2.1 Definitions

A finitist language is determined by its vocabulary. Besides sets of constant, function, and relation symbols, we also specify a set of designated binary (or n-ary, for $n\,{\geq }\,2$ ) relation symbols with distinct relata. For all the examples treated in the present article, the vocabulary contains, at least, two constants $0$ and $1$ , two binary function symbols $+$ and $\cdot $ , and two binary relation symbols $=$ and $<$ , where only $<$ is designated with the first relatum being distinct. The well formed formulae of a finitist language are defined in the same way as those of a first-order language, except that the clauses for (unbounded) quantifiers are replaced by those for quantifiers bounded by designated relation symbols. In our case, the clauses are:

• • if $A(x)$ is a well formed formula and t is a term in which x does not occur, then both $(\forall x\,{<}\,t)A(x)$ and $(\exists x\,{<}\,t)A(x)$ are well formed formulae.

The term “formula” often refers to “well formed formula”. As we consider only theories over classical logic, the negation symbol $\neg $ applies only to atomic formulae, and we define the negation $\sim $ of arbitrary formulae as the usual syntactical operation according to De Morgan’s law. Accordingly, $A\,{\to }\,B$ stands for ${\sim }A\,{\lor }\,B$ .

The logical axioms and rules are the same as those of first-order logic, except that the axioms and rules for bounded quantifiers are as follows, where x is an eigenvariable, namely, x must not occur in B.

Because of our treatment of $\to $ , these rules are equivalent to the following.

Furthermore, though in many cases it is admissible, we consider the following instantiating rule as a logical rule, namely, we assume that all finitist theories are closed under the following rule, where x is an eigenvariable.

It is particularly important to keep this rule in mind when we define theories not only with non-logical axioms but also with non-logical rules.

As we cannot form universal closure, we define partial conservation in terms of provable formulae, not only provable sentences.

All the finitist theories in this article are formulated in extensions of the finitist language of arithmetic. As already discussed in the Introduction, it is important which function symbols are included.

One significant feature of arithmetic with $+$ and $\cdot $ is that it allows us to encode any finite sequence of numbers by a single number. While various ways to achieve this are known, we take the one from [Reference Hájek and Pudlák9, Chapter V], because it seems to be one of the most efficient ways in the literature. It starts with encoding finite sets of numbers. The following lemma is proved in [Reference Hájek and Pudlák9, Chapter V.3.b] or [Reference Walker29, Section 3.2] and shows that $1$ encodes the empty set and that any finite set can be extended with one new element.

Lemma 2.5. There are $\mathfrak {L}(\mathcal {S}^2)$ formulae $\mathrm {Seq}$ and $\in $ such that

$$ \begin{align*} \mathbb{FA}(\mathcal{S}^2) \vdash \mathrm{Seq}(u)\,{\to}\, (\exists v\,{<}\,9{\cdot}u{\cdot}(x{+}1)^2{+}1) \begin{pmatrix} u\,{\leq}\,v\,{\land}\, \mathrm{Seq}(v)\,{\land}\\ (\forall y\,{<}\,v)(y\,{\in}\,v\, {\leftrightarrow}\,y\,{\in}\,u\,{\lor}\,y\,{=}\,x)) \end{pmatrix} \end{align*} $$

with $\mathbb {FA}(\mathcal {S}^2) \vdash (x\,{\in }\,u\,{\to }\,x\,{<}\,u)\,{\land }\, \mathrm {Seq}(1)\,{\land }\, \neg (z\,{\in }\,1)$ .

2.2 Extension by definition and bounded $\mu $ algebra

In the language $\mathfrak {L}(\mathcal {S}^n)$ of the finitist arithmetic $\mathbb {FA}(\mathcal {S}^n)$ , we only have function symbols $0,1,{+},{\cdot },{\#}$ , $\gamma _3,\ldots ,\gamma _n$ . However, similarly as in first-order arithmetic, we may also use additional function symbols in $\mathfrak {L}(\mathcal {S}^n)$ formulae and treat them as mere abbreviations. What we have to keep in mind is that we need a term (in the official language) which bounds the value of such an additional function symbol.

Definition 2.6 (Bounded definable functions)

Let $\mathbb {T}$ expand $\mathbb {FA}(\mathcal {S}^n)$ . A pair of an $\mathcal {S}^n$ term $t(x_0,\ldots ,x_{k-1})$ and an $\mathfrak {L}(\mathcal {S}^n)$ formula $A(x_0,\ldots ,x_{k-1},y)$ , both without free variables other than displayed, is said to define a total k-ary function in $\mathbb {T}$ if

$$\begin{align*}\mathbb{T}\vdash(\exists!y\,{<}\,t(x_0,\ldots,x_{k-1})) A(x_0,\ldots,x_{k-1},y),\end{align*}$$

where $(\exists !y\,{<}\,t)A(y)$ is short for $(\exists y\,{<}\,t)(A(y) \,{\land }\,(\forall z\,{<}\,t)(A(z)\,{\to }\,z\,{=}\,y))$ .

We loosely say that a k-ary function f (with the implicitly intended formula $\mathrm {Gr}_f (x_0,\ldots ,x_{k-1},y)$ for its graph) is bounded definable in $\mathbb {T}$ if there is a term t (called the bounding term) such that the pair of t and $\mathrm {Gr}_f$ defines a total k-ary function in $\mathbb {T}$ .

The first-order analogue of this notion is called bounded definability in [Reference Nemoto and Sato15, Notation 2.9 $(1)$ ].

If a function is bounded definable in $\mathbb {T}$ with a bounding term t, then the use of an additional function symbol f for that function can be justified as follows: a formula of form $B(f(s_0,\ldots ,s_{k-1}))$ can be rewritten as

$$\begin{align*}(\exists y\,{<}\,t(s_0,\ldots,s_{k-1}) (\mathrm{Gr}_f(s_0,\ldots,s_{k-1},y)\,{\land}\, B(y)),\end{align*}$$

and by iterating this process (backward along the construction of the subterms $s_i$ ’s) we can eventually obtain a formula without occurrences of f.

Thus, for the concern of justified use of function symbols, what is important is not the class of function symbols but the bounded $\mu $ algebra generated by it. In combination with Lemma 2.8, the next lemma justifies us to call $\mathbb {FA}(\mathcal {S}^n)$ the finitist arithmetic for $\mathcal {S}^n$ . In what follows, however, we do not add symbols for all the functions in $\mathcal {S}^n$ to $\mathfrak {L}(\mathcal {S}^n)$ as full-fledged symbols, but just as abbreviations (but see also Remark 3.6). For a term s in the expanded language we can define a term t in the original such that $s\,{\leq }\,t$ , by replacing bounded definable functions with their bounding terms. We call this t the bounding term of s.

Proof First, by meta-induction on an $\mathfrak {L}(\mathcal {S}^n)$ formula A, we can show that the characteristic function $\mathrm {char}_A$ of A is in $\mathcal {S}^n$ . The non-trivial case is as follows. Let $A(x_0,\ldots ,x_{k-1})$ be $(\exists y\,{<}\,t(x_0,\ldots ,x_{k-1}))B(y,x_0,\ldots ,x_{k-1})$ . Then

$$\begin{align*}\mathrm{char}_A(x_0,\ldots,x_{k-1}) \,{=}\,\mathrm{char}_<\left( \min\left\{z\,{:}\,\begin{matrix} z\,{=}\,t(x_0,\ldots,x_{k-1}) \,{\lor}\\ \mathrm{char}_B(z,x_0,\ldots,x_{k-1}) \end{matrix} \right\}, t(x_0,\ldots,x_{k-1})\right).\end{align*}$$

Thus, if the pair of t and A defines f, then

$$ \begin{align*} f(x_0,\ldots,x_{k-1}) \,{=}\,\min\{z\,{:}\, z\,{=}\,t(x_0,\ldots,x_{k-1}) \,{\lor}\, \mathrm{char}_A(z,x_0,\ldots,x_{k-1})\,{=}\,0\}.\\[-31pt] \end{align*} $$

Now we are in a position to see the background of our somewhat strange choice of function symbols in $\mathfrak {L}(\mathcal {S}^n)$ . On the one hand, as discussed in the Introduction, the most standard way to slice the class $\mathcal {PR}$ of primitive recursive functions is Grzegorczyk hierarchy $\mathcal {E}^n$ , and $\mathcal {E}^n$ is the bounded $\mu $ algebra generated by $\{{+},{\cdot },\gamma _3,\ldots ,\gamma _n\}$ for $n\,{\geq }\,2$ . Here $\cdot $ is the iteration of $+$ , $\gamma _3$ is that of $\cdot $ , and so on, where $\gamma _3,\ldots ,\gamma _n$ could also be replaced by their binary counterparts $x^y,\ldots $ as they generate the same bounded $\mu $ algebras. On the other hand, in our truth-theoretic study we use arithmetic primarily for the coding of syntax, and in most efficient codings the required recursions and inductions are not along sequence codes (numbers denoting strings), but rather along the length of sequences (the lengths of strings). And, indeed, $\mathcal {S}^n$ was defined in accordance with the requirements for such length-recursions and -inductions. It can alternatively be conceived as the bounded $\mu $ algebra generated by $\{(x_0,\ldots ,x_{k-1}) \mapsto 2^{f(|x_0|,\ldots ,|x_{k-1}|)}\,{:}\, f\,{\in }\,\mathcal {E}^n\}$ , which we call smashed Grzegorczyk hierarchy. From this perspective, we naturally let $\mathcal {S}^1$ be generated by $\{{+},{\cdot }\,\}$ , i.e., $\mathcal {E}^2$ .

A careful reader might consider that, from this perspective, it would be more natural to replace the number-induction $A(0)\,{\land }\,(\forall y\,{<}\,x)(A(y)\,{\to }\, A(y{+}1))\,{\to }\,A(x)$ by the length-induction

$$\begin{align*}B(0)\,{\land}\,(\forall y\,{<}\,x)(B(y)\,{\to}\, B(2{\cdot}y)\,{\land}\,B(2{\cdot}y{+}1) )\,{\to}\,B(x).\end{align*}$$

This is right, but, as far as we accept the schemata unrestrictedly for all formulae, they are equivalent. For the direction from length-induction to number-induction, take $B(y)\,{:\equiv }\, (\forall z\,{<}\,x{-}y)(A(z)\,{\to }\,A(z{+}y))$ and it suffices to see

$$\begin{align*}(\forall y\,{<}\,x)(B(y)\,{\to}\, B(2{\cdot}y)\,{\land}\,B(2{\cdot}y{+}1)),\end{align*}$$

assuming $(\forall y\,{<}\,x)(A(y)\,{\to }\, A(y{+}1))$ . Let $y\,{<}\,x$ and $B(y)$ . For $z\,{<}\,x{-}2{\cdot }y$ , since $z\,{+}\,y\,{<}\,x{-}y$ , by $B(y)$ we have $A(z)\,{\to }\,A(z{+}y)$ and $A(z{+}y)\,{\to }\,A(z{+}2{\cdot }y)$ , which imply $A(z)\,{\to }\,A(z{+}2{\cdot }y)$ . For $z\,{<}\,x{-}2{\cdot }y{-}1$ we have $A(z{+}2{\cdot }y)\,{\to }\,A(z{+}2{\cdot }y{+}1)$ and similarly $A(z)\,{\to }\,A(z{+}2{\cdot }y{+}1)$ . Let us mention that this proof requires an increase of the complexity of formula: known as the implications from $\Sigma ^b_{n+1}\mathsf {-PIND}$ to $\Sigma ^b_{n}\mathsf {-IND}$ , and from $\Sigma ^b_{n}\mathsf {-IND}$ to $\Sigma ^b_{n}\mathsf {-PIND}$ for $n\,{\geq }\,1$ in the notation of [Reference Buss4].

2.3 Bounded recursion

So far, the smash function $\#$ played no particular role. The significance of $\#$ is that, if it is included among the generators of a bounded $\mu $ algebra, then the algebra is closed under some form of recursion.

Theorem 2.11 (Bounded binary recursion)

Let $\mathbb {T}$ expand $\mathbb {FA}(\mathcal {S}^2)$ , let $h,g,p$ be bounded definable functions in $\mathbb {T}$ whose arities are k, $k{+}2$ , and $k{+}1$ respectively, and let $t(y,x_0,\ldots ,x_{k-1})$ be a $\mathbb {T}$ term.

If $\mathbb {T}$ proves $h(x_0,\ldots ,x_{k-1})\,{<}\,t(0,x_0,\ldots ,x_{k-1}) \;{\land }\;p(y,x_0,\ldots ,x_{k-1})\,{\leq }\,y/2$ and

$$ \begin{align*} z\hspace{0.7pt}{<}\hspace{0.7pt}t(p(y{+}1,\hspace{-0.5pt}x_0,\ldots,x_{k-1}),x_0,\ldots,\hspace{-0.5pt}x_{k-1})\ \, {\to}\ \, g(y,\hspace{-0.5pt}x_0,\ldots,x_{k-1},\hspace{-0.5pt}z)\hspace{0.7pt}{<}\hspace{0.7pt} t(y{+}1,\hspace{-0.5pt}x_0,\ldots,x_{k-1}), \end{align*} $$

then there is a ( $k{+}1$ )-ary function f that is bounded definable in $\mathbb {T}$ such that $\mathbb {T}$ proves both $f(0,x_0,\ldots ,x_{k-1}) \,{=}\,h(x_0,\ldots ,x_{k-1})$ and

$$ \begin{align*} f(y{+}1,x_0,\ldots,x_{k-1}) \,{=}\,g(y,x_0,\ldots,x_{k-1},f(p(y{+}1,x_0,\ldots,x_{k-1}),x_0,\ldots,x_{k-1})). \end{align*} $$

Proof For the graph, define $\max(x,y):=(x-y)+y$ and let

$$\begin{align*}&\mathrm{Gr}_f(x_0,\ldots,x_{k-1},y,z) \,{:\equiv}\\ &\quad(\exists u\,{<}\,s(y,x_0,\ldots,x_{k-1}) \,{\#}\,\max(2{\cdot} y,1)) A(u,x_0,\ldots,x_{k-1},y,z),\end{align*}$$

where $s(y,x_0,\ldots ,x_{k-1})\,{\equiv }\, 3^4{\cdot }(y\,{+}\,t(y,x_0,\ldots ,x_{k-1}))^4$ and $A(u,x_0,\ldots ,x_{k-1},y,z)$ is

$$ \begin{align*} &\mathrm{Seq}(u) \,{\land}\, (\forall a,j\,{<}\,u)(\mathrm{pair}(j,a)\,{\in}\,u \,{\to}\,j\,{\leq}\,y) \,{\land}\, \mathrm{pair}(y,z)\,{\in}\,u\\ &\,{\land}\, (\forall j\,{\leq}\,y)(\forall b\,{<}\,u) (\exists c\,{<}\,u) \left(\begin{matrix} j\,{>}\,0\,{\land}\\ \mathrm{pair}(j,b)\,{\in}\,u \end{matrix} \;{\to}\quad\begin{matrix} \mathrm{pair}(p(j,x_0,\ldots,x_{k-1}),c)\,{\in}\,u \\{\land}\,b\,{=}\, g(j{-}1,x_0,\ldots,x_{k-1},c) \end{matrix}\right)\\ &\,{\land}\, (\forall a\,{<}\,u)(\mathrm{pair}(0,a)\,{\in}\,u \,{\to}\,a\,{=}\,h(x_0,\ldots,x_{k-1})). \end{align*} $$

For the uniqueness of z, if $A(u,x_0,\ldots ,x_{k-1},y,z) \,{\land }\,A(v,x_0,\ldots ,x_{k-1},y,z')$ then we have $(\forall a\,{<}\,u)(\forall b\,{<}\,v) (\mathrm {pair}(j,a)\,{\in }\,u \,{\land }\,\mathrm {pair}(j,b)\,{\in }\,v \,{\to }\,a\,{=}\,b)$ by induction on j with $0\,{<}\,j\,{\leq }\,y$ and hence $z\,{=}\,z'$ .

We show $B(p(y,x_0,\ldots ,x_{k-1}))\,{\to }\,B(y)$ for $y\,{>}\,0$ , where

$$\begin{align*}B(y)\,{:\equiv}\, (\exists z\,{<}\,t(y,x_0,\ldots,x_{k-1})) \mathrm{Gr}_f(x_0,\ldots,x_{k-1},y,z).\end{align*}$$

Here and below, for better readability, we omit $x_0,\ldots ,x_{k-1}$ .

Let $z\,{<}\,t(p(y))$ and $u\,{<}\,s(p(y))\,{\#}\, \max(2{\cdot} p(y),1)$ witness $B(p(y))$ . Lemma 2.5 yields $v$ with:

1. (a) $v\,{<}\,9{\cdot }u{\cdot } (\mathrm {pair}(y,g(y{-}1,z)){+}1)^2{+}1$ , and

2. (b) $(\forall w\,{<}\,v)( w\,{\in }\,v\,{\leftrightarrow }\, w\,{\in }\,u \,{\lor }\, w\,{=}\,\mathrm {pair}(y,g(y{-}1,z)))$ .

Obviously $A(v,y,g(y{-}1,z))$ . It remains to bound $v$ . By $z\,{<}\,t(p(y))$ and the premise of the theorem, we have $\mathrm {pair}(y,g(y{-}1,z)) \,{<}\,3{\cdot }(y{+}t(y))^2$ and, by (a), also $v\,{\leq }\,9{\cdot }3^2{\cdot }u{\cdot } (y{+}t(y))^4 \,{=}\, u{\cdot }s(y)$ . Now $u\,{<}\,s(p(y))\,{\#}\,\max(2{\cdot} p(y),1)\,{\leq }\,s(y)\,{\#}\, y$ as s is monotone. Then $v\,{\leq }\,u{\cdot }s(y) \,{<}\, (s(y){\#}y) \,{\cdot }\, (s(y){\#}1)\,{=}\, s(y)\,{\#}(2{\cdot} y)$ by the axioms for $\#$ .

Since $B(0)$ is obvious, we can show $(\forall y\,{\leq }\,x)B(y)$ by induction on x.

In more intuitive terms, $|x|$ is $\lceil \log _2(x{+}1)\rceil $ , the length of the shortest binary expression of x, and $\mathrm {cg}_{n+1}(x,y)\,{=}\,\min \{\gamma _{n+1}(x),y{+}1\}$ . Thus, for $n\,{\geq }\,2$ , we can express the graph $\mathrm {Gr}_{\gamma _{n+1}}$ of $\gamma _{n+1}$ in $\mathfrak {L}(\mathcal {S}^n)$ as follows. (Actually, $\mathrm {Gr}_{\gamma _3}(y,z)$ is known to be expressible only with $+$ and $\cdot $ , without $\#$ . Furthermore, as will be shown in [Reference Sato22], we do not need $+$ and $\cdot $ as functions but only their graphs.)

Definition 2.13. Define $\mathrm {Gr}_{\gamma _n}(x,y)$ for $n\,{\geq }\,3$ as follows:

$$\begin{align*}\mathrm{Gr}_{\gamma_3}(x,y)\,{:\equiv}\, \mathrm{Pow2}(y)\,{\land}\,|y|\,{=}\,x{+}1, \qquad \mathrm{Gr}_{\gamma_{n+4}}(x,y)\,{:\equiv}\, \mathrm{cg}_{n+4}(x,y)\,{=}\,y.\end{align*}$$

Corollary 2.14 (Bounded recursion)

Let $\mathbb {T}$ expand $\mathbb {FA}(\mathcal {S}^3)$ , let $h,g$ be bounded definable functions in $\mathbb {T}$ whose arities are k and $k{+}2$ respectively, and let $t(y,x_0,\ldots ,x_{k-1})$ be a $\mathbb {T}$ term.

If $\mathbb {T}$ proves $h(x_0,\ldots ,x_{k-1})\,{<}\,t(0,x_0,\ldots ,x_{k-1})$ and

$$\begin{align*}z\,{<}\,t(y,x_0,\ldots,x_{k-1}) \,{\to}\, g(y,x_0,\ldots,x_{k-1},z)\,{<}\, t(y{+}1,x_0,\ldots,x_{k-1}),\end{align*}$$

then there is a ( $k{+}1$ )-ary function f that is bounded definable in $\mathbb {T}$ such that $\mathbb {T}$ proves $f(0,x_0,\ldots ,x_{k-1}) \,{=}\,h(x_0,\ldots ,x_{k-1})$ and

$$ \begin{align*} f(y{+}1,x_0,\ldots,x_{k-1}) \,{=}\,g(y,x_0,\ldots,x_{k-1},f(y,x_0,\ldots,x_{k-1})). \end{align*} $$

With bounded binary recursion (Theorem 2.11), we can introduce fundamental operations on (codes of) finite sequences and, hence, on syntactical objects. The following is proved in [Reference Hájek and Pudlák9, Chapter V] or [Reference Walker29, Chapter 4].

Lemma 2.15. There are bounded definable functions $\mathrm {lh}$ , $\mathrm {ev}$ , $\mathrm {slo}$ , and $\mathrm {conc}$ in $\mathbb {FA}(\mathcal {S}^2)$ such that it proves

$$\begin{align*}\begin{matrix} \mathrm{lh}(1)\,{=}\,0 \;{\land}\; \mathrm{Seq}(\mathrm{slo}(x)) \;{\land}\; \mathrm{lh}(\mathrm{slo}(x))\,{=}\,1\; {\land}\; \mathrm{ev}(\mathrm{slo}(x),0)\,{=}\,x \\ {\land}\;(\mathrm{Seq}(u)\,{\land}\,\mathrm{Seq}(v) \,{\to}\, \mathrm{Seq}(\mathrm{conc}(u,v)) \;{\land}\; \mathrm{lh}(\mathrm{conc}(u,v))\,{=}\, \mathrm{lh}(u){+}\mathrm{lh}(u)) \\ {\land}\;(\forall y\,{<}\,\mathrm{lh}(u)) (\mathrm{ev}(u,y)\,{=}\, \mathrm{ev}(\mathrm{conc}(u,v),y)) \\{\land}\; (\forall y\,{<}\,\mathrm{lh}(v)) (\mathrm{ev}(v,y)\,{=}\, \mathrm{ev}(\mathrm{conc}(u,v),\mathrm{lh}(u){+}y))\\ {\land}\; \mathrm{conc}(u,\mathrm{slo}(x))\,{\leq}\,9{\cdot}u {\cdot}(x{+}1)^2 \\{\land}\;(\mathrm{lh}(v)\,{>}\,0\,{\to}\, \mathrm{conc}(v,u)\,{\geq}\,2{\cdot}u) \;{\land}\;(\mathrm{lh}(v)\,{>}\,0\,{\land}\,\mathrm{ev}(v,0)\,{>}\,1\,{\to}\, \mathrm{conc}(u,v)\,{>}\,2{\cdot}u). \end{matrix} \end{align*}$$

2.4 Totality rule and the use of function symbols

It is known that, in first-order arithmetic , the totality of exponentiation $\forall y\exists z\mathrm {Gr}_{\gamma _3}(y,z)$ justifies the use of the function symbol $\gamma _3$ even in the induction schema. However, the straightforward rewriting of $A(\gamma _3(t))$ by $\exists z( \mathrm {Gr}_{\gamma _3}(t,z) \,{\land }\,A(z))$ or by $\forall z( \mathrm {Gr}_{\gamma _3}(t,z) \,{\to }\,A(z))$ does not justify the use of induction for $A(\gamma _3(t))$ . What we need to show is, for any formula A with occurrences of $\gamma _3$ , the existence of a finite fragment $u\,{=}\,\gamma _3{\upharpoonright }k$ (a finite sequence coded by a number) that is long enough to contain all the values of $\gamma _3$ referred to in subformulae of A. A similar idea was used to prove the formalized version of Kleene’s Normal Form Theorem in the function-based second-order arithmetic in [Reference Nemoto and Sato15, Lemma 2.13].

In this subsection, we prove the same result in our finitist setting. As we cannot form totality assertions in the language $\mathfrak {L}(\mathcal {S}^n)$ , we express the totality by a non-logical rule, similarly to the non-logical rule ( $\exists \mbox {-val}$ ) discussed in the Introduction.

Definition 2.17. The rule $(\mathrm{Tot}{\text{-}}\gamma _{n+1})$ is as follows, where x must not occur in A (i.e., x is an eigenvariable) nor in t.

In this subsection we locally interpret $\mathbb {FA}(\mathcal {S}^{n+1})$ in $\mathbb {FA}(\mathcal {S}^n)\,{+}\, (\mathrm{Tot}{\text{-}}\gamma _{n+1})$ . We first define the formulae $\mathrm {FLE}_{n,t}(u)$ and $\mathrm {FLE}_{n,A}(u)$ meaning that u is a long enough fragment of $\gamma _{n+1}$ for all references in a term t or a formula A, respectively.

Definition 2.18. Define $\mathrm {Frag}_{\gamma _{n+1}}(u)\;{:\equiv }\; \mathrm {Seq}(u) \,{\land }\, (\forall x\,{<}\,\mathrm {lh}(u)) \mathrm {Gr}_{\gamma _{n+1}}(x,u(x))$ .

For any $\mathcal {S}^{n+1}$ term t (without added symbols for bounded definable functions) whose variables are among $x_0,\ldots ,x_{k-1}$ , we define the $\mathfrak {L}(\mathcal {S}^n)$ formula $\mathrm {FLE}_{n,t}(u,x_0,\ldots ,x_{k-1})$ by meta-recursion on t as follows.

$$ \begin{align*} &\mathrm{FLE}_{n,t}(u,x_0,\ldots,x_{k-1})\, {:\equiv}\\ &\begin{cases} \mathrm{Frag}_{\gamma_{n+1}}(u), &\!\!\!\!\!\!\!\!\!\!\mbox{ if}\ t\ \mbox{is a variable or constant},\\ {\bigwedge}_{i<\ell}\mathrm{FLE}_{n,t_i}(u,x_0,\ldots,x_{k-1}), &\hspace{-20pt}\!\!\!\!\!\!\!\!\!\!\mbox{ if }t\,{\equiv}\,\mathrm{f}(t_0,\ldots,t_{\ell-1}) \mbox{ for some f from }\mathcal{S}^n,\\ \mathrm{FLE}_{n,s}(u,x_0,\ldots,x_{k-1}) \,{\land}\, s^u(x_0,\ldots,x_{k-1})\,{<}\,\mathrm{lh}(u),\hspace{-40pt} &\!\!\!\!\!\!\!\!\!\!\hspace{55pt}\mbox{ if }t\,{\equiv}\,\gamma_{n+1}(s), \end{cases} \end{align*} $$

where $s^u$ is the result of replacing all the subterms of the form $\gamma _{n+1}(t)$ by $u(t^u)$ in s.

Next, for any $\mathfrak {L}(\mathcal {S}^{n+1})$ formula A whose free variables are among $x_0,\ldots ,x_{k-1}$ , we define the $\mathfrak {L}(\mathcal {S}^n)$ formula $\mathrm {FLE}_{n,A}(u,x_0,\ldots ,x_{k-1})$ by meta-recursion on A as follows.

$$ \begin{align*} & \mathrm{FLE}_{n,A}(u,x_0,\ldots,x_{k-1}) \,{:\equiv}\\ &\begin{cases} {\bigwedge}_{i<2}\mathrm{FLE}_{n,t_i}(u,x_0,\ldots,x_{k-1}), &\hspace{-110pt}\mbox{ if}\ A\ \mbox{is }t_0\,{=}\,t_1,\;t_0\,{<}\,t_1,\; \neg\, t_0\,{=}\,t_1,\;\neg\, t_0\,{<}\,t_1.\\ {\bigwedge}_{i<2}\mathrm{FLE}_{n,B_i}(u,x_0,\ldots,x_{k-1}), &\hspace{-110pt}\mbox{ if}\ A\ \mbox{is }B_0\,{\land}\,B_1 \mbox{ or }B_0\,{\lor}\,B_1,\\ (\exists v\,{\leq}\,u) (\mathrm{FLE}_{n,B}(u,t^v,x_0,\ldots,x_{k-1}) \,{\land}\, \mathrm{FLE}_{n,t}(v,x_0,\ldots,x_{k-1})), \\ & \hspace{-200pt} \mbox{ if}\ A\ \mbox{is }(\forall y\,{<}\,t)B(y,x_0,\ldots,x_{k-1})\mbox{ or }(\exists y\,{<}\,t)B(y,x_0,\ldots,x_{k-1}). \end{cases} \end{align*} $$

For the last clause, it might help to remind that, according to the standard abbreviation in first-order setting, both $y\,{<}\,t$ and $B(y,x_0,\ldots ,x_{k-1})$ are subformulae of $(Q y\,{<}\,t)B(y,x_0,\ldots ,x_{k-1})$ .

Proof (1) We prove this by meta-induction on $n\,{\geq }\,3$ and internal induction on y in $\mathbb {FA}(\mathcal {S}^n)$ . If $x\,{\cdot }\,y\,{=}\,0$ or $y\,{=}\,1$ , this can be seen easily by $\mathrm{cg}_n(0,z) = 1$ and $\mathrm{cg}_n(x,z) \geq \min(x\cdot 2,z)$ . So, assume $\mathrm {cg}_{n+1}(x{+}y{+}1,z)\,{\leq }\,z$ with $x,\, y\,\geq\, 1$ . Then $\mathrm {cg}_{n+1}(x{+}y,z)\,{\leq }\,z$ and so

$$ \begin{align*} \mathrm{cg}_{n+1}(x,z){\cdot} \mathrm{cg}_{n+1}(y{+}1,z) \,&{<}\, \gamma_n(\mathrm{cg}_{n+1}(x,z)){\cdot} \gamma_n(\mathrm{cg}_{n+1}(y,z))\\ \,&{\leq}\, \gamma_n(\mathrm{cg}_{n+1}(x,z) {+} \mathrm{cg}_{n+1}(y,z))\\ \,&{\leq}\, \gamma_n(\mathrm{cg}_{n+1}(x,z) {\cdot} \mathrm{cg}_{n+1}(y,z))\\ \,&{\leq}\, \gamma_n(\mathrm{cg}_{n+1}(x{+}y,z)) \,{=}\,\mathrm{cg}_{n+1}(x{+}y{+}1,z), \end{align*} $$

where the second inequality is obvious for $n\,{=}\,3$ and by the meta-induction hypothesis otherwise since $\gamma _n(x)\,{=}\,\mathrm {cg}_n(x,\gamma _n(x))$ , and where the fourth inequality is by the induction hypothesis.

(2) For uniformity, let $\gamma _2(x)\,{=}\,2{\cdot }x$ . We prove $A(x)$ by induction on x, where

$$\begin{align*}A(x)\;{:\equiv}\;(\forall y\,{<}\,x)(\forall z\,{<}\,x) (\mathrm{Gr}_{\gamma_{n+1}}(6{\cdot}y^2{+}4,z) \,{\to}\,(\exists u\,{<}\,z) (y\,{\leq}\,\mathrm{lh}(u)\,{\land}\, \mathrm{Frag}_{\gamma_{n+1}}(u))). \end{align*}$$

Again $A(0)$ is obvious. Assume $A(x)$ . To show $A(x{+}1)$ , fix $y,z\,{<}\,x{+}1$ and assume $\mathrm {Gr}_{\gamma _{n+1}}(6{\cdot }y^2{+}4,z)$ . If $y\,{=}\,0$ , then the statement is obvious. Thus we may assume $y\,{\neq }\,0$ . By $\mathrm {cg}_{n+1}(6{\cdot }(y{-}1)^2{+}4,z) \,{<}\,z\,{\leq }\,x$ , the induction hypothesis yields $w\,{<}\, \mathrm {cg}_{n+1}(6{\cdot }(y{-}1)^2{+}4,z)$ with $y{-}1\,{\leq }\,\mathrm {lh}(w) \,{\land }\,\mathrm {Frag}_{\gamma _{n+1}}(w)$ . We may assume $\mathrm {lh}(w)\,{=}\,y{-}1$ . Let $u\,{:=}\,w{*}\langle \mathrm {cg}_{n+1}(y{-}1,z)\rangle $ . As $9\,{<}\,2^4\,{\leq }\, \mathrm {cg}_{n+1}(4,z)$ , we have

$$\begin{align*}u\,{\leq}\,9{\cdot}w{\cdot}(\mathrm{cg}_{n+1}(y{-}1,z){+}1)^2 \,{<}\,\mathrm{cg}_{n+1}(4,z){\cdot} \mathrm{cg}_{n+1}(6{\cdot}(y{-}1)^2{+}4,z) {\cdot}(\mathrm{cg}_{n+1}(y,z))^2 \end{align*}$$

and, by (1), $u\,{<}\,\mathrm {cg}_{n+1}(4{+}6{\cdot }(y{-}1)^2{+}4 {+}2{\cdot }y,z) \,{\leq }\,\mathrm {cg}_{n+1}(6{\cdot }y^2{+}4,z) \,{=}\,z$ .

Lemma 2.21. For $n\,{\geq }\,2$ , any $\mathcal {S}^{n+1}$ term t, and any $\mathfrak {L}(\mathcal {S}^{n+1})$ formula A, the following rules are derivable in $\mathbb {FA}(\mathcal {S}^{n}) \,{+}\,(\mathrm{Tot}{\text{-}}\gamma _{n+1})$ , where u is a variable that does not appear in terms t, $t_0,\ldots ,t_{k-1}$ nor in a formula D (but t, $t_0,\ldots ,t_{k-1}$ may occur in D).

Proof We prove the derivability by meta-induction on t and A respectively.

The only non-trivial case for the former rule is where $t\,{\equiv }\,\gamma _{n+1}(s)$ .

The other cases for the former rule are similar to the cases where $A\,{\equiv }\, B\,{\land }\,C$ and $A\,{\equiv }\, B\,{\lor }\,C$ for the latter rule. The cases of bounded quantifiers can be treated similarly.

Lemma 2.23. For $n\,{\geq }\,2$ ,

$$\begin{align*}\mathbb{FA}(\mathcal{S}^n) \vdash \begin{matrix} \mathrm{FLE}_{n,A}(v,x_0,\ldots,x_{k-1}) \,{\land}\,v\,{\leq}\,u\,{\land}\, \mathrm{Frag}_{\gamma_{n+1}}(u) \\{\to}\, (A^v(x_0,\ldots,x_{k-1})\,{\leftrightarrow} \,A^u(x_0,\ldots,x_{k-1})) \end{matrix}.\end{align*}$$

Theorem 2.24. For $n\,{\geq }\,2$ and any $\mathfrak {L}(\mathcal {S}^{n+1})$ formula A whose free variables are among $x_0,\ldots ,x_{k-1}$ , if $\mathbb {FA}(\mathcal {S}^{n+1}) \vdash A(x_0,\ldots ,x_{k-1})$ then

$$\begin{align*}\mathbb{FA}(\mathcal{S}^n)\,{+}\, (\mathrm{Tot}{\text{-}}\gamma_{n+1})\vdash \mathrm{FLE}_{n,A}(u,x_0,\ldots,x_{k-1}) \,{\to}\,A^u(x_0,\ldots,x_{k-1}).\end{align*}$$

Proof We can prove this by meta-induction on the derivation of $A(x_0,\ldots ,x_{k-1})$ in $\mathbb {FA}(\mathcal {S}^{n+1})$ . For better readability, we omit the parameters $x_0,\ldots , x_{k-1}$ . The non-trivial case is modus ponens (MP): A is inferred from B and $B\,{\to }\,A$ . By the meta-induction hypothesis, $\mathbb {FA}(\mathcal {S}^n)\,{+}\, (\mathrm{Tot}{\text{-}}\gamma _{n+1})$ proves both $\mathrm {FLE}_{n,B}(v) \,{\to }\,B^v$ and $\mathrm {FLE}_{n,B\to A}(w) \,{\to }\,(B^w\,{\to }\,A^w)$ . By Lemma 2.23, $\mathbb {FA}(\mathcal {S}^n)\,{+}\, (\mathrm{Tot}{\text{-}}\gamma _{n+1})$ also proves $ v\,{\leq }\,w\,{\land }\,\mathrm {FLE}_{n,B\to A}(w) \,{\to }\, ( \mathrm {FLE}_{n,B}(v)\,{\to }\, (B^v\,{\to }\,A^w))$ . We finally obtain

$$\begin{align*}\mathbb{FA}(\mathcal{S}^n)\,{+}\, (\mbox{Tot-}\gamma_{n+1}) \vdash \begin{matrix} v\,{\leq}\,w\,{\land}\,\mathrm{FLE}_{n,B\to A}(w) \\{\to}\, ( u\,{\leq}\,v\,{\land}\,\mathrm{FLE}_{n,B}(v)\,{\to}\, ( \mathrm{FLE}_{n,A}(u)\,{\to}\,A^u)) \end{matrix} \end{align*}$$

to which we can apply Lemma 2.21 twice.

3.1 Syntax coding

We now show how to encode syntax. Our theories will be formulated in the following finitist languages.

We first assign different numbers to primitive symbols (including parenthesis), for example,

$$ \begin{align*} &\lceil (\rceil\,{:=}\,1; \;\; \lceil )\rceil\,{:=}\,3; \;\; \lceil 0\rceil\,{:=}\,5; \;\; \lceil 1\rceil\,{:=}\,7; \;\; \lceil {+}\rceil\,{:=}\,9; \;\; \lceil {\cdot}\rceil\,{:=}\,11; \;\; \lceil {\#}\rceil\,{:=}\,13; \\ & \lceil {=}\rceil\,{:=}\,15; \;\; \lceil {<}\rceil\,{:=}\,17; \;\;\lceil {\neg}\rceil\,{:=}\,19; \;\; \lceil {\land}\rceil\,{:=}\,21; \;\; \lceil {\lor}\rceil\,{:=}\,23; \;\; \lceil {\forall}\rceil\,{:=}\,25; \;\; \lceil {\exists}\rceil\,{:=}\,27; \\ &\lceil x_j\rceil\,{:=}\,29{+}6j; \;\; \lceil\gamma_{j+3}\rceil\,{:=}\,31{+}6j; \;\; \lceil T_j\rceil\,{:=}\,33{+}6j, \end{align*} $$

and even numbers to the bounded definable functions, which we loosely consider to be in the languages.

Definition 3.2 (Dot notation)

For a function symbol $\mathrm {f}$ of arity $\ell $ (which might be 0, i.e., $\mathrm {f}$ might be constant), let

For any $\mathfrak {L}(\mathcal {S}^n)$ term $t(x_0,\ldots ,x_{k-1})$ , define the $\mathfrak {L}(\mathcal {S}^2)$ term

whose variables are those of $t(x_0,\ldots ,x_{k-1})$ by meta-recursion on t as follows.

Let

. In particular, $\ulcorner x_j\urcorner \,{=}\,\langle \lceil x_j\rceil \rangle $ and

for any constant c.

For any $\mathfrak {L}(\mathcal {S}^n)$ formula $A(x_0,\ldots ,x_{k-1})$ , define an $\mathfrak {L}(\mathcal {S}^2)$ term

(in the expanded language) whose variables are the same as the free variables of $A(x_0,\ldots ,x_{k-1})$ by meta-recursion on A as follows.

Here

stands for $\langle \lceil (\rceil \rangle \,{*}\,\langle \lceil {\neg }\rceil \rangle \,{*}\,t\,{*}\, \langle \lceil )\rceil \rangle $ , and

for $\langle \lceil (\rceil \rangle \,{*}\,s \,{*}\,\langle \lceil \Box \rceil \rangle \,{*}\,t\,{*}\, \langle \lceil )\rceil \rangle $ for $\Box \,{\equiv }\,{\land },{\lor }$ , and

for $\langle \lceil (\rceil \rangle \,{*}\,\langle \lceil Q\rceil \rangle \,{*}\, x \,{*}\,s \,{*}\,t\,{*}\, \langle \lceil )\rceil \rangle $ for $Q\,{\equiv }\,\forall ,\exists $ . Let $\ulcorner A(x_0,\ldots ,x_{k-1})\urcorner $ denote

.

To avoid ambiguity, we apply the dot notation only to minimum units (which might consist of several letters, e.g., $\mathrm {num}$ or $\mathrm {CTerm}_n$ ). Thus, if we apply dot to $\lambda x.A(t(x))$ , we must first introduce $B(x)\,{\equiv }\,A(t(x))$ .

We define $\mathfrak {L}(\mathcal {S}^2)$ formulae for syntactical notions, e.g., $\mathrm {Var}(u)$ , $\mathrm {Term}_n(u)$ , $\mathrm {CTerm}_n(u)$ , $\mathrm {AtForm}_{n,T_{<m}}(u)$ , and $\mathrm {Form}_{n,T_{<m}}(u)$ which mean that u is a code of variable, term, closed term, atomic formula, and formula, respectively, all in the sense of $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ . $\mathrm {Form}^k_{n,T_{<m}}(u, \ulcorner x_0\urcorner ,\ldots ,\ulcorner x_{k-1}\urcorner )$ means additionally that free variables are among $x_0,\ldots ,x_{k-1}$ . Let $\mathrm {Sent}_{n,T_{<m}}(u)\,{:\equiv }\,\mathrm {Form}^0_{n,T_{<m}}(u)$ . We omit “k” and “ ${<}\,m$ ” if $k,m\,{=}\,1$ .

Definition 3.3 (Dyadic numeral $\mathrm {num}(x)$ )

By bounded binary recursion, we define a bounded definable function $\mathrm {num}$ as follows, where $\times $ denotes $\cdot $ for better readability: $\mathrm {num}(0)\,{=}\,\ulcorner 0\urcorner $ ; $\mathrm {num}(1)\,{=}\,\ulcorner 1\urcorner $ ;

;

Definition 3.4. By bounded binary recursion, we have bounded definable functions $\mathrm {sbst}$ and

in $\mathbb {FA}(\mathcal {S}^2)$ for the syntactical operations of substitution and negation, respectively, with the following defining axioms.

Our $\mathrm {sbst}$ does not prevent variable collisions. Moreover, $\mathrm {Term}_n(u)$ does not necessarily imply $\mathrm {Form}_{n,T_{<m}}(w)\,{\land }\,\mathrm {Var}(x) \,{\to }\,\mathrm {Form}_{n,T_{<m}}(\mathrm {sbst}(w,x,u))$ : if a term u contains a variable y then is not a well formed formula (as far as x occurs in $v$ ). However, in what follows, these do not matter for us, as we basically substitute closed terms only. Note also that the bound for $\mathrm {sbst}$ essentially requires $\#$ while $\mathrm {sbst}$ is crucial for a liar sentence.

We can also introduce a uniform version of $\mathrm {sbst}$ . For an assignment a of terms to variables, let the substitution by a applied to $w$ result in $\mathrm {usbs}(w,a)$ . The following definition is by bounded binary recursion on $u{*}\langle a\rangle $ , since $\mathrm {ovw}(a,x,0)\,{\leq }\,a$ implies . Note $\neg \mathrm {Term}_n(0)$ .

Definition 3.5. Define $\mathrm {ovw}(a,x,u)$ by the defining axioms

$$ \begin{align*} \mathrm{ovw}(a,x,u)(x)\,{=}\,u, \qquad \mathrm{lh}(\mathrm{ovw}(a,x,u))\,{=}\,\mathrm{lh}(a),\\ \mbox{ and }(\forall y\,{<}\,\mathrm{lh}(a)) (y\,{\neq}\,x\,{\to}\,\mathrm{ovw}(a,x,u)(y)\,{=}\,a(y)). \end{align*} $$

We introduce $\mathrm {usbs}$ with the same defining axioms as $\mathrm {sbst}$ except:

3.2 Finitist equational truth, lower bounds, and sentence induction

Definition 3.7. $\mathbb {FET}(\mathcal {S}^n)$ with $n\,{\geq }\,2$ is the $\mathfrak {L}(\mathcal {S}^n,T)$ theory generated by the following non-logical axioms:

• ( $\mathbb {FA}(\mathcal {S}^n)$ ): all the axioms of $\mathbb {FA}(\mathcal {S}^n)$ with the schema of induction extended to $\mathfrak {L}(\mathcal {S}^n,T)$ formulae;

• ( $T\mathrm {term}$ ): for any k-ary $\mathcal {S}^n$ function symbol $\mathrm {f}$ (including $k\,{=}\,0$ , i.e., constant symbols);

• ( $T{=}$ ):

and the following non-logical rule, where x must not appear in A nor in t.

The preparatory truth theory $\mathbb {FET}(\mathcal {S}^n)$ only has the axioms and a rule which govern the behaviour of the truth predicate on equations. The motivation behind the rule ( $\exists \mbox {-val}$ ) was explained already in the Introduction.

Proof (i) By the rule ( $\exists \mbox {-val}$ ), it suffices to show , which follows from ( $T{=}$ ).

(ii) We prove this by meta-induction on t. If t is a variable, this is trivial. If $t\,{\equiv }\,\mathrm {f}(t_0,\ldots ,t_{\ell -1})$ (possibly $\ell \,{=}\,0$ ), by (Inst) we substitute $t_i(x_0,\ldots ,x_{k-1})$ and

to $x_i$ and $u_i$ in ( $T\mbox {term}$ ), i.e.,

in which we can replace the premises with

by the meta-induction hypothesis.

(iii) We prove this again by meta-induction on t. If t is a constant or variable, this is by (i). Otherwise, it suffices to see

for a function symbol $\mathrm {f}$ , provided ${\bigwedge }_{i<2k}\mathrm {CTerm}_n(w_i)$ . By ( $\exists \mbox {-val}$ ) we may assume

for fresh $z_i$ . Use (ii) and ( $T{=}$ ).

We may extend (iii) for non-standard terms with non-standardly many variables: by induction on $w$ ,

We next show $\mathbb {FA}(\mathcal {S}^{n+1}) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FET}(\mathcal {S}^n)$ . This yields the lower bounds of all our truth theories, since they all include $\mathbb {FET}(\mathcal {S}^n)$ . By Corollary 2.25 it suffices to show $\mathbb {FA}(\mathcal {S}^{n})\,{+}\, (\mathrm{Tot}{\text{-}}\gamma _{n+1}) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FET}(\mathcal {S}^n)$ .

Lemma 3.9. $\mathbb {FET}(\mathcal {S}^2)$ proves (i)

and (ii)

for $n\,{\geq }\,3$ , where $\mathcal {S}^2$ functions $\mathrm {cgr}_n$ are introduced with the following defining axioms.

Proof We first prove (ii). As $A(0)$ is obvious, by induction, it suffices to show $A(x)\,{\to }\,A(x{+}1)$ , where

Assume $A(x)$ . To show $A(x{+}1)$ , take $y,z\,{<}\,x{+}1$ . Since the claim is obvious for $y\,{=}\,0$ , we assume $y\,{>}\,0$ . What we have to show is

By ( $\exists \mbox {-val}$ ), for fresh $v$ we may assume

(*)

which implies

by ( $T\mathrm {term}$ ). Hence, by ( $T{=}$ ), what we have to show now is

$$\begin{align*}z\,{=}\,\gamma_{n}(v) \,{\leftrightarrow}\, \mathrm{Gr}_{\gamma_{n+1}}(|y|,z). \end{align*}$$

If $z\,{=}\,\gamma _{n}(v)$ , then $v\,{<}\,z\,{\leq }\,x$ , and so the induction hypothesis and ( $*$ ) yield $\mathrm {Gr}_{\gamma _{n+1}}(|y|{-}1,v)$ which implies $(\exists v\,{<}\,z) (\mathrm {Gr}_{\gamma _{n+1}}(|y|{-}1,v) \,{\land }\,z\,{=}\,\gamma _{n}(v))$ , namely, $\mathrm {Gr}_{\gamma _{n+1}}(|y|,z)$ .

Conversely, assume $\mathrm {Gr}_{\gamma _{n+1}}(|y|,z)$ , say $\mathrm {Gr}_{\gamma _{n+1}}(|y|{-}1,w) \,{\land }\,z\,{=}\,\gamma _{n}(w)$ . It suffices to show $w\,{=}\,v$ . The induction hypothesis yields by $w\,{<}\,z\,{\leq }\,x$ . By Lemma 3.8(i) and the assumption ( $*$ ), $w\,{=}\,v$ follows from ( $T{=}$ ).

We can show (i) similarly. Notice

$$\begin{align*}\mathrm{Gr}_{\gamma_3}((1{\#}(y/2)){\cdot}2{-}2,v) \,{\land}\,z\,{=}\,v\,{\#}\,2\,{\to}\, \mathrm{Gr}_{\gamma_3}((1{\#}y){\cdot}2{-}2,z),\end{align*}$$

since $(1{\#}y)\,{=}\,(1{\#}(y/2)){\cdot }2$ for $y\,{>}\,0$ and $\mathrm {Gr}_{\gamma _3}(x,v) \,{\land }\,z\,{=}\,v\,{\#}\,2\,{\to }\, \mathrm {Gr}_{\gamma _3}(x{\cdot }2{+}2,z)$ .

We can now justify sentence induction. In first-order truth theories, induction on the complexity of sentence is a fundamental tool, but it involves a $\Pi _1$ induction formula. Note that does not imply $\mathrm {Sent}_{n,T}(v)$ but only $\mathrm {Sent}_{n,T}(\mathrm {sbst}(v,x, \mathrm {num}(z)))$ , whereas is not guaranteed.

Theorem 3.11 (Sentence induction)

The following rule, with eigenvariables $u,v,y$ , is derivable in $\mathbb {FET}(\mathcal {S}^n)$ .

Proof Define $\mathrm {AssBd}(a,p)$ as follows, meaning that a substitutes variables by either themselves or numerals for numbers ${<}\,p$ :

$$\begin{align*}\mathrm{AssBd}(a,p)\,{:\equiv}\, (\forall x\,{<}\,\mathrm{lh}(a)) (a(x)\,{=}\,x\,{\lor}\,(\exists z\,{\leq}\,p) (a(x)\,{=}\,\mathrm{num}(z))).\end{align*}$$

We have an $\mathcal {S}^3$ term $\mathrm {boa}(p,q)$ which bounds all such a’s of length q, i.e.,

$$\begin{align*}\mathbb{FA}(\mathcal{S}^3) \vdash \mathrm{AssBd}(a,p)\,{\to}\, (\exists b\,{<}\,\mathrm{boa}(p,q))(\forall x\,{<}\,q)(b(x)\,{=}\,a(x)), \end{align*}$$

where a is free. Moreover, we bound the value of $\mathrm {usbs}(v,a)$ by $\gamma _{n+1}$ together with $\mathrm {ht}$ , defined as follows, where $\star \,{\equiv }\,{+},{\cdot },{\#} \mbox { and }\Box \,{\equiv }\,{\land },{\lor } \mbox { and }Q\,{\equiv }\,\forall ,\exists $ .

Fix possibly non-standard p and q. Since we may assume to have $w'$ with $\mathrm {Frag}_{\gamma _{n+1}}(w') \,{\land }\,\mathrm {lh}(w')\,{\geq }\,p$ by Lemma 2.21, induction on $v$ shows

(†)

By $(\mathrm{Tot}{\text{-}}\gamma _{3})$ we may assume the existence of $r\,{=}\,\mathrm {boa}(\gamma _{n+1}(p),q)$ . By definition, this implies $\mathrm {AssBd}(a,\gamma _{n+1}(p))\,{\to }\, (\exists b\,{<}\,r)(\forall x\,{<}\,q)(b(x)\,{=}\,a(x))$ . By induction on $w$ , we now prove

$$\begin{align*}(\forall a\,{<}\,r)\begin{pmatrix}\begin{pmatrix} \mathrm{lh}(a)\,{\leq}\,q\,{\land}\, \mathrm{Form}_{n,T}(w)\,{\land}\, \mathrm{lh}(w)\,{\leq}\,p\\{\land}\,w\,{\leq}\,q\,{\land}\, \mathrm{AssBd}(a,\gamma_{n+1}(p{-}\mathrm{ht}(w))) \end{pmatrix}{\to}\,A(\mathrm{usbs}(w,a))\end{pmatrix}.\end{align*}$$

For example, consider . ( ) yields for some $z\,{\leq }\,\gamma _{n+1}(p{-}\mathrm {ht}(w){+}\mathrm {ht}(u)) \,{=}\,\gamma _{n+1}(p{-}\mathrm {ht}(v))$ . Let $b\,{:=}\,\mathrm {ovw}(a,x,\mathrm {num}(y))\,{<}\,r$ for any $y\,{<}\,z$ . Then we have $\mathrm {AssBd}(b,\gamma _{n+1}(p{-}\mathrm {ht}(v)))$ and $A(\mathrm {usbs}(v,b))$ by induction hypothesis. By $\mathrm {usbs}(v,b)\,{=}\, \mathrm {sbst}(\mathrm {usbs}(v,\mathrm {ovw}(a,x,0)),x,\mathrm {num}(y))$ , since $y\,{<}\,z$ is arbitrary, the third premise of the rule yields $A(\mathrm {usbs}(w,a))$ .

Since p and q are arbitrary, setting $p\,{:=}\,\mathrm {lh}(u)$ , $q\,{:=}\,u$ and $a\,{:=}\,\langle \,\rangle $ , we have $\mathrm {Sent}_{n,T}(u)\,{\to }\,A(u)$ .

4.1 Finitist typed truth

The finitist theory of compositional truth $\mathbb {FCT}(\mathcal {S}^n)$ extends $\mathbb {FET}(\mathcal {S}^n)$ by the axioms which govern the behaviour of the truth predicate on compound formulae generated by propositional connectives and bounded quantifiers. However, in the scope of the truth predicate, only (codes of) those formulae without truth predicate can occur. In other words, the truth predicate cannot occur in the scope of itself. (More precisely, although we have such self-applied formulae in the language, we exclude their codes from the scope of our truth axioms.)

The last four axioms are for (possibly) non-standard sentences, while the schemata (Truth-Negation) and (Truth-Truth) in the Introduction were only for standard ones. The subscript “tp” stands for “typed” truth.

Remark 4.2. In the first-order setting, it is also common to formulate the truth axiom for $\forall $ as follows:

A discussion on the difference between the two alternative formulations can be found in [Reference Wcisło and Łełyk30, Section 1]. However, since there seems to be no way to bound $w$ in truth axioms for bounded quantifiers, we consider only substitutions by numerals.

Within $\mathbb {FCT}(\mathcal {S}^n)$ , we can prove the uniform Tarski biconditional in our setting.

Theorem 4.3. For any $\mathfrak {L}(\mathcal {S}^n)$ formula $A(x_0,\ldots ,x_{k-1})$ whose free variables are among $x_0,\ldots ,x_{k-1}$ ,

Proof We prove the statement by meta-induction on A. We make a case distinction based on the outer-most connective of A. In all the cases, we work in $\mathbb {FCT}(\mathcal {S}^n)$ , and assume for $i\,{<}\,k$ . Then, by Lemma 3.8(ii), for any term $t_i$ .

First consider the cases where A is an atomic formula or the negation of an atomic formula. Take the case of $A\,{\equiv }\,\neg (t_0\,{<}\,t_1)$ . Now ( $T{\neg }\mbox {Atom}$ ) implies that and $\neg (t_0(x_0,\ldots ,x_{k-1}) \,{<}\,t_1(x_0,\ldots ,x_{k-1}))$ are equivalent.

The cases of conjunction and disjunction are easy by ( $T{\land }_{\mathrm {tp}}$ ) and ( $T{\lor }_{\mathrm {tp}}$ ).

Finally, in the case of bounded quantifiers, let

$$\begin{align*}A(x_0,\ldots,x_{k-1})\,{\equiv}\, (\forall y\,{<}\,t(x_0,\ldots,x_{k-1})) B(x_0,\ldots,x_{k-1},y).\end{align*}$$

Then is equivalent to $(\forall y\,{<}\,t_0(x_0,\ldots ,x_{k-1})) B(x_0,\ldots ,x_{k-1},y))$ by ( $T\forall _{\mathrm {tp}}$ ) and by the meta-induction hypothesis.

We now show the formula version of Lemma 3.8(iii) (but without T).

Lemma 4.4. For any $\mathfrak {L}(\mathcal {S}^n)$ formula A whose free variables are among $x_0,\ldots ,x_{k-1}$ ,

Proof By ( $\exists \mbox {-val}$ ), it suffices to show

By ( $T{=}$ ) the premise implies ${\bigwedge }_{i<k}x_i\,{=}\,y_i$ , and so $A(x_0,\ldots ,x_{k-1})\,{\leftrightarrow }\,A(y_0,\ldots ,y_{k-1})$ . Apply Theorem 4.3.

This lemma may also be proved by meta-induction on the formula A. As opposed to the proof above, the proof by meta-induction can be extended to (the codes of) non-standard formulae A similarly to Lemma 3.8(iii) by sentence induction, where the number k of arguments can also be non-standard.

In $\mathbb {FCT}(\mathcal {S}^n)$ , we ignored those formulae in which the truth predicate T has itself (or, more precisely, its code $\ulcorner T\urcorner $ ) in its scope. One way to consider such formulae consistently is to ramify T into several different ones $T_0$ , $T_1$ ,…. This leads us to the next kind of truth theory, the finitist theories of ramified truth.

Definition 4.5 (Finitist ramified truth)

For $n\,{\geq }\,2$ , the $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ theory $\mathbb {FRT}_m(\mathcal {S}^n)$ is generated by the following non-logical axioms:

• ( $\mathbb {FA}(\mathcal {S}^n)$ ): all the axioms of $\mathbb {FA}(\mathcal {S}^n)$ with the schema of induction extended to $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formulae;

• ( $T\mathrm {term}$ ): for any k-ary $\mathcal {S}^n$ function symbol $\mathrm {f}$ (including $k\,{=}\,0$ , i.e., constant symbols);

• ( $T\mathrm {Atom}$ ): ;

• ( $T{\neg }\mathrm {Atom}$ ): ;

• ( $TT^{}_{\mathrm {tp}}$ ): for $i\,{<}\,j$ ;

• ( $T\neg T^{}_{\mathrm {tp}}$ ): for $i\,{<}\,j$ ;

• ( $T{\land }^{}_{\mathrm {tp}}$ ): ;

• ( $T{\lor }^{}_{\mathrm {tp}}$ ): ;

• ( $T\forall ^{}_{\mathrm {tp}}$ ): ;

• ( $T\exists ^{}_{\mathrm {tp}}$ ): ,

and the following non-logical rules: for any $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formula A and $\mathcal {S}^n$ term t in both of which x does not occur.

Lemma 4.6. For any $i,j\,{<}\,m$ ,

Proof Let $\mathrm {SubTerm}(v,u)$ express that $v$ is a subterm of u. For any fixed p, by induction on u we can prove

Let $w$ be the bounding term of u (in the sense defined before Lemma 2.10). By $(\exists \mbox {-val})$ we have p with , which implies the second premise. If then $x\,{\leq }\,p$ and so .

Lemma 4.7. For $j\,{\leq }\,i\,{\leq }\,m$ ,

In $\mathbb {FRT}_{m+1}(\mathcal {S}^n)$ , we can extend the uniform Tarski biconditional (Theorem 4.3) and internal reflection of equality (Lemma 4.4) to $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ .

Theorem 4.8. For any $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formula $A(x_0,\ldots ,x_{k-1})$ whose free variables are among $x_0,\ldots ,x_{k-1}$ ,

Theorem 4.8 implies the following lemma, the internal reflection of equality. This can, as Lemma 4.4, be extended to non-standard formulae A with non-standardly many arguments.

Lemma 4.9. For any $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formula A whose free variables are among $x_0,\ldots ,x_{k-1}$ ,

Note that $\mathbb {FRT}_0(\mathcal {S}^n)$ is identical with $\mathbb {FA}(\mathcal {S}^n)$ and that $\mathbb {FRT}_1(\mathcal {S}^n)$ is identical with $\mathbb {FCT}(\mathcal {S}^n)$ . Also, $\mathbb {FRT}_m(\mathcal {S}^n)$ is included in $\mathbb {FRT}_{m+1}(\mathcal {S}^n)$ along the obvious inclusion. Let us also consider the “limit” of these inclusions.

We may go even further. For any ordinal $\alpha $ in some standard ordinal notation system, we may define $\mathbb {FRT}_\alpha (\mathcal {S}^n)$ . Nonetheless, we do not consider these theories for the following reasons. First, we cannot even state the well-foundedness of $\alpha $ . For example, even if we try to formulate it by a transfinite induction rule, there is no way to express the induction hypothesis without unbounded quantifiers. Second, $\mathbb {FRT}_{<\omega }(\mathcal {S}^n)$ suffices for the following comparison to finitist analogues of two other famous truth theories.

4.2 Finitist self-referential truth

To obtain consistent theories of untyped or self-referential truth, certain naïve truth axioms have to be dropped. Two theories are particularly well known from the first-order setting: Kripke–Feferman truth and Friedman–Sheard truth, as discussed in the Introduction.

Let us start with the finitist theory of Kripke–Feferman truth.

In other words, we define $\mathbb {FKF}(\mathcal {S}^n)$ from $\mathbb {FCT}(\mathcal {S}^n)$ by strengthening the truth axioms for $\land $ , $\lor $ , $\forall $ , $\exists $ to codes of non-typed formulae and by adding ( $TT^{}_{\mathrm {sr}}$ ) and ( $T\neg T^{}_{\mathrm {kf}}$ ). Since the axiom ( $T\neg T^{}_{\mathrm {kf}}$ ) is specific to Kripke–Feferman truth, we write “kf” for its subscript, while “sr” of the others stands for “self-referential”.

Now the uniform Tarski biconditional schema (Theorem 4.8) can be extended to formulae with the truth predicate T, but with a restriction: T appears only positively, i.e., not in the scope of $\neg $ . (Recall that $\neg $ applies only to atomic formulae.)

Lemma 4.12 (Uniform Tarski biconditional for T-positive formulae)

For any $\mathfrak {L}(\mathcal {S}^n,T)$ formula $A(x_0,\ldots ,x_{k-1})$ whose free variables are among $x_0,\ldots ,x_{k-1}$ and in which the relation symbol T occurs only positively,

Before moving to the next truth theory, we explain the $\omega $ -soundness of our finitist theories due to the linguistic restriction. Any finitist formula is absolute between the standard model $\omega $ and any non-standard model. Let us explain this more precisely. First, for any $\mathfrak {L}(\mathcal {S}^n,T)$ structure $(M,T^M)$ , we know that $(M,T^M)\models A(k_0,\ldots ,k_{\ell -1}) \,{\leftrightarrow }\, (\omega ,T^M{\cap }\,\omega )\models A(k_0,\ldots ,k_{\ell -1})$ for any $\mathfrak {L}(\mathcal {S}^n,T)$ formula A and $k_0,\ldots ,k_{\ell -1}\,{\in }\,\omega $ . Hence, if an $\mathfrak {L}(\mathcal {S}^n,T)$ theory $\mathbb {T}$ has a model, then it has a standard model, i.e., a model whose elements are only standard natural numbers. This means that the consistency of $\mathbb {T}$ implies the $\omega $ -soundness of $\mathbb {T}$ .

Hence, the finitist theory of Friedman–Sheard truth, which we introduce next, is $\omega $ -sound (if consistent), contrastingly to the well known fact that the first-order Friedman–Sheard truth theory is $\omega $ -inconsistent.

Definition 4.13 (Finitist Friedman–Sheard truth)

For $n\,{\geq }\,2$ , the $\mathfrak {L}(\mathcal {S}^n,T)$ theory $\mathbb {FFS}(\mathcal {S}^n)$ is generated by the following non-logical axioms:

• ( $\mathbb {FCT}(\mathcal {S}^n)$ ): all the axioms of $\mathbb {FCT}(\mathcal {S}^n)$ ;

• ( $TT{=}$ ): $\mathrm {CTerm}_n(u)\,{\land }\, \mathrm {CTerm}_n(v)\,{\land }\, \mathrm {CTerm}_n(x)\,{\land }\, \mathrm {CTerm}_n(y)$ ;

• ( $T{\land }^{}_{\mathrm {sr}}$ ): ;

• ( $T{\lor }^{}_{\mathrm {sr}}$ ): ;

• ( $T{\neg }^{}_{\mathrm {sr}}$ ): ;

• ( $T\forall ^{}_{\mathrm {sr}}$ ): ;

• ( $T\exists ^{}_{\mathrm {sr}}$ ): ,

with ( $\exists \mbox {-val}$ ) and the following, where B is any $\mathfrak {L}(\mathcal {S}^n,T)$ formula whose free variables are among $y_0,\ldots ,y_{k-1}$ .

We write $\mathbb {FFS}_m(\mathcal {S}^n)\vdash A$ if there is an $\mathbb {FFS}(\mathcal {S}^n)$ proof of A in any path through which the rule ( $T\mbox {-Intr}$ ) is used at most m times and ( $T\mbox {-Elim}$ ) is not used at all.

( $TT{=}$ ) is a special case of ( $TT_{\mathrm {sr}}$ ) in $\mathbb {FKF}(\mathcal {S}^n)$ , with x being a code of equality. Here to $=$ we apply the dot notation iteratedly: is an $\mathcal {S}^2$ term with two arguments, to which we apply Definition 3.2 again. Recall Remark 3.6 for the use of dot notation on the “unofficial” function symbols .

In $\mathbb {FFS}_m(\mathcal {S}^n)\vdash A$ , we count the number of applications of rules only along paths. Thus, $\mathbb {FFS}_m(\mathcal {S}^n)$ is closed under modus ponens (MP): if $\mathbb {FFS}_m(\mathcal {S}^n)\vdash A$ and $\mathbb {FFS}_m(\mathcal {S}^n)\vdash A\,{\to }\,B$ then $\mathbb {FFS}_m(\mathcal {S}^n)\vdash B$ . Not only by this but also by the prohibition of ( $T\mbox {-Elim}$ ), our way of restricting $\mathbb {FFS}_m(\mathcal {S}^n)$ differs from Halbach’s $\mathbf {FS}_m$ in [Reference Halbach11].

We define $\mathbb {FFS}(\mathcal {S}^n)$ from $\mathbb {FCT}(\mathcal {S}^n)$ by strengthening the truth axioms for $\land $ , $\lor $ , $\forall $ , $\exists $ to codes of non-typed formulae and by adding ( $TT{=}$ ) and the truth axiom for $\sim $ as well as the rules (T-Intr) and (T-Elim). The asymmetry between the two rules is not essential because of ( $\exists \mbox {-val}$ ), but has some practical convenience.

Our rules differ from $(\mbox {truth-rule})$ in the Introduction, restricted to sentences, which was employed in Halbach’s [Reference Halbach11] formulation of first-order Friedman–Sheard truth theory and derives our strengthened versions as follows.

However, in our finitist setting, we cannot form $\forall y_0,\ldots ,y_{k-1}B(y_0,\ldots ,y_{k-1})$ nor these derivations. Recall also the weakness of our truth axioms for quantifiers, as mentioned in Remark 4.2.

Another difference from Halbach’s axiomatization of first-order Friedman–Sheard truth theory is that ( $TT{=}$ ) is explicitly included. In the first-order setting, or more precisely, in the presence (or with the justification) of a term for valuation function, ( $TT{=}$ ) and ( $TT{<}$ ), the latter being similarly defined, are provable from the other axioms and rules as follows, where we omit the antecedent “ $\mathrm {CTerm}_n(u)\,{\land }\, \mathrm {CTerm}_n(v)\,{\to }$ ”.

One can see the essential use of

in this proof. Since we do not officially have nor justify $\mathrm {val}$ as a bounded definable function in our setting, we have to simulate the use of

by a further rule as follows.

Proposition 4.14. For $n\,{\geq }\,2$ , the following rule is derivable in $\mathbb {FFS}_0(\mathcal {S}^n)$ , where x does not occur in A.

We postpone the proof until the end of this section. As we do not know if we can derive this rule without ( $TT{=}$ ), we added ( $TT{=}$ ) as an axiom, and it suffices to derive the respective assertions for $<$ , $\neq $ , and $\not <$ . These are necessary for the interpretability in Section 5.2, whose first-order analogue is due to Halbach [Reference Halbach10] and which is, we think, inevitable for any formulation of Friedman–Sheard truth to be called appropriate.

(1) below allows us to exchange and $\mathrm {num}(x)$ in the scope of under $\mathrm {CTerm}(x)$ . By ( $\exists \mbox {-val}$ ), we may assume for fresh z, and replace by and by $\mathrm {num}(\mathrm {num}(z))$ by Lemmata 4.4 and 3.8(ii). However, to replace $\mathrm {num}(\mathrm {num}(z))$ by $\mathrm {num}(x)$ , we need the argument here.

Proof (1) We can derive this as follows.

(2) By Lemma 4.15(1), the following derivation suffices for the third conjunct.

At this point, one can see the reason why we do not employ the standard upper-dot notation to denote $\mathrm {num}(x)$ by , with which it is not easy to distinct and $\mathrm {num}(\mathrm {num}(z))$ . On the other hand, we need extensive use of the other standard notation of lower-dot. For example, we have to be able to denote and the iterated applications in , , , and so on.

Proof of Proposition 4.14. We can consider the following derivation.

We will also consider one more self-referential truth theory, the finitist Kripke–Feferman–Burgess truth theory $\mathbb {FKFB}(\mathcal {S}^n)$ . We do not give the definition of $\mathbb {FKFB}(\mathcal {S}^n)$ here but postpone it to Section 8.

5.1 Ramifying interpretation

To “embed” $\mathbb {FFS}(\mathcal {S}^n)$ in $\mathbb {FRT}_{<\omega }(\mathcal {S}^n)$ , we ramify the truth predicate T into different truth predicates $T_j$ ’s. For this, we introduce the ramifying interpretation $\mathfrak {r}(m)$ of rank m, which replaces the truth predicate T with $T_{m-1}$ and any coded occurrence of T, at the depth j in the (iterated) scope of T, with $T_{m-j-1}$ . Since T may apply to codes of non-standard formulae, we also need the coded version $\mathrm {ram}^m$ of the interpretation.

Definition 5.1 ( $A^{\mathfrak {r}(m)}$ , $\mathrm {ram}^m$ )

For any natural number m, we define an interpretation $\mathfrak {r}(m)$ of $\mathfrak {L}(\mathcal {S}^n,T)$ in $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ by meta-recursion on m as follows. Let $A^{\mathfrak {r}(m)} \,{:\equiv }\,A$ for any atomic $\mathfrak {L}(\mathcal {S}^n)$ formula A;

$$ \begin{align*} (T(t))^{\mathfrak{r}(m)} \;&{:\equiv} \begin{cases} 0\,{=}\,0, & \mbox{ for }m\,{=}\,0,\\ T_{m-1}(\mathrm{ram}^{m-1}(t)), & \mbox{ for }m\,{>}\,0; \end{cases} \end{align*} $$

and let $\mathfrak {r}(m)$ commute with $\neg $ (applied to atomic formulae), $\land $ , $\lor $ , and bounded quantifiers; where by meta-recursion on m and Theorem 2.11 we introduce an $\mathcal {S}^2$ function $\mathrm {ram}^m$ with the following defining axioms.

We easily see $\mathbb {FA}(\mathcal {S}^2) \vdash \ulcorner A^{\mathfrak {r}(m)}\urcorner \,{=}\,\mathrm {ram}^m(\ulcorner A\urcorner )$ for any $\mathfrak {L}(\mathcal {S}^n,T)$ formula A, and generalize it as follows.

Lemma 5.2. For any $\mathfrak {L}(\mathcal {S}^n,T)$ formula B,

Proof We prove (2) by meta-induction on the proof of $\mathbb {FFS}(\mathcal {S}^n)\vdash A$ , because (1) is easier. We focus only on the two non-obvious cases where the last inference is either ( $T\mbox {-Intr}$ ) or ( $T\mbox {-Elim}$ ).

First consider the case of ( $T\mbox {-Intr}$ ), say the last inference is the upper below. The induction hypothesis yields $k,\ell $ such that $\mathbb {FRT}_{m+k}(\mathcal {S}^n) \vdash B^{\mathfrak {r}(m)} (x_0,\ldots ,x_{k-1})$ for $m\,{\geq }\,\ell $ . For such m, we can construct the lower derivation below, which is a proof in $\mathbb {FRT}_{m+1+k}(\mathcal {S}^n)$ . Thus we can take k and $\ell {+}1$ .

Next consider the case of ( $T\mbox {-Elim}$ ), say the last inference is ( $T\mbox {-Elim}$ ) as in Definition 4.13. Then the induction hypothesis yields k and $\ell $ such that for $m\,{\geq }\,\ell $ , where we may assume $\ell \,{>}\,0$ . Then $\mathbb {FRT}_{m+k}(\mathcal {S}^n)\vdash B^{\mathfrak {r}(m-1)}(x_0,\ldots ,x_{k-1})$ by Lemma 5.2 and Theorem 4.8, and we can take $k{+}1$ and $\ell {-}1$ .

5.2 Ignoring interpretation

To interpret $\mathbb {FRT}_{<\omega }(\mathcal {S}^n)$ in $\mathbb {FKF}(\mathcal {S}^n)$ and in $\mathbb {FFS}(\mathcal {S}^n)$ , we can just forget the ramification. As in the definition of $\mathfrak {r}(m)$ , we have to consider that T may apply to codes of non-standard formulae. This is why we need also the coded version of the interpretation.

Definition 5.5. For any natural number m, we define an interpretation $\mathfrak {i}(m)$ of $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ in $\mathfrak {L}(\mathcal {S}^n,T)$ by meta-recursion on m as follows.

Let $A^{\mathfrak {i}(m)} \,{:\equiv }\,A$ for any atomic $\mathfrak {L}(\mathcal {S}^n)$ formula A;

$$ \begin{align*} (T_j(u))^{\mathfrak{i}(m)} \;&{:\equiv}\; T(\mathrm{ign}^{j}(u)) \mbox{ for }j\,{<}\,m; &\qquad (T_j(u))^{\mathfrak{i}(m)} \;&{:\equiv}\; 0\,{=}\,0 \mbox{ for }j\,{\geq}\,m; \end{align*} $$

and let $\mathfrak {i}(m)$ commute with $\neg $ (applied to atomic formulae), $\land $ , $\lor $ , and bounded quantifiers; where by meta-recursion on m and Theorem 2.11, we introduce $\mathrm {ign}^m$ so that $\neg \mathrm {Sent}_{n,T_0,\ldots ,T_{m-1}}(w) \,{\to }\,\mathrm {ign}^m(w)\,{=}\,\ulcorner 0\,{=}\,0\urcorner $ and else

Let us start with interpreting in $\mathbb {FKF}(\mathcal {S}^n)$ . The only axiom of $\mathbb {FRT}_m(\mathcal {S}^n)$ whose ignoring version is missed in $\mathbb {FKF}(\mathcal {S}^n)$ is $(T{\neg }T^{}_{\mathrm {tp}})$ . Therefore, the next lemma is the key for the interpretability.

Proof We proceed by meta-induction on m. We reason in $\mathbb {FKF}(\mathcal {S}^n)$ by sentence induction on $w$ (cf. Theorem 3.11). We may assume $\mathrm {Sent}_{n,T_0,\ldots ,T_{m-1}}(w)$ .

Consider the case where

. Then

. Thus by $(T{=})$ and $(T\neg \mbox {Atom})$ , we have

and so

, to which we apply ( $\exists \mbox {-val}$ ).

In the cases where , or , we reason similarly.

Next consider the case of

with $j\,{<}\,m$ . Then

. Since

, by ( $T\neg T^{}_{\mathrm {kf}}$ ) and ( $TT^{}_{\mathrm {tp}}$ ), we obtain

By meta-induction hypothesis,

, to which we apply ( $\exists \mbox {-val}$ ).

In the case where with $j\,{<}\,m$ , we reason similarly.

The other cases are immediate by induction hypothesis.

Theorem 5.7. For $n\,{\geq }\,2$ and any $\mathfrak {L}(\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formula A whose free variables are among $x_0,\ldots ,x_{k-1}$ , if $\mathbb {FRT}_m(\mathcal {S}^n)\vdash A(x_0,\ldots ,x_{k-1})$ then

$$\begin{align*}\mathbb{FKF}(\mathcal{S}^n) \vdash A^{\mathfrak{i}(m)}(x_0,\ldots,x_{k-1}).\end{align*}$$

Next let us interpret $\mathbb {FRT}_m(\mathcal {S}^n)$ in $\mathbb {FFS}_m(\mathcal {S}^n)$ by $\mathfrak {i}(m)$ . Although this is not necessary for the final result on the $\mathfrak {L}(\mathcal {S}^n)$ conservations (except for the claim we will make in connection with [Reference Sato and Walker24]), we claimed that the axiom $(TT{=})$ should be included in $\mathbb {FFS}(\mathcal {S}^n)$ because we think this interpretability should be feasible. The only axiom of $\mathbb {FRT}_m(\mathcal {S}^n)$ whose ignoring version is missed in $\mathbb {FFS}(\mathcal {S}^n)$ is $(TT^{}_{\mathrm {tp}})$ . Therefore the next lemma is the key for the interpretability.

Proof We prove this statement by meta-induction on m. By applying $(T\mbox {-Intr})$ to the equality axiom, we can extend Lemma 4.4 to $\mathfrak {L} (\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formulae A. Hence, we have to show

. By Lemma 3.8(ii), this is equivalent to

which we now prove by sentence induction on $w$ .

If $w$ does not code any $\mathfrak {L} (\mathcal {S}^n,T_0,\ldots ,T_{m-1})$ formula or if it codes an atomic $\mathfrak {L}(\mathcal {S}^n)$ formula, this follows from ( $TT{=}$ ) and Lemma 4.16(2) by Lemma 3.8(i). Below we assume $\mathrm {Sent}_{n,T_0,\ldots ,T_{m-1}}(w)$ .

If

with $j\,{<}\,m$ , then $ \mathrm {CTerm}_n(v) $ and so we can derive the following.

If

with $v$ being a code of atomic formula, we have the following derivation, where we omit “ $\mathit {AtSent}_{n,T_0,\ldots ,T_{m-1}}(v) \,{\to }$ ”.

If or , we proceed similarly.

Finally assume

or

. For the former, consider the following derivation, where $(\mathrm {IH})$ is the induction hypothesis for $v$ and where we omit “ $\mathrm {Var}(x)\,{\land }\,\mathrm {CTerm}_n(u) \,{\land }\,\mathrm {Sent}_{n,T_0,\ldots ,T_{m-1}}(w) \,{\to }$ ”. Note that another sentence induction shows

.

What we have obtained can be summarized as follows.

1. (1) $\mathbb {FRT}_m(\mathcal {S}^n) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FKF}(\mathcal {S}^n)$ and $\mathbb {FRT}_m(\mathcal {S}^n) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FFS}_m(\mathcal {S}^n)$ for any $m\,{\geq }\,0$ and $n\,{\geq }\,2$ .

2. (2) Thus $\mathbb {FRT}_{<\omega }(\mathcal {S}^n) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FKF}(\mathcal {S}^n)$ and $\mathbb {FRT}_{<\omega }(\mathcal {S}^n) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FFS}(\mathcal {S}^n)$ for $n\,{\geq }\,2$ .

5.3 Summary of this section

We close this section by summarizing the $\mathfrak {L}(\mathcal {S}^n)$ conservations that we have shown.

$$ \begin{align*} &\mathbb{FA}(\mathcal{S}^{n+1}) \;{\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FET}(\mathcal{S}^n)\\&\quad {\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FCT}(\mathcal{S}^n) \,{\equiv}\, \mathbb{FRT}_1(\mathcal{S}^n) \;{\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FRT}_2(\mathcal{S}^n) \;{\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \ldots\\&\quad {\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FRT}_{<\omega}(\mathcal{S}^n) \;{=}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FFS}(\mathcal{S}^n) \;{\leq}_{\mathfrak{L}(\mathcal{S}^n)}\; \mathbb{FKF}(\mathcal{S}^n) \end{align*} $$

These are familiar for experts of axiomatic truth theories, except the first conservation $\mathbb {FA}(\mathcal {S}^{n+1}) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FET}(\mathcal {S}^n)$ .

Our next goal is to show that all these theories are $\mathfrak {L}(\mathcal {S}^n)$ equivalent to $\mathbb {FA}(\mathcal {S}^{n+1})$ . What remains to be shown is $\mathbb {FKF}(\mathcal {S}^n) \,{\leq }_{\mathfrak {L}(\mathcal {S}^n)}\, \mathbb {FA}(\mathcal {S}^{n+1})$ , which we will establish in the next two sections.