A Deflationary Account of the Truth of the Gödel Sentence $\mathcal{G}$

Abstract

We give a negative answer to the question of whether our conviction about the truth of the Gödel sentence $\mathcal{G}$ involves a theory of truth beyond the deflationary theories (Shapiro, J Philos 95:493–521, 1998; Ketland, Mind 108:69–94, 1999; Tennant, Mind 111:551–582, 2002; Ketland, Mind 114:75–88, 2005; Tennant, Mind 114:89–96, 2005; Cieśliński, Mind 119:409–422, 2010). After discussing and dismissing Neil Tennant’s deflationary account of incompleteness, we show how a new deflationary construal of the incompletability of formal systems can be framed in the setting of Peano Arithmetic augmented to include a constructive version of the ω-rule based on Herbrand’s notion of prototype proof.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-10434-8_16

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-10434-8_16

Appendix: Technical Backgrounds and Proofs

1.1 A.1 Peano Arithmetic: Theory and Models

Definition A.1 (Peano Arithmetic).

The language of PA is given by the language of first-order logic with identity enriched with the individual constant $\overline{0}$, the unary functional symbol Succ(_) (the successor) and the two binary functional symbols + and ⋅ . Moreover, the specific deductive apparatus of PA is defined by the following nine proper axioms:

• (1) $x = y \rightarrow (x = z \rightarrow y = z)$

• (2) $x = y \rightarrow Succ(x) = Succ(y)$

• (3) $\overline{0}\neq Succ(x)$

• (4) $Succ(x) = Succ(y) \rightarrow x = y$

• (5) $x + \overline{0} = x$

• (6) $x + Succ(y) = Succ(x + y)$

• (7) $x \cdot \overline{0} = \overline{0}$

• (8) $x \cdot Succ(y) = (x \cdot y) + x$

• (9) For every formula α(x) of PA such that x occurs free in α,

  $\vdash _{\mathsf{PA}}\alpha (\overline{0}) \rightarrow (\forall x(\alpha (x) \rightarrow \alpha (Succ(x)) \rightarrow \forall x\alpha (x))$.

We abridge with $\overline{n}$ the numeral $Succ(Succ\ldots Succ(\overline{0})\ldots )$ resulting from n applications of the successor function to the constant $\overline{0}$. For any pair of terms t and s, t ≠ s is intended to be defined as ¬(t = s).

Definition A.2 (Structure $\mathcal{N}$).

The structure $\mathcal{N} = (\mathbb{N},0,+^{\mathcal{N}},\cdot ^{\mathcal{N}},Succ^{\mathcal{N}})$ is formed by the set of non-negative integers $\mathbb{N} =\{ 0,1,2,\ldots \}$, the distinguished number $0 \in \mathbb{N}$ (which interprets the constant $\overline{0}$), the functional symbols $+^{\mathcal{N}}$ and $\cdot ^{\mathcal{N}}$, respectively, corresponding to the familiar sum and product and the successor function $Succ^{\mathcal{N}}(x)\stackrel{def.}{=}x + ^{\mathcal{N}}1$.

Theorem A.3 (Soundness).

$\mathcal{N}$ is a model of PA and, moreover, PA is sound w.r.t.  $\mathcal{N}$.

Proof.

For establishing that $\mathcal{N}$ is a model of PA (in symbols $\mathcal{N} \vDash \mathsf{PA}$), we have to show that each of the PA axioms is interpreted in $\mathcal{N}$ as a true statement. It is immediate to check that $\mathcal{N}$ satisfies axioms 1–8. As far as the induction principle is concerned (axiom 9), since the domain of $\mathcal{N}$ exactly coincides with the set of naturals $\mathbb{N}$, the inductive mechanism is indeed able to cover the totality of the elements of $\mathcal{N}$, so as to justify the introduction of the universal quantifier.

As far as the soundness property is concerned, the proof consists in showing that, for any formula α, if $\vdash _{\mathsf{PA}}\alpha$, then $\mathcal{N} \vDash \alpha$. We proceed by induction on the length of the PA proof ending with $\vdash _{\mathsf{PA}}\alpha$. The base is clearly provided by the fact that $\mathcal{N} \vDash \mathsf{PA}$. Then, it is easy to see that the logical inference rules transmit the truth from premisses to conclusions. □ 

Remark A.4 (Soundness and Consistency).

Due to the strict bivalence of classical semantics, if a theory is sound w.r.t. a certain model, it is consistent (otherwise the theory would prove a false statement).

Remark A.5 (Standard Model).

The structure $\mathcal{N}$ is said to be the standard model for PA.

1.2 A.2 The Incompleteness Theorems

Definition A.6 (Deductive Independence).

A formula α is said to be independent of PA if $\nvdash _{\mathsf{PA}}\alpha$ and $\nvdash _{\mathsf{PA}}\neg \alpha$.

Definition A.7 (ω-Consistency).

A certain arithmetical theory T is said to be ω-consistent if the following two conditions are mutually excluding:

• For all $n \in \mathbb{N}$, $\vdash _{\mathsf{T}}\alpha (\overline{n})$,

• $\vdash _{\mathsf{T}}\exists x\neg \alpha (x)$.

Remark A.8.

ω-consistency is stronger than consistency so like any ω-consistent theory is also consistent.

The proofs of the incompleteness theorems are here merely sketched; for the technical details the reader is referred to Rautenberg (2000).

Theorem A.9 (First Incompleteness Theorem).

There exists a formula $\mathcal{G}$ such that if PA is ω-consistent, then $\mathcal{G}$ is independent of PA.

Proof.

The proof is developed through the following five points.

• (1) There exists a 1–1 assignment of natural numbers to formulas and demonstrations of PA. $\ulcorner \alpha \urcorner$ and $\overline{\ulcorner \alpha \urcorner }$, respectively, indicate the number associated with α (its Gödelian code) and its corresponding numeral: if $\ulcorner \alpha \urcorner = n$, then $\overline{\ulcorner \alpha \urcorner } = \overline{n}$. In the same way, $\ulcorner \pi \urcorner$ and $\overline{\ulcorner \pi \urcorner }$, respectively, denote the Gödelian code of the proof π and the corresponding numeral.

• (2) It is possible to define a $\Delta _{0}$-formula $\mathit{Dem}_{\mathsf{PA}}(x,y)$ such that $\vdash _{\mathsf{PA}}\mathit{Dem}_{\mathsf{PA}}(\overline{n},\overline{m})$ if, and only if, n encodes a PA demonstration of the formula α with $\ulcorner \alpha \urcorner = m$.

• (3) Consider the predicate $\mathit{Theor}_{\mathsf{PA}}(y)\stackrel{def.}{=}\exists x\mathit{Dem}_{\mathsf{PA}}(x,y)$. Its negation admits a formula $\mathcal{G}$ as a fixed point, i.e.

  $$\displaystyle{\vdash _{\mathsf{PA}}\mathcal{G}\leftrightarrow \neg \mathit{Theor}_{\mathsf{PA}}(\overline{\ulcorner \mathcal{G}\urcorner }).}$$

• (4) $\vdash _{\mathsf{PA}}\mathcal{G}$ implies $\vdash _{\mathsf{PA}}\neg \mathcal{G}$ and so, if PA is consistent, $\nvdash _{\mathsf{PA}}\mathcal{G}$.

• (5) If PA is ω-consistent, then $\nvdash _{\mathsf{PA}}\mathcal{G}$ implies $\nvdash _{\mathsf{PA}}\neg \mathcal{G}$.

Finally, $\mathcal{G}$ is independent of PA. □ 

Theorem A.10 (Second Incompleteness Theorem).

Consider the formula

$$\displaystyle{\mathit{Cons}_{\mathsf{PA}} \equiv \neg \mathit{Theor}_{\mathsf{PA}}(\ulcorner 0 = 1\urcorner )}$$

asserting the consistency of PA: it is independent from PA as well as $\mathcal{G}$.

Proof.

The proof consists in showing that $\mathit{Cons}_{\mathsf{PA}}$ is provably equivalent to $\mathcal{G}$, i.e. $\vdash _{\mathsf{PA}}\!\mathit{Cons}_{\mathsf{PA}} \leftrightarrow \mathcal{G}$. In such a way, $\vdash _{\mathsf{PA}}\!\mathit{Cons}_{\mathsf{PA}}$ and $\vdash _{\mathsf{PA}}\neg \mathit{Cons}_{\mathsf{PA}}$ would, respectively, imply $\vdash _{\mathsf{PA}}\mathcal{G}$ and $\vdash _{\mathsf{PA}}\neg \mathcal{G}$, against the First Incompleteness Theorem. □ 

1.3 A.3 $\Sigma _{1}$-Completeness and Related Results

Definition A.11 (Logical Complexity).

• A formula α belongs to the set $\Delta _{0}$ if it is equivalent to a closed formula α′ in which all the quantifiers, if any, are bounded.

• A formula α belongs to $\Sigma _{1}$ (resp. $\Pi _{1}$) if it is equivalent to a closed formula $\alpha ' \equiv \exists x\beta (x)$ (resp. $\alpha ' \equiv \forall x\beta (x)$) such that $\beta [t/x] \in \Delta _{0}$.

• A formula α belongs to $\Sigma _{n+1}$ (resp. $\Pi _{n+1}$) if it is equivalent to a closed formula $\alpha ' \equiv \exists x\beta (x)$ (resp. $\alpha ' \equiv \forall x\beta (x)$) such that $\beta [t/x] \in \Pi _{n}$ (resp. $\beta [t/x] \in \Sigma _{n}$).

Example A.12.

Both the Gödelian propositions $\mathcal{G}$ and $\mathit{Cons}_{\mathsf{PA}}$ are $\Pi _{1}$-statement.

Remark A.13.

Whereas $\alpha \in \Sigma _{n}$ if, and only if, $\neg \alpha \in \Pi _{n}$, the set of $\Delta _{0}$-formulas is closed under negation.

Proposition A.14.

Let t,s be two closed arithmetical terms:

• (1) If $\mathcal{N} \vDash t = s$ , then $\vdash _{\mathsf{PA}}t = s$,

• (2) If $\mathcal{N} \vDash t\neq s$ , then $\vdash _{\mathsf{PA}}t\neq s$,

• (3) $\vdash _{\mathsf{PA}}\overline{n}\geqslant \overline{m} \rightarrow (\overline{n} = \overline{0} \vee \overline{n} = \overline{1} \vee \ldots \vee \overline{n} = \overline{m})$.

Proof.

The reader can find all the proofs in Rautenberg (2000). □ 

Theorem A.15 ($\Delta _{0}$-Decidability).

If α is a closed $\Delta _{0}$ -formula, then either $\vdash _{\mathsf{PA}}\alpha$ or $\vdash _{\mathsf{PA}}\neg \alpha$ .

Proof.

Let $\alpha \in \Delta _{0}$; we proceed by induction on the number of logical connectives occurring in α.

$\underline{\textit{Base}.}$ :

If no logical connective occurs in α, then $\alpha \equiv t = s$ with t, s closed terms. It is either $\mathcal{N} \vDash t = s$ or $\mathcal{N} \vDash t\neq s$ and so Proposition A.14 gives us the basis.

$\underline{\textit{Step}.}$ :

Proposition A.14 enables us to stress the following conversions

$$\displaystyle\begin{array}{rcl} & & \exists x\leqslant k\alpha (x) \Leftrightarrow \alpha (0) \vee \alpha (1) \vee \ldots \vee \alpha (k) {}\\ & & \forall x\leqslant k\alpha (x) \Leftrightarrow \alpha (0) \wedge \alpha (1) \wedge \ldots \wedge \alpha (k), {}\\ \end{array}$$

for turning any quantified $\Delta _{0}$-formula into an equivalent one without quantifiers. Then it is easy to see that any Boolean composition of decidable propositions is, in turn, decidable. □ 

Corollary A.16 ($\Delta _{0}$-Completeness).

For any closed $\alpha \!\!\!\in \!\!\! \Delta _{0}$ , if $\mathcal{N}\!\!\! \vDash \!\!\!\alpha$ , then $\vdash _{\mathsf{PA}}\alpha$.

Proof.

Let $\mathcal{N} \vDash \alpha$, but $\nvdash _{\mathsf{PA}}\alpha$. For $\alpha \in \Delta _{0}$, by Theorem A.15, it would be $\vdash _{\mathsf{PA}}\neg \alpha$ against the soundness of PA w.r.t. $\mathcal{N}$. □ 

Theorem A.17.

$\mathsf{PA}$ is $\Sigma _{1}$ -complete w.r.t. $\mathcal{N}$ if, and only if, it is $\Delta _{0}$ -decidable.

Proof.

( ⇒ ) Let $\nvdash _{\mathsf{PA}}\alpha$, with α closed and in $\Delta _{0}$. By the $\Sigma _{1}$-completeness, we obtain $\mathcal{N} \nvDash \alpha$ and so $\mathcal{N} \vDash \neg \alpha$. Since $\neg \alpha \in \Delta _{0}$, we perform a further step of $\Sigma _{1}$-completeness so as to obtain $\vdash _{\mathsf{PA}}\alpha$.

($\Leftarrow$) We proceed by absurd: let $\exists x\alpha (x)$ be a closed $\Sigma _{1}$-formula such that $\mathcal{N} \vDash \exists x\alpha (x)$, but $\nvdash _{\mathsf{PA}}\exists x\alpha (x)$. For $\mathcal{N} \vDash \exists x\alpha (x)$, there is an $n \in \mathbb{N}$ such that $\mathcal{N} \vDash \alpha (n)$. Since $\alpha (n) \in \Delta _{0}$, we can apply the just proved $\Delta _{0}$-completeness and obtain $\vdash _{\mathsf{PA}}\alpha (\overline{n})$. As a matter of logic, we finally obtain $\vdash _{\mathsf{PA}}\exists x\alpha (x)$ which contradicts our assumption that $\nvdash _{\mathsf{PA}}\exists x\alpha (x)$. □ 

Corollary A.18 ($\Sigma _{1}$-Completeness).

PA is $\Sigma _{1}$ -complete w.r.t. $\mathcal{N}$.

Proof.

Straightforwardly by Theorems A.15 and A.17. □ 

Corollary A.19.

If $\alpha \in \Pi _{1}$ is independent of PA, then $\mathcal{N} \vDash \alpha$. In particular, we have that $\mathcal{N} \vDash \mathcal{G}$ and $\mathcal{N} \vDash \mathit{Cons}_{\mathsf{PA}}$.

Proof.

By the $\Sigma _{1}$-completeness, we obtain $\mathcal{N} \nvDash \neg \alpha$ from $\nvdash _{\mathsf{PA}}\neg \alpha$, and so $\mathcal{N} \vDash \alpha$. Both the Gödelian propositions $\mathcal{G}$ and $\mathit{Cons}_{\mathsf{PA}}$ instantiate the case just explained so that $\mathcal{N} \vDash \mathcal{G}$ and $\mathcal{N} \vDash \mathit{Cons}_{\mathsf{PA}}$. □ 

1.4 A.4 ω-Logic, Constructive ω-Logic and Some Related Results

Theorem A.20.

For any formula α, $\mathcal{N} \vDash \alpha$ if, and only if, $\vdash _{\mathsf{PA}^{\omega }}\alpha$.

Proof.

(Soundness) It is a matter of extending the proof of Theorem A.3 so as to include the ω-rule. In order to show that any instance of the ω-rule transmits the truth from premisses to the conclusion, it is sufficient to remark that the ω-rule just provides a syntactical rendition of the Tarskian definition of the universal quantifier: if $\mathcal{N} \vDash \alpha (0)$, $\mathcal{N} \vDash \alpha (1)$, $\mathcal{N} \vDash \alpha (2)$ and so on, then $\mathcal{N} \vDash \forall x\alpha (x)$.

(Completeness) We proceed by induction on the logical complexity of α. The $\Delta _{0}$-completeness provides the base of our induction. Then, we distinguish two cases:

• Let $\alpha \equiv \exists x\beta (x) \in \Sigma _{n+1}$. $\mathcal{N} \vDash \exists x\beta (x)$ means that there is an $n \in \mathbb{N}$ such that $\mathcal{N} \vDash \beta (n)$ with $\beta (n) \in \Pi _{n}$. By inductive hypothesis $\vdash _{\mathsf{PA}^{\omega }}\beta (\overline{n})$ and so we can introduce the existential quantifier for finally achieving $\vdash _{\mathsf{PA}^{\omega }}\exists x\beta (x)$.

• Let $\alpha \equiv \forall x\beta (x) \in \Pi _{n+1}$. $\mathcal{N} \vDash \forall x\beta (x)$ means that for all $n \in \mathbb{N}$, $\mathcal{N} \vDash \beta (n)$ with $\beta (n) \in \Sigma _{n}$. By inductive hypothesis we have that for all $n \in \mathbb{N}$, $\vdash _{\mathsf{PA}^{\omega }}\beta (\overline{n})$. Finally, the ω-rule enables us to introduce the universal quantifier so as to obtain $\vdash _{\mathsf{PA}^{\omega }}\forall x\beta (x)$. □ 

Corollary A.21.

PA ^ω is syntactically complete, namely, for any formula α, either $\vdash _{\mathsf{PA}^{\omega }}\alpha$ or $\vdash _{\mathsf{PA}^{\omega }}\neg \alpha$.

Proof.

We show that $\nvdash _{\mathsf{PA}^{\omega }}\alpha$ implies $\vdash _{\mathsf{PA}^{\omega }}\neg \alpha$. Let $\nvdash _{\mathsf{PA}^{\omega }}\alpha$; by Theorem A.20 it is $\mathcal{N} \nvDash \alpha$ and so $\mathcal{N} \vDash \neg \alpha$. Then another application of Theorem A.20 allows us to conclude that $\vdash _{\mathsf{PA}^{\omega }}\neg \alpha$. □ 

Theorem A.22.

For any formula α, if $\nvdash _{\mathsf{PA}^{\omega _{\downarrow }}}\alpha$ , then $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\neg \mathit{Theor}_{\mathsf{PA}}(\overline{\ulcorner \alpha \urcorner })$.

Proof.

Suppose by absurd that there is an $n \in \mathbb{N}$ such that $\vdash _{\mathsf{PA}}\mathit{Dem}_{\mathsf{PA}}(\overline{n},\overline{\ulcorner \alpha \urcorner })$. This latter would imply the existence of a $\mathsf{PA}$ proof π of α such that $\ulcorner \alpha \urcorner = n$. This is in contrast with our hypothesis that $\nvdash _{\mathsf{PA}^{\omega _{\downarrow }}}\!\!\!\!\alpha$ and so we conclude $\nvdash _{\mathsf{PA}}\mathit{Dem}_{\mathsf{PA}}(\overline{n},\overline{\ulcorner \alpha \urcorner })$. Then, the $\Delta _{0}$–decidability allows us to turn $\nvdash _{\mathsf{PA}}\mathit{Dem}_{\mathsf{PA}}(\overline{n},\overline{\ulcorner \alpha \urcorner })$ into $\vdash _{\mathsf{PA}}\!\neg \mathit{Dem}_{\mathsf{PA}}(\overline{n},\overline{\ulcorner \alpha \urcorner })$. The argument just explained is clearly prototypical w.r.t. n (being, in turn, the proof of Theorem A.15 prototypical w.r.t. the formula α) so as, by a step of $\omega _{\downarrow }$ -rule, we can conclude $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\!\!\!\!\forall x\neg \mathit{Dem}_{\mathsf{PA}}(x,\overline{\ulcorner \alpha \urcorner })$, that is, $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\neg \mathit{Theor}_{\mathsf{PA}}(\overline{\ulcorner \alpha \urcorner })$. □ 

Corollary A.23.

$\mathsf{PA}^{\omega _{\downarrow }}$ decides both the Gödelian propositions $\mathcal{G}$ and Cons _PA .

Proof.

Suppose by absurd that $\mathcal{G}$ is not provable in $\mathsf{PA}^{\omega _{\downarrow }}$. By Theorem A.22, we would obtain $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\neg \mathit{Theor}_{\mathsf{PA}}(\overline{\ulcorner \mathcal{G}\urcorner })$ from $\nvdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}$. Now, we know that $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}\leftrightarrow \neg \mathit{Theor}_{\mathsf{PA}}(\overline{\ulcorner \mathcal{G}\urcorner })$ and so we would be able to deduce $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}$ against the fact that we assumed $\nvdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}$. Such an argument leads us to reject $\nvdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}$, that is, to affirm $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathcal{G}$.

The proof of $\vdash _{\mathsf{PA}^{\omega _{\downarrow }}}\mathit{Cons}_{\mathsf{PA}}$ proceeds in an analogous way. □