability predicate (Theorem 5.3).
•

If $n\geq 1$ , then $\mathbf{N}^{+}\mathbf{A}_{0,n}$ is not a provability logic of any provability predicate (Proposition 3.4).
•

If $T$ is $\Sigma_{1}$ -sound and $m\geq 1$ , then $\mathbf{N}^{+}\mathbf{A}_{m,0}$ is not a provability logic of any $\Sigma_{1}$ provability predicate (Proposition 3.5).

2 Preliminaries and Background

Throughout this paper, let $T$ denote a primitive recursively axiomatized consistent extension of Peano arithmetic $\mathsf{PA}$ in the language $\mathcal{L}_{A}$ of first-order arithmetic. We say that $T$ is $\Sigma_{1}$ -ill if $T$ is not $\Sigma_{1}$ -sound. Let $\omega$ be the set of all natural numbers. For each $n\in\omega$ , $\overline{n}$ denotes the numeral of $n$ . In the present paper, we fix a natural Gödel numbering such that if $\alpha$ is a proper sub-expression of a finite sequence $\beta$ of $\mathcal{L}_{A}$ -symbols, then the Gödel number of $\alpha$ is less than that of $\beta$ . For each formula $\varphi$ , let $\ulcorner\varphi\urcorner$ be the numeral of the Gödel number of $\varphi$ . Let $\{\xi_{t}\}_{t\in\omega}$ denote the repetition-free primitive recursive enumeration of all $\mathcal{L}_{A}$ -formulas in ascending order of Gödel numbers. We note that if $\xi_{u}$ is a proper subformula of $\xi_{v}$ , then $u<v$ .

We define the primitive recursive classes of formulas $\Delta_{0}$ , $\Sigma_{1}$ and $\Pi_{1}$ . Let $\Delta_{0}$ be the set of all formulas whose every quantifier is bounded. The classes $\Sigma_{1}$ and $\Pi_{1}$ are defined as the smallest classes satisfying the following conditions:

•

$\Delta_{0}\subseteq\Sigma_{1}\cap\Pi_{1}$ .
•

$\Sigma_{1}$ (resp. $\Pi_{1}$ ) is closed under conjunction, disjunction and existential (resp. universal) quantification.
•

If $\varphi$ is $\Sigma_{1}$ (resp. $\Pi_{1}$ ), then $\neg\varphi$ is $\Pi_{1}$ (resp. $\Sigma_{1}$ ).
•

If $\varphi$ is $\Sigma_{1}$ (resp. $\Pi_{1}$ ) and $\psi$ is $\Pi_{1}$ (resp. $\Sigma_{1}$ ), then $\varphi\to\psi$ is $\Pi_{1}$ (resp. $\Sigma_{1}$ ).

The classes $\Delta_{0}$ , $\Sigma_{1}$ and $\Pi_{1}$ are not closed under logical equivalence, and these classes are primitive recursive. We say that a formula $\varphi$ is $\Delta_{1}(\mathsf{PA})$ if $\varphi$ is $\Sigma_{1}$ and $\varphi$ is $\mathsf{PA}$ -equivalent to some $\Pi_{1}$ formula. Let $\mathrm{Fml}_{\mathcal{L}_{A}}(x)$ and $\Sigma_{1}(x)$ be $\Delta_{1}(\mathsf{PA})$ formulas naturally expressing that “ $x$ is an $\mathcal{L}_{A}$ formula” and “ $x$ is a $\Sigma_{1}$ sentence”, respectively. Let $\mathrm{True}_{\Sigma_{1}}(x)$ be a $\Sigma_{1}$ formula naturally saying that “ $x$ is a true $\Sigma_{1}$ sentence” and we may assume that $\mathrm{True}_{\Sigma_{1}}(x)$ is of the form $\exists y\eta(x,y)$ for some $\Delta_{0}$ formula $\eta(x,y)$ . The formula $\mathrm{True}_{\Sigma_{1}}(x)$ satisfies the condition that for any $\Sigma_{1}$ sentence $\varphi$ , $\mathsf{PA}\vdash\varphi\leftrightarrow\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ (see [3, 4]).

2.1 Provability predicate

We say that a formula $\mathrm{Pr}_{T}(x)$ is a provability predicate of $T$ if for any formula $\varphi$ , $T\vdash\varphi$ if and only if $\mathsf{PA}\vdash\mathrm{Pr}_{T}(\ulcorner\varphi\urcorner)$ . Let $\mathbb{N}$ be the standard model of arithmetic. We note that if $\mathrm{Pr}_{T}(x)$ is a $\Sigma_{1}$ formula, then the condition that $\mathrm{Pr}_{T}(x)$ is a provability predicate of $T$ is equivalent to the condition that for any formula $\varphi$ , $T\vdash\varphi$ if and only if $\mathbb{N}\models\mathrm{Pr}_{T}(\ulcorner\varphi\urcorner)$ .

For each formula $\tau(v)$ representing the set of all axioms of $T$ in $\mathsf{PA}$ , we can construct a formula $\mathrm{Prf}_{\tau}(x,y)$ naturally expressing that “ $y$ is a proof of $x$ from the set of all sentences satisfying $\tau(v)$ ”, and we can define a provability predicate $\mathrm{Pr}_{\tau}(x)$ as $\exists y\mathrm{Prf}_{\tau}(x,y)$ (see [1]). If $\tau(v)$ is a $\Delta_{1}(\mathsf{PA})$ formula, then we can define $\mathrm{Prf}_{\tau}(x,y)$ as a $\Delta_{1}(\mathsf{PA})$ formula. Let $\mathrm{Con}_{\tau}$ denote $\neg\mathrm{Pr}_{\tau}(\ulcorner 0=1\urcorner)$ . In this paper, we assume that $\mathrm{Prf}_{\tau}(x,y)$ is a single-conclusion, that is, $\mathsf{PA}\vdash\forall y\bigl{(}\exists x\mathrm{Prf}_{\tau}(x,y)\to\exists!x\mathrm{Prf}_{\tau}(x,y)\bigr{)}$ holds. We note that $\mathrm{Pr}_{\tau}(x)$ satisfies $\mathsf{PA}\vdash\forall x\bigl{(}\mathrm{Pr}_{\tau}(x)\to\forall z\exists y>z\mathrm{Prf}_{\tau}(x,y)\bigr{)}$ . In addition, $\mathrm{Pr}_{\tau}(x)$ satisfies the following conditions:

•

$\mathsf{PA}\vdash\forall x\forall y\bigl{(}\mathrm{Pr}_{\tau}(x\dot{\to}y)\to\bigl{(}\mathrm{Pr}_{\tau}(x)\to\mathrm{Pr}_{\tau}(y)\bigr{)}\bigr{)}$ .
•

If $\sigma(x)$ is a $\Sigma_{1}$ formula, then $\mathsf{PA}\vdash\forall x\bigl{(}\sigma(x)\to\mathrm{Pr}_{\tau}(\ulcorner\sigma(\dot{x})\urcorner)\bigr{)}$ .

Here, $x\dot{\to}y$ and $\ulcorner\sigma(\dot{x})\urcorner$ are primitive recursive terms corresponding to primitive recursive functions calculating the Gödel number of $\varphi\to\psi$ from those of $\varphi$ and $\psi$ and calculating the Gödel number of $\sigma(\overline{n})$ from $n\in\omega$ , respectively. The first condition is a stronger version of the condition $\mathbf{D2}$ . If $\mathrm{Pr}_{\tau}(x)$ is $\Sigma_{1}$ , then $\mathrm{Pr}_{\tau}(x)$ satisfies the condition $\mathbf{D3}$ . Here, the conditions $\mathbf{D2}$ and $\mathbf{D3}$ are introduced in Section 1. From now on, we fix a $\Delta_{1}(\mathsf{PA})$ formula $\tau(x)$ representing $T$ in $\mathsf{PA}$ . Let $\mathrm{Proof}_{T}(x,y)$ and $\mathrm{Prov}_{T}(x)$ be the $\Delta_{1}(\mathsf{PA})$ formula $\mathrm{Prf}_{\tau}(x,y)$ and the $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{\tau}(x)$ respectively. Let $\mathrm{Con}_{T}$ denote $\neg\mathrm{Prov}_{T}(\ulcorner 0=1\urcorner)$ .

2.2 Provability logic

The language of modal propositional logic $\mathcal{L}_{\Box}$ consists of propositional variables, the logical constant $\bot$ , the logical connectives $\neg,\wedge,\vee,\to$ , and the modal operator $\Box$ . Let $\mathsf{MF}$ be the set of all $\mathcal{L}_{\Box}$ -formulas. For each $A\in\mathsf{MF}$ , let $\mathsf{Sub}(A)$ denote the set of all subformulas of $A$ . For each $A\in\mathsf{MF}$ and $n\in\omega$ , we define $\Box^{n}A$ inductively as follows: Let $\Box^{0}A$ be $A$ and let $\Box^{n+1}A$ be $\Box\Box^{n}A$ . The axioms of the basic modal logic $\mathbf{K}$ consists of propositional tautologies in the language $\mathcal{L}_{\Box}$ and the distribution axiom scheme $\Box(A\to B)\to(\Box A\to\Box B)$ . The inference rules of modal logic $\mathbf{K}$ consists of Modus Ponens $(\mathrm{MP})\ \dfrac{A\quad A\to B}{B}$ and Necessitation $(\mathrm{Nec})\ \dfrac{A}{\Box A}$ .

For each provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ , we say that a mapping $f$ from $\mathsf{MF}$ to a set of $\mathcal{L}_{A}$ -sentences is an arithmetical interpretation based on $\mathrm{Pr}_{T}(x)$ if $f$ satisfies the following clauses:

•

$f(\bot)$ is $0=1$ ,
•

$f(\neg A)$ is $\neg f(A)$ ,
•

$f(A\circ B)$ is $f(A)\circ f(B)$ for $\circ\in\{\wedge,\vee,\to\}$ , and
•

$f(\Box A)$ is $\mathrm{Pr}_{T}(\ulcorner f(A)\urcorner)$ .

For any provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ , let $\mathsf{PL}(\mathrm{Pr}_{T})$ denote the set of all $\mathcal{L}_{\Box}$ -formulas $A$ such that for any arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , $T\vdash f(A)$ . We call the set $\mathsf{PL}(\mathrm{Pr}_{T})$ provability logic of $\mathrm{Pr}_{T}(x)$ .

A well-known result in the study of provability logics is Solovay’s arithmetical completeness theorem. The modal logic $\mathbf{GL}$ is obtained by adding axiom scheme $\Box(\Box A\to A)\to\Box A$ to $\mathbf{K}$ .

Theorem 2.1 (Solovay [13]).

If $T$ is $\Sigma_{1}$ -sound, then $\mathsf{PL}(\mathrm{Prov}_{T})=\mathbf{GL}$ .

Fitting, Marek, and Truszczyński [2] introduced the pure logic of necessitation $\mathbf{N}$ , which is obtained by removing the distribution axiom $\Box(A\to B)\to(\Box A\to\Box B)$ from $\mathbf{K}$ . In Kurahashi [6], it is proved that some extensions of $\mathbf{N}$ can be a provability logic of a provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ . The modal logics $\mathbf{N4}$ is obtained by adding the axiom scheme $\Box A\to\Box\Box A$ . The following are obtained in [6].

•

For each $L\in\{\mathbf{N},\mathbf{N4}\}$ , there exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $\mathsf{PL}(\mathrm{Pr}_{T})=L$ .
•

$\mathbf{N}=\bigcap\{\mathsf{PL}(\mathrm{Pr}_{T})\mid\mathrm{Pr}_{T}(x)$ is a provability predicate of $T\}$
•

$\mathbf{N4}=\bigcap\{\mathsf{PL}(\mathrm{Pr}_{T})\mid\mathrm{Pr}_{T}(x)$ is a provability predicate of $T$ satisfying $\mathbf{D3}\}$

2.3 The logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$

Fitting, Marek, and Truszczyński introduced a relational semantics for $\mathbf{N}$ .

Definition 2.2 ( $\mathbf{N}$ -frames and $\mathbf{N}$ -models).

•

A tuple $(W,\{\prec_{B}\}_{B\in\mathsf{MF}})$ is called an $\mathbf{N}$ -frame if $W$ is non-empty set and $\prec_{B}$ is a binary relation on $W$ for every $B\in\mathsf{MF}$ .
•

An $\mathbf{N}$ -frame $(W,\{\prec_{B}\}_{B\in\mathsf{MF}})$ is finite if $W$ is finite.
•

A triple $(W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash)$ is called an $\mathbf{N}$ -model if $(W,\{\prec_{B}\}_{B\in\mathsf{MF}})$ is an $\mathbf{N}$ -frame and $\Vdash$ is a binary relation between $W$ and $\mathsf{MF}$ satisfying the usual conditions for propositional connectives and the following condition:

$x\Vdash\Box B\iff\forall y\in W(x\prec_{B}y\Longrightarrow y\Vdash B).$
•

A formula $A$ is valid in an $\mathbf{N}$ -model $(W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash)$ if $x\Vdash A$ for every $x\in W$ .
•

A formula $A$ is valid in an $\mathbf{N}$ -frame $\mathcal{F}=(W,\{\prec_{B}\}_{B\in\mathsf{MF}})$ if $A$ is valid in any $\mathbf{N}$ -model $(\mathcal{F},\Vdash)$ based on $\mathcal{F}$ .

Fitting, Marek, and Truszczyński proved that $\mathbf{N}$ is sound and complete with respect to the above semantics. In addition, $\mathbf{N}$ has the finite frame property with respect to this semantics.

Theorem 2.3 ([2, Theorem 3.6 and Theorem 4.10]).

For any $\mathcal{L}_{\Box}$ -formula $A$ , the following are equivalent:

1.

$\mathbf{N}\vdash A$ .
2.

$A$ is valid in all $\mathbf{N}$ -frames.
3.

$A$ is valid in all finite $\mathbf{N}$ -frames.

For each $m,n\in\omega$ , the logic $\mathbf{N}\mathbf{A}_{m,n}$ is obtained by adding the axiom scheme

\mathrm{Acc}_{m,n}:\Box^{n}A\to\Box^{m}A

to $\mathbf{N}$ , and the logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is obtained by adding the inference rule

(\mathrm{Ros}^{\Box})\dfrac{\neg\Box A}{\neg\Box\Box A}

to $\mathbf{N}\mathbf{A}_{m,n}$ . Extensions $\mathbf{N}\mathbf{A}_{m,n}$ and $\mathbf{N}^{+}\mathbf{A}_{m,n}$ of $\mathbf{N}$ are introduced in Kurahashi and Sato [10]. It is known that $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is exactly $\mathbf{N}\mathbf{A}_{m,n}$ for $m\geq 1$ or $n\leq 1$ .

Proposition 2.4 ([10, Proposition 4.2]).

Let $m\geq 1$ or $n\leq 1$ . The Rule $\mathrm{Ros}^{\Box}$ is admissible in $\mathbf{N}\mathbf{A}_{m,n}$ . Consequently, $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is exactly $\mathbf{N}\mathbf{A}_{m,n}$ .

Kurahashi and Sato also introduced the notion $(m,n)$ -accessibility.

Definition 2.5.

Let $\mathcal{F}=(W,\{\prec_{B}\}_{B\in\mathsf{MF}})$ be an $\mathbf{N}$ -frame.

•

Let $x,y\in W$ , $B\in\mathsf{MF}$ and $k\in\omega$ . We define $x\prec_{B}^{k}y$ as follows:

x\prec_{B}^{k}y:\iff\begin{cases}x=y&\text{if}\ k=0\\ \exists w\in W(x\prec_{\Box^{k-1}A}w\prec_{A}^{k-1}y)&\text{if}\ k\geq 1.\end{cases}

•

We say that the frame $\mathcal{F}$ is $(m,n)$ -accessible if for any $x,y\in W$ , if $x\prec_{B}^{m}y$ , then $x\prec_{B}^{n}y$ for every $\Box^{m}B\in\mathsf{MF}$ .

For $m,n\in\omega$ , the logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is sound and complete and has the finite frame property with respect to $(m,n)$ -accessible $\mathbf{N}$ -frames.

Fact 2.6 (The finite frame property of $\mathbf{N}^{+}\mathbf{A}_{m,n}$ [10, Theorem 5.1]).

Let $A\in\mathsf{MF}$ and $m,n\in\omega$ . The following are equivalent:

1.

$\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ .
2.

$A$ is valid in all $(m,n)$ -accessible $\mathbf{N}$ -frames.
3.

$A$ is valid in all finite $(m,n)$ -accessible $\mathbf{N}$ -frames.

Fact 2.7 (The dicidability of $\mathbf{N}^{+}\mathbf{A}_{m,n}$ [10, Corollary 6.10]).

For each $m,n\in\omega$ , the logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$ is decidable.

From the proof of [10, Theorem 5.1], it is obvious that we can primitive recursively construct a finite $(m,n)$ -accessible $\mathbf{N}$ -model which falsifies a formula $A$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\nvdash A$ .

3 Main results

The following is the main result of this paper:

Theorem 3.1.

Suppose $m\neq n$ .

•

For each $m,n\geq 1$ , there exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}=\mathsf{PL}(\mathrm{Pr}_{T})$ .
•

Suppose $T$ is $\Sigma_{1}$ -ill. For each $m\geq 1$ , there exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,0}=\mathsf{PL}(\mathrm{Pr}_{T})$ .
•

For each $n\geq 1$ , there exists no provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $\mathbf{N}^{+}\mathbf{A}_{0,n}\subseteq\mathsf{PL}(\mathrm{Pr}_{T})$ .
•

Suppose $T$ is $\Sigma_{1}$ -sound. For each $m\geq 1$ , there exists no $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $\mathbf{N}^{+}\mathbf{A}_{m,0}\subseteq\mathsf{PL}(\mathrm{Pr}_{T})$ .

Table 1 in Section 1 summarizes the situation in Theorem 3.1. Our strategy to prove the first clause of Theorem 3.1 depends on whether $T$ is $\Sigma_{1}$ -sound or not. In fact, we prove the following theorems.

Theorem 3.2.

Suppose $m\neq n$ . If $T$ is $\Sigma_{1}$ -sound and $m,n\geq 1$ , then there exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ if and only if $T\vdash f(A)$ .

Theorem 3.3.

Suppose $m\neq n$ . If $T$ is $\Sigma_{1}$ -ill, $m\geq 1$ and $n\geq 0$ , then there exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ if and only if $T\vdash f(A)$ .

Theorem 3.2 and 3.3 imply the first and second clauses of Theorem 3.1. In Section 4, we prove Theorem 3.2 by distinguishing the two more cases $n>m\geq 1$ and $m>n\geq 1$ and Theorem 3.3 with respect to the case $m>n\geq 1$ . In Section 5, we prove Theorem 3.3 with respect to the cases $n>m\geq 1$ and $m>n=0$ .

We show the third clause of Theorem 3.1. For any $n\in\omega$ , provability predicate $\mathrm{Pr}_{T}(x)$ , and formula $\varphi$ , we define $\mathrm{Pr}_{T}^{n}(\ulcorner\varphi\urcorner)$ inductively as follows: Let $\mathrm{Pr}_{T}^{0}(\ulcorner\varphi\urcorner)$ be $\varphi$ and $\mathrm{Pr}_{T}^{n+1}(\ulcorner\varphi\urcorner)$ be $\mathrm{Pr}_{T}(\ulcorner\mathrm{Pr}_{T}^{n}(\ulcorner\varphi\urcorner)\urcorner)$ . The following proposition is a generalization of the fact $\mathsf{PL}(\mathrm{Pr}_{T})\nsupseteq\mathbf{KT}\ (=\mathbf{K}+\Box A\to A)$ for any $\mathrm{Pr}_{T}(x)$ (cf. [12]).

Proposition 3.4.

For each $n\geq 1$ , there exists no provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $\Box^{n}p\to p\in\mathsf{PL}(\mathrm{Pr}_{T})$ .

Proof.

Suppose that there is a provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ satisfying $\Box^{n}p\to p\in\mathsf{PL}(\mathrm{Pr}_{T})$ . By the fixed-point lemma, we have a sentence $\sigma$ such that $\mathsf{PA}\vdash\neg\mathrm{Pr}_{T}^{n}(\ulcorner\sigma\urcorner)\leftrightarrow\sigma$ . Let $f$ be an arithmetical interpretation satisfying $f(p)=\sigma$ . Then, it follows that $T\vdash f(\Box^{n}p)\to f(p)$ , that is, we obtain $T\vdash\mathrm{Pr}_{T}^{n}(\ulcorner\sigma\urcorner)\to\sigma$ . By the law of excluded middle, $T\vdash\sigma$ holds. We also have $T\vdash\mathrm{Pr}_{T}^{n}(\ulcorner\sigma\urcorner)$ and $T\vdash\neg\mathrm{Pr}_{T}^{n}(\ulcorner\sigma\urcorner)$ , which is a contradiction. ∎

Proposition 3.4 implies the third clause of Theorem 3.1. Lastly, we prove the fourth clause of Theorem 3.1.

Proposition 3.5.

Let $T$ be $\Sigma_{1}$ -sound. For each $m\geq 1$ , there exists no $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $p\to\Box^{m}p\in\mathsf{PL}(\mathrm{Pr}_{T})$ .

Proof.

Suppose that there is a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $p\to\Box^{m}p\in\mathsf{PL}(\mathrm{Pr}_{T})$ . Let $\sigma$ be a $\Pi_{1}$ sentence satisfying $\mathsf{PA}\vdash\sigma\leftrightarrow\neg\mathrm{Pr}_{T}(\ulcorner\sigma\urcorner)$ and $f$ be an arithmetical interpretation such that $f(p)=\sigma$ . We obtain $T\vdash\sigma\to\mathrm{Pr}_{T}^{m}(\ulcorner\sigma\urcorner)$ , and $T\vdash\neg\mathrm{Pr}_{T}(\ulcorner\sigma\urcorner)\to\mathrm{Pr}_{T}^{m}(\ulcorner\sigma\urcorner)$ . Since $T$ is $\Sigma_{1}$ -sound, we have $\mathbb{N}\models\neg\mathrm{Pr}_{T}(\ulcorner\sigma\urcorner)\to\mathrm{Pr}_{T}^{m}(\ulcorner\sigma\urcorner)$ . We can easily prove that $T\nvdash\sigma$ . Thus, we obtain $\mathbb{N}\not\models\mathrm{Pr}_{T}(\ulcorner\sigma\urcorner)$ , and $\mathbb{N}\models\mathrm{Pr}_{T}^{m}(\ulcorner\sigma\urcorner)$ . Therefore, it follows from the $\Sigma_{1}$ -soundness of $T$ that $T\vdash\sigma$ . This contradicts $T\nvdash\sigma$ . ∎

Proposition 3.5 implies the fourth clause of Theorem 3.1.

Also, Proposition 3.5 is only applicable to $\Sigma_{1}$ provability predicates, so we do not know whether $\mathbf{N}^{+}\mathbf{A}_{m,0}$ becomes a provability logic of some $\mathrm{Pr}_{T}(x)$ which is not $\Sigma_{1}$ if $T$ is $\Sigma_{1}$ -sound. We propose the following problem.

Problem 3.6.

Let $T$ be $\Sigma_{1}$ -sound and $m\geq 1$ . Is there a $\Sigma_{2}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that $\mathbf{N}^{+}\mathbf{A}_{m,0}=\mathsf{PL}(\mathrm{Pr}_{T})$ ?

Kurahashi proved that if $\mathrm{Pr}_{T}(x)$ satisfies $\mathbf{D2}$ and $p\to\Box^{m}p\in\mathsf{PL}(\mathrm{Pr}_{T})$ , then $\Box^{m}\bot\in\mathsf{PL}(\mathrm{Pr}_{T})$ (cf. [8, Lemma 3.12]). Hence, $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,0}=\mathsf{PL}(\mathrm{Pr}_{T})$ does not satisfy $\mathbf{D2}$ because $\mathbf{N}^{+}\mathbf{A}_{m,0}\nvdash\Box^{m}\bot$ .

4 Proof of Theorem 3.2

First, we introduce some notions, which are used in throughout of this paper. We call an $\mathcal{L}_{A}$ -formula propositionally atomic if it is either atomic or of the form $Qx\psi$ , where $Q\in\{\forall,\exists\}$ . For each propositionally atomic formula $\varphi$ , we prepare a propositional variable $p_{\varphi}$ . Let $I$ be a primitive recursive injection from $\mathcal{L}_{A}$ -formulas into propositional formulas, which is defined as follows:

•

$I(\varphi)$ is $p_{\varphi}$ for each propositionally atomic formula $\varphi$ ,
•

$I(\neg\varphi)$ is $\neg I(\varphi)$ ,
•

$I(\varphi\circ\psi)$ is $I(\varphi)\circ I(\psi)$ for $\circ\in\{\wedge,\vee\to\}$ .

Let $X$ be a finite set of formulas. An $\mathcal{L}_{A}$ -formula is called a tautological consequence (t.c.) of $X$ if $\bigwedge_{\psi\in X}I(\psi)\to I(\varphi)$ is a tautology. For each $n\in\omega$ , let $P_{T,n}:=\{\varphi\mid\mathbb{N}\models\exists y\leq\overline{n}\ \mathrm{Proof}_{T}(\ulcorner\varphi\urcorner,y)\}$ . We see that if $\varphi$ is a t.c. of $P_{T,n}$ , then $\varphi$ is provable in $T$ . We can formalize the above notions in the language of $\mathsf{PA}$ .

Next, we prepare a $\mathsf{PA}$ -provably recursive function $h$ , which is originally introduced in [9]. The function $h$ is defined by the recursion theorem as follows:

•

$h(0)=0$ .
•

$h(s+1)=\begin{cases}i&\text{if}\ h(s)=0\\ &\quad\&\ i=\min\{j\in\omega\setminus\{0\}\mid\neg S(\overline{j})\ \text{is a t.c.~{}of}\ P_{T,s}\},\\ h(s)&\text{otherwise}.\end{cases}$

Here, $S(x)$ is the $\Sigma_{1}$ formula $\exists y(h(y)=x)$ . The function $h$ satisfies the property that for each $s$ and $i$ , $h(s+1)=i$ implies $i\leq s+1$ , which guarantees that the function $h$ is $\mathsf{PA}$ -provably recursive (See [9], p. 603). The following proposition holds for $h$ .

Proposition 4.1 (Cf. [9, Lemma 3.2.]).

1.

$\mathsf{PA}\vdash\forall x\forall y(0<x<y\land S(x)\to\neg S(y))$ .
2.

$\mathsf{PA}\vdash\neg\mathrm{Con}_{T}\leftrightarrow\exists x(S(x)\land x\neq 0)$ .
3.

For each $i\in\omega\setminus\{0\}$ , $T\nvdash\neg S(\overline{i})$ .
4.

For each $l\in\omega$ , $\mathsf{PA}\vdash\forall x\forall y(h(x)=0\land h(x+1)=y\land y\neq 0\to x\geq\overline{l})$ .

We are ready to prove Theorem 3.2.

4.1 The case $n>m\geq 1$

In this subsection, we prove the following theorem, which is Theorem 3.2 with respect to the case $n>m\geq 1$ .

Theorem 4.2.

Suppose that $T$ is $\Sigma_{1}$ -sound and $n>m\geq 1$ . There exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ if and only if $T\vdash f(A)$ .

Proof.

Let $T$ be $\Sigma_{1}$ -sound and $n>m\geq 1$ . First, we prepare the $\Sigma_{1}$ formula $\mathrm{Prov}_{T}^{\Sigma}(x)\equiv\mathrm{Prov}_{T}(x)\wedge\bigr{(}\Sigma_{1}(x)\to\mathrm{True}_{\Sigma_{1}}(x)\bigl{)}$ . Here, $\Sigma_{1}(x)$ is a $\Delta_{1}(\mathsf{PA})$ formula and $\mathrm{True}_{\Sigma_{1}}(x)$ is a $\Sigma_{1}$ formula, and they are introduced in Section 2. We show that the $\Sigma_{1}$ formula $\mathrm{Prov}_{T}^{\Sigma}(x)$ is a $\Sigma_{1}$ provability predicate of $T$ proving $\Sigma_{1}$ -reflection principle of $\mathrm{Prov}_{T}^{\Sigma}(x)$ .

Claim 4.1.

For any formula $\varphi$ , $T\vdash\varphi$ if and only if $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ .

Proof.

$(\Leftarrow)$ : Suppose $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ . Then, $\mathsf{PA}\vdash\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ , and we get $T\vdash\varphi$ . $(\Rightarrow)$ : Assume $T\vdash\varphi$ . Then, $\mathsf{PA}\vdash\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ holds. If $\varphi$ is a $\Sigma_{1}$ sentence, $T\vdash\varphi$ implies $\mathbb{N}\models\varphi$ by the $\Sigma_{1}$ soundness of $T$ , and we get $\mathsf{PA}\vdash\varphi$ . Then, we obtain $\mathsf{PA}\vdash\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ . If $\varphi$ is not a $\Sigma_{1}$ sentence, then $\mathsf{PA}\vdash\neg\Sigma_{1}(\ulcorner\varphi\urcorner)$ holds. In either case, it follows that $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ . ∎

Claim 4.2.

For any $\Sigma_{1}$ sentence $\sigma$ , $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\sigma\urcorner)\to\sigma$ .

Proof.

Let $\sigma$ be a $\Sigma_{1}$ sentence. Since $\mathsf{PA}\vdash\Sigma_{1}(\ulcorner\sigma\urcorner)$ , we obtain $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\sigma\urcorner)\to\mathrm{True}_{\Sigma_{1}}(\ulcorner\sigma\urcorner)$ . It follows that $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\sigma\urcorner)\to\sigma$ . ∎

Let $\langle A_{k}\rangle_{k\in\omega}$ denote a primitive recursive enumeration of all $\mathbf{N}^{+}\mathbf{A}_{m,n}$ -unprovable formulas. For each $k\in\omega$ , we can primitive recursively construct a finite $(m,n)$ -accessible $\mathbf{N}$ -model $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ which falsifies $A_{k}$ (see Fact 2.6 and Fact 2.7). We may assume that the sets $\{W_{k}\}_{k\in\omega}$ are pairwise disjoint subsets of $\omega$ and $\bigcup_{k\in\omega}W_{k}=\omega\setminus\{0\}$ . We may also assume that for each $i\in\omega\setminus\{0\}$ , we can primitive recursively find a unique $k\in\omega$ satisfying $i\in W_{k}$ . We define a $(m,n)$ -accessible $\mathbf{N}$ -model $\mathcal{M}=\bigl{(}W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash\bigr{)}$ as follows:

•

$W$ is $\bigcup_{k\in\omega}W_{k}=\omega\setminus\{0\}$ ,
•

$x\prec_{B}y$ if and only if $x,y\in W_{k}$ and $x\prec_{k,B}y$ for some $k\in\omega$ ,
•

$x\Vdash p$ if and only if $x\in W_{k}$ and $x\Vdash_{k}p$ for some $k\in\omega$ .

We may assume that each $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ is $\Delta_{1}(\mathsf{PA})$ represented in $\mathsf{PA}$ and we see that basic properties of $\mathcal{M}$ are proved in $\mathsf{PA}$ .

Next, we define a $\mathsf{PA}$ -provably recursive function $g_{0}$ outputting all $T$ -provable formulas step by step. The definition of $g_{0}$ consists of Procedure 1 and Procedure 2, and starts with Procedure 1. In Procedure 1, we define the values $g_{0}(0),g_{0}(1),\ldots$ by referring to the values $h(0),h(1),\ldots$ and $T$ -proofs based on $\mathrm{Proof}_{T}(x,y)$ . At the first time $h(s+1)\neq 0$ , the definition of $g_{0}$ switches to Procedure 2 at Stage $s$ .

We define the $\mathsf{PA}$ -provably recursive function $g_{0}$ by using the recursion theorem. In the definition of $g_{0}$ , we use the $\Sigma_{1}$ formula $\mathrm{Pr}_{g_{0}}(x)\equiv$ $\exists y(g_{0}(y)=x\wedge\mathrm{Fml}_{\mathcal{L}_{A}}(x))$ and the arithmetical interpretation $f_{0}$ based on $\mathrm{Pr}_{g_{0}}(x)$ such that $f_{0}(p)\equiv\exists x(S(x)\wedge x\neq 0\wedge x\Vdash p)$ . Here, for each $A\in\mathsf{MF}$ , $f_{0}(A)$ is $\mathsf{PA}$ -provably recursively computed from $A$ . Also, $A$ is $\mathsf{PA}$ -provably recursively computed from $f_{0}(A)$ because $f_{0}$ is injection. The definition of $g_{0}$ is as follows:

Procedure 1.

Stage s:

•

If $h(s+1)=0$ , then we consider the following cases:

Case A:

$s$ is a $T$ -proof of a formula $\varphi$ .

Case A-1:

If $\varphi$ is not a $\Sigma_{1}$ sentence, then $g_{0}(s)=\varphi$ .

Case A-2:

$\varphi$ is a $\Sigma_{1}$ sentence and $\varphi$ is not of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ .

Case A-2-1:

If there exists an $l<s$ such that $l$ is a witness of $\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ , then $g_{0}(s)=\varphi$ .

Case A-2-2:

Otherwise. $g_{0}(s)=0$ .

Case A-3:

$\varphi$ is a $\Sigma_{1}$ sentence and $\varphi$ is of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for some $\psi$ .

We can find a formula $\rho$ and $r\geq 1$ such that $\varphi\equiv\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)$ , where $\rho$ is not of the form of $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ . Here, $r$ denotes the maximum number of outermost consecutive applications of $\mathrm{Pr}_{g_{0}}(x)$ in $\varphi$ .

Case A-3-1:

If for each $0\leq i\leq r-1$ , there exists an $l_{i}<s$ such that $g_{0}(l_{i})=\mathrm{Pr}_{g_{0}}^{i}(\ulcorner\rho\urcorner)$ , then $g_{0}(s)=\varphi$ .

Case A-3-2:

Otherwise. Let $g_{0}(s)=0$ .

Case B:

$s$ is not a $T$ -proof of any formula $\varphi$ . Let $g_{0}(s)=0$ .

Then, go to Stage $s+1$ .
•

If $h(s+1)\neq 0$ , then go to Procedure 2.

Procedure 2. Suppose $s$ and $i\neq 0$ satisfy $h(s)=0$ and $h(s+1)=i$ . Let $k$ be a number such that $i\in W_{k}$ . Let $\{\xi_{t}\}_{t\in\omega}$ denote the primitive recursive enumeration of all $\mathcal{L}_{A}$ -formulas introduced in Section 2. Define

g_{0}(s+t)=\begin{cases}\xi_{t}&\text{if}\ \xi_{t}\equiv f_{0}(B)\ \&\ i\Vdash_{k}\Box B\ \text{for some}\ \Box B\in\mathsf{Sub}(A_{k}),\\ 0&\text{otherwise}.\end{cases}

We have completed the definition of $g_{0}$ .

Claim 4.3.

For any formula $\varphi$ , $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{0}}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ .

Proof.

We distinguish the following two cases.

Case 1:

$\varphi$ is not a $\Sigma_{1}$ sentence.

Since $\mathsf{PA}\vdash\neg\Sigma_{1}(\ulcorner\varphi\urcorner)$ , we obtain

\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner).

(1)

We argue in $\mathsf{PA}+\mathrm{Con}_{T}$ : By Proposition 4.1, the definition of $g_{0}$ never goes to Procedure 2. By the definition of Procedure 1 of $g_{0}$ , it follows that $\mathrm{Pr}_{g_{0}}(\ulcorner\varphi\urcorner)$ if and only if $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ . Then, it concludes that $\mathrm{Pr}_{g_{0}}(\ulcorner\varphi\urcorner)$ if and only if $\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ by (1).

Case 2:

$\varphi$ is a $\Sigma_{1}$ sentence.

We argue in $\mathsf{PA}+\mathrm{Con}_{T}$ : By Proposition 4.1, the construction of $g_{0}$ never switches to Procedure 2.

$(\to)$ : Suppose $\mathrm{Pr}_{g_{0}}(\ulcorner\varphi\urcorner)$ . Then, $g_{0}(s)=\varphi$ for some $s$ . Since the definition of $g_{0}$ at Stage $s$ is in Procedure 1, $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ holds. We consider the following two cases.

•

$\varphi$ is output in Case A-2-1:

There exists an $l<s$ such that $l$ is a witness of $\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ . Hence, we have proved that $\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ holds.
•

$\varphi$ is output in Case A-3-1: Let $\rho$ and $r\geq 1$ be such that $\varphi\equiv\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)$ , where $\rho$ is not of the form of $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ . It follows that for each $i\leq r-1$ , there exists an $l_{i}<s$ such that $g_{0}(l_{i})=\mathrm{Pr}_{g_{0}}^{i}(\ulcorner\rho\urcorner)$ . In particular, since $g_{0}(l_{r-1})=\mathrm{Pr}_{g_{0}}^{r-1}(\ulcorner\rho\urcorner)$ , it follows that $\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)$ holds. Hence, $\mathrm{True}_{\Sigma_{1}}(\ulcorner\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)\urcorner)$ holds. Therefore, we obtain $\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ .

$(\leftarrow):$ Suppose $\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ . Then, we obtain $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ and $\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ . We consider the following two cases.

•

$\varphi$ is not of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ .

Let $l$ be a witness of $\mathrm{True}_{\Sigma_{1}}(\ulcorner\varphi\urcorner)$ . Since we have $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ , there exists some $s>l$ such that $s$ is a $T$ -proof of $\varphi$ . The number $s$ satisfies $g_{0}(s)=\varphi$ .
•

$\varphi$ is of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for some $\psi$ .

Let $\rho$ and $r\geq 1$ be such that $\varphi\equiv\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)$ , where $\rho$ is not of the form of $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ . Since $\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\varphi\urcorner)$ is true, there exists some $l_{r-1}$ such that

$g_{0}(l_{r-1})=\mathrm{Pr}_{g_{0}}^{r-1}(\ulcorner\rho\urcorner).$ (2)

We show that $\varphi$ is output by $g_{0}$ in Case A-3-1. If $r-1=0$ , then $\varphi$ is $\mathrm{Pr}_{g_{0}}(\ulcorner\rho\urcorner)$ and $g_{0}(l_{r-1})=\rho$ by (2). Since $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ , there exists a $T$ -proof $s$ of $\varphi$ such that $s>l_{r-1}$ . Since $\rho$ is not of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for any $\psi$ , we obtain $g_{0}(s)=\mathrm{Pr}_{g_{0}}(\ulcorner\rho\urcorner)=\varphi$ . Suppose $r-1>0$ . Since $\mathrm{Pr}_{g_{0}}^{r-1}(\ulcorner\rho\urcorner)$ is of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for some $\psi$ , $\mathrm{Pr}_{g_{0}}^{r-1}(\ulcorner\rho\urcorner)$ is output in Case A-3-1. Then, for each $i\leq l-2$ there exists an $l_{i}$ such that $g_{0}(l_{i})=\mathrm{Pr}_{g_{0}}^{i}(\ulcorner\rho\urcorner)$ . By (2), for each $i\leq l-1$ there exists an $l_{i}$ satisfying $g_{0}(l_{i})=\mathrm{Pr}_{g_{0}}^{i}(\ulcorner\rho\urcorner)$ . Since we obtain $\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ , there is a $T$ -proof $s$ of $\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)$ such that $s>l_{i}$ for all $i\leq r-1$ . Hence, we obtain $g_{0}(s)=\mathrm{Pr}_{g_{0}}^{r}(\ulcorner\rho\urcorner)\equiv\varphi$ .

∎ By Claim 4.3, for any $\varphi$ , we obtain $\mathbb{N}\models\mathrm{Pr}_{g_{0}}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\varphi\urcorner)$ . Since $\mathrm{Prov}_{T}^{\Sigma}(x)$ is a $\Sigma_{1}$ provability predicate of $T$ by Claim 4.1, so is $\mathrm{Pr}_{g_{0}}(x)$ .

Claim 4.4.

For any formula $\varphi$ , $\mathsf{PA}\vdash\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ .

Proof.

For each $k\geq 1$ , we obtain $\mathsf{PA}\vdash\mathrm{Prov}_{T}^{\Sigma}(\ulcorner\mathrm{Pr}_{g_{0}}^{k}(\ulcorner\varphi\urcorner)\urcorner)\to\mathrm{Pr}_{g_{0}}^{k}(\ulcorner\varphi\urcorner)$ by Claim 4.2. By Claim 4.3, $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{0}}^{k+1}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{k}(\ulcorner\varphi\urcorner)$ . Since $n>m\geq 1$ , it follows that $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ .

Next, we show that $\mathsf{PA}+\neg\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ . We argue in $\mathsf{PA}+\neg\mathrm{Con}_{T}$ : By Proposition 4.1.2, there exists some $i\neq 0$ satisfying $S(\overline{i})$ . Let $s$ and $k$ be such that $h(s)=0$ and $h(s+1)=i\in W_{k}$ . Then, the definition of $g_{0}$ switches to Procedure 2 at Stage $s$ . Suppose that $\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)$ holds, namely, $\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)$ is output by $g_{0}$ . We show that $\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ holds, that is, $\mathrm{Pr}_{g_{0}}^{m-1}(\ulcorner\varphi\urcorner)$ is output by $g_{0}$ . We distinguish the following two cases.

•

$\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)$ is output in Procedure 1:

Since $n>m\geq 1$ , $\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)$ is of the form $\mathrm{Pr}_{g_{0}}(\ulcorner\psi\urcorner)$ for some $\psi$ and is output at Case A-3-1 in Procedure 1. Hence, for each $i\leq n-1$ , there exists an $l_{i}$ such that $g_{0}(l_{i})=\mathrm{Pr}_{g_{0}}^{i}(\ulcorner\varphi\urcorner)$ . Since $0\leq m-1<n-1$ holds, we obtain $g_{0}(l_{m-1})=\mathrm{Pr}_{g_{0}}^{m-1}(\ulcorner\varphi\urcorner)$ .
•

$\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)$ is output in Procedure 2:

Let $\xi_{u}\equiv\mathrm{Pr}_{g_{0}}^{m-1}(\ulcorner\varphi\urcorner)$ . It follows that $\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)\equiv f_{0}(B)$ and $i\Vdash_{k}\Box B$ for some $\Box B\in\mathsf{Sub}(A_{k})$ . Then, by $\mathrm{Pr}_{g_{0}}^{n-1}(\ulcorner\varphi\urcorner)\equiv f_{0}(B)$ , we obtain some $C\in\mathsf{Sub}(A_{k})$ such that $f_{0}(C)=\varphi$ and $B\equiv\Box^{n-1}C$ . Since $i\Vdash_{k}\Box B$ and $i\Vdash_{k}\Box^{n}C\to\Box^{m}C$ , we get $i\Vdash_{k}\Box^{m}C$ . It follows from $m\leq n-1$ and $B\in\mathsf{Sub}(A_{k})$ that $\Box^{m}C\in\mathsf{Sub}(A_{k})$ holds. Also, we have $f_{0}(\Box^{m-1}C)\equiv\mathrm{Pr}_{g_{0}}^{m-1}(\ulcorner f_{0}(C)\urcorner)\equiv\xi_{u}$ . It concludes $g_{0}(s+u)=\xi_{u}\equiv\mathrm{Pr}_{g_{0}}^{m-1}(\ulcorner\varphi\urcorner)$ .

Therefore, we obtain $\mathsf{PA}+\neg\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ . By the law of excluded middle, it concludes that $\mathsf{PA}\vdash\mathrm{Pr}_{g_{0}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{0}}^{m}(\ulcorner\varphi\urcorner)$ . ∎

Claim 4.5.

Let $i\in W_{k}$ and $B\in\mathsf{Sub}(A_{k})$ .

1.

If $i\Vdash_{k}B$ , then $\mathsf{PA}\vdash S(\overline{i})\to f_{0}(B)$ .
2.

If $i\nVdash_{k}B$ , then $\mathsf{PA}\vdash S(\overline{i})\to\neg f_{0}(B)$ .

Proof.

We prove Clauses 1 and 2 simultaneously by induction on the construction of $B\in\mathsf{Sub}(A_{k})$ . We prove the base case of the induction. When $B$ is $\bot$ , Clauses 1 and 2 trivially hold. Suppose that $B\equiv p$ .

1.

If $i\Vdash_{k}p$ , then $\mathsf{PA}\vdash S(\overline{i})\to\exists x\bigl{(}S(x)\wedge x\neq 0\wedge x\Vdash p\bigr{)}$ holds. Hence, we get $\mathsf{PA}\vdash S(\overline{i})\to f_{0}(p)$ .
2.

Suppose $i\nVdash_{k}p$ . By Proposition 4.1.1, $\mathsf{PA}\vdash S(\overline{i})\to\forall x(S(x)\wedge x\neq 0\to x=\overline{i})$ . Thus, we obtain $\mathsf{PA}\vdash S(\overline{i})\to\forall x(S(x)\wedge x\neq 0\to x\nVdash p)$ and $\mathsf{PA}\vdash S(\overline{i})\to\neg f_{0}(p)$ is proved.

Next, we prove the induction cases. Since the cases of $\neg,\ \wedge,\ \vee$ and $\to$ are easily proved, we only consider the case $B\equiv\Box C$ .

1.

Suppose $i\Vdash_{k}\Box C$ . We argue in $\mathsf{PA}+S(\overline{i})$ : Let $s$ be such that $h(s)=0$ and $h(s+1)=i$ . Let $\xi_{t}\equiv f_{0}(C)$ . Since $\Box C\in\mathsf{Sub}(A_{k})$ and $i\Vdash_{k}\Box C$ hold, we obtain $g_{0}(s+t)=\xi_{t}$ . Thus, $\mathrm{Pr}_{g_{0}}(\ulcorner f_{0}(C)\urcorner)$ holds, that is, it follows that $f_{0}(\Box C)$ .
2.
Suppose $i\nVdash_{k}\Box C$ . Then, there exists a $j\in W_{k}$ such that $i\prec_{k,C}j$ and $j\nVdash_{k}C$ . By the induction hypothesis, we obtain $\mathsf{PA}\vdash S(\overline{j})\to\neg f_{0}(C)$ . Let $p$ be a proof of $S(\overline{j})\to\neg f_{0}(C)$ in $T$ . We argue in $\mathsf{PA}+S(\overline{i})$ : Let $s$ be such that $h(s)=0$ and $h(s+1)=i$ . Suppose, towards a contradiction, that $f_{0}(C)$ is output by $g_{0}$ .
1. (i)
  
  $f_{0}(C)$ is output in Procedure 1:
  
  There exists an $l<s$ such that $l$ is a proof of $f_{0}(C)$ in $T$ . Hence, $f_{0}(C)\in P_{T,s-1}$ holds. By Proposition 4.1.4, $p<s$ holds, and we obtain $S(\overline{j})\to\neg f_{0}(C)\in P_{T,s-1}$ . Since $f_{0}(C)\to\neg S(\overline{j})$ is a t.c. of $P_{T,s-1}$ , it follows that $\neg S(\overline{j})$ is a t.c. of $P_{T,s-1}$ . We obtain $h(s)\neq 0$ , which is a contradiction.
2. (ii)
  
  $f_{0}(C)$ is output in Procedure 2:
  
  Then, $f_{0}(C)\equiv f_{0}(C^{\prime})\ \text{and}\ i\Vdash_{k}\Box C^{\prime}\ \text{for some}\ \Box C^{\prime}\in\mathsf{Sub}(A_{k})$ holds. Since $f_{0}(C)\equiv f_{0}(C^{\prime})$ and $f_{0}$ is an injection, we obtain $C\equiv C^{\prime}$ . Hence, $i\Vdash_{k}\Box C$ holds. This is a contradiction.

We have shown that $\mathsf{PA}+S(\overline{i})$ proves $f_{0}(C)$ is never output by $g_{0}$ . Therefore, we obtain $\mathsf{PA}+S(\overline{i})\vdash\neg f_{0}(\Box C)$ . ∎

We complete our proof of Theorem 4.2. The implication $(\Rightarrow)$ is obvious from Claim 4.4. We prove the implication $(\Leftarrow)$ . Suppose $\mathbf{N}^{+}\mathbf{A}_{m,n}\nvdash A$ . Then, $A\equiv A_{k}$ for some $k\in\omega$ and $i\nVdash_{k}A$ for some $i\in W_{k}$ . Hence, we obtain $\mathsf{PA}\vdash S(\overline{i})\to\neg f_{0}(A)$ by Claim 4.5. Thus, it follows from Proposition 4.1.3 that $T\nvdash f_{0}(A)$ . ∎

4.2 The case $m>n\geq 1$

In this subsection, we verify Theorem 4.3, which is Theorem 3.2 and Theorem 3.3 with respect to the case $m>n\geq 1$ . In a proof of Theorem 4.3, we need not suppose that $T$ is $\Sigma_{1}$ -sound. However, the proof of Theorem 4.3 is not applicable to the case $m>n=0$ when $T$ is $\Sigma_{1}$ -ill, so we prove this case in Section 5.

Theorem 4.3.

Suppose that $m>n\geq 1$ . There exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ if and only if $T\vdash f(A)$ .

Proof.

Let $m>n\geq 1$ . Let $\langle A_{k}\rangle_{k\in\omega}$ be a primitive recursive enumeration of all $\mathbf{N}^{+}\mathbf{A}_{m,n}$ -unprovable formulas. For each $k\in\omega$ , we can primitive recursively construct a finite $(m,n)$ -accessible $\mathbf{N}$ -model $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ falsifying $A_{k}$ . As in Subsection 4.1, let $\mathcal{M}=\bigl{(}W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash\bigr{)}$ be the $(m,n)$ -accessible $\mathbf{N}$ -model defined as a disjoint union of these finite $(m,n)$ -accessible $\mathbf{N}$ -models.

By the recursion theorem, we define a $\mathsf{PA}$ -provably recursive function $g_{1}$ corresponding to the case $m>n\geq 1$ . In the definition of $g_{1}$ , we use the $\Sigma_{1}$ formula $\mathrm{Pr}_{g_{1}}(x)\equiv$ $\exists y(g_{1}(y)=x\wedge\mathrm{Fml}_{\mathcal{L}_{A}}(x))$ and the arithmetical interpretation $f_{1}$ based on $\mathrm{Pr}_{g_{1}}(x)$ such that $f_{1}(p)\equiv\exists x(S(x)\wedge x\neq 0\wedge x\Vdash p)$ . In Procedure 1, $g_{1}$ outputs all $T$ -provable formulas. In Procedure 2, to ensure the condition $\mathsf{PA}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ , $g_{1}$ outputs $\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)$ if $g_{1}$ already outputs $\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ .

Procedure 1.

Stage $s$ :

•

If $h(s+1)=0$ ,

$g_{1}(s)=\begin{cases}\varphi&\text{if}\ s\ \text{is a}\ T\text{-proof of}\ \varphi,\\ 0&\text{otherwise}.\end{cases}$

Then, go to Stage $s+1$ .
•

If $h(s+1)\neq 0$ , go to Procedure 2.

Procedure 2.

Suppose $s$ and $i\neq 0$ satisfy $h(s)=0$ and $h(s+1)=i$ . Let $k$ be such that $i\in W_{k}$ . Define

g_{1}(s+t)=\begin{cases}\xi_{t}&\text{if}\ \xi_{t}\equiv f_{1}(B)\ \&\ i\Vdash_{k}\Box B\ \text{for some}\ \Box B\in\mathsf{Sub}(A_{k})\\ &\quad\text{or}\ \xi_{t}\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)\ \&\ g_{1}(l)=\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)\\ &\quad\quad\quad\text{for some}\ \varphi\ \text{and}\ l<s+t,\\ 0&\text{otherwise}.\end{cases}

We have finished the definition of $g_{1}$ .

Claim 4.6.

$\mathsf{PA}+\mathrm{Con}_{T}\vdash\forall x\forall y\Bigl{(}\mathrm{Fml}_{\mathcal{L}_{A}}(x)\to\bigl{(}\mathrm{Proof}_{T}(x,y)\leftrightarrow g_{1}(y)=x\bigr{)}\Bigr{)}$ .

Proof.

We argue in $\mathsf{PA}+\mathrm{Con}_{T}$ : By Proposition 4.1.2, we have $h(s)=0$ for any number $s$ , and the definition of $g_{1}$ continues to be in Procedure 1. Thus, it follows that for any formula $\varphi$ and any number $p$ , $g_{1}(p)=\varphi$ if and only if $p$ is a $T$ -proof of $\varphi$ . ∎

By Claim 4.6, for any formula $\varphi$ , we obtain $\mathbb{N}\models\mathrm{Pr}_{g_{1}}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Prov}_{T}(\ulcorner\varphi\urcorner)$ , and we see that $\mathrm{Pr}_{g_{1}}(x)$ is a $\Sigma_{1}$ provability predicate of $T$ .

Claim 4.7.

For any $\mathcal{L}_{A}$ -formula $\varphi$ , $\mathsf{PA}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ .

Proof.

First, we show $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ . For any formula $\psi$ , $\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)$ is a $\Sigma_{1}$ sentence. Then, $\mathsf{PA}\vdash\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)\to\mathrm{Prov}_{T}(\ulcorner\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)\urcorner)$ . By Claim 4.6, $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Prov}_{T}(\ulcorner\psi\urcorner)\leftrightarrow\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)$ . Hence, we obtain $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)\to\mathrm{Pr}_{g_{1}}(\ulcorner\mathrm{Pr}_{g_{1}}(\ulcorner\psi\urcorner)\urcorner)$ . In particular, it follows from $m>n\geq 1$ that $\mathsf{PA}+\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ .

Next, we show that $\mathsf{PA}+\neg\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ . We argue in $\mathsf{PA}+\neg\mathrm{Con}_{T}+\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)$ : By Proposition 4.1, $h(s)=0$ and $h(s+1)=i$ hold for some $i\neq 0$ and number $s$ , and the definition of $g_{1}$ swithches to Procedure 2 at Stage $s$ . Let $\xi_{v}\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)$ and we prove that $\xi_{v}$ is output by $g_{1}$ in Procedure 2. Since $\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)$ holds, $\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ is output by $g_{1}$ . We consider the following two cases.

•

$\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ is output in Procedure 1:

Then, $g_{1}(l)=\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ holds for some $l<s$ . Since $l<s+v$ , $g_{1}(s+v)$ = $\xi_{v}$ holds by the definition of $g_{2}$ in Procedure 2.
•

$\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ is output in Procedure 2:

Then, $g_{1}(s+u)=\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ , where $\xi_{u}\equiv\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ . Since $m-1>n-1$ , the Gödel number of $\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)$ is larger than that of $\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ . Hence, $v>u$ holds by the definition of $\{\xi_{t}\}_{t\in\omega}$ . Then, we obtain $g_{1}(s+v)=\xi_{v}$ .

Thus, it follows that $\mathsf{PA}+\neg\mathrm{Con}_{T}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ . By the law of excluded middle, it concludes that $\mathsf{PA}\vdash\mathrm{Pr}_{g_{1}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{1}}^{m}(\ulcorner\varphi\urcorner)$ .

∎

Claim 4.8.

Let $i\in W_{k}$ and $B\in\mathsf{Sub}(A_{k})$ .

1.

If $i\Vdash_{k}B$ , then $\mathsf{PA}\vdash S(\overline{i})\to f_{1}(B)$ .
2.

If $i\nVdash_{k}B$ , then $\mathsf{PA}\vdash S(\overline{i})\to\neg f_{1}(B)$ .

Proof.

We prove Clauses 1 and 2 simultaneously by induction on the construction of $B\in\mathsf{Sub}(A_{k})$ . We only give a proof of the case $B\equiv\Box C$ . We can prove Clause 1 similarly as in the proof of Claim 4.5. We prove Clause 2.

Suppose $i\nVdash_{k}\Box C$ . Then, there exists a $j\in W_{k}$ such that $i\prec_{k,C}j$ and $j\nVdash_{k}C$ . By the induction hypothesis, we obtain $\mathsf{PA}\vdash S(\overline{j})\to\neg f_{1}(C)$ . Let $p$ be a proof of $S(\overline{j})\to\neg f_{1}(C)$ in $T$ . Let $m^{\prime}\geq 0$ and a formula $D\in\mathsf{Sub}(C)$ be such that $C\equiv\Box^{m^{\prime}}D$ , where $D$ is not of the form $\Box G$ for any $G$ . Here, if $C$ is a boxed formula, $m^{\prime}$ is the maximum number of the outermost consecutive boxes of $C$ , and otherwise $m^{\prime}=0$ . We show that $\mathsf{PA}+S(\overline{i})$ proves that $f_{1}(C)$ is never output by $g_{1}$ . We distinguish the following two cases.

Case 1:

$m^{\prime}\geq m-1$ .

It follows from $m^{\prime}\geq m-1$ that we can find $E\in\mathsf{Sub}(C)$ such that $C\equiv\Box^{m-1}E$ . Since $i\nVdash_{k}\Box C$ , we obtain $i\nVdash_{k}\Box^{m}E$ . Thus, $i\nVdash_{k}\Box^{n}E$ holds by $i\Vdash_{k}\Box^{n}E\to\Box^{m}E$ . Since $n\leq m-1$ , by the induction hypothesis

\mathsf{PA}\vdash S(\overline{i})\to\neg f_{1}(\Box^{n}E).

(3)

We argue in $\mathsf{PA}+S(\overline{i})$ : Let $s$ be such that $h(s)=0$ and $h(s+1)=i$ . Suppose, towards a contradiction, that $f_{1}(C)$ is output by $g_{1}$ . We consider the following three cases.

(i)

$f_{1}(C)$ is output in Procedure 1:

For some $l<s$ , $l$ is a proof of $f_{1}(C)$ in $T$ and $f_{1}(C)\in P_{T,s-1}$ holds. By Proposition 4.1.4, we obtain $p<s$ . Thus, $S(\overline{j})\to\neg f_{1}(C)\in P_{T,s-1}$ holds, which implies that $\neg S(\overline{j})$ is a t.c. of $P_{T,s-1}$ . Hence, it follws that $h(s)\neq 0$ . This is a contradiction.
(ii)

$f_{1}(C)\equiv f_{1}(C^{\prime})\ \text{and}\ i\Vdash_{k}\Box C^{\prime}\ \text{for some}\ \Box C^{\prime}\in\mathsf{Sub}(A_{k})$ holds:

Since $f_{1}$ is an injection, we obtain $C\equiv C^{\prime}$ . Then, $i\Vdash_{k}\Box C$ . This contradicts $i\nVdash_{k}\Box C$ .
(iii)

$f_{1}(C)\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)$ and $g_{1}(l)=\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ for some $\varphi$ and $l$ : Since $C\equiv\Box^{m-1}E$ , we obtain $f_{1}(C)\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner f(E)\urcorner)$ , so $\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner f_{1}(E)\urcorner)$ holds. Then, $f_{1}(E)\equiv\varphi$ and $f_{1}(\Box^{n-1}E)\equiv\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner f_{1}(E)\urcorner)\equiv\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ . Since $g_{1}(l)=f_{1}(\Box^{n-1}E)$ , it follows that $\mathrm{Pr}_{g_{1}}(\ulcorner f_{1}(\Box^{n-1}E)\urcorner)$ holds. Hence, we obtain $f_{1}(\Box^{n}E)$ , which contradicts (3).

Case2:

$m^{\prime}<m-1$

We argue in $\mathsf{PA}+S(\overline{i})$ : Let $s$ be such that $h(s)=0$ and $h(s+1)=i$ . Suppose, towards a contradiction, that $f_{1}(C)$ is output by $g_{1}$ . If $f_{1}(C)$ is output at the cases corresponding to (i) and (ii) in the proof of Case 1, then we can obtain a contradiction similarly. If $f_{1}(C)\equiv\mathrm{Pr}_{g_{1}}^{m-1}(\ulcorner\varphi\urcorner)$ and $g_{1}(l)=\mathrm{Pr}_{g_{1}}^{n-1}(\ulcorner\varphi\urcorner)$ for some $\varphi$ and $l$ , then $C$ must be of the form $\Box^{m-1}F$ for some $F$ . Since $m^{\prime}<m-1$ , this is a contradiction.

∎ We finish our proof of Theorem 4.3. By Claim 4.7, we obtain the first clause. By Proposition 4.1.3 and Claim 4.8, the second clause holds.

∎

5 Proof of Theorem 3.3

In Theorem 4.3, we have proved Theorem 3.3 with respect to the case $m>n\geq 1$ . In this section, we prove Theorem 3.3 for the remaining cases $n>m\geq 1$ and $m>n=0$ . Before proving Theorem 3.3, we introduce a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{\sigma}(x)$ and a $\mathsf{PA}$ -provably recursive function $h^{\prime}$ in stead of the function $h$ defined in Section 4. Throughout this section, we assume that $T$ is a $\Sigma_{1}$ -ill theory.

Since $T$ is $\Sigma_{1}$ -ill, it is well-known that we obtain a $\Delta_{1}(\mathsf{PA})$ formula $\sigma(x)$ representing $T$ in $\mathsf{PA}$ such that $\mathbb{N}\models\forall x(\tau(x)\leftrightarrow\sigma(x))$ and $T\vdash\neg\mathrm{Con}_{\sigma}$ (cf. [11, Theorem 8 (b)]). Remark that we get $\mathbb{N}\models\forall x\bigl{(}\mathrm{Pr}_{T}(x)\leftrightarrow\mathrm{Pr}_{\sigma}(x)\bigr{)}$ . In particular, since $T$ is consistent, $\mathbb{N}\models\mathrm{Con}_{\sigma}$ holds. The set of sentences defined by $\sigma(x)$ in $\mathbb{N}$ is exactly $T$ . On the other hand, the behaviors of $\sigma(x)$ and $\tau(x)$ may be different in $\mathsf{PA}$ , so let $T_{\sigma}$ denote the set of sentences defined by $\sigma(x)$ in $\mathsf{PA}$ . For each $n\in\omega$ , let $P_{\sigma,n}:=\{\varphi\mid\mathbb{N}\models\exists y\leq\overline{n}\ \mathrm{Prf}_{\sigma}(\ulcorner\varphi\urcorner,y)\}$ . In $\mathsf{PA}$ , we can see that if $\varphi$ is a t.c. of $P_{\sigma,n}$ , then $\varphi$ is provable from $T_{\sigma}$ .

In our proofs presented in this section, we use a $\mathsf{PA}$ -provably recursive function $h^{\prime}$ instead of the function $h$ introduced in a previous section. Let $\bigl{\{}\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}\bigr{\}}_{k\in\omega}$ be a primitive recursive enumeration of $\mathbf{N}$ -models such that the sets $\{W_{k}\}_{k\in\omega}$ are pairwise disjoint subsets of $\omega$ and $\bigcup_{k\in\omega}W_{k}=\omega\setminus\{0\}$ . Let $\{A_{k}\}_{k\in\omega}$ be a primitive recursive enumeration of $\mathcal{L}_{\Box}$ -formulas. We construct the function $h^{\prime}$ by using the enumerations $\bigl{\{}\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}\bigr{\}}_{k\in\omega}$ and $\{A_{k}\}_{k\in\omega}$ . The function $h^{\prime}$ is defined by using the recursion theorem as follows:

•

$h^{\prime}(0)=0$ .
•

$h^{\prime}(s+1)=\begin{cases}i&\text{if}\ h^{\prime}(s)=0\\ &\quad\&\ i=\min\{j\in\omega\setminus\{0\}\mid\neg S^{\prime}(\overline{j})\ \text{is a t.c.~{}of}\ P_{\sigma,s}\\ &\quad\quad\text{or}\ \exists k\exists B[j\in W_{k}\ \&\ B\in\mathsf{Sub}(A_{k})\ \text{s.t.}\\ &\quad\quad\quad\forall x\varphi_{B}(x)\ \text{and}\ \varphi_{B}(\overline{j})\to\neg S^{\prime}(\overline{j})\ \text{are a t.c.'s~{}of}\ P_{\sigma,s}]\}\\ h^{\prime}(s)&\text{otherwise}.\end{cases}$

Here, $S^{\prime}(x)$ and $\varphi_{B}(x)$ are formulas

\exists y\bigl{(}h^{\prime}(y)=x\bigr{)}\ \text{and}\ \bigl{(}x\neq 0\wedge\exists y(x\in W_{y}\wedge B\in\mathsf{Sub}(A_{y})\wedge x\nVdash_{y}B)\bigr{)}\to\neg S^{\prime}(x),

respectively. We have finished the definition of $h^{\prime}$ .

$\mathsf{PA}$ -provably recursiveness of the function $h^{\prime}$ is guaranteed by the following claim.

Claim 5.1.

For any $j\neq 0$ , if $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=j$ , then $j\leq s+1$ .

Proof.

Suppose $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=j\in W_{k}$ for some $k$ . If $P_{\sigma,s}$ is not propositionally satisfiable, then $\neg S^{\prime}(\overline{1})$ is a t.c. of $P_{\sigma,s}$ . Hence, $j=1\leq s+1$ holds.

Then, we assume that $P_{\sigma,s}$ is propositionally satisfiable. We distinguish the following two cases.

•

$\neg S^{\prime}(\overline{j})$ is a t.c. of $P_{\sigma,s}$ :

Since $S^{\prime}(\overline{j})$ is propositionally atomic, $S^{\prime}(\overline{j})$ is a subformula of a formula contained in $P_{\sigma,s}$ . Thus, we obtain $j\leq s$ .
•

$\varphi_{B}(\overline{j})\to\neg S^{\prime}(\overline{j})$ is a t.c. of $P_{\sigma,s}$ for some $B\in\mathsf{Sub}(A_{k})$ :
Then, we further consider the following two cases.
- –
  
  $\neg S^{\prime}(\overline{j})$ is a t.c. of $P_{\sigma,s}$ :
  
  Since $S^{\prime}(\overline{j})$ is a subformula of a formula contained in $P_{\sigma,s}$ , we obtain $j\leq s$ .
- –
  
  $\neg S^{\prime}(\overline{j})$ is not a t.c. of $P_{\sigma,s}$ :
  
  Since $j=0\to\varphi_{B}(\overline{j})$ is a tautology, $j=0\to\neg S^{\prime}(\overline{j})$ is a t.c. of $P_{\sigma,s}$ . If $j=0$ were not a subformula of any formula contained in $P_{\sigma,s}$ , then $\neg S^{\prime}(\overline{j})$ would be a t.c. of $P_{\sigma,s}$ because $j=0$ is distinct from $S^{\prime}(\overline{j})$ . Thus, $j=0$ is a subformula of a formula which is in $P_{\sigma,s}$ , and we obtain $j\leq s$ .
In either case, $j\leq s+1$ holds.

∎

We obtain the following proposition corresponding to Proposition 4.1.

Proposition 5.1.

1.

$\mathsf{PA}\vdash\forall x\forall y\bigl{(}0<x<y\wedge S^{\prime}(x)\to\neg S^{\prime}(y)\bigr{)}$ .
2.

$\mathsf{PA}\vdash\neg\mathrm{Con}_{\sigma}\leftrightarrow\exists x\bigl{(}S^{\prime}(x)\wedge x\neq 0\bigr{)}$ .
3.

For each $i\in\omega\setminus\{0\}$ , $T\nvdash\neg S^{\prime}(\overline{i})$ .
4.

For each $l\in\omega$ , $\mathsf{PA}\vdash\forall x\forall y\bigl{(}h^{\prime}(x)=0\wedge h^{\prime}(x+1)=y\wedge y\neq 0\to x\geq l\bigr{)}$ .

Proof.

1.

Immediate from the definition $h^{\prime}$ .
2.
We work in $\mathsf{PA}$ : $(\leftarrow)$ : Let $i\neq 0$ be such that $S^{\prime}(\overline{i})$ . Let $k$ and $s$ be such that $i\in W_{k}$ , $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . We distinguish the following two cases.
- •
  
  $\neg S^{\prime}(\overline{i})$ is a t.c. of $P_{\sigma,s}$ :
  
  Then, $\neg S^{\prime}(\overline{i})$ is provable in $T_{\sigma}$ . Since $S^{\prime}(\overline{i})$ is a true $\Sigma_{1}$ sentence, $S^{\prime}(\overline{i})$ is provable in $T_{\sigma}$ by the $\Sigma_{1}$ -completeness. It follows that $T_{\sigma}$ is inconsistent.
- •
  
  $\forall x\varphi_{B}(x)$ and $\varphi_{B}(\overline{i})\to\neg S^{\prime}(\overline{i})$ are t.c.’s of $P_{\sigma,s}$ for some $B\in\mathsf{Sub}(A_{k})$ : Then, $\forall x\varphi_{B}(x)$ and $\varphi_{B}(\overline{i})\to\neg S^{\prime}(\overline{i})$ are provable in $T_{\sigma}$ . Hence, $\varphi_{B}(\overline{i})$ is provable in $T_{\sigma}$ , and so is $\neg S^{\prime}(\overline{i})$ . Since $S^{\prime}(\overline{i})$ is a true $\Sigma_{1}$ sentence, $S^{\prime}(\overline{i})$ is provable in $T_{\sigma}$ . It concludes that $T_{\sigma}$ is inconsistent.
In all cases, we obtain that $T_{\sigma}$ is inconsistent.
$(\to)$ : If $T_{\sigma}$ is inconsistent, then $\neg S^{\prime}(\overline{i})$ is $T_{\sigma}$ -provable for some $i\neq 0$ . Let $s$ be a proof of $\neg S^{\prime}(\overline{i})$ . Then, $\neg S^{\prime}(\overline{i})$ is a t.c. of $P_{\sigma,s}$ , and we obtain $h^{\prime}(s+1)\neq 0$ .
3.

Suppose that there exists an $i\in\omega\setminus\{0\}$ such that $T\vdash\neg S^{\prime}(\overline{i})$ . Let $p$ be a proof of $\neg S^{\prime}(\overline{i})$ in $T$ . Then, $\neg S^{\prime}(\overline{i})$ is a t.c. of $P_{\sigma,p}$ and we obtain $\mathbb{N}\models\exists x\bigl{(}S^{\prime}(x)\wedge x\neq 0\bigr{)}$ . By clause 2, we obtain $\mathbb{N}\models\neg\mathrm{Con}_{\sigma}$ . This contradicts the consistency of $T$ .
4.

Since $\mathbb{N}\models\mathrm{Con}_{\sigma}$ , for any $l\in\omega$ , we obtain $\mathbb{N}\models h^{\prime}(\overline{l})=0$ . Thus, $\mathsf{PA}\vdash h^{\prime}(\overline{l})=0$ , and clause 4 is easily obtained.

∎

We are ready to prove Theorem 3.3.

5.1 The case $n>m\geq 1$

In this subsection, we verify the following theorem, which is Theorem 3.3 with respect to the case $n>m\geq 1$ .

Theorem 5.2.

Suppose that $T$ is $\Sigma_{1}$ -ill and $n>m\geq 1$ . There exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash A$ if and only if $T\vdash f(A)$ .

Proof.

Let $n>m\geq 1$ . Let $\langle A_{k}\rangle_{k\in\omega}$ denote a primitive recursive enumeration of all $\mathbf{N}^{+}\mathbf{A}_{m,n}$ -unprovable formulas. As in the proof of Theorem 3.2, for each $k\in\omega$ , we obtain a primitive recursively constructed finite $(m,n)$ -accessible $\mathbf{N}$ -model $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ falsifying $A_{k}$ and let $\mathcal{M}=\bigl{(}W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash\bigr{)}$ be the $(m,n)$ -accessible $\mathbf{N}$ -model defined as a disjoint union of these models. We prepare the $\mathsf{PA}$ -provably recursive function $h^{\prime}$ constructed from the enumerations $\langle A_{k}\rangle_{k\in\omega}$ and $\bigl{\{}\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}\bigr{\}}_{k\in\omega}$ .

By the recursion theorem, we define a $\mathsf{PA}$ -provably recursive function $g_{2}$ corresponding to Theorem 3.3 with respect to the case $n>m\geq 1$ . In the definition of $g_{2}$ , we use the $\Sigma_{1}$ formula $\mathrm{Pr}_{g_{2}}(x)\equiv$ $\exists y(g_{2}(y)=x\wedge\mathrm{Fml}_{\mathcal{L}_{A}}(x))$ and the arithmetical interpretation $f_{2}$ based on $\mathrm{Pr}_{g_{2}}(x)$ such that $f_{2}(p)\equiv\exists x(S^{\prime}(x)\wedge x\neq 0\wedge x\Vdash p)$ . In Procedure 1, $g_{2}$ outputs all formulas which are provable from $T_{\sigma}$ . Also, if $g_{2}$ does not output $\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner f_{2}(B)\urcorner)$ in Procedure 2, then $\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(B)\urcorner)$ is not output by $g_{2}$ in Procedure 2.

Procedure 1.

Stage $s$ :

•

If $h^{\prime}(s+1)=0$ ,

$g_{2}(s)=\begin{cases}\varphi&\text{if}\ s\ \text{is a}\ T_{\sigma}\text{-proof of}\ \varphi,\\ 0&\text{otherwise}.\end{cases}$

Then, go to Stage $s+1$ .
•

If $h^{\prime}(s+1)\neq 0$ , go to Procedure 2.

Procedure 2.

Suppose $s$ and $i\neq 0$ satisfy $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Let $k$ be a number such that $i\in W_{k}$ . Define

g_{2}(s+t)=\begin{cases}0&\text{if}\ \xi_{t}\equiv f_{2}(B)\ \&\ i\nVdash_{k}\Box B\ \text{for some}\ \Box B\in\mathsf{Sub}(A_{k})\\ &\quad\text{or}\ \xi_{t}\equiv\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(B)\urcorner)\ \&\ \ g_{2}(s+u)=0,\\ &\quad\quad\text{where}\ \xi_{u}\equiv\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner f_{2}(B)\urcorner)\ \text{for some}\ B\in\mathsf{MF}\ \text{and}\ u<t,\\ \xi_{t}&\text{otherwise}.\end{cases}

The construction of $g_{2}$ has been finished.

The following claim guarantees that $\mathrm{Pr}_{g_{2}}(x)$ is a $\Sigma_{1}$ provability predicate of $T$ .

Claim 5.2.

For any formula $\varphi$ , $\mathsf{PA}+\mathrm{Con}_{\sigma}\vdash\mathrm{Pr}_{g_{2}}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Pr}_{\sigma}(\ulcorner\varphi\urcorner)$ .

Proof.

We work in $\mathsf{PA}+\mathrm{Con}_{\sigma}$ : By Proposition 5.1.2, the definition of $g_{2}$ never switches to Procedure 2. Hence, for any $s$ , $g_{2}(s)=\varphi$ if and only if $s$ is a proof of $\varphi$ in $T_{\sigma}$ . ∎

Claim 5.3.

For any $C\in\mathsf{MF}$ ,

\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge x\Vdash_{y}\Box C)\bigr{)}\to f_{2}(\Box C).

Proof.

Let $m^{\prime}\geq 1$ and $D$ be such that $\Box C\equiv\Box^{m^{\prime}}D$ , where $D$ is not a boxed formula. Here, $m^{\prime}\geq 1$ denotes the maximum number of the outermost consecutive boxes of $\Box C$ . We distinguish the following two cases.

Case 1:

$m^{\prime}<n$ .

We work in $\mathsf{PA}$ : Let $i\neq 0$ and $k$ be such that $S^{\prime}(\overline{i})$ , $i\in W_{k}$ , and $i\Vdash_{k}\Box C$ . Let $s$ be such that $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Then, the definition of $g_{2}$ goes to Procedure 2 at Stage $s$ . We show that $f_{2}(C)$ is output by $g_{2}$ in Procedure 2. Suppose, towards a contradiction, that $f_{2}(C)$ is not output by $g_{2}$ in Procedure 2. Let $\xi_{t}\equiv f_{2}(C)$ . Then, $g_{2}(s+t)=0$ . We consider the following two cases.

•

$\xi_{t}\equiv f_{2}(E)$ and $i\nVdash_{k}\Box E$ for some $\Box E\in\mathsf{Sub}(A_{k})$ : Then, we obtain $C\equiv E$ . Hence, $i\nVdash\Box C$ holds, which is a contradiction.
•

$\xi_{t}\equiv\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(E)\urcorner)$ and $g_{2}(s+u)=0$ , where $\xi_{u}\equiv\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner f_{2}(E)\urcorner)$ for some $E\in\mathsf{MF}$ and $u<t$ : Since $f_{2}(C)\equiv\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(E)\urcorner)$ , we obtain $C\equiv\Box^{n-1}E$ . Since $\Box C\equiv\Box^{n}E$ , this contradicts the assumption that $m^{\prime}<n$ .

Therefore, $f_{2}(C)$ is output by $g_{2}$ in Procedure 2.

Case 2:

$m^{\prime}\geq n$ .

Since $m^{\prime}-m\geq n-m$ , we can uniquely find numbers $r<n-m$ and $q\geq 1$ such that $m^{\prime}-m=q(n-m)+r$ . Hence, $\Box^{m^{\prime}}D\equiv\Box^{m}\Box^{q(n-m)}\Box^{r}D$ . Let $\Box^{r}D\equiv E$ . Then, it follows that $\Box^{m^{\prime}}D\equiv\Box^{q(n-m)}\Box^{m}E$ .

Subclaim 1.

For each $0\leq j\leq q$ , we have $i\Vdash_{k}\Box^{(q-j)(n-m)}\Box^{m}E$ .

Proof.

We prove the statement by induction on $j\leq q$ . If $j=0$ , then we obtain $i\Vdash_{k}\Box^{q(n-m)}\Box^{m}E$ by $i\Vdash_{k}\Box C\equiv\Box^{m^{\prime}}D$ . We prove on $j+1\leq q$ . By the induction hypothesis, we obtain $i\Vdash_{k}\Box^{(q-j)(n-m)}\Box^{m}E$ , and we get $i\Vdash_{k}\Box^{n}\Box^{(q-j-1)(n-m)}E$ . Then, it follows from $\mathrm{Acc}_{m,n}$ that $i\Vdash_{k}\Box^{m}\Box^{(q-(j+1))(n-m)}E$ .

∎

Subclaim 2.

For each $0\leq j\leq q$ , $f_{2}(\Box^{j(n-m)}\Box^{m-1}E)$ is output by $g_{2}$ in Procedure 2.

Proof.

For each $0\leq j\leq q$ , let $t_{j}$ be such that $\xi_{t_{j}}\equiv f_{2}(\Box^{j(n-m)}\Box^{m-1}E)$ . We prove $g_{2}(s+t_{j})=\xi_{t_{j}}$ by induction on $j\leq q$ . We prove the base case $j=0$ . Suppose, towards a contradiction, that $g_{2}(s+t_{0})=0$ . Since $r<n-m$ , we have $m-1+r<n-1$ . Then, $\xi_{t_{0}}\equiv\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner f_{2}(E)\urcorner)$ is not of the form $\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(G)\urcorner)$ for any $G$ . Hence, by $g_{2}(s+t_{0})=0$ , it follows that $\xi_{t_{0}}\equiv f_{2}(F)$ and $i\nVdash_{k}\Box F$ for some $\Box F\in\mathsf{Sub}(A_{k})$ . Since $f_{2}(\Box^{m-1}E)\equiv f_{2}(F)$ , it follows that $\Box^{m-1}E\equiv F$ . Hence, we obtain $i\nVdash_{k}\Box^{m}E$ . This contradicts Subclaim 1. Thus, we get $g_{2}(s+t_{0})=\xi_{t_{0}}$ .

Next, we prove the induction case $j+1\leq q$ . By the induction hypothesis, we obtain $g_{2}(s+t_{j})=\xi_{t_{j}}$ . Suppose, towards a contradiction, that $g_{2}(s+t_{j+1})=0$ . We consider the following two cases.

•

$\xi_{t_{j+1}}\equiv f_{2}(F)$ and $i\nVdash_{k}\Box F$ hold for some $\Box F\in\mathsf{Sub}(A_{k})$ : We get $\Box^{(j+1)(n-m)}\Box^{m-1}E\equiv F$ . Thus, we obtain $i\nVdash_{k}\Box^{(j+1)(n-m)}\Box^{m}E$ , which contradicts Subclaim 1.
•

$\xi_{t_{j+1}}\equiv\mathrm{Pr}_{g_{2}}^{n-1}(\ulcorner f_{2}(F)\urcorner)$ and $g_{2}(s+u)=0$ , where $\xi_{u}\equiv\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner f_{2}(F)\urcorner)$ for some $F\in\mathsf{MF}$ and $u<t_{j+1}$ :

Then, we obtain $\Box^{n-1}F\equiv\Box^{(j+1)(n-m)}\Box^{m-1}E$ . It follows that $\Box^{m-1}F\equiv\Box^{j(n-m)}\Box^{m-1}E$ , so we obtain $\xi_{u}\equiv\xi_{t_{j}}$ and $u=t_{j}$ . Hence, we obtain $g_{2}(s+t_{j})=g_{2}(s+u)=0$ . This contradicts $g_{2}(s+t_{j})=\xi_{t_{j}}$ .

We have proved $g_{2}(s+t_{j+1})=\xi_{t_{j+1}}$ . ∎

By Subclaim 2, we obtain $f_{2}(C)$ is output by $g_{2}$ , that is, $f_{2}(\Box C)$ holds.

∎

Claim 5.4.

Let $B\in\mathsf{MF}$ .

1.

$\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge B\in\mathsf{Sub}(A_{y})\wedge x\Vdash_{y}B)\bigr{)}\to f_{2}(B)$ .
2.

$\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge B\in\mathsf{Sub}(A_{y})\wedge x\nVdash_{y}B)\bigr{)}\to\neg f_{2}(B)$ .

Proof.

We prove Clauses 1 and 2 simultaneously by induction on the construction of $B\in\mathsf{Sub}(A_{k})$ . First, we prove the base step of the induction. In the case $B\equiv\bot$ , Clauses 1 and 2 are easily proved. Suppose $B\equiv p$ . Clause 1 holds trivially. We only give a proof of Clause 2.

By Proposition 5.1.1, we obtain

	$\displaystyle\mathsf{PA}\vdash x\neq 0\wedge S^{\prime}(x)\wedge x\in W_{y}\wedge x\nVdash_{y}p$	$\displaystyle\to\bigl{(}S^{\prime}(z)\wedge z\neq 0\to z=x\bigr{)}$
		$\displaystyle\to\forall z\bigl{(}S^{\prime}(z)\wedge z\neq 0\to z\nVdash p\bigr{)}.$

Then, we get

\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge p\in\mathsf{Sub}(A_{y})\wedge x\nVdash_{y}p)\bigr{)}\to\neg f_{2}(p).

Next, we prove the induction step. The cases of $\neg$ , $\wedge$ , $\vee$ , and $\to$ are easily proved. We prove the case $B\equiv\Box C$ .

1.

We obtain Clause 1 by Claim 5.3.

If $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash\Box C$ , then $\mathsf{PA}\vdash\forall x\forall y(x\in W_{y}\to x\Vdash_{y}\Box C)$ holds by formalizing the soundness of $\mathbf{N}^{+}\mathbf{A}_{m,n}$ . Hence, Clause 2 trivially holds.

Therefore, we may assume that $\mathbf{N}^{+}\mathbf{A}_{m,n}\nvdash\Box C$ . Then, there exists a $j\in\omega$ such that $\Box C\equiv A_{j}$ . Let $\bigl{(}W_{j},\{\prec_{j,D}\}_{D\in\mathsf{MF}},\Vdash_{j}\bigr{)}$ be a countermodel of $A_{j}$ and $r\in W_{j}$ be such that $r\nVdash_{j}\Box C$ . Then, there exists an $l\in W_{j}$ such that $r\prec_{j,C}l$ and $l\nVdash_{j}C$ . Thus, we obtain

\mathsf{PA}\vdash\overline{l}\neq 0\wedge\exists y(\overline{l}\in W_{y}\wedge C\in\mathsf{Sub}(A_{y})\wedge\overline{l}\nVdash_{y}C),

which implies that $\mathsf{PA}\vdash\varphi_{C}(\overline{l})\to\neg S^{\prime}(\overline{l})$ . Also, since

\mathsf{PA}\vdash\exists x\neg\varphi_{C}(x)\leftrightarrow\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge C\in\mathsf{Sub}(A_{y})\wedge x\nVdash_{y}C)\bigr{)},

it follows from the induction hypothesis that $\mathsf{PA}\vdash\exists x\neg\varphi_{C}(x)\to\neg f_{2}(C)$ . Let $p$ and $q$ be $T$ -proofs of $\varphi_{C}(\overline{l})\to\neg S^{\prime}(\overline{l})$ and $\exists x\neg\varphi_{C}(x)\to\neg f_{2}(C)$ respectively.

We work in $\mathsf{PA}$ : Let $i\neq 0$ and $k$ be such that $S^{\prime}(\overline{i})$ , $i\in W_{k}$ , $\Box C\in\mathsf{Sub}(A_{k})$ and $i\nVdash_{k}\Box C$ . Let $s$ be such that $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Suppose, towards a contradiction, that $f_{2}(C)$ is output by $g_{2}$ . We consider the following two cases.

•

$f_{2}(C)$ is output in Procedure 1:

Then, we obtain $f_{2}(C)\in P_{\sigma,s-1}$ and it follows from Proposition 5.1 that $q\leq s-1$ . Hence, $\exists x\neg\varphi_{C}(x)\to\neg f_{2}(C)\in P_{\sigma,s-1}$ holds, which implies that $\forall x\varphi_{C}(x)$ is a t.c. of $P_{\sigma,s-1}$ . Since $p\leq s-1$ by Proposition 5.1, $\varphi_{C}(\overline{l})\to\neg S^{\prime}(\overline{l})$ is a t.c. of $P_{\sigma,s-1}$ . We also have $l\in W_{j}$ and $C\in\mathsf{Sub}(A_{j})$ . Hence, it follows that $h^{\prime}(s)\neq 0$ , which is a contradiction.
•

$f_{2}(C)$ is output in Procedure 2:

Let $\xi_{u}\equiv f_{2}(C)$ . Since $\Box C\in\mathsf{Sub}(A_{k})$ , $\xi_{u}\equiv f_{2}(C)$ and $i\nVdash_{k}\Box C$ , we obtain $g_{2}(s+u)$ =0, which means that $f_{2}(C)$ is never output in Procedure 2. This is a contradiction.

Therefore, $f_{2}(C)$ is not output by $g_{2}$ , that is, $\neg f_{2}(\Box C)$ holds.

∎

Claim 5.5.

For any formula $\varphi$ , $T\vdash\mathrm{Pr}_{g_{2}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{2}}^{m}(\ulcorner\varphi\urcorner)$ .

Proof.

We consider the following two cases.

Case 1:

$\varphi$ is not of the form $f_{2}(A)$ for any $A\in\mathsf{MF}$ .

We work in $T$ : By the definition of $\sigma(x)$ , $\neg\mathrm{Con}_{\sigma}$ holds. Then, there exist some $i\neq 0$ and $s$ such that $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ by Proposition 5.1.2. Let $\xi_{u}\equiv\mathrm{Pr}_{g_{2}}^{m-1}(\ulcorner\varphi\urcorner)$ . Then, $\xi_{u}$ is not of the form $f_{2}(B)$ for any $B\in\mathsf{MF}$ . Thus, we obtain $g_{2}(s+u)=\xi_{u}$ . It follows that $\mathrm{Pr}_{g_{2}}^{m}(\ulcorner\varphi\urcorner)$ holds.

Case 2:

$\varphi$ is of the form $f_{2}(A)$ for some $A\in\mathsf{MF}$ .

We further distinguish the following two cases.

•

$\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash\Box^{n}A$ :

Then, we obtain $\mathbf{N}^{+}\mathbf{A}_{m,n}\vdash\Box^{m}A$ by $\mathrm{Acc}_{m,n}$ . This implies

$\mathsf{PA}\vdash\forall x\forall y(x\in W_{y}\to x\Vdash_{y}\Box^{m}A).$ (4)

We work in $T$ : Since $\neg\mathrm{Con}_{\sigma}$ holds, we get $S^{\prime}(\overline{i})$ for some $i\neq 0$ . Let $i\in W_{k}$ . Then, $i\Vdash_{k}\Box^{m}A$ holds by (4). Hence, it follows from Claim 5.3 that $f_{2}(\Box^{m}A)$ holds, which means that $\mathrm{Pr}_{g_{2}}^{m}(\ulcorner f_{2}(A)\urcorner)$ holds.
•

$\mathbf{N}^{+}\mathbf{A}_{m,n}\nvdash\Box^{n}A$ :

Let $j$ be such that $\Box^{n}A\equiv A_{j}$ and $\bigl{(}W_{j},\{\prec_{j,B}\}_{B\in\mathsf{MF}},\Vdash_{j}\bigr{)}$ be a countermodel of $A_{j}$ . Thus we obtain $r\nVdash_{j}\Box^{n}A$ for some $r\in W_{j}$ , and $l\nVdash_{j}\Box^{n-1}A$ holds for some $l\in W_{j}$ . As in the proof of Claim 5.4, we obtain $\mathsf{PA}\vdash\varphi_{\Box^{n-1}A}(\overline{l})\to\neg S^{\prime}(\overline{l})$ . From Claim 5.4.2, it follows that $\mathsf{PA}\vdash\exists x\neg\varphi_{\Box^{n-1}A}(\overline{l})\to\neg f_{2}(\Box^{n-1}A)$ . Let $p$ and $q$ be $T$ -proofs of $\varphi_{\Box^{n-1}A}(\overline{l})\to\neg S^{\prime}(\overline{l})$ and $\exists x\neg\varphi_{\Box^{n-1}A}(\overline{l})\to\neg f_{2}(\Box^{n-1}A)$ respectively. We prove $T\vdash\neg\mathrm{Pr}_{g_{2}}^{m}(\ulcorner f_{2}(A)\urcorner)\to\neg\mathrm{Pr}_{g_{2}}^{n}(\ulcorner f_{2}(A)\urcorner)$ .

We work in $T$ : It follows that $\neg\mathrm{Con}_{\sigma}$ holds. Let $i\neq 0$ and $s$ be such that $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Set $\xi_{t}\equiv f_{2}(\Box^{n-1}A)$ and $\xi_{u}\equiv f_{2}(\Box^{m-1}A)$ for $u<t$ . Suppose, towards a contradiction, that $\neg\mathrm{Pr}_{g_{2}}(\ulcorner f_{2}(\Box^{m-1}A)\urcorner)$ and $\mathrm{Pr}_{g_{2}}(\ulcorner f_{2}(\Box^{n-1}A)\urcorner)$ hold. Then, $f_{2}(\Box^{n-1}A)$ is output by $g_{2}$ . Since $\xi_{u}$ is not output by $g_{2}$ , we obtain $g_{2}(s+u)$ =0. Thus, we get $g_{2}(s+t)=0$ . Therefore, $f_{2}(\Box^{n-1}A)$ is output by $g_{2}$ in Procedure 1, and $f_{2}(\Box^{n-1}A)\in P_{\sigma,s-1}$ holds. Also, since $q\leq s-1$ , we obtain $\exists x\neg\varphi_{\Box^{n-1}A}(x)\to\neg f_{2}(\Box^{n-1}A)\in P_{\sigma,s-1}$ . Thus, $\forall x\varphi_{\Box^{n-1}A}(x)$ is a t.c. of $P_{\sigma,s-1}$ . By $p\leq s-1$ , $\varphi_{\Box^{n-1}A}(\overline{l})\to\neg S^{\prime}(\overline{l})$ is a t.c. of $P_{\sigma,s-1}$ . Hence, it follows from $\Box^{n-1}A\in\mathsf{Sub}(A_{j})$ and $l\in W_{j}$ that $h^{\prime}(s)\neq 0$ . This is a contradiction. It concludes that $\neg\mathrm{Pr}_{g_{2}}^{m}(\ulcorner f_{2}(A)\urcorner)$ implies $\neg\mathrm{Pr}_{g_{2}}^{n}(\ulcorner f_{2}(A)\urcorner)$ .

In either case, we have proved $T\vdash\mathrm{Pr}_{g_{2}}^{n}(\ulcorner\varphi\urcorner)\to\mathrm{Pr}_{g_{2}}^{m}(\ulcorner\varphi\urcorner)$ . ∎ We finish our proof of Theorem 5.2. Clause 1 and the implication $(\Rightarrow)$ of Clause 2 trivially hold by Claim 5.2 and Claim 5.5.

We prove the implication $(\Leftarrow)$ of Clause 2. Suppose $\mathbf{N}^{+}\mathbf{A}_{m,n}\nvdash A$ . Then, there exists a $k\in\omega$ such that $A\equiv A_{k}$ . Let $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ be a countermodel of $A_{k}$ and $j\in W_{k}$ be $j\nVdash_{k}A_{k}$ . Hence, we obtain

\mathsf{PA}\vdash\overline{j}\neq 0\wedge\exists y\bigl{(}\overline{j}\in W_{y}\wedge A\in\mathsf{Sub}(A_{y})\wedge\overline{j}\nVdash_{y}A\bigr{)}

and it follows that $\mathsf{PA}\vdash\varphi_{A}(\overline{j})\to\neg S^{\prime}(\overline{j})$ . By using the contrapositive of Claim 5.4.2, we get $\mathsf{PA}\vdash f_{2}(A)\to\forall x\varphi_{A}(x)$ , which implies $\mathsf{PA}\vdash f_{2}(A)\to\varphi_{A}(\overline{j})$ . Then, we obtain $\mathsf{PA}\vdash f_{2}(A)\to\neg S^{\prime}(\overline{j})$ . From Proposition 5.1.3, $T\nvdash f_{2}(A)$ holds. ∎

5.2 The case $m\geq 1$ and $n=0$

In this subsection, we verify the following theorem, which is Theorem 3.3 with respect to the case $m>n=0$ .

Theorem 5.3.

Suppose that $T$ is $\Sigma_{1}$ -ill and $m\geq 1$ . There exists a $\Sigma_{1}$ provability predicate $\mathrm{Pr}_{T}(x)$ of $T$ such that

1.

for any $A\in\mathsf{MF}$ and arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ , if $\mathbf{N}^{+}\mathbf{A}_{m,0}\vdash A$ , then $T\vdash f(A)$ , and
2.

there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_{T}(x)$ such that $\mathbf{N}^{+}\mathbf{A}_{m,0}\vdash A$ if and only if $T\vdash f(A)$ .

In fact, we can prove Theorem 3.3 with respect to the more general case $m>n\geq 0$ in the same way as the proof of Theorem 5.3.

Proof.

Let $m\geq 1$ . We prepare a primitive recursive enumeration of all $\mathbf{N}^{+}\mathbf{A}_{m,0}$ -unprovable formulas $\langle A_{k}\rangle_{k\in\omega}$ . As in the proof of Theorem 3.2, for each $k\in\omega$ , we can primitive recursively construct a finite $(m,0)$ -accessible $\mathbf{N}$ -model $\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}$ which falsifies $A_{k}$ and let $\mathcal{M}=\bigl{(}W,\{\prec_{B}\}_{B\in\mathsf{MF}},\Vdash\bigr{)}$ be the $(m,0)$ -accessible $\mathbf{N}$ -model defined as a disjoint union of these models. We prepare the $\mathsf{PA}$ -provably recursive function $h^{\prime}$ constructed from the enumerations $\langle A_{k}\rangle_{k\in\omega}$ and $\bigl{\{}\bigl{(}W_{k},\{\prec_{k,B}\}_{B\in\mathsf{MF}},\Vdash_{k}\bigr{)}\bigr{\}}_{k\in\omega}$ .

By the recursion theorem, we define a $\mathsf{PA}$ -provably recursive function $g_{3}$ corresponding to the case $m\geq 1$ and $n=0$ . In the definition of $g_{3}$ , we use the $\Sigma_{1}$ -formula $\mathrm{Pr}_{g_{3}}(x)\equiv$ $\exists y(g_{3}(y)=x\wedge\mathrm{Fml}_{\mathcal{L}_{A}}(x))$ and the arithmetical interpretation $f_{3}$ based on $\mathrm{Pr}_{g_{3}}(x)$ such that $f_{3}(p)\equiv\exists x(S^{\prime}(x)\wedge x\neq 0\wedge x\Vdash p)$ . The function $g_{3}$ is defined as follows:

Procedure 1.

Stage $s$ :

•

If $h^{\prime}(s+1)=0$ ,

$g_{3}(s)=\begin{cases}\varphi&\text{if}\ s\ \text{is a}\ T_{\sigma}\text{-proof of}\ \varphi\\ 0&\text{otherwise}.\end{cases}$

Then, go to Stage $s+1$ .
•

If $h^{\prime}(s+1)\neq 0$ , go to Procedure 2.

Procedure 2.

Suppose $s$ and $i\neq 0$ satisfy $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Let $k$ be a number such that $i\in W_{k}$ . Define

g_{3}(s+t)=\begin{cases}0&\text{if}\ \xi_{t}\equiv f_{3}(B)\ \&\ i\nVdash_{k}\Box B\ \text{for some}\ \Box B\in\mathsf{Sub}(A_{k})\\ \xi_{t}&\text{otherwise}.\end{cases}

We finish the construction of $g_{3}$ .

Claim 5.6.

For any formula $\varphi$ , $\mathsf{PA}+\mathrm{Con}_{\sigma}\vdash\mathrm{Pr}_{g_{3}}(\ulcorner\varphi\urcorner)\leftrightarrow\mathrm{Pr}_{\sigma}(\ulcorner\varphi\urcorner)$ .

Proof.

This is proved as in the proof of Claim 5.2. ∎

Claim 5.6 guarantees that $\mathrm{Pr}_{\sigma}(x)$ is a $\Sigma_{1}$ provability predicate.

Claim 5.7.

Let $B\in\mathsf{MF}$ .

1.

$\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge B\in\mathsf{Sub}(A_{y})\wedge x\Vdash_{y}B)\bigr{)}\to f_{3}(B)$ .
2.

$\mathsf{PA}\vdash\exists x\bigl{(}x\neq 0\wedge S^{\prime}(x)\wedge\exists y(x\in W_{y}\wedge B\in\mathsf{Sub}(A_{y})\wedge x\nVdash_{y}B)\bigr{)}\to\neg f_{3}(B)$ .

Proof.

By induction on the construction of $B\in\mathsf{Sub}(A_{k})$ , we prove clauses 1 and 2 simultaneously. We only give a proof of the case $B\equiv\Box C$ for some $C\in\mathsf{MF}$ .

1.

We work in $\mathsf{PA}$ : Let $i\neq 0$ and $k$ be such that $S^{\prime}(\overline{i})$ , $i\in W_{k}$ , $\Box C\in\mathsf{Sub}(A_{k})$ and $i\Vdash_{k}\Box C$ . Let $s^{\prime}$ be such that $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Set $\xi_{t}\equiv f_{3}(C)$ . Since $f_{3}$ is injective, we obtain $g_{3}(s+t)=\xi_{t}$ , so $f_{3}(\Box C)$ holds.
2.

This is verified as in the proof of clause 2 in Claim 5.4.

∎

Claim 5.8.

For any formula $\varphi$ , $T\vdash\varphi\to\mathrm{Pr}_{g_{3}}^{m}(\ulcorner\varphi\urcorner)$ .

Proof.

We prove the contrapositive $T\vdash\neg\mathrm{Pr}_{g_{3}}^{m}(\ulcorner\varphi\urcorner)\to\neg\varphi$ . We work in $T$ : Suppose $\neg\mathrm{Pr}_{g_{3}}^{m}(\ulcorner\varphi\urcorner)$ holds. Since we obtain $\mathrm{Pr}_{\sigma}(\ulcorner 0=1\urcorner)$ , there exists an $i\neq 0$ such that $S^{\prime}(\overline{i})$ . Let $k$ and $s$ be such that $i\in W_{k}$ , $h^{\prime}(s)=0$ and $h^{\prime}(s+1)=i$ . Set $\xi_{t}\equiv\mathrm{Pr}_{g_{3}}^{m-1}(\ulcorner\varphi\urcorner)$ . Since $\xi_{t}$ is not output by $g_{3}$ in Procedure 2, we get $g_{3}(s+t)=0$ . Hence, it follows that $\xi_{t}\equiv f_{3}(B)$ and $i\nVdash_{k}\Box B$ for some $\Box B\in\mathsf{Sub}(A_{k})$ . Since $f_{3}(B)\equiv\mathrm{Pr}_{g_{3}}^{m-1}(\ulcorner\varphi\urcorner)$ , there exists a $C\in\mathsf{Sub}(A_{k})$ such that $B\equiv\Box^{m-1}C$ and $f_{3}(C)\equiv\varphi$ . Thus, we obtain $i\nVdash_{k}\Box^{m}C$ , which implies $i\nVdash_{k}C$ by $\mathrm{Acc}_{m,0}$ . It follows from clause 2 of Claim 5.7 that $\neg f_{3}(C)$ holds. Hence, it concludes $\neg\varphi$ holds.

∎

We complete our proof of Theorem 5.3. By Claim 5.8, we obtain the first clause. Also, the second clause is proved from Proposition 5.1.3 and Claim 5.7 as in the proof of Theorem 5.2. ∎

Acknowledgement

The author would like to thank Taishi Kurahashi for his valuable comments.

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Arithmetical completeness for some extensions of the pure logic of necessitation

Abstract

1 Introduction

Problem 1.1 ([6, Problem 2.2]).

2 Preliminaries and Background

2.1 Provability predicate

2.2 Provability logic

Theorem 2.1 (Solovay [13]).

2.3 The logic 𝐍+​𝐀m,n\mathbf{N}^{+}\mathbf{A}_{m,n}

Definition 2.2 (𝐍\mathbf{N}-frames and 𝐍\mathbf{N}-models).

Theorem 2.3 ([2, Theorem 3.6 and Theorem 4.10]).

Proposition 2.4 ([10, Proposition 4.2]).

Definition 2.5.

Fact 2.6 (The finite frame property of 𝐍+​𝐀m,n\mathbf{N}^{+}\mathbf{A}_{m,n} [10, Theorem 5.1]).

Fact 2.7 (The dicidability of 𝐍+​𝐀m,n\mathbf{N}^{+}\mathbf{A}_{m,n} [10, Corollary 6.10]).

3 Main results

Theorem 3.1.

Theorem 3.2.

Theorem 3.3.

Proposition 3.4.

Proof.

Proposition 3.5.

Proof.

Problem 3.6.

4 Proof of Theorem 3.2

Proposition 4.1 (Cf. [9, Lemma 3.2.]).

4.1 The case n>m≥1n>m\geq 1

Theorem 4.2.

Proof.

Claim 4.1.

Proof.

Claim 4.2.

Proof.

Case A:

Case A-1:

Case A-2:

Case A-2-1:

Case A-2-2:

Case A-3:

Case A-3-1:

Case A-3-2:

Case B:

Claim 4.3.

Proof.

Case 1:

Case 2:

Claim 4.4.

Proof.

Claim 4.5.

Proof.

4.2 The case m>n≥1m>n\geq 1

Theorem 4.3.

Proof.

Claim 4.6.

Proof.

Claim 4.7.

Proof.

Claim 4.8.

Proof.

Case 1:

Case2:

5 Proof of Theorem 3.3

Claim 5.1.

Proof.

Proposition 5.1.

Proof.

5.1 The case n>m≥1n>m\geq 1

Theorem 5.2.

Proof.

Claim 5.2.

Proof.

Claim 5.3.

Proof.

Case 1:

Case 2:

Subclaim 1.

Proof.

Subclaim 2.

Proof.

Claim 5.4.

2.3 The logic $\mathbf{N}^{+}\mathbf{A}_{m,n}$

Definition 2.2 ( $\mathbf{N}$ -frames and $\mathbf{N}$ -models).

Fact 2.6 (The finite frame property of $\mathbf{N}^{+}\mathbf{A}_{m,n}$ [10, Theorem 5.1]).

Fact 2.7 (The dicidability of $\mathbf{N}^{+}\mathbf{A}_{m,n}$ [10, Corollary 6.10]).

4.1 The case $n>m\geq 1$

4.2 The case $m>n\geq 1$

5.1 The case $n>m\geq 1$

5.2 The case $m\geq 1$ and $n=0$