2 Arithmetization of Yablo sentences

Let’s start by going over the results pertaining to Yablo sentences obtained in the standard arithmetical setting.

2.1 Existence of Yablo sequences

One might ask how we actually know that Yablo sequences exist in formal theories. This is a legitimate question since we’re moving from the paradox as formulated hand-wavily in natural language to its properly defined formalized counterpart. It is possible to construct a Yablo sequence within a given theory, but in order to do so, we need to use a general version of the diagonal lemma (for formulae with two free variables in the language containing the truth predicate).

Diagonal Lemma 1. Let $\mathtt {T}$ be a first-order theory in the language $\mathcal {L}_{Tr}$ (a language of arithmetic extended by a truth predicate), containing Robinson’s arithmetic Q. Then, for any $\mathcal {L}_{Tr}$ -formula $\varphi (x,y)$ there is an $\mathcal {L}_{Tr}$ -formula $\psi (x)$ such that:

$$ \begin{align*} \mathtt{T}\vdash \: \psi(x) \equiv \varphi(x, \overline{\ulcorner \psi(x) \urcorner}). \end{align*} $$

The expression ${Qx \: P(\overline {\ulcorner \varphi (\dot {x}) \urcorner })}$ , where $Q \in \{ \forall , \exists \}$ is to be read as follows: for all natural numbers x (there exists a natural number x such that), the result of substituting a numeral denoting x for a variable free in $\varphi $ has property P. When the intended meaning is clear from the context, we sometimes omit the dot notation.

Definition 1 (Yablo Formula)

$Y(x)$ is a Yablo formula in a theory $\mathtt {T}$ iff it satisfies the Yablo condition, i.e.,

$$\begin{align*}\mathtt{T} \vdash \forall x (Y(x) \equiv \forall w> x \neg Tr(\overline{\ulcorner Y(\dot{w}) \urcorner})).\end{align*}$$

This also gives rise to a natural way of defining sentences belonging to a Yablo sequence.

Definition 2 (Yablo Sentence)

$\varphi $ is a Yablo sentence in a theory $\mathtt {T}$ iff it is obtained by substituting a numeral for x in Yablo formula $Y(x)$ .

This enables us to prove that we are not dealing with fictitious entities, but with properly defined formal objects. Next, we prove the following theorem.

Theorem 1 (Existence of Yablo Formula [Reference Priest13])

Let $\mathtt {T}$ be a theory in a language $\mathcal {L}_{Tr}$ containing Peano Arithmetic $\mathtt {PA}$ . Then there exists a Yablo formula in $\mathtt {T}$ .

Proof. Let $\varphi (x,y) = \forall w>x \: \neg Tr(sub(y, \ulcorner y \urcorner , name(w)))$ . By the Diagonal Lemma, there is a formula $Y(x)$ such that:

$$\begin{align*}\mathtt{T} \vdash Y(x) \equiv \forall w>x \: \neg Tr(sub(\overline{\ulcorner Y(x) \urcorner}, \ulcorner y \urcorner, name(w))).\end{align*}$$

Therefore by the fact that $sub$ represents the appropriate function, we have

$$\begin{align*}\mathtt{T} \vdash Y(x) \equiv \forall w>x \: \neg Tr(\overline{\ulcorner Y(\dot{w}) \urcorner}).\end{align*}$$

Thus we may conclude that $Y(x)$ is a Yablo formula. Hence, Yablo sentences exist.

2.2 Consistency of Yablo sequences

An interesting question one might ask is exactly what principles about Yablo sentences lead to the inconsistency of a formal theory. Despite the fact that on the level of natural language it is not difficult to derive contradiction from the definition of Yablo sequence, the formal counterpart (to be specified below) is only $\omega $ -inconsistent and consistent. Most of the results from this section were originally obtained by Ketland [Reference Ketland4]; but Theorem 4 is new.

Definition 3 ( $\omega $ -consistency)

Let $\mathtt {T}$ be a first-order theory in the arithmetical language $\mathcal {L}$ . $\mathtt {T}$ is $\omega $ -consistent if there is no $\varphi (x) \in Frm_{\mathcal {L}}$ such that simultaneously:

$$ \begin{align*} \forall n \in \omega \:\: \mathtt{T}\vdash \neg\varphi(\overline{n}) \\ \mathtt{T} \vdash \exists x \varphi(x). \end{align*} $$

If there is such a formula $\varphi $ , then we say that T is an $\omega $ -inconsistent theory.

Definition 4 ( $\mathtt {PA_F}$ )

Let $\mathcal {L}_F$ be a language of arithmetic extended by one monadic predicate F:

$$\begin{align*}\mathtt{PA_F}:= \mathtt{PA} \cup \{F(\overline{n}) \equiv \forall x>\overline{n} \: \neg F(x) \: : n \in \omega \}.\end{align*}$$

Lemma 1. $\mathtt {PA_F}$ is $\omega $ -inconsistent.

Proof. Work in $\mathtt {PA_F}$ . Fix an $n\in \omega $ and assume $F(\overline {n})$ . By the definition of $\mathtt {PA_F}$ , we have

(⋆) $$ \begin{align} \forall x> \overline{n} \: \neg F(x).\end{align} $$

In particular, (⋆) entails $ \forall x> \overline {n+1} \: \neg F(x)$ . This sentence is however equivalent to $F(\overline {n+1})$ . But from (⋆), $\neg F(\overline {n+1})$ follows. Contradiction. So $\neg F(\overline {n})$ .

Since the choice of n was arbitrary, this means

(1) $$ \begin{align} \forall n \in \omega \,\,\, \,\, \mathtt{PA_F} \vdash \: \neg F(\overline{n}). \end{align} $$

But by the definition of $\mathtt {PA_F}$ we then have

$$\begin{align*}\forall n \in \omega \,\,\,\,\, \mathtt{PA_F} \vdash \exists x> \overline{n} \: F(x).\end{align*}$$

In particular,

(2) $$ \begin{align} \mathtt{PA_F} \vdash \exists x \: F(x). \end{align} $$

(1) with (2) jointly mean that $\mathtt {PA_F}$ is $\omega $ -inconsistent, as advertised.

Lemma 2. $\mathtt {PA_F}$ is consistent.

Proof. Consider any nonstandard model $\mathcal {M}$ of $\mathtt {PA}$ . Pick a nonstandard element $a \in M$ (M is the domain of $\mathcal {M})$ and let $A=\{a\}$ . Extend the language of $\mathtt {PA}$ with a monadic predicate F and put $F^{\mathcal {M}}=A$ . Since a is nonstandard, we get

$$\begin{align*}\forall n \in \omega \: \,\, (\mathcal{M}, A) \models \neg F(n).\end{align*}$$

But since A is non-empty, $(\mathcal {M}, A) \models \exists x \: F(x)$ . Moreover, because there is a non-standard witness for the formula $F(x)$ , we have

$$\begin{align*}\forall n \in \omega \: \,\,(\mathcal{M}, A) \models \: \exists x>n \: F(x).\end{align*}$$

Hence we obtain $\forall n \in \omega \: \,\, (\mathcal {M}, A) \models \: F(n) \equiv \forall x>n \: \neg F(x)$ (because both sides of the equivalence are false in the model). But because $\mathcal {M}$ is already a model of $\mathtt {PA}$ , the last statement lets us conclude that $(\mathcal {M}, A) \models \: \mathtt {PA_F}$ , which by soundness means that $\mathtt {PA_F}$ is consistent.

Definition 5 (Local Arithmetical Disquotation)

$$ \begin{align*}AD & = \{Tr(\overline{\ulcorner \varphi \urcorner}) \equiv \varphi \: : \varphi\in Sent_{\mathcal{L}}\}.\end{align*} $$

Definition 6 (Local Yablo Disquotation)

$$ \begin{align*} YD & = \{Tr(\overline{\ulcorner Y(\overline{n}) \urcorner}) \equiv Y(\overline{n})\: : Y(\overline{n}) \text{ belongs to the Yablo sequence}\}.\end{align*} $$

Definition 7 ( $\mathtt {PA_D}$ )

Let $\mathtt {PAT^-}$ be a theory obtained from $\mathtt {PA}$ by extending the language of arithmetic with a truth predicate $Tr$ . Further, let $\mathtt {PAT}$ be obtained from $\mathtt {PAT^-}$ by allowing the induction scheme to apply also to formulae containing $Tr$ .Footnote ² Let $\mathtt {PA_D} = \mathtt {PAT} \cup AD \cup YD$ and let $\mathtt {PA_D^-}$ be $\mathtt {PA_D}$ with the induction axiom scheme restricted to formulae without the truth predicate.

Theorem 2. $\mathtt {PA_D^-}$ is $\omega $ -inconsistent.

Proof. By Definition 1 (of Yablo formulae) and Theorem 1 we obtain

$$\begin{align*}\forall n \in \omega \:\,\, \mathtt{PA_D} \vdash Y(\overline{n}) \equiv \forall x> \overline{n} \: \neg Tr(\overline{\ulcorner Y (\dot{x}) \urcorner}).\end{align*}$$

By the fact that $\mathtt {PA_D}$ contains $YD$ we get

$$\begin{align*}\forall n \in \omega \: \mathtt{PA_D} \vdash Tr(\overline{\ulcorner Y(\overline{n}) \urcorner}) \equiv \forall x> \overline{n} \: \neg Tr(\overline{\ulcorner Y (\dot{x}) \urcorner}).\end{align*}$$

Let $F(x):= Tr(\overline {\ulcorner Y (\dot {x}) \urcorner })$ . With this definition we get

$$\begin{align*}\forall n \in \omega \: \,\, \mathtt{PA_D} \vdash F(\overline{n}) \equiv \forall x> \overline{n} \neg F(x).\end{align*}$$

This means $\mathtt {PA_D}$ contains $\mathtt {PA_F}$ , and so applying Lemma 1 to this fact ends the proof.

It immediately follows the following corollary.

Corollary 1. $\mathtt {PA_D}$ is $\omega $ -inconsistent.

However, we have the following theorem.

Theorem 3. $\mathtt {PA_D^-}$ is consistent.

Proof. Consider any nonstandard model $\mathcal {M}$ of $\mathtt {PA}$ . Take

$$\begin{align*}Th_{\mathcal{L}}(\mathcal{M})=\{\ulcorner \varphi \urcorner : \mathcal{M} \models \varphi \text{ and } \varphi \in Sent_{\mathcal{L}}\}.\end{align*}$$

Further, let us denote $t(x):=\ulcorner Y(\dot {x}) \urcorner $ (t assigns to any number n, the code of the n-th Yablo sentence; the value of t depends on the value of x which occurs as a free variable in the formula $Y(x)$ ). The formula $t(x)=y$ is true of infinitely many standard numbers, and so by overspill there are nonstandard numbers b and c such that $t^{\mathcal {M}}(b) = c$ , i.e., $c = \ulcorner Y(b) \urcorner $ . Let $S= Th_{\mathcal {L}}(\mathcal {M}) \cup \{c\}$ . We extend the language of $\mathtt {PA}$ with the truth predicate $Tr$ and put $Tr^{\mathcal {M}} = S$ . We then have

$$ \begin{align*} \forall n \in \omega \:\:\,\, (\mathcal{M}, S) \models \exists x>n \:\: Tr(\ulcorner Y(\dot{x}) \urcorner), \end{align*} $$

since we have $(\mathcal {M}, S) \models Tr(c)$ and hence $\forall n \in \omega \:\,\, (\mathcal {M}, S) \models \: b>n \wedge Tr(\ulcorner Y(b) \urcorner )$ . S contains the codes of all true arithmetical sentences, so

$$ \begin{align*} (\mathcal{M}, S) \models AD. \end{align*} $$

The codes of standard sentences $Y(\overline {n})$ do not belong to S—such sentences are neither arithmetical (because they contain the truth predicate) nor coded by c (because c codes a non-standard Yablo sentence). Thus we conclude that

$$ \begin{align*} \forall n \in \omega \: \,\,(\mathcal{M}, S) \models \neg Tr(\ulcorner Y(\bar{n}) \urcorner). \end{align*} $$

Additionally, by Definition 1 and Theorem 1 we have

$$ \begin{align*} \forall n \in \omega \:\,\, \mathtt{PAT^-} \vdash Y(\overline{n}) \equiv \forall x>n \: \neg Tr(\ulcorner Y(\dot{x}) \urcorner). \end{align*} $$

Since $(\mathcal {M},S)$ is a model of $\mathtt {PAT^-}$ , we obtain

$$ \begin{align*} \forall n \in \omega \: (\mathcal{M}, S) \models Y(\overline{n}) \equiv \forall x>n \: \neg Tr(\ulcorner Y(\dot{x}) \urcorner). \end{align*} $$

But from this it follows that

(3) $$ \begin{align} \forall n \in \omega \: (\mathcal{M}, S) \models \neg Y(\overline{n}). \end{align} $$

Therefore we may conclude that

$$ \begin{align*} (\mathcal{M}, S) \models YD, \end{align*} $$

which ends the proof.

Note that in what follows some of the arguments are model-theoretic: in such contexts, we sometimes talk about particular numbers without assuming that they are standard, and so without assuming that there are standard numerals. For this reason we’ll follow the shady practice of not using the bar notation for numerals in such contexts.

Theorem 4. $\mathtt {PA_D}$ is consistent.

Proof. We will show that $\mathtt {PA_D}$ is finitely satisfiable. Consider any finite subset $\Delta $ of $\mathtt {PA_D}$ . There is only a finite number of elements of YD (that is, instances of disquotation schema for Yablo sentences) in $\Delta $ . Hence, there is a greatest m such that there is an element of YD for $Y(m)$ in $\Delta $ . Without loss of generality, we may assume that for all $n \leq m \Delta $ contains elements of YD for sentences $Y(n)$ . Now take a model $\mathcal {M} \models \mathtt {PA}$ and expand it by

$$ \begin{align*} Tr^{\mathcal{M}} := \{\ulcorner Y(m) \urcorner\} \cup \{\ulcorner \varphi \urcorner: \: \varphi \in Sent_{\mathcal{L}} \cap \Delta \: \: \: \text{and} \: \: \: \mathcal{M} \models \varphi \}. \end{align*} $$

By this definition we have

$$ \begin{align*} (1) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models Tr(\ulcorner Y(m) \urcorner), \end{align*} $$

$$ \begin{align*} (2) \: \: \: \: \forall k < m \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models \neg Tr(\ulcorner Y(k) \urcorner), \end{align*} $$

$$ \begin{align*} (3) \: \: \: \: \forall k> m \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models \neg Tr(\ulcorner Y(k) \urcorner), \end{align*} $$

and for all arithmetical $\varphi \in \Delta $ that hold in $\mathcal {M}$ :

$$ \begin{align*} (4) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models Tr(\ulcorner \varphi \urcorner). \end{align*} $$

From $(1)$ we easily get that: $ \forall k<m \: \: (\mathcal {M}, Tr^{\mathcal {M}}) \models \exists x>k \: Tr(\ulcorner Y(\dot {x}) \urcorner ), $ and hence by the definition of Yablo sentences:

$$ \begin{align*} (5) \: \: \: \: \forall k < m \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models \neg Y(k). \end{align*} $$

By $(3)$ we obtain: $ (\mathcal {M}, Tr^{\mathcal {M}}) \models \forall x>m \: \neg Tr(\ulcorner Y(\dot {x}) \urcorner ), $ and hence, again, by the definition of Yablo sentences:

$$ \begin{align*} (6) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}}) \models Y(m). \end{align*} $$

$(1)$ and $(6)$ give us then:

$$ \begin{align*} (\mathcal{M}, Tr^{\mathcal{M}})\models Y(m) \equiv Tr(\ulcorner Y(m) \urcorner), \end{align*} $$

whereas by $(2)$ and $(5)$ we have

$$ \begin{align*} \forall k < m \:\: (\mathcal{M}, Tr^{\mathcal{M}})\models Y(k) \equiv Tr(\ulcorner Y(k) \urcorner). \end{align*} $$

These two statements are obviously equivalent to: $\forall n \leq m \: \: (\mathcal {M}, Tr^{\mathcal {M}})\models Y(n) \equiv Tr(\ulcorner Y(n) \urcorner )$ , and it follows that:

$$ \begin{align*} (*) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}})\models \Delta \cap YD. \end{align*} $$

From (4) and the definition of $Tr^{\mathcal {M}}$ we immediately obtain: $ (\mathcal {M}, Tr^{\mathcal {M}})\models \varphi \equiv Tr(\ulcorner \varphi \urcorner )$ for all arithmetical $\varphi \in \Delta $ , which means that:

$$ \begin{align*} (**) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}})\models \Delta \cap AD. \end{align*} $$

Let $Ind(\psi )$ denote the instance of the induction scheme for the $\mathcal {L}_{Tr}$ formula $\psi $ . Since $\mathcal {M}\models \mathtt {PA}$ , clearly for all arithmetical $\psi $ , if $Ind(\psi )\in \Delta $ , $\mathcal {M}\models Ind(\psi )$ .

For $\psi $ containing $Tr$ such that $Ind(\psi )\in \Delta $ , consider that the only atomic $\psi $ that we care about is of the form $Tr(k)$ . The instance of induction for $Tr(k)$ will have the following form:

$$ \begin{align*} Tr(0) \wedge \forall x\, (Tr(x) \rightarrow Tr(x+1)) \rightarrow \forall x\, Tr(x). \end{align*} $$

Clearly, the antecedent is false: 0 doesn’t code a true sentence, and it is false that if a number codes a true sentence then so does its successor. It is folklore result that if a predicate is inductive, the induction scheme can be extended to all formulae containing it.Footnote ³

So, by the reasoning above we can even make a stronger claim that for any $\psi \in \mathcal {L}_{Tr}$ ,

$$ \begin{align*} (***) \: \: \: \: (\mathcal{M}, Tr^{\mathcal{M}})\models Ind(\psi). \end{align*} $$

It follows from $(*)$ , $(**)$ and $(***)$ that $(\mathcal {M}, Tr^{\mathcal {M}})\models \Delta $ . By compactness, this means that $\mathtt {PA_D}$ is satisfiable and therefore consistent.

Corollary 2. $\mathtt {PA_D}$ is a conservative extension of $\mathtt {PA}$ .

Proof. Let us consider an arithmetical sentence $\varphi $ such that $\mathtt {PA} \nvdash \varphi $ . This entails that $\mathtt {PA} \cup \{\neg \varphi \}$ is consistent. Then there is a nonstandard model $\mathcal {M}$ of $\mathtt {PA}$ such that $\mathcal {M} \models \neg \varphi $ . Since in the proof of Theorem 4 it did not actually matter what arithmetical extension of $\mathtt {PA}$ we started with, we can repeat the reasoning with $Th(\mathcal {M})\supset \mathtt {PA_D}\cup \{\neg \varphi \}$ , and so there is a model $\mathcal {M'}$ elementarily equivalent to $\mathcal {M}$ such that $(\mathcal {M'}, Tr^{\mathcal {M'}}) \models \: \mathtt {PA_D}$ . Obviously, since by the construction: $\neg \varphi \in Tr^{\mathcal {M'}}$ , we have $(\mathcal {M'}, Tr^{\mathcal {M'}})\not \models \: \varphi $ , and so $\mathtt {PA_D} \nvdash \varphi $ .

It is however still possible to derive a contradiction from the Yablo sequence, but in order to achieve this we would have to use a generalized version of the truth principle governing the Yablo sequence—it would be necessary to add to our arithmetical theory principles that would be strong enough to prove a version of the disquotation schema uniform for all Yablo sentences.

Definition 8 (Uniform Yablo Disquotation)

(UYD) $$ \begin{align} & \forall x (Tr(\overline{\ulcorner Y(\dot{x}) \urcorner}) \equiv Y(x)). \end{align} $$

Theorem 5. Let $\mathtt {S=PAT + UYD}$ . S is inconsistent.

Proof. We are working in $\mathtt {S}$ . By the definition of Yablo formula and Theorem 1, we have

$$\begin{align*}\forall x \: (Y(x) \equiv \forall w> x \neg Tr(\overline{\ulcorner Y(\dot{w}) \urcorner})).\end{align*}$$

From (UYD) we then derive

(4) $$ \begin{align} \forall x \: (Y(x) \equiv \forall w> x \: \neg Y(\dot{w})). \end{align} $$

From this and unraveling of $Y(w)$ we infer

$$\begin{align*}\forall x \: (Y(x) \equiv \forall w> x \exists z>w \: Tr(\overline{\ulcorner Y(\dot{z}) \urcorner})).\end{align*}$$

By (UYD) again we get

(5) $$ \begin{align} \: \: \: \forall x \: (Y(x) \equiv \forall w> x \: \exists z>w \: Y(z)). \end{align} $$

By extendability ( $\forall x \exists \, y \: x<y$ ) and transitivity of $<$ (both clearly provable in $\mathtt {S}$ ) and (5) we get

(6) $$ \begin{align} \forall x \: (Y(x) \equiv \exists w> x \: Y(w)). \end{align} $$

By (4) and (6) we have

$$\begin{align*}\forall x \: ((\forall w> x \neg Y(w)) \equiv (\exists w > x Y(w))).\end{align*}$$

Contradiction.

Definition 9 ( $\omega $ -rule)

The $\omega $ -rule is the following infinitary inference rule defined for arithmetical formulae $\varphi $ :

$$ \begin{align*} \frac{\varphi(0), \: \varphi(\overline{1}), \: \varphi(\overline{2}),\: \ldots, \: \varphi(\overline{n}), \: \ldots}{\forall x \:\varphi(x)}. \end{align*} $$

Definition 10 ( $\mathtt {T}^\omega $ )

Let $\mathtt {T}$ be an axiomatizable first-order theory in the language $\mathcal {L}_{Tr}$ . If $\alpha $ is an ordinal, a sequence $(\varphi _0, \ldots , \varphi _\alpha )$ of formulae is a derivation in $\omega $ -logic ( $\omega $ -derivation) from $\mathtt {T}$ if and only if, for each ordinal $\beta \leq \alpha $ :

1. 1. $\varphi _\beta $ is an axiom of $\mathtt {T}$ or

2. 2. there are $\gamma < \beta $ and $\delta < \beta $ , such that $\varphi _\delta = (\varphi _\gamma \rightarrow \varphi _\beta )$ , or

3. 3. (3) $\varphi _\beta = \forall x \psi (x)$ for some $\psi (x) \in Frm_{\mathcal {L}_{Tr}}$ with exactly one free variable and there is an injective function $f: \omega \rightarrow \beta $ such that $\forall n \in \omega \: \: \varphi _{f(n)}= \psi (n)$ (this condition means that $\varphi _\beta $ has been introduced by means of the $\omega $ -rule).

We say that $\varphi $ is a theorem of $\mathtt {T}$ in $\omega $ -logic if there is an $\omega $ -derivation of $\varphi $ from $\mathtt {T}$ . We also call such $\varphi $ an $\omega $ -consequence of $\mathtt {T}$ and denote it as follows: $\mathtt {T}^\omega \vdash \varphi $ . $\mathtt {T}^\omega $ is a set of sentences that are theorems of $\mathtt {T}$ in $\omega $ -logic.

Theorem 6. Let $\mathtt {PA}^{\omega -}_{\mathtt {D}} = (\mathtt {PAT^-} \cup AD \cup YD)^\omega $ . $\mathtt {PA}^{\omega -}_{\mathtt {D}}$ is inconsistent.

Proof. One way to see why this holds is by reference to Theorem 2: adding the $\omega $ -rule to an $\omega $ -inconsistent system results in an inconsistent system. But we find the slow-motion version of the argument somewhat informative, so here it is.

Since $\mathtt {PA}^{\omega -}_{\mathtt {D}}$ is an extension of $\mathtt {PAT^-}$ , by Theorem 1 Yablo sentences exist in this theory. We are working in $\mathtt {PA}^{\omega -}_{\mathtt {D}}$ . Fix $n \in \omega $ and suppose $Y(\overline {n})$ for reductio. By the definition of Yablo sentence, we therefore have $\forall x \: (x>\overline {n} \Rightarrow \neg Tr(\overline {\ulcorner Y(\dot {x}) \urcorner }))$ . Hence

(7) $$ \begin{align} \neg Tr(\overline{\ulcorner Y(\overline{n+1}) \urcorner}) \end{align} $$

as well as $\forall x \: (x>\overline {n+1} \Rightarrow \neg Tr(\overline {\ulcorner Y(\dot {x}) \urcorner }))$ . But the last sentence is equivalent to $Y(\overline {n+1})$ . By the appropriate disquotation instance we therefore get

$$\begin{align*}Tr(\overline{\ulcorner Y(\overline{n+1}) \urcorner}),\end{align*}$$

which contradicts (7). Since the choice of n was arbitrary, we have

$$ \begin{align*} \forall n \in \omega \: \: \: \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash \neg Y(\overline{n}). \end{align*} $$

In particular,

$$ \begin{align*} (*) \: \: \: \: \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash \neg Y(\overline{23}). \end{align*} $$

By the fact that we have a disquotation instance for every Yablo sentence in our theory we also have

$$ \begin{align*} \forall n \in \omega \: \: \: \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash \neg Tr(\overline{\ulcorner Y(\overline{n}) \urcorner}). \end{align*} $$

It is worth noting that so far we’ve only used the resources of $\mathtt {PA}_{\mathtt {D}}$ in this proof. Now we’ll need to use the specific means of $\mathtt {PA}^{\omega -}_{\mathtt {D}}$ . By applying the $\omega $ -rule we therefore obtain

$$ \begin{align*} \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash \forall x \: \neg Tr(\overline{\ulcorner Y(\dot{x}) \urcorner}). \end{align*} $$

In particular we have

$$ \begin{align*} \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash \forall x \: (x> \overline{23} \rightarrow \neg Tr(\overline{\ulcorner Y(\dot{x}) \urcorner})). \end{align*} $$

But this means we have

$$ \begin{align*} (**) \: \: \: \: \mathtt{PA}^{\omega-}_{\mathtt{D}} \vdash Y(\overline{23}). \end{align*} $$

It immediately follows from $(*)$ and $(**)$ that $\mathtt {PA}^{\omega -}_{\mathtt {D}}$ is inconsistent.

Let’s sum up the situation so far. We have a paradox in natural language. Formalizing it, as long as we work in a disquotational theory of truth without any other assumptions, even those theories which prove the existence of Yablo sentences are still consistent (albeit $\omega $ -inconsistent). To obtain a contradiction, we need either inferential rules that go beyond the standard means of first-order logic (namely: the $\omega $ -rule), or a stronger uniform principle of disquotation for Yablo sentences.

Since the paradox somehow involves the concept of infinity (as the essential role of the $\omega $ -rule suggests), we’ll turn our attention to a formalization of Yablo’s paradox in a setting that takes a somewhat different approach to infinity. It is the framework developed by the late Marcin Mostowski and others in [Reference Mostowski9, Reference Mostowski, Rojszczak, Cachro and Kurczewsk10, Reference Mostowski and Zdanowski12], meant as a formalization of the concept of potential infinity. We’ll describe how one could go about avoiding the paradox by taking the distinction between potential and actual infinity seriously.

3 Potentially infinite domains and sl-semantics

In this section we consider the so-called $sl$ -semantics of $FM$ -domains. This framework was motivated by considerations in computational foundations of mathematics and the search for the semantics under which first-order sentences would be interpreted in potentially infnite domains.

Potentially infinite domains are in this framework understood as sequences of finite models increasing without any finite bound. Once we turn to actually finite domains, we need to overcome a small technical obstacle concerning the arithmetical language: it seems that the reference of some singular terms should come out undefined. One way to go would be to employ the apparatus of partial functions. A simpler method, however, uses relational symbols instead of function symbols. This is the one that we’ll employ.

Let $R \subseteq \omega ^r$ be an r-ary relation on natural numbers. By $R^{(n)}$ we denote ${R \cap \{0, 1, \ldots , n-1\}^r}$ , the restriction of this relation to first n natural numbers. For any model $\mathcal {A}$ over some fixed signature $\sigma = (R_1, \ldots , R_k)$ (in particular, one can take the signature to comprise the relational counterpart of standard arithmetical functions, such as addition or multiplication) we define the $FM$ -domain of $\mathcal {A}$ as follows:

$$\begin{align*}FM(\mathcal{A}) = \{\mathcal{A}_n: \:\: n = 1, 2, \ldots\},\end{align*}$$

where

$$\begin{align*}\mathcal{A}_n = (\{0, 1, \ldots, n-1\}, R_1^{(n)}, \ldots, R_k^{(n)}).\end{align*}$$

That is, $FM(\mathcal {A})$ is the set of all finite initial segments of natural numbers, with the arithmetical relations appropriately restricted.

For the purposes of this section, by $\mathbb {N}$ we denote the standard model of arithmetic $(\omega , +, \times , 0, s, <)$ , where the arithmetical functions are interpreted as relations, so we have that

$$ \begin{align*} FM(\mathbb{N}) = \{\mathbb{N}_n: \:\: n = 1, 2, \ldots\}, \end{align*} $$

where

$$ \begin{align*} \mathbb{N}_n = (\{0, 1, \ldots, n-1\}, +^{(n)}, \times^{(n)}, 0^{(n)}, s^{(n)}, <^{(n)}). \end{align*} $$

(Note that this means that numerals are replaced with predicates, so that if in a finite point a numerical would fail to refer, its predicative counterpart refers to the empty set, and if it wouldn’t, its predicative counterpart refers to an appropriate singleton. Thus, in the standard arithmetical language each n-ary symbol is replaced by the corresponding $(n+1)$ -ary one; we’ll keep referring to it as $\mathcal {L}$ with no threat of misunderstanding.)

Ultimately, we’ll be evaluating formulas in $FM(\mathbb {N})$ ; but in order to do that, we first need to evaluate them in the finite segments that $FM(\mathbb {N})$ contains. Satisfaction at finite models involved is understood according to the standard notion of satisfaction. Satisfaction in $FM(\mathbb {N})$ is understood, intuitively, as satisfaction in all sufficiently large segments from $FM(\mathbb {N})$ .

Definition 11 ( $sl(FM(\mathbb {N}))$ )

For any $\varphi \in Sent_{\mathcal {L}}$ we say that $\varphi $ is $sl$ -true in $FM(\mathbb {N})$ (true in sufficiently large models, hence the abbreviation $sl$ ):

$$\begin{align*}FM(\mathbb{N}) \models_{sl} \varphi \text{ if and only if } \exists m \: \forall k \:\: (k \geq m \:\: \Rightarrow \:\: \mathbb{N}_k \models \varphi).\end{align*}$$

Let us then denote

$$ \begin{align*} sl(FM(\mathbb{N})) = \{\varphi \in Sent_{\mathcal{L}} : \:\: FM(\mathbb{N}) \models_{sl} \varphi\}. \end{align*} $$

$sl(FM(\mathbb {N}))$ is called the sl-theory of $FM(\mathbb {N})$ . More generally we could say that for a given class $\mathcal {K}$ of finite models:

$$ \begin{align*} sl(\mathcal{K}) = \{\varphi \in Sent_{\mathcal{L}_{\sigma}} : \:\: \exists n \: \forall \mathcal{M} \in \mathcal{K} \:\: (card(\mathcal{M}) \geq n \:\: \Rightarrow \:\: \mathcal{M} \models \varphi) \}. \end{align*} $$

Now we can use the notion to define a consequence relation by saying:

$$ \begin{align*} \Gamma \models_{sl} \varphi \text{ iff } \forall \mathcal{K}\,\left[\Gamma \subseteq sl(\mathcal{K}) \Rightarrow \varphi\in sl(\mathcal{K})\right]. \end{align*} $$

Obviously, for a given vocabulary, for any class $\mathcal {K}$ of finite models and for any set of sentences $\Delta $ we say that $\mathcal {K} \models _{sl} \Delta $ if and only if $\mathcal {K} \models _{sl} \varphi $ for any $\varphi \in \Delta $ .

3.1 Yablo sequences under $sl$ -semantics

We consider the language obtained by adjoining the truth predicate $Tr$ to the arithmetical language $\mathcal {L}$ and we modify $(FM(\mathbb {N}))$ by adding to its every element $\mathbb {N}_k$ an interpretation $T_k$ of the truth predicate  $Tr$ .

Definition 12 ( $FM(\mathbb {N})^T$ )

Let $\mathcal {K} = \{ (\mathbb {N}_k, T_k): k \in \omega \: \: \text {and} \: \: T_k \subseteq \{0, \ldots , k-1\} \}$ . An $FM(\mathbb {N})^T$ -domain is any subset of $\mathcal {K}$ such that for any natural m it contains exactly one model of the cardinality m.

We obviously have to ensure that Yablo sentences exist in $sl((FM(\mathbb {N})^T))$ . For this, we need to state some important facts concerning the $sl$ -theory of $FM$ -domains, leaving them without a proof. Further details may be found in [Reference Mostowski9–Reference Mostowski and Zdanowski12].

Theorem 7 (Representability of substitution and naming)

There exist formulae Sub (x,y,z) and Name (x) such that for any $FM(\mathbb {N})^T$ -domain the following hold:

• • $FM(\mathbb {N})^T \models _{sl}$ Sub $(\ulcorner \varphi (x) \urcorner , \ulcorner x \urcorner , \ulcorner t \urcorner ) = \ulcorner \varphi (t) \urcorner $ for any formula $\varphi $ and any term t.

• • $FM(\mathbb {N})^T \models _{sl}$ Name $(n) = \ulcorner \overline {n} \urcorner $ for any natural number n.

From the existence of the formulae $\textbf {Sub}(x,y,z)$ and $\textbf {Name}(x)$ we get the following theorem.

Theorem 8 (FM Diagonal Lemma)

The following hold for any $FM(\mathbb {N})^T$ :

• • For any $\mathcal {L}_{Tr}$ -formula $\varphi (x)$ with exactly one free variable there is an $\mathcal {L}_{Tr}$ -sentence $\psi $ such that $FM(\mathbb {N})^T \models _{sl} \psi \equiv \varphi (\ulcorner \psi \urcorner )$ .

• • For any $\mathcal {L}_{Tr}$ -formula $\varphi (x, y)$ with exactly two free variables there is an $\mathcal {L}_{Tr}$ -formula $\psi (x)$ such that $FM(\mathbb {N})^T \models _{sl} \psi (x) \equiv \varphi (x, \ulcorner \psi (x) \urcorner )$ .

Theorem 9 (FM-Undefinability of Truth)

There is no arithmetical formula $\psi (x)$ such that for all arithmetical sentences $\varphi $ :

$$ \begin{align*}FM(\mathbb{N}) \models_{sl} \psi(\ulcorner \varphi \urcorner) \equiv \varphi.\end{align*} $$

Corollary 3 (Existence of Yablo sentences in $FM(\mathbb {N})^T$ -domains)

There exists a formula $Y(x)$ such that for any $FM(\mathbb {N})^T$ -domain we have

$$\begin{align*}\forall n \in \omega \: \: FM(\mathbb{N})^T \models_{sl} Y(n) \equiv \forall x \: (x>n \rightarrow \neg Tr(\ulcorner Y(\dot{x}) \urcorner)),\end{align*}$$

i.e., Yablo sentences exist in $sl(FM(\mathbb {N})^T)$ .

Proof. Obvious by the fact that the diagonal lemma is $sl$ -true in $FM(\mathbb {N})^T$ .

Theorem 10 (Yablo sentences are false in the limit)

For any class $\mathcal {K}$ of finite models, if $\mathcal {K} \models _{sl} AD+YD$ , then for all $n \in \omega \mathcal {K} \models _{sl} \neg Y(n)$ . In fact, AD isn’t essential (it’s only added to ensure that $Tr$ behaves like a truth predicate): for any $n\in \omega $ we have $YD \models _{sl} \neg Y(n)$ .

Proof. Let us fix a class $\mathcal {K}$ . For the sake of contradiction, suppose that there is a Yablo sentence that is not false at sufficiently large models, that is which for any size of a model is true at some model in $\mathcal {K}$ of at least that size, that is,

$$ \begin{align*} \exists n \forall k \exists \mathcal{M} \in \mathcal{K} \: \: (card(\mathcal{M})\geq k \: \wedge \: \mathcal{M}\models Y(n)). \end{align*} $$

Let us take take such n. Let us fix k and take $\mathcal {M} \in \mathcal {K}$ (without loss of generality we could assume that $\mathcal {M} = \mathbb {N}^T_k$ for some natural k) with $card(\mathcal {M})\geq k$ such that $\ulcorner Y(n+1) \urcorner \in |\mathcal {M}|$ and:

1. (i) $\mathcal {M} \models Y(n)$ .

2. (ii) $\mathcal {M} \models Y(n+1) \equiv \forall x (x>n+1 \rightarrow \neg Tr(\ulcorner Y(\dot {x})\urcorner ))$ .

3. (iii) $\mathcal {M} \models Y(n+1) \equiv Tr(\ulcorner Y(n+1)\urcorner )$ .

4. (iv) $\mathcal {M}\models Y(n) \equiv \forall x \, (x> n \rightarrow \neg Tr(\ulcorner Y (\dot {x}) \urcorner ))$ (this last condition means that $\mathcal {M}$ is sufficiently large to satisfy the defining formula of Yablo sentences).

(i) will be satisfied by the assumption of the proof, (ii) and (iv) follow from Corollary 3 and (iii) results from the assumption that $\mathcal {K}\models _{sl}YD$ once we have enough numbers to code all the formulas needed for the claims to hold.

From (i) and the fact that $\mathcal {M}$ models and therefore the defining formula for Yablo sentences we obtain

(8) $$ \begin{align} \: \: \: \: \mathcal{M} \models \neg Tr(\ulcorner Y(n+1)\urcorner), \end{align} $$

as well as

(9) $$ \begin{align} \: \: \: \: \mathcal{M} \models \forall x (x>n+1 \rightarrow \neg Tr(\ulcorner Y(\dot{x})\urcorner)). \end{align} $$

Now, from (8), by (iii) we have that:

$$ \begin{align*} \mathcal{M} \models \neg Y(n+1), \end{align*} $$

and from (9), by (ii) we have that:

$$ \begin{align*} \mathcal{M} \models Y(n+1), \end{align*} $$

which gives a contradiction that ends the proof.

We will now show a construction of a class $\mathcal {K}$ such that $\mathcal {K} \models _{sl} YD+AD$ , which means that Theorem 10 holds non-vacuously. It turns out that $FM(\mathbb {N})^{Y}$ is such a class.

Definition 13 ( $FM(\mathbb {N})^{Y}$ and $sl(FM(\mathbb {N})^Y)$ )

We fix a formula $Y(x)$ satisfying the condition specified in Corollary 3. A family of models $FM(\mathbb {N})^{Y}$ is an $FM(\mathbb {N})^{T}$ -domain $\{\mathbb {N}^Y_k: \: \: k \in \omega \}$ such that $\mathbb {N}^Y_k = (\mathbb {N}_k, T_k)$ , where $T_k = TA_k \cup TY_k$ , and:

$$ \begin{align*} TA_k &= \{\ulcorner \varphi \urcorner: \: \: \varphi \in Sent_{\mathcal{L}} \: \: \text{and} \: \: \mathbb{N}_k \models \varphi \} \cap |\mathbb{N}_k| \\ TY_k &= \{\ulcorner Y(m) \urcorner: \: \: \ulcorner Y(m)\urcorner \in |\mathbb{N}_k| \wedge \ulcorner Y(m+1)\urcorner \notin |\mathbb{N}_k| \}. \end{align*} $$

If $\vert \mathcal {M}\vert =k$ , we sometimes write $T_{\mathcal {M}}$ instead of $T_k$ .

Naturally, $sl(FM(\mathbb {N})^Y) = \{ \varphi : \: \: FM(\mathbb {N})^Y \models _{sl} \varphi \}$ .

It is worth noting that the family $FM(\mathbb {N})^Y$ is not a proper potentially-infinite domain in the sense defined by Mostowski [Reference Mostowski11], i.e., for any natural numbers $r<s \mathbb {N}^Y_r$ is not a submodel of $\mathbb {N}^Y_s$ , since the interpretation of the truth predicate $Tr$ varies between subsequent models from the domain.

Observation 1. For any class $\mathcal {K}$ , if $\mathcal {K} \models _{sl} YD$ , then for sufficiently large $\mathcal {M} \in \mathcal {K}$ , there is exactly one $n \in \omega $ s.t. $\ulcorner Y(n) \urcorner \in T_{\mathcal {M}}$ .

Theorem 11. $\forall n \in \omega \: \: \: FM(\mathbb {N})^Y \models _{sl} \: \neg Y(n)$ .

Proof. We claim that for any n there exists m such that for any $k> m$ :

$$ \begin{align*} \mathbb{N}^Y_k \models \neg Y(n). \end{align*} $$

Indeed, let us fix n and take m such that $\ulcorner Y(n+1) \urcorner \in |\mathbb {N}^Y_m|$ and for any $x<n+1$ and any $k>m$ we have

$$ \begin{align*} \mathbb{N}^Y_k \models \: Y(x) \: \equiv \: \forall w>x \:\: \neg Tr(\ulcorner Y(\dot{w}) \urcorner). \end{align*} $$

Let $k>m$ . Then obviously $\ulcorner Y(n+1) \urcorner \in |\mathbb {N}^Y_k|$ . Let j be the greatest number such that $\ulcorner Y(j) \urcorner \in |\mathbb {N}^Y_k|$ . Such a number exists since our $FM$ -domain is infinite and every model in it is finite. Obviously, $n<j$ . From the definition of the class $FM(\mathbb {N})^Y$ and by the choice of j we obtain

$$ \begin{align*} \mathbb{N}^Y_k \models Tr(\ulcorner Y(j) \urcorner), \end{align*} $$

and for any $x<j$ we get

$$ \begin{align*} \mathbb{N}^Y_k \models \: \exists w (w>x \: Tr(\ulcorner Y(\dot{w})\urcorner)). \end{align*} $$

Thus, since $n<j$ , we obtain

$$ \begin{align*} \mathbb{N}^Y_k \models \neg Y(n), \end{align*} $$

which ends the proof.

Corollary 4. $sl(FM(\mathbb {N})^Y)$ is $\omega $ -inconsistent.

Proof. We have just shown that

$$ \begin{align*} \forall n \in \omega \: \: (\neg Y(n) \in sl(FM(\mathbb{N})^Y)). \end{align*} $$

Intuitively speaking, for each sufficiently large finite segment, the last Yablo sentence whose code exists in it will be satisfied, and so the existential claim some Yablo sentence is true will be satisfied in all sufficiently large segments, without the n-th Yablo sentence is true being satisfied in all sufficiently large segments (because in different segments different objects will be the last Yablo sentences). The argument for $\omega $ -inconsistency relies on this tension.

We obviously also have that for all sufficiently large k there exists $j<k$ such that by the definition of $FM(\mathbb {N})^Y)$ we have $\mathbb {N}^Y_k \models Y(j)$ , so by existential generalization we obtain that there is m such that for all $k>m \mathbb {N}^Y_k \models \exists x Y(x)$ and thus:

$$ \begin{align*} \exists x Y(x) \in sl(FM(\mathbb{N})^Y).\\[-35pt] \end{align*} $$

So, it seems, there is a finitistic approach to arithmetic, according to which all Yablo sentences are false. The cost of this move, however, isn’t negligible: the set of arithmetical formulae true in the intended model is $\omega $ -inconsistent. Therefore, we pursue the topic further, looking at another way to think finitistically about the issue.

4 Modal interpretation of quantifiers in potentially infinite domains

Thus we also have that $(\mathcal {K}, M ) \models _{m} \forall x \varphi (x)$ iff for all $N \in \mathcal {K}$ s.t.

$R(M,N)$ and for all $a \in N (\mathcal {K}, N) \models _{m} \varphi [a]$ .

The intuition behind this semantics is as follows. ‘ $\exists x \: \varphi (x)$ ’ reads ‘there could be enough objects so that for some a, $\varphi (a)$ ’ and ‘ $\forall x \: \varphi (x)$ ’ reads ‘however many more objects there could be, it still would be the case that for any a, $\varphi (a)$ ’.

Now, let $msl(FM(\mathbb {N})) = \{\varphi : \exists n \: \forall k \: k \geq n \Rightarrow (FM(\mathbb {N}), \mathbb {N}_k) \models _{m} \varphi \}.$ That is, intuitively, $msl(FM(\mathbb {N}))$ is the set of those formulas, which are true in all sufficiently large models, where the notion of truth involves the modal reading of quantifiers.

As an example, let $\varphi = \exists x \: \forall y \: x \geq y$ —that is, $\varphi $ says: there exists a maximal element. While, as we remember, $\varphi \in sl(FM(\mathbb {N}))$ , things are different under m-semantics— $\varphi $ is false in every possible world of the FM-domain.

Before we move on, let us emphasize that just as with $sl$ -semantics, we work with a relational arithmetical language. While in the case of $sl$ -semantics this wasn’t too important, it becomes crucial when we turn to $msl$ -semantics.

For otherwise, we need to treat, say, addition and successor as total functions. This being the case, for each finite initial segment we’d need to identify the candidates for the values of functions which intuitively should surpass the capabilities of that segment. The least unnatural way to do this would be to plug in loops at ends of segments, so that $s(max(\mathbb {N}_k))=max(\mathbb {N}_k),$ etc. But then we would run into problems. For instance, take $\varphi := \exists x \: x+x = x \wedge x \neq 0$ . If we evaluate atomic sentences in the elements of our FM-domain, then for any k we have

$$ \begin{align*}\mathbb{N}_k \models_m \varphi \: \text{iff} \: \: \exists j \geq k \: \exists a < j \: \mathbb{N}_j \models a+a = a \wedge a \neq 0.\end{align*} $$

However, the above would come out true. After all, let $a=max(\mathbb {N}_j)$ . Then, $a+a=\underbrace {s\dots s}_{a}a=a$ and we have $\mathbb {N}_j\models a\neq 0 \wedge a+a=a$ and so $\mathbb {N}_k\models _m \varphi $ for any k. Thus, we would have $\varphi \in msl(FM(\mathbb {N}))$ , while $\mathtt {PA}\vdash \neg \varphi $ .

The underlying cause of the issue is that when we work with a functional language we cannot think about the initial segments as submodels of larger initial segments, because the functions are not preserved when we move to superstructures. The problem disappears when we abandon function symbols and use a relational language instead. So this is what we’ll do in what follows.

In particular, we’ll be working towards a theorem according to which the resulting arithmetic is the classical arithmetic, unlike in the previous case. We’ll start with two lemmata.

Lemma 4. For any quantifier-free $\varphi (x_1, \dots , x_n)$ and for any choice of parameters $a_1, \dots a_n\in \mathbb {N}$ , if we have $a_1, \dots , a_n \in \mathbb {N}_k$ , then it holds that:

$$\begin{align*}\mathbb{N}_k \models \varphi[a_1, \dots, a_n] \text{ iff } \mathbb{N}\models \varphi[a_1, \dots, a_n].\end{align*}$$

Proof. The proof is by induction on formula complexity.

For the basic case of quantifier-free formulae, the claim holds by Lemma 4. For boolean connectives, the equivalence is trivial. The only interesting case is for ${\varphi := \exists x \psi (x)}$ .

$\supseteq $ : Suppose $\varphi \in Th(\mathbb {N})$ , that is, $\mathbb {N}\models \exists x \psi (x)$ . Then there is a witness $a\in \mathbb {N}$ , such that $\psi [a]\in Th(\mathbb {N})$ . By IH, $\psi [a]\in msl(FM(\mathbb {N}))$ . This means

(10) $$ \begin{align}\exists k \, \forall l\geq k \, \mathbb{N}_l \models_m \psi [a]\end{align} $$

and from this it follows that:

(11) $$ \begin{align}\exists k \forall l\geq k\, \exists j\geq l \,\exists a < j \, \mathbb{N}_j\models_m \psi[a], \end{align} $$

which means that $\varphi \in msl(FM(\mathbb {N}))$ .

$\subseteq $ : Say $\varphi := \exists x\, \psi (x)\in msl(FM(\mathbb {N}))$ . So (11) holds as well and there is an a such that (10) also holds. This step essentially depends on the language being relational; thanks to this assumption for any $k_1<k_2$ we have $\mathbb {N}_{k_1}\subseteq \mathbb {N}_{k_2}\subseteq \mathbb {N}$ (we mean the submodel relation here) and by the m-semantics witnesses of existential claims remain such witnesses in supermodels). This means $\psi [a]\in msl(FM(\mathbb {N}))$ , and so by the IH, $\psi [a]\in Th(\mathbb {N})$ , and therefore $\varphi \in Th(\mathbb {N})$ .

5 Yablo sequences in potentially infinite domains under the modal interpretation of quantifiers

The semantics in sufficiently large models in potentially infinite domains under modal interpretation of quantifiers presented above entails that Yablo sentences stay paradoxical even for the finitist, if she interprets the quantifiers in the modal manner. That is, we’ll be arguing that not even local Yablo Disquotation can be included in an $msl$ -theory. Let’s start with a lemma.

Proof. First note that the Diagonal Lemma holds for the extended language in the $msl$ -theory (because its arithmetical part is $Th(\mathbb {N})$ ). So Yablo sentences belong to it. Now, suppose some Yablo sentence is in the $msl$ -theory, that is,

$$\begin{align*}\exists n \, \left[Y(n)\in msl(FM(\mathbb{N}^Y))\right].\end{align*}$$

This means

$$\begin{align*}\exists l\, \forall k \geq l \: \mathbb{N}_k \models_m Y(n).\end{align*}$$

Pick an l witnessing this. By the definition of Yablo sentences this entails

$$\begin{align*}\forall k\geq l\,\mathbb{N}_k \models_m \forall x\, (x> n \rightarrow \neg Tr(Y(x))). \end{align*}$$

By the semantics, this means

$$\begin{align*}\forall k \geq l \forall p \geq k \forall a < p \: \: \mathbb{N}_p \models_m a> n \rightarrow \neg Tr(Y(a)).\end{align*}$$

But then:

(12) $$ \begin{align}\forall p \geq l \: \forall a \in (n,p) \: \mathbb{N}_p \models_m \neg Tr(Y(a)).\end{align} $$

So, by Yablo Disquotation, if we take a p sufficiently large to satisfy Tarski biconditionals for Yablo sentences $Y(a)$ for $n<a<p$ (such p exist by the fact that $YD\subseteq msl(FM(\mathbb {N})^Y)$ ), we have that models of size p fail to satisfy Yablo sentences for numbers between n and p:

$$\begin{align*}\forall p \geq l \: \forall a \in (n,p) \: \mathbb{N}_p \models_m \neg Y(a).\end{align*}$$

Now, fix p and a. By the definition of $Y(x)$ the above means:

$$\begin{align*}\mathbb{N}_p \models_m \exists x> a \, Tr(Y(x)).\end{align*}$$

By our definition of $\models _m$ this is equivalent to:

$$\begin{align*}\exists q \geq p \, \exists b < q\, \mathbb{N}_q \models_m b> a \wedge Tr(Y(b)).\end{align*}$$

Hence,

$$\begin{align*}\exists q \geq p \, \exists b \in (a,q)\, \mathbb{N}_q \models_m Tr(Y(b)).\end{align*}$$

This, however, contradicts (12), which completes the argument.

With this lemma at hand, we can proceed to the theorem which tells us that not only Yablo sentences are not in any $msl$ -theory, but also that no $msl$ -domain can (modally) satisfy the local Yablo Disquotation principles either.

Proof. Suppose there is an $msl(FM(\mathbb {N})^Y)$ which contains all Local Yablo Disquotation sentences.

By Lemma 5, we know that $\forall n \: Y(n) \not \in msl(FM(\mathbb {N})^Y)$ . We therefore have

$$\begin{align*}\forall n\, \forall l \: \exists k \geq l \: \mathbb{N}_k \not\models_m Y(n).\end{align*}$$

By the definition of the Yablo sentences we infer

$$\begin{align*}\forall n \forall l \exists p \geq l \exists a> n \: \mathbb{N}_p \models_m Tr(Y(a)).\end{align*}$$

Which, by Yablo Disquotation, yields:

$$\begin{align*}\forall n \forall l \exists p \geq l \exists a> n \: \mathbb{N}_p \models_m Y(a).\end{align*}$$

The claim holds for any n and l. For us, it is enough to look at $n = l = 0$ . By the definition of $Y(a)$ and m-semantics we obtain

$$\begin{align*}\exists p, a> 0 \forall q \geq p \: \mathbb{N}_q \models_m \forall x > a \: \neg Tr(Y(x)).\end{align*}$$

Pick an $a>0$ witnessing the above claim. By the definition of $msl$ -theory we now have

$$\begin{align*}Y(a) \in msl(FM(\mathbb{N})^Y),\end{align*}$$

which is impossible by Lemma 5.

So, when we consider Yablo sequences with different treatments of infinity in the background, the following observations come to mind:

1. 1. In the standard setting, without potential infinity, Local Arithmetical Disquotation and Local Yablo Disquotation are consistent, yet $\omega $ -inconsistent with the background arithmetical theory. Once $\omega $ -rule or Uniform Yablo Disquotation is introduced, the theory is inconsistent.

2. 2. Under $sl$ -semantics, Yablo sentences are all false (in the limit), yet the $sl$ -theory of a given $FM$ -domain is consistent, but $\omega $ -inconsistent. This is a particular case of a general flaw of $sl$ -semantics, because $sl(FM(\mathbb {N}))$ itself is $\omega $ -inconsistent.

3. 3. Under $msl$ -semantics, i.e., semantics in sufficiently large models in potentially infinite domains under the modal interpretation of quantifiers, even adding only the Local Arithmetical Disquotation and Yablo Disquotation results in an inconsistent theory. Uniform Yablo Disquotation or $\omega $ -rule is not needed to ensure this.

6 A remark on semantics and ontology

One question that might come to mind in this context is this. If for the first-order relational language in the limit we simply get the true arithmetic, what’s the point? Why isn’t this just syntactic sugar? Doesn’t this mean that the finitistic and potentialist parlance is a bit of cheating and that the entire framework gives rise to actualism with respect to the structure that we approximate?

Well, while the semantics—as desired—legitimates the (prima facie infinitistic) theory of the standard model of arithmetic, at the ontological level the difference remains: the ontology is indeed finitistic and we evaluate formulae in finite models. While it is hard to capture this idea in terms of a first-order language, it becomes more transparent when we look at a second-order language.

If we take a relational second-order relational language, where in the standard model quantifiers range over the powerset of $\mathbb {N}$ and in m-semantics at each segment they only range over its powerset, the set of formulae belonging to the $msl$ -theory will be quite different from those true in the standard model. Namely, it will be those formulae which are true in the standard model where second-order quantifiers are taken to range over finite sets.

Formally, let us extend the definition of m-satisfaction for the standard $FM$ -domain, including the second-order domain of quantification. Let $\mathcal {P}(\mathbb {N}_k)$ denote the powerset of the set $|\mathbb {N}_k| = \{0, \ldots , k-1\}$ , i.e.,

$$ \begin{align*}\mathcal{P}(\mathbb{N}_k) = \mathcal{P}(\{0, \ldots, k-1\}).\end{align*} $$

Now add the following clause for the existential second-order quantification to the definition of m-semantics. $(\mathbb {N}_k, \mathcal {P}(\mathbb {N}_k)) \models _m \exists X \phi (X)$ iff

$$ \begin{align*}\exists n \geq k \exists A \in P(N_n) (\mathbb{N}_n, \mathcal{P}(\mathbb{N}_n)) \models_m \phi[A].\end{align*} $$

Further, we extend the definition of the $msl$ -theory of the $FM$ -domain for the standard second-order model of arithmetic, i.e., to the structure $\mathbb {N}_2 := (\mathbb {N}, \mathcal {P}(\mathbb {N}))$ in the obvious way. Observe that for the expanded models, it still holds that if $k<n$ , then $(\mathbb {N}_k, \mathcal {P}(\mathbb {N}_k))$ is a submodel of $(\mathbb {N}_n, \mathcal {P}(\mathbb {N}_n))$ .

Interestingly, the following now holds.

Proof. We just need to extend the proof for the formulae containing second-order quantifiers. The case of atomic second-order formulae is trivial. Suppose $\psi = \exists X \varphi (X)$ .

$(\Rightarrow )$ By assumption, for sufficiently large $k \psi $ holds under m-semantics in every $(\mathbb {N}_k, \mathcal {P}(\mathbb {N}_k))$ . So, by definition, for each such k there is an $n> k$ and $A \subset |\mathbb {N}_n|$ with

$$ \begin{align*}(\mathbb{N}_n, \mathcal{P}(\mathbb{N}_n)) \models \varphi[A].\end{align*} $$

This means that we almost always (i.e., for all but finitely many finite models—the ones given by the sufficiently large k) have finite witnesses for existential second-order claims. By the extended m-semantics, witnesses of existential claims remain such witnesses in supermodels, thus $\varphi [A] \in msl(FM((\mathbb {N}_2)))$ . Since $A \in \mathcal {P}_{fin}(\mathbb {N})$ , by the Inductive Hypothesis,

$$ \begin{align*}(\mathbb{N}, \mathcal{P}_{fin}(\mathbb{N})) \models \varphi[A],\end{align*} $$

so by existential generalization

$$ \begin{align*}(\mathbb{N}, \mathcal{P}_{fin}(\mathbb{N})) \models \exists X \varphi(X).\end{align*} $$

$(\Leftarrow )$ Assume $(\mathbb {N}, \mathcal {P}_{fin}(\mathbb {N})) \models \exists X \varphi (X)$ , which means that there is a finite set of natural number A such that

$$ \begin{align*}\mathbb{N} \models \varphi[A].\end{align*} $$

By the inductiveFootnote ⁵ assumption, for sufficiently large k we have

$$ \begin{align*}(\mathbb{N}_k, \mathcal{P}(\mathbb{N}_k)) \models_m \varphi[A],\end{align*} $$

thus giving that for all but finitely many k:

$$ \begin{align*}\exists A \in \mathcal{P}(\mathbb{N}_k) \: (\mathbb{N}_k, \mathcal{P}(\mathbb{N}_k)) \models_m \varphi[A],\end{align*} $$

and therefore (since $k <n$ and $A \subseteq |\mathbb {N}_k|$ , then also $A \subseteq |\mathbb {N}_n|$ ), we have that for all but finitely many k there is an $n> k$ with

$$ \begin{align*}\exists A \in \mathcal{P}(\mathbb{N}_n) \: (\mathbb{N}_n, \mathcal{P}(\mathbb{N}_n)) \models_m \varphi[A].\end{align*} $$

This, by the extended definition of m-semantics, means that $\exists X \varphi (X) \in msl(FM((\mathbb {N}_2)))$ .

This fact is best understood as: in our potentialist-finitistic framework, the finitist ontology is hidden in the realm of the domain of second-order quantification, not in the semantics of first-order formulas, which—as desired—gives rise to true arithmetic, and thus vindicates the infinitistic idealization for arithmetic on potentialist-finitistic grounds. We can adopt a purely instrumentalist approach to the extension of m-semantics to the level of subsets of natural numbers and treat it just as an explanatory tool needed to answer a question about location of finitism in the potentialist framework that we propose. The answer then should be that although within the potentialist semantics for first-order arithmetic we essentially quantify over all natural numbers (with the modal understanding of this!), when it comes to collections of numbers, we can only express and recognize the truths concerning finite sets. This is exactly as it should be. The ontology of $FM$ -domains is finitistic in the sense that the domain does not provide a single infinite universe, but a family of its approximations—finite initial segments, the universes of which (and subsets of these) are precisely the finite sets of natural numbers, so if we allow quantification over not only numbers, but also of their collections, i.e., if we turn to second-order semantics, it is quite natural that we will be able to capture the properties of finite sets only. What seems to require an explanation then is why—under modal potentialist semantics—can we capture all the properties of individual natural numbers (i.e., in the first-order setting)? The answer lies in the question itself—if one shares finitistic intuitions, formalizes them in a model-theoretic manner and adopts correct potentialist semantics, then there is indeed no difference from the perspective of first-order number theory. The reason why this is the case is that—in a sense specified by the modal semantics—a position of an extremely local object such as an individual number in a finite structure that can be indefinitely extended is not distinct from the position of such object in the entire structure, provided we can make references to individual objects only. In a very loose sense, it does not matter whether we look at a number from the perspective of an infinite set or from the perspective of a finite, yet unbounded one. Consequently, finitism isn’t too revolutionary in the context of first-order arithmetic, since the modal finitistic semantics vindicates the corresponding infinitistic idealization.

However, finitistically available sets of numbers, provided we can refer to such sets using quantification (i.e., in the second-order setting), can have very different properties in an infinite structure and in its finite approximations, even if allow the finite segments to be indefinitely extendable, since in the system of approximations there simply are no infinite sets—they are never objects in the $FM$ -domains framework (whereas individual numbers are always finite objects, irregardless of whether they are located in $\mathbb {N}$ or in the elements of $FM(\mathbb {N})$ ).

We claim that this is the correct (ontological) commitment of (arithmetical) finitism. Any (finite) number of individual objects can be added to a given finite universe and it will still be finite, but there is no way to finitely extend the universe of a finite model so that the model becomes infinite, and this affects the properties of subsets of a given structure rather than individual elements of the structure. To illustrate the consequences of this, observe that for each particular number n one can always extend a finite initial segment A of $\mathbb {N}$ to B so that n is not the maximal element of B even if it was a maximal element of A. However, for no subset X of A can we find an extension to B so that X was not injectable into its proper subset in A, but is injectable in its proper subset in B.Footnote ⁶ So while it seems that for first-order context extreme finitism is untenable, and one might thus think that our potentialist account of arithmetic ideologically reduces to actualism, it is our results concerning the second-order framework that suggest that there is an open door for an argument against the claim that arithmetical potentialism is actualism in sheep’s clothing. There still can be a difference between the family of approximations of a structure and the structure itself, but semantically it has to be captured by higher-order means. Whether these means are allowed, is a completely different question that we do not engage with here.

Interestingly, our approach somehow contrasts with the so-called predicativist approach to arithmetic, under which reference to all subsets of an infinite set is not legitimate, and only some infinite subsets of natural numbers are accepted if they can be referred to, where what type of predicativism we’re dealing with largely depends on what means of reference are admissible.Footnote ⁷ Under the modal semantics, the theory of sets of natural numbers seems to be more restrictive, since, in a sense, only finite sets are allowed. Consequently, there will be sentences of second-order arithmetic true under $msl$ -semantics that are false in predicative second-order arithmetic. This, however, does not present a problem, since $msl$ -semantics is provided to give account not of predicativism in arithmetic, but of finitism and potentialism therein. The contrast between predicativism and our formalism thus plays its role in reflecting on the difference between these philosophical approaches to arithmetic, and illustrating yet another (apart from the vindication of infinitistic idealization above) possible weak point in finitism—at the second-order level it is a position even more restrictive than predicativism. We leave it for further work to investigate if this is an accurate argument.

All in all, our reading of the results concerning the first- and second-order theories of arithmetic under $msl$ -semantics provides the following philosophical standpoints:

• • It is second-order potentialist semantics (rather than the first-order one) that captures the finitistic ontology of $FM$ -domains.

• • At the first-order level, the potentialist semantics vindicates the infinistic idealization of natural numbers,

• • For this reason, a revisionary finitistic approach to first-order arithmetic is unjustified.

• • Nevertheless, the difference between second-order $msl$ theory of natural numbers and standard second-order true arithmetic allows for arguing that potentialist systems of approximations to a given structure do not reduce to hidden actualism.

• • However, the second-order $msl$ -theory of $\mathbb {N}$ is very restrictive.