2.1 Determinateness

We describe and axiomatize $\mathfrak {D}$ , the set of determinate sentences in more detail. First, we declare all atomic sentences of the base language $\mathcal {L}_0$ determinate:

1. D1

That is, all closed identity statements are determinate. If there were further predicate symbols in the base language, analogous axioms would be added. The axioms below will enable us to prove that all sentences of the base language $\mathcal {L}_0$ are determinate.

If and only if $\phi $ is determinate, that is, in $\mathfrak {D}$ , we add a copy $\mathrm {T}\ulcorner \phi \urcorner $ of $\phi $ to $\mathfrak {D}$ . This is stated for arbitrary closed terms t, not only numerals $\ulcorner \phi \urcorner $ :

1. D2

The expression ${t}^{\circ }$ stands for the value of the term t. We cannot have a function symbol ${}^{\circ }$ in our language, and thus this function needs to be expressed by a suitable formula.

A negated sentence is determinate iff the sentence is:

1. D4

Whether a conjunction is determinate depends on the determinateness of its conjuncts. Clearly, if both conjuncts are determinate, then so is their conjunction; and if both are indeterminate, so is their conjunction. But is a conjunction determinate if only one conjunct is?

Our choice here is motivated by the function of truth as a generalizing device. Typically, a universal generalization would have the form $\forall x\, (\phi (x)\rightarrow \mathrm {T} x)$ . The formula $\phi (x)$ could express that x is a provable sentence of Peano arithmetic or that x is sentence of the form $\mathrm {T}\ulcorner \mathrm {T}\cdots \ulcorner 0\! =\!0\urcorner \urcorner \cdots \urcorner $ . We consider the simple example $\forall x\, (x\! =\!\ulcorner 0\! =\! 0\urcorner \rightarrow \mathrm {T} x)$ whose antecedent is satisfied by a single sentence only; we ‘generalize’ via semantic ascent over all sentences of the form $0\! =\! 0$ . Of course, generalization via semantic ascent is not needed for a single sentence; but ‘generalizing’ over a single sentence demonstrates the principle. If $\mathfrak {D}$ contains the sentences required for generalizing via semantic ascent, then $\forall x\, (x=\ulcorner 0\! =\! 0\urcorner \rightarrow \mathrm {T} x)$ should be determinate. The determinateness of universal quantified sentences depends on their instances. Let $\lambda $ be an indeterminate sentence. Then the instance

(1) $$ \begin{align} \ulcorner \lambda\urcorner\! =\!\ulcorner 0\! =\! 0\urcorner\,\rightarrow\, \mathrm{T} \ulcorner \lambda\urcorner \end{align} $$

has a determinate antecedent and an indeterminate consequent, by axioms D1 and D2. If the instance (1) were indeterminate, so would be the universal generalization $\forall x\, (x\! =\!\ulcorner 0\! =\! 0\urcorner \rightarrow \mathrm {T} x)$ by any reasonably determinateness rules for quantification. Hence, our axioms for $\mathrm {D}$ should declare (1) determinate.

The determinateness of (1) should follow from the determinateness of the antecedent $\ulcorner \lambda \urcorner \! =\!\ulcorner 0\! =\! 0\urcorner $ . The falsity of the determinate antecedent renders the consequent irrelevant; and the truth value of (1) does not depend on the consequent. Thus, we stipulate that a sentence $\phi \rightarrow \psi $ can inherit its determinateness from a determinate and false antecedent. Equally, we also postulate that it is determinate if it has a true and determinate consequent.

This method of generalization applies equally if we do not only ‘generalize’ over a single sentence such as $0 = 0 $ , but, for instance, over all sentences provable in $\mathsf {PA}$ . Moreover, iterated semantic ascent permits generalization over all sentences $0\! =\! 0$ , $\mathrm {T}\ulcorner 0\! =\! 0\urcorner $ , $\mathrm {T}\ulcorner \mathrm {T}\ulcorner 0\! =\! 0\urcorner \urcorner $ , and so on.

In our official language negation and conjunction are our only connectives. $\phi \rightarrow \psi $ is conceived as an abbreviation of $\neg (\phi \wedge \neg \psi )$ . In the presence of D4 we can then state our determinateness axiom for binary connectives in the following way:

• $\phi \wedge \psi $ is determinate iff:

• $\phi $ and $\psi $ are both determinate, or

• one of the conjuncts is false and determinate.

This is expressed in the following axiom:

1. D5

By our conventions above, $\mathrm {F} x$ abbreviates , namely, falsehood of x; also note that $\mathrm {D} x$ and are equivalent by the axiom D4. By axiom T4, $\mathrm {F}$ can be replaced with $\neg \mathrm {T}$ .

Another way to argue for our treatment of binary connectives would be to argue that sentences such as $\forall x\, (x=\ulcorner 0\! =\! 0\urcorner \rightarrow \mathrm {T} x)$ and its instance (1) are only about the determinate sentence $0\! =\! 0$ and should be treated just like $\mathrm {T}\ulcorner 0\! =\! 0\urcorner $ . The latter is determinate by axiom D1. Thus, it may be argued, semantic ascent applies to (1) and the universally quantified sentence in the same way it applies to $\mathrm {T}\ulcorner 0\! =\! 0\urcorner $ . However, making precise the underlying notion of aboutness is notoriously challenging (see Picollo [Reference Picollo33, Reference Picollo34]); we do not attempt to pursue this line here.

Universally quantified sentences are treated as conjunctions of their instances:

1. D6

Injecting the notion of determinateness into the object language makes it possible to apply the truth and determinateness predicates to sentences containing $\mathrm {D}$ . We treat both, truth and determinateness, as completely type-free: $\mathrm {D}$ and $\mathrm {T}$ can be applied meaningfully to any sentence of the language. This is in contrast to traditional conceptions of groundedness, which apply to sentences with truth, while truth cannot be applied to sentences containing groundedness claims. Therefore, in our setting, the question arises whether atomic sentences of the form $\mathrm {D} t$ are determinate.

Treating $\mathrm {D}$ like an elementary predicate of $\mathcal {L}_0$ , that is, a predicate from $\mathcal {L}_0$ would suggest the axiom . This would mean that we have $\mathrm {T}\ulcorner \mathrm {D} t\urcorner \leftrightarrow \mathrm {D} t$ for all terms t. However, when it comes to complex sentences, we axiomatize $\mathrm {D}$ in terms of $\mathrm {T}$ in D5 and D6. Hence, it looks unsafe to declare all sentence $\mathrm {D} t$ determinate; $\mathrm {D}$ should be treated with the same caution as $\mathrm {T}$ : If, and only if $\phi $ is determinate, we stipulate that $\mathrm {D}\ulcorner \phi \urcorner $ is determinate. Generalizing this to terms other than numerals gives the following axiom analogous to D2:

1. D3

This concludes the list of determinateness axioms.

2.2 Disquotation

We still need to add axioms stipulating that a sentence $\mathrm {T}\ulcorner \phi \urcorner $ in $\mathfrak {D}$ is always obtained by semantic ascent and thus a ‘copy’ of $\phi $ . Formally, we require that $\mathrm {D}\ulcorner \phi \urcorner \rightarrow (\mathrm {T}\ulcorner \phi \urcorner \leftrightarrow \phi )$ is a theorem of our theory $\mathsf {CD}$ for every sentence of $\mathcal {L}$ . We also require that our theory proves a more general, ‘uniform’ version of the disquotation schema; that is, we generalize it by quantifying over the closed terms and postulate for every formula $\phi (x_1,\ldots , x_n)$ with at most $x_1,\ldots ,x_n$ the following:

1. DDS

In this uniform determinate disquotation schema the expression stands for a complex term expressing that $t_1,\ldots ,t_n$ are formally substituted for the free variables in $\phi (x_1,\ldots ,x_n)$ . This permits us to bind the variables $t_1,\ldots ,t_n$ in . This concludes the list of axioms for our theory $\mathsf {CD}$ .

2.3 Alternative axiomatizations

In the presence of the compositional axioms T4 –T6, we need to stipulate DDS only for atomic $\phi $ . Whatever our policy on disquotation is, we can always focus on the atomic instances; T4 –T6 will ensure that also all instances of $T\ulcorner \phi \urcorner \leftrightarrow \phi $ will be provable for all sentences $\phi $ built from those atomic formulae. Because we aim at an axiomatization that is as lean as possible for technical reasons, our official axiomatization does not feature schema DDS, but only disquotational axioms concerning atomic sentences. In particular, schema DDS can be replaced with the following three axioms:

1. T1

2. T2

3. T3

T2 can be derived from the following instance of DDS and axiom D3:

(2)

Thus the three axioms imply DDS in the presence of the other axioms as is established in Lemma 3.1.

The principles T3, (2), and T2 show that the truth and determinateness predicates interact in a serious manner. In particular, $\mathrm {D}$ is not only a formalization of a metatheoretic notion that has been injected into the object language; predicates of the object language—and especially the truth predicate—can be meaningfully applied to sentences with $\mathrm {D}$ .

From (2) we can prove one direction of the T-sentences $\phi \rightarrow \mathrm {T}\ulcorner \phi \urcorner $ for all sentences without $\mathrm {T}$ , including those with $\mathrm {D}$ . Some sentences without $\mathrm {T}$ but with $\mathrm {D}$ —such as $\mathrm {D}\ulcorner \lambda \urcorner $ for some indeterminate $\lambda $ —are not determinate, and thus DDS will not give us the T-schema for some sentences with $\mathrm {D}$ . Therefore, we consider the addition of the right-to-left direction to T2:

1. T2 ⁺ .

This sentence is not provable from the disquotation schema DDS, and is not covered by our policy about $\mathfrak {D}$ . With $\textsf {T2}^{+}$ we can derive all T-sentences $\mathrm {T}\ulcorner \phi \urcorner \leftrightarrow \phi $ as long as $\phi $ is $\mathrm {T}$ -free.

Determinateness is one of many predicates such as analyticity, knowledge, necessity, and logical validity that are deeply intertwined with truth. Having the T-sentences $\mathrm {T}\ulcorner \phi \urcorner \leftrightarrow \phi $ also for $\phi $ containing such predicates may be desirable; but we know already that this is not be possible for all such $\phi $ (see Halbach [Reference Halbach20]), as axioms like $\textsf {T2}^{+}$ may engender inconsistency, depending on the axioms for the other predicates. We cannot think of a better strategy than to consider these axioms on a case-by-case basis and to endorse axioms like $\textsf {T2}^{+}$ , as long as they do not yield undesirable consequences. With $\textsf {T2}^{+}$ we leave the safety of disquotation for determinate sentences only. In the case of determinateness we show that axiom $\textsf {T2}^{+}$ does not affect the -soundness of our theory and thus $\textsf {T2}^{+}$ can be added as a further optional axiom.

The quantifier $\forall t$ in D2, D3, T2, T3, and $\textsf {T2}^{+}$ ranges over all closed terms, including those that do not denote a sentence. It may be desirable to restrict the quantifier in these axioms to terms denoting sentences. This would allow us to make any stipulations about sentences $\mathrm {T} t$ and $\mathrm {D} t$ where t fails to denote a sentence. We have chosen the version above for their simplicity.

2.4 Extensionality

In this section we discuss two additional axioms that are irrelevant to most of the metamathematical properties of our system. However, at least the first of these axioms is conceptually important. The issue goes beyond our specific system, and even beyond axiomatic theories, because semantic theories of truth are affected as well.

We expect truth to be an extensional notion unlike necessity, apriority, or being known. The compositional axioms prove the extensionality of truth for sentences in the following sense: Substituting a subformula in a sentence $\phi $ with another subformula with the same truth value does not affect truth or falsity of $\phi $ .

One would expect extensionality at the level of terms as well. That is, if the terms s and t have the same value, a sentence $\phi (t)$ ought to be true if, and only if $\phi (s)$ is true. However, this principle fails to be provable from the axioms we have listed so far. Only restricted versions can be proved, for instance, if the values of s and t are determinate or $\chi $ is purely arithmetical. Without axiom R1, we can refute $\mathrm {T}\ulcorner \mathrm {T} s\urcorner \wedge \neg \mathrm {T}\ulcorner \mathrm {T} t\urcorner $ under the assumption $s=t$ only for determinate s and t, but not for all terms. Here, we call a term determinate iff its value is. Thus, we state extensionality with respect to terms as an axiom:

1. R1

Axiom R1 is closely related to the question whether identity is a logical constant. Using R1, we can prove the truth of $\forall x\,\forall y\, (x\! =\! y\rightarrow (\phi (x)\rightarrow \phi (y)))$ for all formulae $\phi $ , which is a logical truth if identity is a logical constant. If identity is a logical constant and our theory is to prove all logically valid sentences, then we have to add R1 (or some other axiom).Footnote ³

So far we have focused on truth. For determinateness we also stipulate extensionality:

1. R2

Whether determinateness should also be extensional may be less clear. For our purposes the extensionality of determinateness is not decisive. Omitting R2 would not affect the results below.

4.1 The model

Let us define $\mathcal {L}$ -formulae $\mathcal {D}_{i} ( x )$ ( $1 \leq i \leq 6$ ) as follows:

We thereby define $\mathcal {D} ( x ) :\Leftrightarrow \bigvee _{1 \leq i \leq 6} \mathcal {D}_{i} ( x )$ . $\mathcal {D}$ describes the closure condition of a determinateness predicate $\mathrm {D}$ (relative to a fixed interpretation of the truth predicate $\mathrm {T}$ ).

By $( \mathbb {N}, X, Y )$ let us denote the $\mathcal {L}$ -structure with domain

, the set of natural numbers, in which $\mathrm {D}$ is interpreted by X, $\mathrm {T}$ is interpreted by Y, and all the other symbols receive the standard interpretations. Both $\mathrm {D}$ and $\mathrm {T}$ occur only positively in $\mathcal {D}$ . Hence, for each

, $\mathcal {D}$ induces the following monotone operator

so that

In general, given a monotone operator

, we say that

is $\Gamma $ -closed if $\Gamma ( X ) \subset X$ , and that

is a $\Gamma $ -fixed point if $\Gamma ( X ) = X$ .

Given an $\mathcal {L}_0$ -formula or $\mathcal {L}_0$ -term e, let us denote its standard interpretation by $e^{\mathbb {N}}$ and its Gödel number by $\# e$ . For instance, denote the set of the Gödel numbers of all $\mathcal {L}$ -sentences.

For each ordinal $\alpha $ , we define sets $D_{\alpha }$ and $T_{\alpha }$ as follows:

It is obvious from the definition that $T_{\lambda } = D_{\lambda } \cap T_{\lambda }$ for all limit ordinals $\lambda $ . We can also show by induction on ordinals that

for all ordinals $\xi $ .

The next two propositions are obvious by definition.

Proof For example, we have and $( ( \# a )^{\circ } )^{\mathbb {N}} = a^{\mathbb {N}}$ , from which claim 2 follows, since we have

$$ \begin{align*} a^{\mathbb{N}} \in D_{\xi} \quad \Leftrightarrow \quad ( \mathbb{N}, D_{\xi}, T_{\xi} ) \models \mathcal{D}_{2} ( \# \mathrm{T} a ) \quad \Leftrightarrow \quad \# \mathrm{T} a \in D_{\xi + 1}.\\[-33pt] \end{align*} $$

Let us define a formula $x \approx y$ as follows:

Namely, for natural numbers $n, m \in \mathbb {N}$ , $n \approx ^{\mathbb {N}} \! m$ means that either $n = m$ or n and m code numerically equivalent $\mathcal {L}$ -sentences in the sense that they are substitution instances of the same $\mathcal {L}$ -formula with closed terms with the same values. Since all the existential quantifiers in the definition can actually be bounded by the values of some primitive recursive functions on x and y, the relation $\approx ^{\mathbb {N}}$ is primitive recursive in the value function (or the index function $[ \, \cdot \, ]$ of the primitive recursive functions) and thus provably recursive in $\mathsf {PA}$ . The extensionality axioms R1 and R2 say that truth and determinateness are invariant across numerically equivalent $\mathcal {L}$ -sentences, namely, sentences $\phi $ and $\psi $ with $\# \phi \approx ^{\mathbb {N}} \# \psi $ .Footnote ⁷ The following proposition is shown by a straightforward induction on ordinals.

The next is the main lemma of this subsection.

Proof By simultaneous induction on $\xi $ . The base step is trivial. Let $\xi $ be a limit ordinal and $\eta \geq \xi $ . For each $n \in D_{\xi }$ , there is some $\zeta < \xi $ such that $n \in D_{\zeta }$ , for which we have $D_{\zeta } \cap T_{\xi } = D_{\zeta } \cap T_{\zeta } = D_{\zeta } \cap T_{\eta }$ and $D_{\zeta } \subset D_{\eta }$ by the induction hypothesis.

Let $\xi $ be a successor ordinal. Claim 1 is shown by sub-induction on $\eta $ ; we can assume $\eta> \xi $ . First, suppose $\eta $ is a limit. If $n \in D_{\xi } \cap T_{\xi }$ , then $n \in T_{\eta }$ by definition. Let $n \in D_{\xi } \cap T_{\eta }$ for the converse. By definition, there is some $\zeta < \eta $ such that $n \in D_{\zeta } \cap T_{\zeta }$ . If $\zeta < \xi $ , then $D_{\zeta } \cap T_{\zeta } \subset T_{\xi }$ by the induction hypothesis for 1; otherwise, $\xi \leq \zeta < \eta $ and thus $D_{\xi } \cap T_{\zeta } = D_{\xi } \cap T_{\xi }$ by the sub-induction hypothesis. Next, suppose $\eta $ is a successor, and take any $n = \# \phi \in D_{\xi }$ ( ) for an $\mathcal {L}$ -sentence $\phi $ . If $\phi $ is an $\mathcal {L}_0$ -atomic, then the claim follows from Lemmata 4.2.1 and 4.3.1. If $\phi = \mathrm {T} a$ for some closed $\mathcal {L}_0$ -term a, then we have $a^{\mathbb {N}} \in D_{\xi - 1}$ by Lemma 4.2.2, and thus we obtain

$$\begin{align*}n \in T_{\xi} \stackrel{ 4.3.2 }{ \quad \Leftrightarrow \quad } a^{\mathbb{N}} \in T_{\xi - 1} \stackrel{ \mathrm{IH} }{ \quad \Leftrightarrow \quad } a^{\mathbb{N}} \in T_{\eta - 1} \stackrel{ 4.3.2 }{ \quad \Leftrightarrow \quad } n \in T_{\eta}, \end{align*}$$

where ‘IH’ denotes the induction hypothesis. If $\phi = \mathrm {D} a$ , then we have $a^{\mathbb {N}} \in D_{\xi - 1}$ by Lemma 4.2.3, and thus $n \in T_{\xi }$ and $a^{\mathbb {N}} \in D_{\eta - 1}$ by Lemma 4.3.3 and the induction hypothesis for 2, respectively, the latter of which implies $n \in T_{\eta }$ again by Lemma 4.3.3; hence, both $n \in T_{\xi }$ and $n \in T_{\eta }$ hold. We move on to the cases for complex $\phi $ s. Let $\phi = \psi \wedge \theta $ . By Lemma 4.2.5, $n \in D_{\xi }$ implies

$$ \begin{align*} & \bigl( \# \psi, \# \theta \in D_{\xi - 1} \bigr) \vee \bigl( \# \neg \psi \in D_{\xi - 1} \cap T_{\xi - 1} \bigr) \vee \bigl( \# \neg \theta \in D_{\xi - 1} \cap T_{\xi - 1} \bigr). & \end{align*} $$

If the first disjunct holds, then we have

$$ \begin{align*} n \in T_{\xi} \stackrel{ 4.3.5 }{ \ \ \Leftrightarrow \ \ } \# \psi, \# \theta \in T_{\xi} \stackrel{\mathrm{IH}}{ \ \ \Leftrightarrow \ \ } \# \psi, \# \theta \in T_{\xi - 1} \stackrel{\mathrm{IH}}{ \ \ \Leftrightarrow \ \ } \# \psi, \# \theta \in T_{\eta} \stackrel{ 4.3.5 }{ \ \ \Leftrightarrow \ \ } n \in T_{\eta}. \end{align*} $$

If the second disjunct holds, then we have $\# \neg \psi \in T_{\xi } \cap T_{\eta }$ by the induction hypothesis for 1, from which we can infer

$$ \begin{align*} \# \neg \psi \in T_{\xi} \cap T_{\eta} \stackrel{4.3.4}{ \quad \Leftrightarrow \quad } \# \psi \not\in T_{\xi}\ \mathrm{and}\ \# \psi \not\in T_{\eta} \stackrel{4.3.5}{ \quad \Rightarrow \quad } \# \phi \not\in T_{\xi}\ \mathrm{and}\ \# \phi \not\in T_{\eta}. \end{align*} $$

The case where the third disjunct holds can be similarly treated. The claim for the remaining cases where $\phi $ is of the form $\neg \psi $ or $\forall x \psi ( x )$ can be shown similarly.

Claim 2 for a successor $\xi $ is also shown by sub-induction on $\eta $ . As before, we can assume $\eta> \xi $ , and the limit case is obvious. Let $\eta $ be a successor ordinal and take any $n = \# \phi \in D_{\xi }$ for an $\mathcal {L}$ -sentence $\phi $ . If $\phi $ is $\mathcal {L}_0$ -atomic, then the claim trivially obtains by Lemma 4.2.1. If either $\varphi = \mathrm {T} a$ or $\varphi = \mathrm {D} a$ , then we have

$$ \begin{align*} n \in D_{\xi} \stackrel{\mathrm{4.2.2\ or\ 3}}{ \ \ \quad \Leftrightarrow \quad \ \ } a^{\mathbb{N}} \in D_{\xi - 1} \stackrel{\mathrm{IH}}{ \ \ \quad \Rightarrow \quad \ \ } a^{\mathbb{N}} \in D_{\eta - 1} \stackrel{\mathrm{4.2.2\ or\ 3}}{ \ \ \quad \Leftrightarrow \quad \ \ } n \in D_{\eta}. \end{align*} $$

The claim for the other cases for complex $\phi $ s can be straightforwardly shown. For instance, if $\phi = \psi \wedge \theta $ , then we have

$$ \begin{align*} & n \in D_{\xi} & & \stackrel{\mathrm{4.2.5}}{ \ \Leftrightarrow \ } & & \bigl( \# \psi, \# \theta \in D_{\xi - 1} \bigr) \vee \bigl( \# \neg \psi \in D_{\xi - 1} \cap T_{\xi - 1} \bigr) \vee \bigl( \# \neg \theta \in D_{\xi - 1} \cap T_{\xi - 1} \bigr) &\\ & & & \stackrel{\mathrm{IH}}{ \ \Rightarrow \ } & & \bigl( \# \psi, \# \theta \in D_{\eta - 1} \bigr) \vee \bigl( \# \neg \psi \in D_{\eta - 1} \cap T_{\eta - 1} \bigr) \vee \bigl( \# \neg \theta \in D_{\eta - 1} \cap T_{\eta - 1} \bigr) &\\ & && \stackrel{\mathrm{4.2.5}}{ \ \Leftrightarrow \ } & & n \in D_{\eta}. & \end{align*} $$

The other cases can be treated in a similar way.

Let denote the least uncountable ordinal. By Lemma 4.5, we have $D_{\xi } \subset D_{\eta }$ and $D_{\xi } \cap T_{\xi } \subset D_{\eta } \cap T_{\eta }$ for all $\xi \leq \eta $ . Hence, by the standard cardinality consideration, we have the following nice properties of and :

(6)

in particular, is a -fixed point. Let us denote and by $\mathit {D}_{\infty }$ and $\mathit {T}_{\infty }$ , respectively. We finally give an -model of $\mathsf {CD}$ .

Proof By definition, it immediately follows that $( \mathbb {N}, \mathit {D}_{\infty }, \mathbb {T}_{\infty } )$ is a model of T1, $\textsf {T2}^{+}$ , and T4–T6. To see $( \mathbb {N}, \mathit {D}_{\infty }, \mathbb {T}_{\infty } ) \models \textsf {T3}$ , take any

and suppose $( {n}^{\circ } )^{\mathbb {N}} \in \mathit {D}_{\infty }$ . Then, there exist some closed $\mathcal {L}_0$ -term a and $\mathcal {L}$ -sentence $\phi $ such that $n = \# a$ and $( {n}^{\circ } )^{\mathbb {N}} = a^{\mathbb {N}} = \# \phi \in \mathit {D}_{\infty }$ (

). Hence, we have

Since $\mathit {D}_{\infty }$ is a $\Gamma _{\mathcal {D} [ \mathit {T}_{\infty } ]}$ -fixed point, $( \mathbb {N}, \mathit {D}_{\infty }, \mathbb {T}_{\infty } )$ satisfies D1–D4 (which does not depend on the interpretation of $\mathrm {T}$ ). To show $( \mathbb {N}, \mathit {D}_{\infty }, \mathbb {T}_{\infty } ) \models \textsf {D5}$ , take any $\mathcal {L}$ -sentences $\phi $ and $\psi $ . By Lemma 4.6.1, we have $\mathit {D}_{\infty } \cap \mathbb {T}_{\infty } = \mathit {D}_{\infty } \cap \mathit {T}_{\infty }$ and thus obtain

$$ \begin{align*} \# \phi \wedge \psi \in \mathit{D}_{\infty} & \stackrel{\mathrm{4.2.5}}{ \quad \Leftrightarrow \quad } ( \# \phi, \# \psi \in \mathit{D}_{\infty} ) \vee ( \# \neg \phi \in \mathit{D}_{\infty} \cap \mathit{T}_{\infty} ) \vee ( \# \neg \psi \in \mathit{D}_{\infty} \cap \mathit{T}_{\infty} )\\ & \stackrel{ \ }{ \quad \Leftrightarrow \quad } ( \# \phi, \# \psi \in \mathit{D}_{\infty} ) \vee ( \# \neg \phi \in \mathit{D}_{\infty} \cap \mathbb{T}_{\infty} ) \vee ( \# \neg \psi \in \mathit{D}_{\infty} \cap \mathbb{T}_{\infty} ). \end{align*} $$

We can similarly show $( \mathbb {N}, \mathit {D}_{\infty }, \mathbb {T}_{\infty } ) \models \textsf {D6}$ using $\mathit {D}_{\infty } \cap \mathit {T}_{\infty } = \mathit {D}_{\infty } \cap \mathbb {T}_{\infty }$ .

4.2 The minimality of D

In this section, we will show that $\mathit {D}_{\infty }$ ( ) has a certain ‘minimality’ property.

Let us define $\mathcal {L}$ -formulae $\mathcal {T}_{j} ( x )$ ( $1 \leq j \leq 6$ ) as follows:

We define $\mathcal {T} ( x ) :\Leftrightarrow \bigvee _{1 \leq j \leq 6} \mathcal {T}_{j} ( x )$ . Note that $\mathcal {T}$ says nothing about the Gödel numbers of $\mathcal {L}$ -sentences of the form $\neg \mathrm {D} a$ . Both $\mathrm {D}$ and $\mathrm {T}$ occur only positively in $\mathcal {T}$ . Hence, for each given

, it induces a monotone operator

such that

note that

for all

.

Proof It suffices to show the following under the supposition of (a) and (b): for all $\mathcal {L}$ -sentences $\phi $ and ordinals $\alpha $ ,

(7) $$ \begin{align} \# \phi &\in D_{\alpha} \rightarrow \# \phi \in X, \end{align} $$

(8) $$ \begin{align} \# \phi &\in D_{\alpha} \cap T_{\alpha} \rightarrow \# \phi \in Y, \end{align} $$

(9) $$ \begin{align} \# \phi &\in D_{\alpha} \setminus T_{\alpha} \rightarrow \# \neg \phi \in Y. \end{align} $$

They are shown by simultaneous induction on $\alpha $ . The base step is trivial. If $\alpha $ is a limit, the claim follows from the induction hypothesis using Lemma 4.5. Let $\alpha $ be a successor. The claim for the case where $\phi $ is an $\mathcal {L}_0$ -atomic is obvious. We will go through the remaining cases.

Let $\phi = \mathrm {T} a$ and suppose $\# \phi \in D_{\alpha }$ . We have $a^{\mathbb {N}} \in D_{\alpha - 1}$ (

) by Lemma 4.2.2. Then, we have $a^{\mathbb {N}} \in X$ by the induction hypothesis and thus $\# \phi \in X$ by (a); hence (7) holds for $\phi $ . The other claims (8) and (9) are obtained as follows:

Let $\phi = \mathrm {D} a$ and suppose $\# \phi \in D_{\alpha }$ . We have $a^{\mathbb {N}} \in D_{\alpha - 1}$ by Lemma 4.2.3. Then, we get $a^{\mathbb {N}} \in X$ by the induction hypothesis, which implies $\# \phi \in Y$ by (b); hence, (7) and (8) hold for $\phi $ . The other claim (9) trivially holds, since $\# \phi \not \in T_{\alpha }$ would imply $a^{\mathbb {N}} \not \in D_{\alpha - 1}$ by Lemma 4.3.3, which is absurd.

Let $\phi = \neg \psi $ and suppose $\# \phi \in D_{\alpha }$ . We have $\# \psi \in D_{\alpha - 1}$ by Lemma 4.2.4. Then, it follows that $\# \psi \in X$ by the induction hypothesis and thus $\# \phi \in X$ by (a). The other claims (8) and (9) are obtained as follows:

$$ \begin{align*} & \# \phi \in T_{\alpha} & & \stackrel{4.3.4}{ \Leftrightarrow } && \# \psi \not\in T_{\alpha} && \stackrel{4.5.1}{ \Leftrightarrow } && \# \psi \not\in T_{\alpha - 1} & & \stackrel{\mathrm{IH}}{ \Rightarrow } && \# \phi \in Y; & \\ & \# \phi \not\in T_{\alpha} && \stackrel{4.3.4}{ \Leftrightarrow } && \# \psi \in T_{\alpha} && \stackrel{4.5.1}{ \Leftrightarrow } & & \# \psi \in T_{\alpha - 1} && \stackrel{\mathrm{IH}}{ \Rightarrow } & & \# \psi \in Y && \stackrel{\mathrm{(b)}}{ \Rightarrow } & & \# \neg \phi \in Y. & \end{align*} $$

Let $\phi = \psi \wedge \theta $ and suppose $\# \phi \in D_{\alpha }$ . By Lemma 4.2.5, there are three cases to be separately considered. First assume $\# \psi , \# \theta \in D_{\alpha - 1}$ . Then, we get $\# \psi \in X$ and $\# \theta \in X$ by the induction hypothesis, and thus $\# \phi \in X$ by (a); hence, (7) holds for $\phi $ . We next obtain (8) as follows:

$$ \begin{align*} \# \phi \in T_{\alpha} \stackrel{4.3.5}{ \ \ \Leftrightarrow \ \ } \# \psi, \# \theta \in T_{\alpha} \stackrel{4.5.1}{ \ \ \Leftrightarrow \ \ } \# \psi, \# \theta \in T_{\alpha - 1} \stackrel{\mathrm{IH}}{ \ \ \Rightarrow \ \ } \# \psi, \# \theta \in Y \stackrel{\mathrm{(b)}}{ \ \ \Rightarrow \ \ } \# \phi \in Y. \end{align*} $$

For the remaining claim (9), we infer

$$ \begin{align*} \# \phi \not\in T_{\alpha} \stackrel{4.3.5}{ \ \ \ \Leftrightarrow \ \ \ } \# \psi \not\in T_{\alpha} \ \mathrm{or} \ \# \theta \not\in T_{\alpha} & \stackrel{4.5.1}{ \ \ \ \Leftrightarrow \ \ \ } \# \psi \not\in T_{\alpha - 1} \ \mathrm{or} \ \# \theta \not\in T_{\alpha - 1}\\ & \stackrel{\mathrm{IH}}{ \ \ \ \Rightarrow \ \ \ } \# \neg \psi \in Y \ \mathrm{or} \ \# \neg \theta \in Y \stackrel{\mathrm{(b)}}{ \ \ \ \Rightarrow \ \ \ } \# \neg \phi \in Y. \end{align*} $$

Second assume $\# \neg \psi \in D_{\alpha - 1} \cap T_{\alpha - 1}$ . We have $\# \neg \psi \in X \cap Y$ by the induction hypothesis, which implies $\# \phi \in X$ and $\# \neg \phi \in Y$ by (a) and (b); hence, (7) and (9) hold for $\phi $ . Note that $\# \phi \in T_{\alpha }$ can never be the case under the current assumption, since it would imply $\# \neg \psi \not \in T_{\alpha }$ by Lemmata 4.3.4 and 4.3.5 and thus $\# \neg \psi \not \in T_{\alpha - 1}$ by Lemma 4.5.1; hence, (8) trivially holds for $\phi $ . The case where $\# \neg \theta \in D_{\alpha - 1} \cap T_{\alpha - 1}$ can be similarly treated.

The remaining case where $\phi = \forall x \psi ( x )$ can be similarly treated to the last case, and we omit the details.

Remark 4.11. As we have remarked, $\mathit {D}_{\infty }$ is $\Gamma _{\mathcal {D} [ \mathit {T}_{\infty } ]}$ -closed. We can also show that $\mathit {T}_{\infty }$ is $\Gamma _{\mathcal {T} [ \mathit {D}_{\infty } ]}$ -closed; the proof is routine. Hence, it follows from Lemma 4.10 that $\mathit {D}_{\infty }$ and $\mathit {T}_{\infty }$ are simultaneously inductively defined by the following monotone operators from $\mathcal {P} ( \mathbb {N} ) \times \mathcal {P} ( \mathbb {N} )$ to $\mathcal {P} ( \mathbb {N} )$ :

$$ \begin{align*} & \Gamma_{\mathcal{D}} ( X, Y ) = \Gamma_{\mathcal{D} [ Y ]} ( X ) & & \mathrm{and} & & \Gamma_{\mathcal{T}} ( X, Y ) = \Gamma_{\mathcal{T} [ X ]} ( Y ). & \end{align*} $$

Lemma 4.12. Let . Suppose the same conditions (a) and (b) as in Lemma 4.10, as well as the following additional condition:

$$\begin{align*} \kern-28pt(\mathrm{c}) \qquad\qquad & \mathrm{for\ all }\ \mathcal{L}\text{-}\mathrm{sentences }\ \phi, \mathrm{ if }\ \# \phi \in X, \mathrm{ then\ either }\ \# \phi \not\in Y \mathrm{ or }\ \# \neg \phi \not\in Y. \end{align*}$$

Then the following holds.

1. 1. $\mathit {D}_{\infty } \cap Y = \mathit {T}_{\infty }$ .

2. 2. $\mathit {D}_{\infty }$ is the least $\Gamma _{\mathcal {D} [ Y ]}$ -closed set.

The next theorem immediately follows from Lemmata 4.1, 4.9, and 4.12.

This theorem says that every sentence in $\mathit {D}_{\infty }$ is determinate in any -model of $\mathsf {CD}$ , the truth and the falsity are invariable on $\mathit {D}_{\infty }$ across all -models of $\mathsf {CD}$ , and $\mathit {D}_{\infty }$ is the least fixed point of $\Gamma _{\mathcal {D} [ Y ]}$ (and thus is the least set Z with $( \mathbb {N}, Z, Y ) \models \textsf {D1}-\textsf {D6}$ ) whenever $( \mathbb {N}, X, Y ) \models \mathsf {CD}$ for some X. This mathematical fact, as well as the philosophical story that motivated $\mathsf {CD}$ , suggests an axiom expressing such a minimality property of $\mathrm {D}$ , which yields a new theory $\mathsf {CD}_{\mu }$ . This axiom and the resulting system will be briefly explained in Section 9, but their full analysis is left for the Part II.

5.1 The systems $\mathsf {KF}$ and $\mathsf {CT}$

For the sake of the reader’s convenience, we repeat the definition of the system $\mathsf {KF}$ of Feferman [Reference Feferman8] (in our notation).

Definition 5.1. The $\mathcal {L}_{\mathrm {T}}$ -system $\mathsf {KF}$ is defined as $\mathsf {PA}$ with full induction in $\mathcal {L}_{\mathrm {T}}$ augmented with the following axioms:

1. K1

2. K2

3. K3

4. K4

5. K5

6. K6

7. K7

8. R3

Note that, while $\mathrm {F} x$ and $\neg \mathrm {T} x$ are equivalent in $\mathsf {CD}$ , they are not equivalent in $\mathsf {KF}$ , and we have to distinguish them when working in $\mathsf {KF}$ . It is easily shown that $\mathsf {R}3$ is redundant and provable from the other axioms (see [Reference Cantini5, Lemma 3.1]). We will occasionally consider two extra axioms $\mathrm {Cons}$ and $\mathrm {Comp}$ , which are defined as follows:

$$ \begin{align*} \kern-90pt\mathrm{Cons}{:} \qquad\qquad\qquad\qquad\qquad\ & \forall x\, \bigl( {{\mathrm{Sent}}_{\mathrm{T}}}(x)\rightarrow\neg(\mathrm{T} x \wedge \mathrm{F} x ) \bigr), \end{align*} $$

$$ \begin{align*} \kern-96.5pt\mathrm{Comp}{:} \qquad\qquad\qquad\qquad\qquad & \!\forall x\, \bigl( {{\mathrm{Sent}}_{\mathrm{T}}} ( x ) \rightarrow ( \mathrm{T} x \vee \mathrm{F} x ) \bigr). \end{align*} $$

For each $\varphi \in \mathcal {L}_{\mathrm {T}}$ , let $\varphi ^{c}$ denote Cantini’s ‘dual’ translation of $\varphi $ : namely, $\mathrm {T}^{c} a := \neg \mathrm {F} a$ for each $\mathcal {L}_0$ -term a, and c preserves the $\mathcal {L}_0$ -vocabulary (as well as the logical connectives and quantifiers); see [Reference Cantini5, Section 4]. c is a relative interpretation of $\mathsf {KF} + \mathrm {Cons}$ in $\mathsf {KF} + \mathrm {Comp}$ and vice versa. Cantini [Reference Cantini5] showed that $\mathsf {KF} + \mathrm {Cons}$ and $\mathsf {KF} + \mathrm {Comp}$ are both proof-theoretically equivalent to $\mathsf {KF}$ .

5.2 Reduction of $\mathsf {KF}$ to $\mathsf {CD}_{\mathsf {0}}$

The translation $\natural $ of $\mathcal {L}_{\mathrm {T}}$ into $\mathcal {L}$ replaces every atomic formula $\mathrm {T}^{\natural } a $ (for a term a) with $\mathrm {T} a \wedge \mathrm {D} a $ ; $\natural $ does not change anything else.

In what follows, we will occasionally write instead of , instead of , and similarly for other formulae.

Proof We will only exhibit the proofs that $\natural $ preserves the axioms K2, K5, and K7. For K2, take any

. Then we have

The other conjunct can be shown similarly. For K5, take any $x, y \in {{\mathrm {Sent}}_{\mathrm {T}}}$ ; then we have

For K7, take any v and x such that

; then we have

The remaining cases are similarly and even more easily shown.

Because $\natural $ does not affect arithmetical sentences, the next corollary follows.

Finally, since $\mathsf {KF}$ and $\mathsf {RA}_{< \varepsilon _{0}}$ are known to have the same arithmetical theorems by results due to Cantini [Reference Cantini5] and Feferman [Reference Feferman8], $\mathsf {RA}_{< \varepsilon _{0}}$ is arithmetically conservative over $\mathsf {CD}$ .

5.3 Reduction of $\mathsf {CT} [ \! [ \mathsf {KF} ] \! ]$ to $\mathsf {CD}_{\mathsf {2}}^{+}$

Let k be an $\mathcal {L}_0$ -definable function that represents the arithmetization of the translation $\natural $ of $\mathcal {L}_{\mathrm {T}}$ in $\mathcal {L}$ : that is, for all ,

(10)

With the standard coding, k is primitive recursive and thus definable in $\mathsf {PA}$ , and $\mathsf {PA}$ proves . Furthermore, with the notation of [Reference Cantini5], $\mathsf {PA}$ also proves $\forall z \bigl ( \mathrm {Free} ( z, x ) \leftrightarrow \mathrm {Free} ( z, k x ) \bigr )$ , where $\mathrm {Free} ( a, b )$ says that a is a code of a variable that is free in the formula coded by b; in particular, we have .

We thereby extend $\natural $ to a translation of $\mathcal {L}_{\mathrm {T}}^{+}$ to $\mathcal {L}$ by stipulating the following:

$$\begin{align*}\mathbb{T}^{\natural} x \ := \ \mathrm{T} k x. \end{align*}$$

We will use the same symbol $\natural $ for the two translations for simplicity.

Proof It immediately follows from the definition of $\natural $ and the last lemma that the $\natural $ -translations of the axioms $\mathbb {T}1$ for all $\mathcal {L}_0$ -atomics and $\mathbb {T}2$ – $\mathbb {T}4$ are provable in $\mathsf {CD}_{\mathsf {2}}$ . $\mathsf {R}4^{\natural }$ readily follows from R1 and Lemma 5.7 (though this step is actually redundant because $\mathsf {R}4$ is derivable from the other axioms of $\mathsf {CT} [ \! [ \mathsf {KF} ] \! ]$ ). For the remaining case, i.e., the axiom $\mathbb {T}1$ for an atomic of the form $\mathrm {T} x$ , take any

. Then we have

note that the use of $\textsf {T2}^{+}$ in the third equivalence is crucial here.

Now, the following fact is essentially due to Cantini [Reference Cantini5].

Hence, $\mathsf {RT}_{< \varepsilon _{\varepsilon _{0}}}$ is arithmetically conservative over $\mathsf {CD}^{+}$ .

6.1 Semi-formal system $\mathsf {CD}^{\infty }$

In this subsection, we will introduce a Tait-style semi-formal system $\mathsf {CD}^{\infty }$ .

Throughout this Section 6, capital Greek letters $\Gamma $ and will be used as variables ranging over finite sets of $\mathcal {L}$ -sentences, and we will only consider $\mathcal {L}$ -sentences in negation-normal form (or in what is sometimes called ‘Tait form’), which are constructed from closed $\mathcal {L}_0$ -literals by conjunction, disjunction, and universal and existential quantifiers; we will abuse the notation and denote the negation operation on negation-normal forms, which is often written as ‘ $\sim $ ’ in the literature, also by $\neg $ , e.g., $\neg ( \mathrm {T} a \wedge \mathrm {D} a ) = \neg \mathrm {T} a \vee \neg \mathrm {D} a$ . Following the convention, for a finite set $\Gamma $ of $\mathcal {L}$ -sentences and an $\mathcal {L}$ -sentence A, we will write $\Gamma , A$ for $\Gamma \cup \{ A \}$ .

The derivation relation in the semi-formal system $\mathsf {CD}^{\infty }$ for ordinals and and a set $\Gamma $ of $\mathcal {L}$ -sentences, which intuitively means that at least one $\phi \in \Gamma $ is verified by a derivation of length with cut-rank p, is defined as follows.

Axioms. Let a, b, and c be any closed $\mathcal {L}_0$ -terms, $\Gamma $ any set of $\mathcal {L}$ -sentences, any ordinal, and any finite ordinal.

1. (Ax1) If A is a true closed $\mathcal {L}_0$ -literal, then .

2. (Ax2) If $a^{\mathbb {N}} \approx ^{\mathbb {N}} b^{\mathbb {N}}$ (see p.13 for the definition of $\approx $ ), then .

3. (Ax3) If $a^{\mathbb {N}} \approx ^{\mathbb {N}} b^{\mathbb {N}}$ , then .

4. (D1) If and , then .

Logical Rules. Let A and B be $\mathcal {L}$ -sentences, $C ( x )$ an $\mathcal {L}$ -formula with at most one free variable, an ordinal, and .

1. (∨1) If for some ordinal , then .

2. (∨2) If for some ordinal , then .

3. (∧) If and for some ordinals , then .

4. (∃) If for some closed $\mathcal {L}_0$ -term a and ordinal , then .

5. (∀) If there exists some ordinal for each closed $\mathcal {L}_0$ -term a such that , then .

6. (cut) If and for some ordinals and $\mathcal {L}$ -sentence A with complexity $< p$ , then .

Rules for positive occurrences of $\mathrm {D}$ . For ordinals and and a closed term a, holds, if one of the following holds for some ordinal Footnote ⁸ :

• $( \textsf {D}{2}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $( \textsf {D}{3}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $( \textsf {D}{4}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $( \textsf {D}{5}_{\infty }^{\mathit {pos}} )$ There are some closed terms b and c such that and .

• $( \textsf {D}{6}_{\infty }^{\mathit {pos}} )$ There are some closed terms b and c such that and .

Rules for negative occurrences of $\mathrm {D}$ . For ordinals and and a closed term a, holds, if one of the following holds for some ordinal :

• $( \textsf {D}{2}_{\infty }^{\mathit {neg}}) $ There is a closed term b such that and .

• $( \textsf {D}{3}_{\infty }^{\mathit {neg}}) $ There is a closed term b such that and .

• $( \textsf {D}{4}_{\infty }^{\mathit {neg}}) $ There is a closed term b such that and .

• $( \textsf {D}{5}_{\infty }^{\mathit {neg}}) $ There are some closed terms b and c such that and .

• $( \textsf {D}{6}_{\infty }^{\mathit {neg}}) $ There are some closed terms b and c such that and .

Rules for positive occurrences of $\mathrm {T}$ . For ordinals and and a closed term a, holds, if one of the following holds for some ordinal :

• $(\textsf {T}{1}_{\infty }^{\mathit {pos}} )$ There are some closed terms b and c such that and .

• $(\textsf {T}{2}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $(\textsf {T}{3}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $(\textsf {T}{4}_{\infty }^{\mathit {pos}} )$ There is a closed term b such that and .

• $(\textsf {T}{5}_{\infty }^{\mathit {pos}} )$ There are some closed terms b and c such that and .

• $(\textsf {T}{6}_{\infty }^{\mathit {pos}} )$ There are some closed terms b and c such that and .

Rules for a negative occurrence of $\mathrm {T}$ . For ordinals and and a closed term a, holds, if one of the following holds for some ordinal :

• $( \textsf {T}{1}_{\infty }^{\mathit {neg}} )$ There are some closed terms b and c such that and .

• $( \textsf {T}{3}_{\infty }^{\mathit {neg}} )$ There is a closed term b such that and .

• $( \textsf {T}{4}_{\infty }^{\mathit {neg}} )$ There is a closed term b such that and .

• $( \textsf {T}{5}_{\infty }^{\mathit {neg}} )$ There are some closed terms b and c such that and .

• $( \textsf {T}{6}_{\infty }^{\mathit {neg}} )$ There are some closed terms b and c such that and .

Note that there is no rule for a negative occurrence of $\mathrm {T}$ corresponding to $(\textsf {T}{2}_{\infty }^{\mathit {pos}} )$ .

The next lemma is nearly obvious from the design of $\mathsf {CD}^{\infty }$ , and the proof is routine.

The following three lemmata can also be shown standardly.

Note that means that there is a derivation of $\psi $ in $\mathsf {CD}^{\infty }$ in which $( \mathrm {cut} )$ is only applied to closed $\mathcal {L}$ -literals; such a derivation corresponds to what Cantini [Reference Cantini5] calls a quasi-normal derivation in his semi-formal system $\mathrm {TKF}^{\infty }$ .

6.2 Reduction of $\mathsf {CD}$ to $\mathsf {KF}$

Let $\sigma $ , $\tau $ , and $\xi $ be ordinals. Recall that we only consider $\mathcal {L}$ -sentences in negation-normal form throughout the Section 6. Given an $\mathcal {L}$ -sentence $\phi $ , we write $\models \phi [ \sigma , \tau , \xi ]$ when $\phi $ is true under the following interpretation:

• • the quantifiers range over ;

• • the $\mathcal {L}_0$ -vocabulary receives the standard interpretations;

• • each negative occurrence of $\mathrm {D}$ in $\phi $ is interpreted by $D_{\sigma + 1}$ ;

• • each positive occurrence of $\mathrm {D}$ in $\phi $ is interpreted by $D_{\tau + 1}$ ;

• • each (either negative or positive) occurrence of $\mathrm {T}$ in $\phi $ is interpreted by $T_{\xi + 1}$ .

We extend this notation to finite sets $\Gamma $ of $\mathcal {L}$ -sentences and write $\models \Gamma [ \sigma , \tau , \xi ]$ when $\models \phi [ \sigma , \tau , \xi ]$ for some $\phi \in \Gamma $ . By Lemma 4.5.2, the sequence $\langle D_{\xi } \mid \xi \in On \rangle $ is a monotonically increasing, where $On$ denotes the class of ordinals, and thus the next lemma can be shown in the standard manner.

The next is the main lemma of the present section.

Proof The claim is shown by induction on derivation. Throughout this proof, a, b, c, and d always denote closed $\mathcal {L}_0$ -terms.

Suppose that the last inference is made by either of the following axioms:

where $\mathit {A}$ is a true closed $\mathcal {L}_0$ -literal, $a^{\mathbb {N}}\! \approx ^{\mathbb {N}} b^{\mathbb {N}}$ , and $c^{\mathbb {N}}$ is the Gödel number of a closed $\mathcal {L}_0$ -atomic. The claim for the first case is obvious; the claim for the second case follows from Proposition 4.4.2, and the claim for the third case follows from Proposition 4.2.1. Next, suppose the last inference is made by the axiom (Ax2):

where $a^{\mathbb {N}}\! \approx ^{\mathbb {N}} b^{\mathbb {N}}$ . Take any ordinal $\sigma $ and $\xi $ . Since

by Lemma 4.5.1,

implies $a^{\mathbb {N}} \not \in D_{\sigma + 1}$ and thus $b^{\mathbb {N}} \not \in D_{\sigma + 1}$ by Proposition 4.4.1. Hence, either

or $b^{\mathbb {N}} \not \in D_{\sigma + 1}$ holds, and thus

.

We move on to the induction step. The case where we use a logical rule in the last inference, except for cut, can be treated in the usual manner. For instance, suppose the last inference is made by $( \forall )$ , namely, (our variant of) the

-rule:

where

for each a. Take any $\sigma $ and $\xi $ with

. It follows from the induction hypothesis that for each closed term a, we have the following:

If

for any a, then we have

by Lemma 6.6. Otherwise, for all closed terms a, we have

and thus

by Lemma 6.6, which implies

.

Next, the claim for the cases where the last inference is made by one of the rules for determinateness can be straightforwardly shown using Lemmata 4.2 and 4.5. For example, suppose the last inference is of the following form:

where

and

. Take any $\sigma $ and $\xi $ with

. If we have $\models \Gamma \cup \{ \neg \mathrm {D} a \} [ \sigma , \sigma + 2^{\gamma }, \xi ]$ , then we get

by Lemma 6.6. Otherwise, we have the following by the induction hypothesis:

Since $D_{\sigma + 1} \supset D_{\sigma }$ and $D_{\sigma } \cap T_{\xi + 1} = D_{\sigma } \cap T_{\sigma }$ by Lemma 4.5, this implies

Hence, by Lemma 4.2.5, we obtain $a^{\mathbb {N}} \not \in D_{\sigma + 1}$ and thus

. Let us see one more example. Suppose the last inference is of the following form:

where

and

. Take any $\sigma $ and $\xi $ with

. As before, if $\models \Gamma \cup \{ \mathrm {D} a \} [ \sigma , \sigma + 2^{\gamma }, \xi ]$ , then the claim follows by Lemma 6.6. Otherwise, by the induction hypothesis, we have

Since

, it follows from Lemma 4.5 that

which implies

by Lemma 4.2.6 and thus

. We leave the other cases to the reader.

Now, let us assume that the last inference is made by a truth rule. Suppose the last inference is made by either $(\textsf {T}{4}_{\infty }^{\mathit {pos}} )$ or $( \textsf {T}{3}_{\infty }^{\mathit {neg}} )$ :

where

and

. Take any $\sigma $ and $\xi $ with

. We will prove the claim only for the former case; the latter case can be similarly shown. If ${\models \neg \mathrm {T} b [ \sigma , \sigma + 2^{\gamma }, \xi ]}$ , then $b^{\mathbb {N}} \not \in T_{\xi + 1}$ and thus $a^{\mathbb {N}} \in T_{\xi + 1}$ by Lemma 4.3.4, which implies

. Otherwise, we have $\models \Gamma \cup \{ \mathrm {T} a \} [ \sigma , \sigma + 2^{\gamma }, \xi ]$ by the induction hypothesis, and the claim follows by Lemma 6.6 as before. Next, suppose that we make either of the following inferences at the last step:

where

and

. Take any $\sigma $ and $\xi $ with

. For the former case, if $\models \mathrm {T} {b}^{\circ } \wedge \mathrm {D} {b}^{\circ } [ \sigma , \sigma + 2^{\gamma }, \xi ]$ , then, by Lemma 4.5, we have

and thus, by Lemmata 4.2.2, 4.3.2, and 4.5, we obtain

which implies

; otherwise, we have $\models \Gamma \cup \{ \mathrm {T} a \} [ \sigma , \sigma + 2^{\gamma }, \xi ]$ by the induction hypothesis, and the claim follows by Lemma 6.6 as before. Similarly, for the latter case, if $\models \neg \mathrm {T} {b}^{\circ } \wedge \mathrm {D} {b}^{\circ } [ \sigma , \sigma + 2^{\gamma }, \xi ]$ , then, by Lemma 4.5, we have

and thus, again by Lemmata 4.2.2, 4.3.2, and 4.5, we obtain

which implies

; otherwise, we have $\models \Gamma \cup \{ \neg \mathrm {T} a \} [ \sigma , \sigma + 2^{\gamma }, \xi ]$ by the induction hypothesis, and the claim follows by Lemma 6.6 as before. Thirdly, suppose that the last inference is made by the rule $(\textsf {T}{2}_{\infty }^{\mathit {pos}} )$ :

where

and

. Take any $\sigma $ and $\xi $ with

. Suppose $\models \mathrm {D} {b}^{\circ } [ \sigma , \sigma + 2^{\gamma }, \xi ]$ . We have $( {b}^{\circ } )^{\mathbb {N}} \in D_{\sigma + 2^{\gamma } + 1}$ and thus $( {b}^{\circ } )^{\mathbb {N}} \in D_{\xi }$ by Lemma 4.5.2. Hence, we get $a^{\mathbb {N}} \in T_{\xi + 1}$ by Lemma 4.3.3 and thus

. Otherwise, the claim follows by Lemma 6.6 as before. The remaining cases for the other truth rules can be dealt with similarly.

Finally, let us assume that the last inference is made by $( \mathrm {cut} )$ . We only consider the crucial case where the cut formulae are of the forms $\mathrm {D} a$ and $\neg \mathrm {D} a$ :

where

. Let $\gamma = \max \{ \gamma _{0}, \gamma _{1} \}$ and take any $\sigma $ and $\xi $ with

. Suppose

for contradiction. Since

, it would follow that

$$ \begin{align*} & \not\models \Gamma [ \sigma, \sigma + 2^{\gamma}, \xi ] & & \mathrm{and} & & \not\models \Gamma [ \sigma + 2^{\gamma}, \sigma + 2^{\gamma} + 2^{\gamma}, \xi ] & \end{align*} $$

by Lemma 6.6. Hence, by the induction hypothesis, we would have

$$ \begin{align*} & \models \mathrm{D} a [ \sigma, \sigma + 2^{\gamma}, \xi ] & & \mathrm{and} & & \models \neg \mathrm{D} a [ \sigma + 2^{\gamma}, \sigma + 2^{\gamma} + 2^{\gamma}, \xi ], & \end{align*} $$

which means $a^{\mathbb {N}} \!\in D_{\sigma + 2^{\gamma } + 1}$ and $a^{\mathbb {N}}\! \not \in D_{\sigma + 2^{\gamma } + 1}$ ; a contradiction. The remaining case in which the cut formulae are closed $\mathcal {L}_{\mathrm {T}}$ -literals can be straightforwardly treated.

Combining this with Lemma 6.5, we obtain the next corollary.

Now, the argument of this section can be adequately formalized within $\mathsf {KF}$ (or $\mathsf {RA}_{< \varepsilon _{0}}$ ), in which we can formalize the construction of $D_{\xi }$ s and $T_{\xi }$ s for sufficiently large ordinals ( $< \varepsilon _{0}$ ) and prove (the arithmetization of) Corollary 6.8 in terms of them. Hence, in particular, we obtain the following.

Combining this corollary with Corollary 5.6, we obtain our main theorem.

In fact, our proofs establish stronger results: All the systems are proof-theoretically reducible to each other in the sense of Feferman [Reference Feferman7, Reference Feferman9].

7.1 Total and consistent predicates

We begin with the following definition:

$$\begin{align*}D^{+} ( x ) \ :\Leftrightarrow \ ( \mathrm{F} x \leftrightarrow \neg \mathrm{T} x ). \end{align*}$$

Thus, $D^{+} ( x )$ is equivalent to $( \mathrm {T} x \vee \mathrm {F} x ) \wedge ( \neg \mathrm {T} x \vee \neg \mathrm {F} x )$ . We obviously have

(11) $$ \begin{align} \begin{aligned} & \mathsf{KF} + \mathrm{Cons} \vdash ( \forall x \in {{\mathrm{Sent}}_{\mathrm{T}}} ) \bigl( D^{+} ( x ) \leftrightarrow ( \mathrm{T} x \vee \mathrm{F} x ) \bigr),\\ & \mathsf{KF} + \mathrm{Comp} \vdash ( \forall x \in {{\mathrm{Sent}}_{\mathrm{T}}} ) \bigl( D^{+} ( x ) \leftrightarrow ( \neg \mathrm{T} x \vee \neg \mathrm{F} x ) \bigr). \end{aligned} \end{align} $$

Proof We work within $\mathsf {KF} + \mathrm {Cons}$ ; recall that we have $D^{+} ( x ) \leftrightarrow ( \mathrm {T} x \vee \mathrm {F} x )$ for all $x \in {{\mathrm {Sent}}_{\mathrm {T}}}$ in $\mathsf {KF} + \mathrm {Cons}$ . For claim 1, take any $x, y \in {{\mathrm {Sent}}_{\mathrm {T}}}$ ; then we have

Claim 2 can be shown similarly to 1 using Lemma 7.1.2 and K3. For claim 3, take any v and x with

; then we have

Claim 4 can be shown similarly to 3 using Lemma 7.1.2 and K3.

Proof Take any

. The following four equivalences are provable in $\mathsf {KF}$ :

The claim follows from these by logic.

7.2 Reduction of $\mathsf {CD}^{+}$ to $\mathsf {CT} [ \! [ \mathsf {KF} + \mathrm {Cons} ] \! ]$

In the present subsection, we give an interpretation of $\mathsf {CD}^{+}$ in $\mathsf {CT} [ \! [ \mathsf {KF} + \mathrm {Cons} ] \! ]$ that preserves $\mathcal {L}_0$ . For this purpose, we need a few preliminary definitions.

By the recursion theorem we obtain a recursive function f that satisfies the following condition:

(12)

Here is an arithmetical representation of the syntactic operation of applying the function f to terms so that is provable in $\mathsf {PA}$ for each .Footnote ⁹ We can show that f is provably total in $\mathsf {PA}$ and thus f is a $\mathsf {PA}$ -definable function. Obviously we have $f x \in {\mathrm {Fml}_{\mathrm {T}}}$ for every (provably in $\mathsf {PA}$ ), and, as in the case for the function k, we have $\forall z \bigl ( \mathrm {Free} ( z, x ) \leftrightarrow \mathrm {Free} ( z, f x ) \bigr )$ ; hence, we have $f x \in {{\mathrm {Sent}}_{\mathrm {T}}}$ for every (provably in $\mathsf {PA}$ ).

The next is shown by routine induction on the complexity of x (cf. Lemma 5.7).

Now, we define a translation $\sharp $ of $\mathcal {L}$ into $\mathcal {L}_{\mathrm {T}}^{+}$ as follows: for each term a,

$$ \begin{align*} & \mathrm{D}^{\sharp} a \ := \ \mathrm{T} f a \vee \mathrm{F} f a & & \mathrm{and} & & \mathrm{T}^{\sharp} a \ := \ \mathbb{T} f a; & \end{align*} $$

$\sharp $ preserves all the $\mathcal {L}_0$ -vocabulary. By (11), we have

(13)

We will show that $\sharp $ is a relative interpretation of $\mathsf {CD}$ in $\mathsf {CT} [ \! [ \mathsf {KF} + \mathrm {Cons} ] \! ]$ .

Proof The claim is readily observed from the following equivalences:

Proof We work within $\mathsf {CT} [ \! [ \mathsf {KF} + \mathrm {Cons} ] \! ]$ . Claim 1 is obvious by $\mathbb {T}1$ . For claim 2, take any

, and then we infer

Claims 3 and 4 readily follow from $\mathbb {T}2$ and $\mathbb {T}3$ , respectively, together with the definition (12) of f. For claim 5, let v and x be such that

. Then, we have

Proof If

then $f ( x ) = \ulcorner 0 \!\neq \! 0\urcorner $ and thus $\mathbb {T} f ( x ) \wedge \mathrm {T} f ( x )$ . Let

. The claim is shown by induction on the complexity of x. For the base step, assume

. The case where $x \in {\mathrm {AtSent}_{0}}$ is obvious. Next, if

for some

, then we have

Lastly, let

for some

. Then, we have

For the induction step, we first let

and assume $\mathrm {D}^{\sharp } x$ . We have $\mathrm {D}^{\sharp } y$ by Lemma 7.5.3 and thus $D^{+} f ( y )$ by (13). Hence, we have

note that the assumption $\mathrm {D}^{\sharp } x$ is crucial here. The other cases can be shown similarly, but without the need to use the assumption $\mathrm {D}^{\sharp } x$ .