G\"{o}del's incompleteness theorem and the Anti-Mechanist Argument: revisited

TL;DR

This paper revisits G"{o}del's incompleteness theorem and its implications for the Anti-Mechanist Argument, clarifying misunderstandings and discussing recent developments in the philosophical and mathematical debates.

Contribution

It provides supplementary explanations and classifications of misunderstandings related to G"{o}del's theorem in the context of Anti-Mechanist Arguments, building on Krajewski's work.

Findings

Clarification of misunderstandings about G"{o}del's theorem

Analysis of G"{o}del's Disjunctive Thesis and Undemonstrability of Consistency

Discussion of recent advances affecting these issues

Abstract

This is a paper for a special issue of the journal "Studia Semiotyczne" devoted to Stanislaw Krajewski's paper [30]. This paper gives some supplementary notes to Krajewski's [30] on the Anti-Mechanist Arguments based on G\"{o}del's incompleteness theorem. In Section 3, we give some additional explanations to Section 4-6 in Krajewski's [30] and classify some misunderstandings of G\"{o}del's incompleteness theorem related to Anti-Mechanist Arguments. In Section 4 and 5, we give a more detailed discussion of G\"{o}del's Disjunctive Thesis, G\"{o}del's Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7-8 in Krajewski's [30], describing how recent advances bear on these issues.

Equations

$$(\varphi (0)\wedge \forall x(\varphi (x)\to \varphi (\mathbf{S}x)))\to \forall x\varphi (x),$$

Full text

This paper is the research result of the Humanities and Social Sciences of Ministry of Education Planning Fund project “Research on Gödel’s incompleteness theorem” (project no: 17YJA72040001). I would like to thank the fund support by the Humanities and Social Sciences of Ministry of Education Planning Fund in China. I would like to thank Peter Koellner for sending me his drafts of [27, 28] prior to publication, and the referee for detailed helpful comments.

1. Introduction

Gödel’s incompleteness theorem is one of the most remarkable and profound discoveries in the 20th century, an important milestone in the history of modern logic. Gödel’s incompleteness theorem has wide and profound influence on the development of logic, philosophy, mathematics, computer science and other fields, substantially shaping mathematical logic as well as foundations and philosophy of mathematics from 1931 onward. The impact of Gödel’s incompleteness theorem is not confined to the community of mathematicians and logicians, and it has been very popular and widely used outside mathematics.

Gödel’s incompleteness theorem raises a number of philosophical questions concerning the nature of mind and machine, the difference between human intelligence and machine intelligence, and the limit of machine intelligence. It is well known that Turing proposed a convincing analysis of the vague and informal notion of “computable” in terms of the precise mathematical notion of “computable by a Turing machine”. So we can replace the vague notion of computation with the mathematically precise notion of a Turing machine. In this paper, following Koellner in [27], we stipulate that the notion “the mind cannot be mechanized” means that the mathematical outputs of the idealized human mind outstrip the mathematical outputs of any Turing machine.111In this paper, we will not consider the performance of actual human minds, with their limitations and defects; but only consider the idealized human mind and look at what it can do in principle (see [27], p. 338.). A popular interpretation of Gödel’s first incompleteness theorem ($\sf G1$) is that $\sf G1$ implies that the mind cannot be mechanized. The Mechanistic Thesis claims that the mind can be mechanized. In this paper, we will not examine the broad question of whether the mind can be mechanized, which has been extensively discussed in the literature (e.g.  Penrose [44], Chalmers [5], Lucas [37], Lindström [35], Feferman [13], Shapiro [53, 54], Koellner [26, 27, 28] and Krajewski [30]). Instead we will only examine the question of whether $\sf G1$ implies that the mind cannot be mechanized.

This is a paper for a special issue of Semiotic Studies devoted to Krajewski’s paper [30]. We first give a summary of Krajewski’s work in [30]. In [30], Krajewski gave a detailed analysis of the alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem. Following Gödel himself and other leading logicians, Krajewski refuted the Anti-Mechanist Arguments (the Lucas Argument and the Penrose Argument), and claimed that they are not implied by Gödel’s incompleteness theorem alone. Moreover, Krajewski [30] demonstrated the inconsistency of Lucas’s arithmetic and the semantic inadequacy of Penrose’s arithmetic. Krajewski [30] also discussed two consequences of Gödel’s incompleteness theorem directly related to Anti-Mechanist Arguments: our consistency is not provable (Gödel’s Undemonstrability of Consistency Thesis), and we cannot define the natural numbers. The discussion in Krajewski’s paper is mainly from the philosophical perspective. However, the discussion in this paper is mainly from the logical perspective based on some recent advances on the study of Gödel’s incompleteness theorem and Gödel’s Disjunctive Thesis. Basically, we agree with Krajewski’s analysis of the Anti-Mechanist Arguments and his conclusion that Gödel’s incompleteness theorem alone does not imply that the Anti-Mechanist Arguments hold. However, some discussions in [30] are vague. Moreover, in the recent work on Gödel’s Disjunction Thesis one finds precise versions which can actually be proved. The motivation of this paper is to give some supplementary notes to Krajewski’s recent paper [30] on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem.

This paper is structured as follows. In Section 2, we review some notions and facts we will use in this paper. In Section 3, we give some supplementary notes to Section 5-6 in Krajewski’s [30] and classify some misunderstandings of Gödel’s incompleteness theorem related to Anti-Mechanist Arguments. In Section 4, we give a more detailed discussion of Gödel’s Disjunctive Thesis as in Section 7 in Krajewski’s [30] based on recent advances of the study on Gödel’s Disjunctive Thesis in the literature. In Section 5, we give a more precise discussion of Gödel’s Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 8 in Krajewski’s paper.

2. Preliminaries

In this section, we review some basic notions and facts used in this paper. Our notations are standard. For textbooks on Gödel’s incompleteness theorem, we refer to [9, 39, 33, 55, 3]. There are some good survey papers on Gödel’s incompleteness theorem in the literature (see [56, 1, 29, 60, 8]).

In this paper, we focus on first order theory based on countable language, and always assume the arithmetization of the base theory with a recursive set of non-logical constants. For a given theory $T$, we use $L(T)$ to denote the language of $T$. For more details about arithmetization, we refer to [39]. Under the arithmetization, any formula or finite sequence of formulas can be coded by a natural number (called the Gödel number of the syntactic item). In this paper, $\ulcorner\phi\urcorner$ denotes the numeral representing the Gödel number of $\phi$.

We say a set of sentences $\Sigma$ is recursive if the set of Gödel numbers of sentences in $\Sigma$ is recursive.222For ease of exposition, we will pass back and forth between the two. A theory $T$ is decidable if the set of sentences provable in $T$ is recursive; otherwise it is undecidable. A theory $T$ is recursively axiomatizable if it has a recursive set of axioms, i.e. the set of Gödel numbers of axioms of $T$ is recursive. A theory $T$ is finitely axiomatizable if it has a finite set of axioms. A theory $T$ is essentially undecidable iff any recursively axiomatizable consistent extension of $T$ in the same language is undecidable. We say a sentence $\phi$ is independent of $T$ if $T\nvdash\phi$ and $T\nvdash\neg\phi$. A theory $T$ is incomplete if there is a sentence $\phi$ in $L(T)$ which is independent of $T$; otherwise, $T$ is complete (i.e., for any sentence $\phi$ in $L(T)$, either $T\vdash\phi$ or $T\vdash\neg\phi$). Informally, an interpretation of a theory $T$ in a theory $S$ is a mapping from formulas of $T$ to formulas of $S$ that maps all axioms of $T$ to sentences provable in $S$. If $T$ is interpretable in $S$, then all sentences provable (refutable) in $T$ are mapped, by the interpretation function, to sentences provable (refutable) in $S$. Interpretability can be accepted as a measure of strength of different theories. For the precise definition of interpretation, we refer to [59] for more details.

Theorem 2.1** ([57], Theorem 7, p. 22).**

Let $T_{1}$ and $T_{2}$ be two consistent theories such that $T_{2}$ is interpretable in $T_{1}$. If $T_{2}$ is essentially undecidable, then $T_{1}$ is also essentially undecidable.

Robinson Arithmetic $\mathbf{Q}$ was introduced in [57] by Tarski, Mostowski and Robinson as a base axiomatic theory for investigating incompleteness and undecidability.

Definition 2.2**.**

Robinson Arithmetic $\mathbf{Q}$ is defined in the language $\{\mathbf{0},\mathbf{S},+,\cdot\}$ with the following axioms:

$\mathbf{Q}_{1}$**: **

$\forall x\forall y(\mathbf{S}x=\mathbf{S}y\rightarrow x=y)$;

$\mathbf{Q}_{2}$**: **

$\forall x(\mathbf{S}x\neq\mathbf{0})$;

$\mathbf{Q}_{3}$**: **

$\forall x(x\neq\mathbf{0}\rightarrow\exists y(x=\mathbf{S}y))$;

$\mathbf{Q}_{4}$**: **

$\forall x\forall y(x+\mathbf{0}=x)$;

$\mathbf{Q}_{5}$**: **

$\forall x\forall y(x+\mathbf{S}y=\mathbf{S}(x+y))$;

$\mathbf{Q}_{6}$**: **

$\forall x(x\cdot\mathbf{0}=\mathbf{0})$;

$\mathbf{Q}_{7}$**: **

$\forall x\forall y(x\cdot\mathbf{S}y=x\cdot y+x)$.

The theory $\mathbf{PA}$ consists of axioms $\mathbf{Q}_{1}$-$\mathbf{Q}_{2}$, $\mathbf{Q}_{4}$-$\mathbf{Q}_{7}$ in Definition 2.2 and the following axiom scheme of induction:

[TABLE]

where $\phi$ is a formula with at least one free variable $x$.

Let $\mathfrak{N}=\langle\mathbb{N},+,\times\rangle$ denote the standard model of $\mathbf{PA}$. We say $\phi\in L(\mathbf{PA})$ is a true sentence of arithmetic if $\mathfrak{N}\models\phi$. We define that $\mathit{Th(\mathbb{N},+,\cdot)}$ is the set of sentence $\phi$ in $L(\mathbf{PA})$ such that $\mathfrak{N}\models\phi$. Similarly, we have the definition of $\mathit{Th(\mathbb{Z},+,\cdot)}$, $\mathit{Th(\mathbb{Q},+,\cdot)}$ and $\mathit{Th(\mathbb{R},+,\cdot)}$.

We introduce a hierarchy of $L(\mathbf{PA})$-formulas called the arithmetical hierarchy (see [39, 20]). Bounded formulas ($\Sigma^{0}_{0}$, or $\Pi^{0}_{0}$, or $\Delta^{0}_{0}$ formula) are built from atomic formulas using only propositional connectives and bounded quantifiers (in the form $\forall x\leq y$ or $\exists x\leq y$). A formula is $\Sigma^{0}_{n+1}$ if it has the form $\exists x\phi$ where $\phi$ is $\Pi^{0}_{n}$. A formula is $\Pi^{0}_{n+1}$ if it has the form $\forall x\phi$ where $\phi$ is $\Sigma^{0}_{n}$. Thus, a $\Sigma^{0}_{n}$-formula has a block of $n$ alternating quantifiers, the first one being existential, and this block is followed by a bounded formula. Similarly for $\Pi^{0}_{n}$-formulas. A formula is $\Delta^{0}_{n}$ if it is equivalent to both a $\Sigma^{0}_{n}$ formula and a $\Pi^{0}_{n}$ formula.

A theory $T$ is said to be $\omega$-consistent if there is no formula $\phi(x)$ such that $T\vdash\exists x\phi(x)$ and for any $n\in\omega$, $T\vdash\neg\phi(\bar{n})$. A theory $T$ is $1$-consistent if there is no such formula $\phi(x)$ which is $\Delta^{0}_{1}$. A theory $T$ is sound iff for any formula $\phi$, if $T\vdash\phi$, then $\mathfrak{N}\models\phi$; a theory $T$ is $\Sigma^{0}_{1}$-sound iff for any $\Sigma^{0}_{1}$ formula $\phi$, if $T\vdash\phi$, then $\mathfrak{N}\models\phi$.

In the following, unless stated otherwise, let $T$ be a recursively axiomatizable consistent extension of $\mathbf{PA}$. There is a formal arithmetical formula $\mathbf{Proof}_{T}(x,y)$ (called Gödel’s proof predicate) which represents the recursive relation $Proof_{T}(x,y)$ saying that $y$ is the Gödel number of a proof in $T$ of the formula with Gödel number $x$. Since we will discuss general provability predicates based on proof predicates, now we give a general definition of proof predicate which is a generalization of properties of Gödel’s proof predicate $\mathbf{Proof}_{T}(x,y)$.

Definition 2.3**.**

We say a formula $\mathbf{Prf}_{T}(x,y)$ is a proof predicate of $T$ if it satisfies the following conditions:333We can say that each proof predicate represents the relation “$y$ is the code of a proof in $T$ of a formula with Gödel number $x$”.

1. (1)

$\mathbf{Prf}_{T}(x,y)$ is $\Delta^{0}_{1}(\mathbf{PA})$;444We say a formula $\phi$ is $\Delta^{0}_{1}(\mathbf{PA})$ if there exists a $\Sigma^{0}_{1}$ formula $\alpha$ such that $\mathbf{PA}\vdash\phi\leftrightarrow\alpha$, and there exists a $\Pi^{0}_{1}$ formula $\beta$ such that $\mathbf{PA}\vdash\phi\leftrightarrow\beta$. 2. (2)

$\mathbf{PA}\vdash\forall x(\mathbf{Prov}_{T}(x)\leftrightarrow\exists y\mathbf{Prf}_{T}(x,y))$; 3. (3)

for any $n\in\omega$ and formula $\phi,\mathbb{N}\models\mathbf{Proof}_{T}(\ulcorner\phi\urcorner,\overline{n})\leftrightarrow\mathbf{Prf}_{T}(\ulcorner\phi\urcorner,\overline{n})$; 4. (4)

$\mathbf{PA}\vdash\forall x\forall x^{\prime}\forall y(\mathbf{Prf}_{T}(x,y)\wedge\mathbf{Prf}_{T}(x^{\prime},y)\rightarrow x=x^{\prime})$.

We define the provability predicate $\mathbf{Pr}_{T}(x)$ from a proof predicate $\mathbf{Prf}_{T}(x,y)$ by $\exists y\,\mathbf{Prf}_{T}(x,y)$, and the consistency statement $\mathbf{Con}(T)$ from a provability predicate $\mathbf{Pr}_{T}(x)$ by $\neg\mathbf{Pr}_{T}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$.

**D1: **

If $T\vdash\phi$, then $T\vdash\mathbf{Pr}_{T}(\ulcorner\phi\urcorner)$;

**D2: **

If $T\vdash\mathbf{Pr}_{T}(\ulcorner\phi\rightarrow\varphi\urcorner)\rightarrow(\mathbf{Pr}_{T}(\ulcorner\phi\urcorner)\rightarrow\mathbf{Pr}_{T}(\ulcorner\varphi\urcorner))$;

**D3: **

$T\vdash\mathbf{Pr}_{T}(\ulcorner\phi\urcorner)\rightarrow\mathbf{Pr}_{T}(\ulcorner\mathbf{Pr}_{T}(\ulcorner\phi\urcorner)\urcorner)$.

$\mathbf{D1}$-$\mathbf{D3}$ is called the Hilbert-Bernays-Löb derivability condition. Note that $\mathbf{D1}$ holds for any provability predicate $\mathbf{Pr}_{T}(x)$. We say that provability predicate $\mathbf{Pr}_{T}(x)$ is standard if it satisfies $\mathbf{D2}$ and $\mathbf{D3}$. In this paper, unless stated otherwise, we assume that $\mathbf{Con}(T)$ is the canonical arithmetic sentence expressing the consistency of $T$ and $\mathbf{Con}(T)$ is formulated via a standard provability predicate.

The reflection principle for $T$, denoted by $\mathbf{Rfn}_{T}$, is the schema $\mathbf{Pr}_{T}(\ulcorner\phi\urcorner)\rightarrow\phi$ for every sentence $\phi$ in $L(T)$. The reflection principle for $T$ restricted to a class of sentences $\Gamma$ will be denoted by $\Gamma$-$\mathbf{Rfn}_{T}$.

Let $\alpha(x)$ be a formula in $L(T)$. We can similarly define the provability predicate and consistency statement w.r.t.  formula $\alpha(x)$ as follows. Define the formula $\mathbf{Prf}_{\alpha}(x,y)$ saying “$y$ is the Gödel number of a proof of the formula with Gödel number $x$ from the set of all sentences satisfying $\alpha(x)$”. Define the provability predicate $\mathbf{Pr}_{\alpha}(x)$ of $\alpha(x)$ as $\exists y\mathbf{Prf}_{\alpha}(x,y)$ and the consistency statement $\mathbf{Con}_{\alpha}(T)$ as $\neg\mathbf{Pr}_{\alpha}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$. We say that formula $\alpha(x)$ is a numeration of $T$ if for any $n$, $T\vdash\alpha(\overline{n})$ iff $n$ is the Gödel number of some sentence in $T$.

3. Some notes on Gödel-based Anti-Mechanist Arguments

There has been a massive amount of literature on the Anti-Mechanist Arguments due primarily to Lucas and Penrose (see Lucas [36], Penrose [44]) which claim that $\sf G1$ shows that the human mind cannot be mechanized. The Anti-Mechanist Argument began with Nagel and Newman in [41] and continued with Lucas’s publication in [36]. Nagel and Newman’s argument was criticized by Putnam in [47] and earlier by Gödel (see Feferman [13]), while Lucas’s argument was much more widely criticized in the literature. See Feferman [13] for a historical account and Benacerraf [2] for an influential criticism of Lucas. Penrose proposed a new argument for the Anti-Mechanist Argument in [45, 46]. Penrose’s new argument is the most sophisticated and promising Anti-Mechanist Argument which has been extensively discussed and carefully analyzed in the literature (see Chalmers [5], Feferman [12], Lindström [34, 35], and Shapiro [53, 54], Gaifman [14] and Koellner [26, 27, 28], etc).

Most philosophers and logicians believe that variants of the arguments of Lucas and Penrose are not fully convincing. However, they do not agree so well on what is wrong with arguments of Lucas and Penrose. One strength of Krajewski’s paper [30] is that it provides a detailed review of the history of Anti-Mechanist Arguments based on Gödel’s incompleteness theorem (see Section 3 in [30]) and an analysis of these Gödel-based Anti-Mechanist Arguments (e.g.  Lucas’s argument in Section 4 and Penrose’s argument in Section 6 in [30]). In this section, based on Krajewski’s work, we give some supplementary notes of Krajewski’s Section 5-6 in [30].

For us, the Gödel-based Anti-Mechanist Argument comes from some misinterpretations of Gödel’s incompleteness theorem. To understand the source of these misinterpretations or illusions, we should first have correct interpretations of Gödel’s incompleteness theorem. In the following, we first review some important facts about Gödel’s incompleteness theorem which are helpful to clarify some misinterpretations of Gödel’s incompleteness theorem.

Gödel proved his incompleteness theorem in [15] for a certain formal system $\mathbf{P}$ related to Russell-Whitehead’s Principia Mathematica and based on the simple theory of types over the natural number series and the Dedekind-Peano axioms (see [1], p. 3). Gödel’s original first incompleteness theorem ([15, Theorem VI]) says that for formal theory $T$ formulated in the language of $\mathbf{P}$ and obtained by adding a primitive recursive set of axioms to the system $\mathbf{P}$, if $T$ is $\omega$-consistent, then $T$ is incomplete. The following theorem is a modern reformulation of Gödel’s first incompleteness theorem.

Theorem 3.1** (Gödel’s first incompleteness theorem $(\sf G1)$).**

If $T$ is a recursively axiomatized extension of $\mathbf{PA}$, then there exists a Gödel sentence $\mathbf{G}$ such that:

1. (1)

if $T$ is consistent, then $T\nvdash\mathbf{G}$; 2. (2)

if $T$ is $\omega$-consistent, then $T\nvdash\neg\mathbf{G}$.

Thus if $T$ is $\omega$-consistent, then $\mathbf{G}$ is independent of $T$ and hence $T$ is incomplete. If $T$ is consistent, Gödel sentence $\mathbf{G}$ is a true $\Pi^{0}_{1}$ sentence of arithmetic. Gödel’s proof of $\sf G1$ is constructive: one can effectively find a true $\Pi^{0}_{1}$ sentence $\mathbf{G}$ of arithmetic such that $\mathbf{G}$ is independent of $T$ assuming $T$ is $\omega$-consistent. Gödel  calls this the “incompletability or inexhaustability of mathematics”. Note that only assuming that $T$ is consistent, we can show that $\mathbf{G}$ is a true sentence of arithmetic unprovable in $T$. But it is not enough to show that $T\nvdash\neg\mathbf{G}$ only assuming that $T$ is consistent. To show that $T\nvdash\neg\mathbf{G}$, we need a stronger condition such as “$T$ is 1-consistent” or “$T$ is $\Sigma^{0}_{1}$-sound”.

Let $T$ be a recursively axiomatized extension of $\mathbf{PA}$. After Gödel, Rosser constructed Rosser sentence $\mathbf{R}$ (a $\Pi^{0}_{1}$ sentence) and showed that if $T$ is consistent, then $\mathbf{R}$ is independent of $T$. Rosser improved Gödel’s $\sf G1$ in the sense that Rosser proved that $T$ is incomplete only assuming that “$T$ is consistent” which is weaker than “$T$ is 1-consistent”.

In this paper, let $\langle M_{n}:n\in\omega\rangle$ be the list of Turing machines and $\mathit{Th(M_{n})}$ be the set of sentences produced by the Turing machine $M_{n}$. Let $C=\{n:\mathit{Th(M_{n})}$ is a consistent theory} and $S=\{n:\mathit{Th(M_{n})}$ is a sound theory}. The following proposition on inconsistency and unsoundness is from [30].

Proposition 3.2**.**

1. (1)

If $F$ is a partial recursive function such that $C\subseteq dom(F)$ and $F(n)\notin\mathit{Th(M_{n})}$ for any $n\in C$, then $\{F(n):n\in dom(F)\}$ is inconsistent. 2. (2)

If $F$ is a partial recursive function such that $S\subseteq dom(F)$ and $F(n)\notin\mathit{Th(M_{n})}$ for any $n\in S$, then $\{F(n):n\in dom(F)\}$ is inconsistent.

A natural question is: whether there exists such a function $F$ with these properties. However, the effective version of Gödel’s first incompleteness theorem ($\sf EG1$) tells us that there exists a partial recursive function $F$ such that for any $n\in\omega$, if $\mathit{Th(M_{n})}$ is consistent, then $F(n)$ is defined and $F(n)$ is the Gödel number of a true arithmetic sentence which is not provable in $\mathit{Th(M_{n})}$. Thus there exists such a function $F$ with the properties as stated in Proposition 3.2.

One popular interpretation of $\sf EG1$ is: for any Turing machine $M_{n},F(n)$ picks up the true sentence of arithmetic not produced by $M_{n}$. However, this is a misinterpretation of $\sf EG1$ which in fact says that for such a partial recursive function $F$, if $\mathit{Th(M_{n})}$ is consistent, then $F(n)$ is the Gödel number of a true sentence of arithmetic which is not provable in $\mathit{Th(M_{n})}$. A natural question is: whether there exists an effective procedure such that we can decide whether $\mathit{Th(M_{n})}$ is consistent. The answer is negative since $C$ is a complete $\Pi^{0}_{1}$ set as Koellner points out in [27].

Krajewski [30] claimed that $C$ and $S$ are not recursive. However, as Krajewski [30] commented, Proposition 3.2 on inconsistency and unsoundness does not require that for $n\in dom(F)$, $F(n)$ is the code of a true arithmetic sentence. But we do not see that $C$ or $S$ is not recursive from Proposition 3.2. However, if we add the condition that for $n\in dom(F),F(n)\in\mathbf{Truth}\setminus\mathit{Th(M_{n})}$, then we can show that $C$ and $S$ are not recursive. Let us take $C$ for example and show that $C$ is not recursive.

Proposition 3.3**.**

$C$* is not recursive.555In fact, $C$ is a complete $\Pi^{0}_{1}$ set as Koellner points out in [27].*

Proof.

Suppose $C$ is recursive. Let $A=\{F(n):n\in C\}$. Then $A$ is recursive enumerable. Suppose $A=\mathit{Th(M_{m})}$ for some $m$. Note that $A\subseteq\textbf{Truth}$, and so $A$ is consistent. By the definition of $C,m\in C$ and hence $F(m)\in A$. But, on the other hand, $F(m)\notin\mathit{Th(M_{m})}=A$ which leads to a contradiction. ∎

Since $C$ is undecidable, it is impossible to effectively distinguish the case that $\mathit{Th(M_{n})}$ is consistent and the case that $\mathit{Th(M_{n})}$ is not consistent.

In fact, Theorem 3.2 can be generalized in the following form:

Theorem 3.4**.**

Let $P$ be any property about first order theory (i.e.  consistency, soundness, 1-consistency, etc). Let $C=\{n:\mathit{Th(M_{n})}$ has property $P$}. Suppose $F$ is a partial recursive function satisfying the following conditions:

1. (1)

$C\subseteq dom(F)$, 2. (2)

for each $n\in C$, $F(n)\notin\mathit{Th(M_{n})}$.

Then, $\{F(n):n\in dom(F)\}$ does not have property $P$.

Proof.

Let $A=\{F(n):n\in dom(F)\}$. Suppose $A$ has property $P$. Since $F$ is partial recursive, $A$ is recursively enumerable. Suppose $A=\mathit{Th(M_{k})}$ for some $k$. Since $A$ has property $P$, we have $k\in C$. Thus, $F(k)\notin\mathit{Th(M_{k})}=A$ which contradicts that $F(k)\in A$. ∎

Gödel announced the second incompleteness theorem $(\sf G2)$ in an abstract published in October 1930: no consistency proof of systems such as Principia, Zermelo-Fraenkel set theory, or the systems investigated by Ackermann and von Neumann is possible by methods which can be formulated in these systems (see [65], p. 431). For a theory $T$, recall that $\mathbf{Con}(T)$ is the canonical arithmetic sentence expressing the consistency of $T$ under Gödel’s recursive arithmetization of $T$. The following is a modern reformulation of $\sf G2$:

Theorem 3.5**.**

Let $T$ be a recursively axiomatized extension of $\mathbf{PA}$. If $T$ is consistent, then $T\nvdash\mathbf{Con}(T)$.

From $\sf G2$, we cannot get that $\mathbf{Con}(T)$ is independent of $T$ only assuming that $T$ is consistent. It is provable in $T$ that if $T$ is consistent, then $T\vdash\mathbf{Con}(T)\leftrightarrow\mathbf{G}$ and thus $T\nvdash\mathbf{Con}(T)$. However, it is not provable in $T$ that if $T$ is consistent, then $T+\mathbf{Con}(T)$ is also consistent.666See [3, Theorem 4, p. 97] for a modal proof in $\mathbf{GL}$ of this fact using the arithmetic completeness theorem for $\mathbf{GL}$. So it is not enough to show that $T\nvdash\neg\mathbf{Con}(T)$ only assuming that $T$ is consistent. But we could prove that $\mathbf{Con}(T)$ is independent of $T$ by assuming that $T$ is 1-consistent which is stronger than the condition “$T$ is consistent”.777It is an easy fact that if $T$ is 1-consistent and $S$ is not a theorem of $T$, then $\mathbf{Pr}_{T}(\ulcorner S\urcorner)$ is not a theorem of $T$. Let 1-$\mathbf{Con}(T)$ be the sentence in $L(\mathbf{PA})$ expressing that $T$ is 1-consistent. Fact 3.6 is a summary of these results.

Fact 3.6**.**

Let $T$ be a recursively axiomatized consistent extension of $\mathbf{PA}$.

1. (1)

$T\vdash\mathbf{Con}(T)\rightarrow\mathbf{Con}(T+\neg\mathbf{Con}(T))$; 2. (2)

$T\nvdash\mathbf{Con}(T)\rightarrow\mathbf{Con}(T+\mathbf{Con}(T))$; 3. (3)

$T\vdash\mathbf{Con}(T)\rightarrow\mathbf{Con}(T+\mathbf{R})$;888Recall that $\mathbf{R}$ is the Rosser sentence. 4. (4)

$T\vdash 1$-$\mathbf{Con}(T)\rightarrow\mathbf{Con}(T+\mathbf{Con}(T))$.

An illusion of the application of Gödel’s incompleteness theorem is that we can add consistencies (or Out-Gödeling) forever: from $\mathbf{Con}(T)$, we have $\mathbf{Con}(T+\mathbf{Con}(T))$, then $\mathbf{Con}(T+\mathbf{Con}(T+\mathbf{Con}(T)))$ and so on. However, by Fact 3.6, this does not hold. For the iteration of adding the consistency statement (or Out-Gödeling), we need a stronger condition: $T$ is 1-consistent. The following fact shows the difference between $\mathbf{Con}(T)$ and 1-$\mathbf{Con}(T)$.

Fact 3.7** ([56]).**

Let $T$ be a recursively axiomatized consistent extension of $\mathbf{PA}$. Then $T\vdash\mathbf{Con}(T)\leftrightarrow\Pi^{0}_{1}$-$\mathbf{Rfn}_{T}$ and $T\vdash 1$-$\mathbf{Con}(T)\leftrightarrow\Sigma^{0}_{1}$-$\mathbf{Rfn}_{T}$.

As a corollary of Fact 3.7, 1-$\mathbf{Con}(T)\vdash$ l-$\mathbf{Con}(T+\mathbf{Con}(T))$ (see Proposition 3 in [43]). Thus, if we assume 1-$\mathbf{Con}(T)$, then we can prove $\mathbf{Con}(T)$, $\mathbf{Con}(T+\mathbf{Con}(T))$, $\mathbf{Con}(T+\mathbf{Con}(T+\mathbf{Con}(T)))$ and we can continue forever (note that the assumption 1-$\mathbf{Con}(T)$ is stronger than all these statements).

In summary, the differences between Rosser sentence and Gödel sentence, as well as between $\mathbf{Con}(T)$ and 1-$\mathbf{Con}(T)$ are very important. However, these differences are often overlooked in informal philosophical discussions of Gödel’s incompleteness theorem.

4. Gödel’s disjunctive thesis

The focus of Krajewski’s paper [30] is not about Gödel’s Disjunctive Thesis even if he gives a very brief discussion of Gödel’s Disjunctive Thesis related to the Anti-Mechanist Arguments in Section 7. In this section, we give a more detailed discussion of Gödel’s Disjunctive Thesis and its relevance to the Mechanistic Thesis based on recent advances on the study of Gödel’s Disjunctive Thesis. This section is a summary of Koellner’s papers [27] and [28], and we follow Koellner’s presentation very closely.

Gödel did not argue that his incompleteness theorem implies that the mind cannot be mechanized. Instead, Gödel argued that his incompleteness theorem implies a weaker conclusion: Gödel’s Disjunctive Thesis ($\sf GD$).

**The first disjunct: **

The mind cannot be mechanized.

**The second disjunct: **

There are absolutely undecidable statements.999In the sense that there are mathematical truths that cannot be proved by the idealized human mind.

**Gödel’s Disjunctive Thesis ($\sf GD$): **

Either the first disjunct or the second disjunct holds.101010The original version of $\sf GD$ was introduce by Gödel in [17] (see p. 310): “So the following disjunctive conclusion is inevitable: either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified (where the case that both terms of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives)”.

Gödel’s Disjunctive thesis ($\sf GD$) concerns the limit of mathematical knowledge and the possibility of the existence of mathematical truths that are inaccessible to the idealized human mind. The first disjunct expresses an aspect of the power of the idealized human mind, while the second disjunct expresses an aspect of its limitations.111111We refer to [24], a recent comprehensive research volume about $\sf GD$, for more discussions of the status of $\sf GD$.

What about Gödel’s view toward the first disjunct and the second disjunct? For Gödel, the first disjunct is true and the second disjunct is false; that is the mind cannot be mechanized and human mind is sufficiently powerful to capture all mathematical truths. Gödel’s incompleteness theorem shows certain weaknesses and limitations of one given Turing machine. For Gödel, mathematical proof is an essentially creative activity and his incompleteness theorem indicates the creative power of human reason. Gödel believes that the distinctiveness of the human mind when compared to a Turing machine is evident in its ability to come up with new axioms and develop new mathematical theories. Gödel shared Hilbert’s belief expressed in 1926 in the words: “in mathematics there is no ignoramuses, we should know and we must know” (see [48], p. 192). Based on his rationalistic optimism, Gödel believed that we are arithmetically omniscient and the second disjunct is false.121212For more discussions of the status of the second disjunct, we refer to [24]. However, Gödel admits that he cannot give a convincing argument for either the first disjunct or the second disjunct. Gödel thinks that the most he can claim to have established is his Disjunctive Thesis. For Gödel, $\sf GD$ is a “mathematically established fact” of great philosophical interest which follows from his incompleteness theorem, and it is “entirely independent from the standpoint taken toward the foundation of mathematics” (Gödel, [17]).131313In the literature there is a consensus that Gödel’s argument for $\sf GD$ is definitive, but until now we have no compelling evidence for or against any of the two disjuncts (Horsten and Welch, [24]). In the following, we give a concise overview of the current progress on Gödel’s disjunctive thesis based on Koellner’s work in [26, 27, 28].

Let $\mathbf{K}$ be the set of sentences in $L(\mathbf{PA})$ that the idealized human mind can know. Let $\mathbf{Truth}$ be the set of sentences in $L(\mathbf{PA})$ which are true in the standard model of arithmetic and Prov be the set of sentences in $L(\mathbf{PA})$ which are provable in $\mathbf{PA}$. Gödel refers to Truth as objective mathematics and $\mathbf{K}$ as subjective mathematics. Recall that a theory $T$ in $L(\mathbf{PA})$ is sound if $T\subseteq\mathbf{Truth}$. In this paper, we assume that $\mathbf{K}$ is sound. However, from $\sf G1$, we have $\textbf{Prov}\subsetneq\textbf{Truth}$ since Gödel’s sentence is a true sentence of arithmetic not provable in $\mathbf{PA}$.141414Let us take Fermat’s last theorem for another example. People have shown that Fermat’s last theorem is a true sentence of arithmetic but, as far as I know, it is still an open problem whether Fermat’s last theorem is provable in $\mathbf{PA}$. So Fermat’s last theorem belongs to $\mathbf{K}$ but it is open whether it belongs to Prov.

Note that $\sf GD$ concerns the concepts of relative provability, absolute provability, and truth. Before we present the analysis of $\sf GD$, let us first examine two key notions about provability: relative provability and absolute provability. The notion of relative provability is well understood and we have a precise definition of relative provability in a formal system. But the notion of absolute provability is much more ambiguous and we have no unambiguous formal definition of absolute provability as far as we know. The notion of absolute provability is intended to be intensionally different from the notion of relative provability in that absolute provability is not conceptually connected to a formal system. In contrast to the notion of relative provability, there is little agreement on what principles of the notion “absolute provability” should be adopted. In this paper, we identify the notion of “relatively provable with respect to a given formal system $F$” with the notion of “producible by a Turing machine $M$” (where $M$ is the Turing machine corresponding to $F$)151515Note that sentences relatively provable with respect to a given formal system $F$ can be enumerated by a Turing machine. and we identify the notion of “absolute provability” with the notion of “what the idealized human mind can know”.161616Williamson [64] makes the similar definition that a mathematical hypothesis is absolutely decidable if and only if either it or its negation can in principle be known by a normal mathematical process; otherwise it is absolutely undecidable. Under this assumption, $\mathbf{K}$ is just the set of sentences that are absolutely provable.

In this paper, we assume without loss of generality that $\mathbf{Q}\subseteq\mathit{Th(M_{n})}$ such that both $\sf G1$ and $\sf G2$ apply to $\mathit{Th(M_{n})}$. For a natural number $n$, we say that a statement $\phi$ is relatively undecidable w.r.t.  theory $\mathit{Th(M_{n})}$ for some $n$ if $\phi\notin\mathit{Th(M_{n})}$ and $\neg\phi\notin\mathit{Th(M_{n})}$. We say that a statement $\phi$ is absolutely undecidable if $\phi\notin\mathbf{K}$ and $\neg\phi\notin\mathbf{K}$. Let us first examine what the incompleteness theorem tells us about the relationship between $\mathit{Th(M_{n})},\mathbf{K}$ and $\mathbf{Truth}$.

Note that $\sf G1$ tells us that for any sufficiently strong consistent theory $F$ containing $\mathbf{Q}$, there are statements which are relatively undecidable with respect to $F$. But as Gödel argued, these statements are not absolutely undecidable; instead one can always pass to higher systems in which the sentence in question is provable (see [18], p. 35). For example, from $\sf G2$, $\mathbf{Con}(\mathbf{PA})$ is not provable in $\mathbf{PA}$; but $\mathbf{Con}(\mathbf{PA})$ is provable in second order arithmetic ($\mathbf{Z_{2}}$). Since $\sf G2$ applies to $\mathbf{Z_{2}}$, the $\Pi^{1}_{0}$-truth $\mathbf{Con}(\mathbf{Z_{2}})$ is not provable in $\mathbf{Z_{2}}$. But $\mathbf{Con}(\mathbf{Z_{2}})$ is provable in $\mathbf{Z_{3}}$ (third order arithmetic) which captures the $\Pi^{1}_{0}$-truth that was missed by $\mathbf{Z_{2}}$. This pattern continues up through the orders of arithmetic and up through the hierarchy of set-theoretic systems; at each stage a missing $\Pi^{1}_{0}$-truth is captured at the next stage (see [27], p. 347).

Now let us examine the question of whether the incompleteness theorem shows that $\sf GD$ holds. From the literature, we have found a natural framework $\mathbf{EA_{T}}$ in which we can show that if the concepts of relative provability, absolute provability and truth satisfy some principles, then one can give a rigorous proof of $\sf GD$, vindicating Gödel’s claim that $\sf GD$ is a mathematically established fact (see Koellner [27], p. 355).

Now we introduce two systems of epistemic arithmetic: $\mathbf{EA}$ and $\mathbf{EA_{T}}$. For the presentation of $\mathbf{EA}$ and $\mathbf{EA_{T}}$, we closely follow Koellner’s discussion in [26, 27]. The first is designed to deal with $\mathit{Th(M_{e})}$ and $\mathbf{K}$, and the second is designed to deal with $\mathit{Th(M_{e})},\mathbf{K}$ and $\mathbf{Truth}$. For $\mathbf{EA_{T}}$, we only require a typed truth predicate.171717A typed truth predicate is one that applies only to statements that do not themselves involve the truth predicate. In contrast, a type-free truth predicate is one which also applies to statements that themselves involve the truth predicate. The principles governing typed truth predicates are perfectly straightforward and uncontroversial, while the principles governing type-free truth predicates are much more delicate. See [27] The basic system $\mathbf{EA}$ of epistemic arithmetic has axioms of arithmetic and axioms of absolute provability, and the extended system $\mathbf{EA_{T}}$ has additional axioms of typed truth.181818These systems were first introduced by Myhill [40], Reinhardt [50, 49, 51] and Shapiro [52], and then investigated by many others (e.g. Horsten [23], Leitgeb [32], Carlson [4], Koellner [26, 27] and others). In $\mathbf{EA}$ and $\mathbf{EA_{T}}$, $\mathbf{K}$ is treated as an operator rather than a predicate. From results in Gödel [16], Myhill [40], Montague [38], Thomason [58], and others, if one formulates a theory of absolute provability with $\mathbf{K}$ as a predicate then inconsistency may come (see [27]). The basic axioms of absolute provability are:191919The basic conditions we will impose on knowability are: (1) if the idealized human mind knows $\phi$ and $\phi\rightarrow\psi$ then the idealized human mind knows $\psi$; (2) if the idealized human mind knows $\phi$ then $\phi$ is true; (3) if the idealized human mind knows $\phi$ then the idealized human mind knows that the idealized human mind knows $\phi$.

**K1: **

Universal closures of formulas of the form $\mathbf{K}\phi$ where $\phi$ is a first-order validity.

**K2: **

Universal closures of formulas of the form $(\mathbf{K}(\phi\rightarrow\psi)\wedge\mathbf{K}\phi)\rightarrow\mathbf{K}\psi$.

**K3: **

Universal closures of formulas of the form $\mathbf{K}\phi\rightarrow\phi$.

**K4: **

Universal closures of formulas of the form $\mathbf{K}\phi\rightarrow\mathbf{K}\mathbf{K}\phi$.202020$\mathbf{K1}$-known as logical omniscience-says that $\mathbf{K}$ holds of all first-order logical validities; $\mathbf{K2}$ says that $\mathbf{K}$ is closed under modus ponens, and so distributes across logical derivations; $\mathbf{K3}$ says that $\mathbf{K}$ is correct; and $\mathbf{K4}$ says that $\mathbf{K}$ is absolutely self-reflective (see [27]).

The language $L({\mathbf{EA}})$ is $L({\mathbf{PA}})$ expanded to include an operator $\mathbf{K}$ that takes formulas of $L({\mathbf{EA}})$ as arguments. The axioms of arithmetic are simply those of $\mathbf{PA}$, only now the induction scheme is taken to cover all formulas in $L({\mathbf{EA}})$. For a collection $\Gamma$ of formulas in $L({\mathbf{EA}})$, let $\mathbf{K}\Gamma$ denote the collection of formulas $\mathbf{K}\phi$ where $\phi\in\Gamma$. The system $\mathbf{EA}$ is the theory axiomatized by $\Sigma\cup\mathbf{K}\Sigma$, where $\Sigma$ consists of the axioms of $\mathbf{PA}$ in the language $L({\mathbf{EA}})$ and the basic axioms of absolute provability. The language $L({\mathbf{EA_{T}}})$ of $\mathbf{EA_{T}}$ is the language $L({\mathbf{EA}})$ augmented with a unary predicate $T$. The system $\mathbf{EA_{T}}$ is the theory axiomatized by $\Sigma\cup\mathbf{K}\Sigma$, where $\Sigma$ consists of the axioms of $\mathbf{PA}$ in the language $L({\mathbf{EA_{T}}})$, the basic axioms of absolute provability (in the language $L({\mathbf{EA_{T}}})$), and the Tarskian axioms of truth for the language $L({\mathbf{EA}})$.

From the incompleteness theorem, Gödel made the following two claims about the relationship between $\mathit{Th(M_{e})},\mathbf{K}$ and $\mathbf{Truth}$.

**Claim One: **

For any $e\in\mathbb{N}$, $\mathbf{K}(\mathit{Th(M_{e})}\subseteq\mathbf{Truth})\rightarrow\mathit{Th(M_{e})}\subsetneqq\mathbf{K}$.212121The informal proof of Claim One is as follows: Suppose $\mathbf{K}(\mathit{Th(M_{e})}\subseteq\mathbf{Truth})$. Since it is knowable that $\mathit{Th(M_{e})}$ is consistent, it is knowable that there is a true sentence of arithmetic which is not provable in $\mathit{Th(M_{e})}$. So $\mathit{Th(M_{e})}\subsetneq\mathbf{K}$.

**Claim Two: **

Either $\neg\exists e(\mathit{Th(M_{e})}=\mathbf{K})$ or $\exists\phi(\phi\in\mathbf{Truth}\wedge\phi\notin\mathbf{K}\wedge\neg\phi\notin\mathbf{K})$.222222The informal proof of Claim Two is as follows: Suppose $\mathit{Th(M_{e})}=\mathbf{K}$ for some $e$. Since $\mathit{Th(M_{e})}$ is R.E. but Truth is not arithmetic, $\mathbf{K}\subsetneq\mathbf{Truth}$. So we can find some $\phi\in\mathbf{Truth}$ but $\phi\notin\mathbf{K}$ and $\neg\phi\notin\mathbf{K}$.

Gödel’s Claim One is formalizable and provable in $\mathbf{EA_{T}}$. In fact, something stronger is provable in $\mathbf{EA}$ as the following theorem shows:

Theorem 4.1** (Reinhardt, [49]).**

Assume that $S$ includes $\mathbf{EA}$. Suppose $F(x)$ is a formula with one free variable.

1. (1)

If for each sentence $\phi$, $S\vdash\mathbf{K}(F(\ulcorner\phi\urcorner)\rightarrow\phi)$. Then there is a sentence $\varphi$ such that $S\vdash\mathbf{K}\varphi\wedge\mathbf{K}\neg F(\ulcorner\varphi\urcorner)$. 2. (2)

If for each sentence $\phi$, $S\vdash\mathbf{K}(\mathbf{K}\phi\rightarrow F(\ulcorner\phi\urcorner))$. Then $S\vdash\mathbf{K}\neg\mathbf{K}(\mathbf{Con}(F))$.

From the following theorem, $\sf GD$ is also formalizable and provable in $\mathbf{EA_{T}}$ which confirms Gödel’s claim that $\sf GD$ is a mathematically established fact.232323It is a little delicate to formalize $\sf GD$ in $\mathbf{EA_{T}}$ since $\mathbf{K}$ is formalized as an operator in $\mathbf{EA_{T}}$ and so we are prohibited from quantifying into it. For the details, we refer to Reinhardt [51] and Koellner [26, 27].

Theorem 4.2** (Reinhardt, [51]).**

Assume $\mathbf{EA_{T}}$. Then $\sf GD$ holds.

Following Reinhardt, we should distinguish three levels of the mechanistic thesis.

1. (1)

The weak mechanistic thesis ($\sf WMT$): $\exists e(\mathbf{K}=\mathit{Th(M_{e})})$; 2. (2)

The strong mechanistic thesis ($\sf SMT$): $\mathbf{K}\exists e(\mathbf{K}=\mathit{Th(M_{e})})$; 3. (3)

The super strong mechanistic thesis ($\sf SSMT$): $\exists e\,\mathbf{K}(\mathbf{K}=\mathit{Th(M_{e})})$.

Note that $\sf WMT$ is just the first disjunct which says that there is a Turing machine which coincides with the idealized human mind in the sense that the two have the same outputs. Note that $\sf SMT$ says that the idealized human mind knows that there is a Turing machine which coincides with the idealized human mind. Note that $\sf SSMT$ says that there is a particular Turing machine such that the idealized human mind knows that that particular machine coincides with the idealized human mind.

Suppose $\sf WMT$ holds. Then there exists an $e^{\ast}$ such that in fact $\mathbf{K}=\mathit{Th(M_{e^{\ast}})}$. It might seem at first that if we know that there is such an $e^{\ast}$ then we will be able to find, in a computable way, the indices $e$ such that $\mathbf{K}=\mathit{Th(M_{e})}$. But this is an illusion, as demonstrated by Rice’s Theorem, which we shall now explain.

In recursion theory, the sets $\mathit{Th(M_{e})}$ are known as computably enumerable sets. Each such set is the domain of a partial computable function $\varphi_{e}$. Rice’s Theorem states that for any class $C$ of partial computable functions, $\{e:\varphi_{e}\in C\}$ is computable iff either $C=\emptyset$ or $C$ is the class of all partial computable functions. Now consider the set of indices that we are interested in, namely, $\{e:\mathbf{K}=dom(\varphi_{e})\}$, that is, $\{e:\varphi_{e}\in C\}$ where $C=\{\varphi_{e}:\mathbf{K}=dom(\varphi_{e})\}$. It follows immediately from Rice’s theorem that $\{e:\mathbf{K}=dom(\varphi_{e})\}$ is not computable.

The following theorem shows that we can prove in $\mathbf{EA_{T}}$ that there does not exist a particular Turing machine such that the idealized human mind knows that that particular Turing machine coincides with the idealized human mind.

Theorem 4.3** (Reinhardt, [49]).**

$\mathbf{EA_{T}}+\sf SSMT$* is inconsistent.*

The following theorem shows that, from the viewpoint of $\mathbf{EA_{T}}$ it is possible that the idealized human mind is in fact a Turing machine. From Theorem 4.3, it just cannot know which one.

Theorem 4.4** (Reinhardt [50]).**

$\mathbf{EA_{T}}+\sf WMT$* is consistent.*

From Theorem 4.4, the first disjunct is not provable in $\mathbf{EA_{T}}$. But Gödel did think that one day we would be in a position to prove the first disjunct, and what was missing, as he saw it, was an adequate resolution of the paradoxes involving self-applicable concepts like the concept of truth. Gödel thought that “[i]f one could clear up the intensional paradoxes somehow, one would get a clear proof that mind is not machine”.242424This quotation is from Hao Wang’s reconstruction of his conversations with Gödel. See Wang [61], p. 187.

The following technical theorem from Carlson shows that, from the point of view of $\mathbf{EA_{T}}$, it is possible that the idealized human mind knows that it is a Turing machine: it just cannot know which one.

Theorem 4.5** (Carlson, [4]).**

$\mathbf{EA_{T}}+\sf SMT$* is consistent.*

Now we give a summary for the question whether Gödel’s incompleteness theorems imply the first disjunct. The incompleteness theorems imply that $\neg\exists e\,\mathbf{K}(\mathbf{K}=\mathit{Th(M_{e})})$. But from Theorem 4.4, it does not follow that $\neg\exists e(\mathbf{K}=\mathit{Th(M_{e})})$; and from Theorem 4.5, it does not even follow that $\neg K\exists e\,(\mathbf{K}=\mathit{Th(M_{e})})$. The difference between $\exists e\,\mathbf{K}$ and $\mathbf{K}\exists e$ before $\mathbf{K}=\mathit{Th(M_{e})}$ is essential. Assuming the principles embodied in $\mathbf{EA_{T}}$, it is possible to know that we are a Turing machine (i.e. $\mathbf{K}\exists e(\mathbf{K}=\mathit{Th(M_{e})})$); it is just not possible for there to be a Turing machine such that we know that we are that Turing machine (i.e. $\exists e\,\mathbf{K}(\mathbf{K}=\mathit{Th(M_{e})}))$.

Penrose proposed a new argument for the first disjunct in [45, 46]. Penrose’s new argument is the most sophisticated and promising argument for the first disjunct. It has been extensively discussed and carefully analyzed in the literature (see Chalmers [5], Feferman [12], Lindström [34, 35], and Shapiro [53, 54], Gaifman [14] and Koellner [26, 28], etc). The question of whether Penrose’s new argument establishes the first disjunct is quite subtle. Penrose’s new argument involves treating truth as type-free, and so for the analysis and formalization of Penrose’s new argument, we need to employ type-free notions of truth. However, we now have many type-free theories of truth and there is no consensus as to which option is best. Koellner was the first to discuss Penrose’s new argument in the context of type-free truth. And he shows that when one shifts to a type-free notion of truth then one can treat $\mathbf{K}$ as a predicate (as a contrast, in the context of $\mathbf{EA}$ and $\mathbf{EA_{T}}$, $\mathbf{K}$ cannot be treated as a predicate).

In the literature, Koellner proposed the framework $\mathbf{DTK}$ which employs Feferman’s type-free theory of determinate truth $\mathbf{DT}$ and some additional axioms governing $\mathbf{K}$ to the axioms of $\mathbf{DT}$.252525For the details of the system $\mathbf{DT}$ and $\mathbf{DTK}$, see [26, 28]. The following results about the system $\mathbf{DTK}$ are due to Koellner. From [26, 28], $\mathbf{DTK}$ is consistent (see [26, Theorem 7.14.1]) and $\mathbf{DTK}$ proves $\sf GD$ (see [26, Theorem 7.15.3]). However, the particular argument Penrose gives for the first disjunct fails in the context of $\mathbf{DTK}$ (see [28, Theorem 4.1]). Moreover, even if we restrict the first and second disjunct to arithmetic statements, $\mathbf{DTK}$ can neither prove nor refute either the first disjunct or the second disjunct (see [26, Theorem 7.16.1-7.16.2]). From the point of view of $\mathbf{DTK}$, it is in principle impossible to prove or refute either disjunct. Koellner concluded that “Since the statements that “the mind cannot be mechanized” and “there are absolutely undecidable statements” are independent of the natural principles governing the fundamental concepts and, moreover, are independent of any plausible principles in sight, it seems likely that these statements are themselves “absolutely undecidable”” (Koellner, [28], p. 469).262626Koellner concluded in [28] with a disjunctive conclusion of his own: “Either the statements that “the mind cannot be mechanized” and “there are absolutely undecidable statements” are indefinite (as the philosophical critique maintains) or they are definite and the above results and considerations provide evidence that they are about as good examples of “absolutely undecidable” propositions as one might find” ([28], p. 480).

In our previous discussion of $\sf GD$, the first disjunct and the second disjunct, we identified absolutely undecidability with knowability of the idealized human mind and define that $\phi$ is absolutely undecidable if $\phi\notin\mathbf{K}$ and $\neg\phi\notin\mathbf{K}$. Under this framework, the second disjunct is equivalent to “$\mathbf{K}$ is not complete”. Under the assumption that $\mathbf{K}\subseteq\textbf{Truth}$, the second disjunct is equivalent to “$\mathbf{K}\subsetneq\textbf{Truth}$”. However, $\sf G1$ only tells us that $\textbf{Prov}\subsetneq\textbf{Truth}$, and it does not tell us that $\mathbf{K}\subsetneq\textbf{Truth}$.

Another natural informal definition of absolutely undecidability is: $\phi$ is absolutely undecidable if there is no consistent extension $T$ of $\mathbf{ZFC}$ with well-justified axioms such that $\phi$ is provable in $T$. In this paper, we focus on whether Gödel’s incompleteness theorem implies that the human mind cannot be mechanized. In philosophy of set theory, there are extensive discussions about wether there exists an absolutely undecidable statement in set theory. For a detailed discussion of the question of absolutely undecidability in set theory and especially whether the Continuum Hypothesis is absolutely undecidable, we refer to Koellner [25].

5. Gödel’s Undemonstrability of Consistency Thesis and the definability of natural numbers

In Section 8, Krajewski [30] discussed two consequences of Gödel’s incompleteness theorem directly related to the Anti-Mechanist Arguments: Gödel’s Undemonstrability of Consistency Thesis and the undefinability of natural numbers. For us, Krajewski’s discussion on these two consequences is mainly philosophical and not very precise. In this section, we want to give a more precise logical analysis of Gödel’s Undemonstrability of Consistency Thesis and the undefinability of natural numbers.

Let us first examine the definability of natural numbers. As a consequence of Gödel’s incompleteness theorem, Krajewski [30] claimed that we can not define the natural numbers in the sense that there is not a complete axiomatic system which fully characterizes all truths about natural numbers. We give some supplementary notes to make this point more precise.

Firstly, whether a theory about natural numbers is complete depends on the language of the theory. In the languages $\,L(\mathbf{0},\mathbf{S})$, $L(\mathbf{0},\mathbf{S},<)$ and $L(\mathbf{0},\mathbf{S},<,+)$, there are, respectively, recursively axiomatized complete arithmetic theories (see Section 3.1-3.2 in [9]). For example, Presburger arithmetic is a complete theory of the arithmetic of addition in the language $L(\mathbf{0},\mathbf{S},+)$ (see Theorem 3.2.2 in [39], p. 222). However, if a recursively axiomatized theory contains enough information about addition and multiplication, then it is incomplete and hence it must miss some truths about arithmetic. For example, any recursively axiomatized consistent extension of $\mathbf{Q}$ is incomplete. Thus, in Krajewski’s sense, we can not define the natural numbers in any recursively axiomatized consistent extension of $\mathbf{Q}$.

Secondly, if we discuss the definability of a set with respect to a structure, then the definability of natural numbers depends on the structure we talk about. It is well known that $\mathbb{N}$ is definable in $(\mathbb{Z},+,\cdot)$ and $(\mathbb{Q},+,\cdot)$ (see Chapter XVI in [10]), and $\mathit{Th(\mathbb{N},+,\cdot)}$ is interpretable in $\mathit{Th(\mathbb{Z},+,\cdot)}$ and $\mathit{Th(\mathbb{Q},+,\cdot)}$. Since $\mathit{Th(\mathbb{N},+,\cdot)}$ is undecidable,272727I.e.  there does not exist an effective algorithm such that given any sentence $\phi$ in $L(\mathbf{PA})$, we can effectively decide whether $\langle\mathbb{N},+,\cdot\rangle\models\phi$ or not. by Theorem 2.1, $\mathit{Th(\mathbb{Z},+,\cdot)}$ and $\mathit{Th(\mathbb{Q},+,\cdot)}$ are all undecidable and hence not recursive axiomatizable. But $\mathit{Th(\mathbb{R},+,\cdot)}$ is a decidable, recursively axiomatizable theory (even if not finitely axiomatizable) and $\mathit{Th(\mathbb{R},+,\cdot)}=\mathbf{RCF}$ (the theory of real closed field) (see [10], p. 320-321). As a corollary, $\mathbb{N}$ is not definable in the structure $\langle\mathbb{R},+,\cdot\rangle$ (if $\mathbb{N}$ is definable in $\langle\mathbb{R},+,\cdot\rangle$, then $\mathit{Th(\mathbb{N},+,\cdot)}$ is interpretable in $\mathit{Th(\mathbb{R},+,\cdot)}$ and thus, by Theorem 2.1, $\mathit{Th(\mathbb{R},+,\cdot)}$ is undecidable which leads to a contradiction). In summary, if we consider matters of definability relative to the base structure, then whether the set of natural numbers is definable depends on the base structure: $\mathbb{N}$ is definable in $\langle\mathbb{Z},+,\cdot\rangle$ and $\langle\mathbb{Q},+,\cdot\rangle$, but $\mathbb{N}$ is not definable in $\langle\mathbb{R},+,\cdot\rangle$.

Now we examine Gödel’s Undemonstrability of Consistency Thesis (i.e.  ${\sf G2}$). The intensionality of Gödel sentence and the consistency sentence has been widely discussed in the literature (e.g.  Feferman [11], Halbach-Visser [21, 22], Visser [59]). Halbach and Visser examined the sources of intensionality in the construction of self referential sentences of arithmetic in [21, 22] and argued that corresponding to the three stages of the construction of self referential sentences of arithmetic, there are at least three sources of intensionality: coding, expressing a property and self-reference. Visser [59] located three sources of indeterminacy in the formalization of a consistency statement for a theory $T$:

1. (I)

the choice of a proof system; 2. (II)

the choice of a way of numbering; 3. (III)

the choice of a specific formula numerating the axiom set of $T$.

In summary, the intensional nature ultimately traces back to the various parameter choices that one has to make in arithmetizing the provability predicate. That is the source of both the intensional nature of the Gödel sentence and the consistency sentence.

For a consistent theory $T$, we say that $\sf G2$ holds for $T$ if the consistency statement of $T$ is not provable in $T$. However, this definition is vague, and whether $\sf G2$ holds for $T$ depends on how we formulate the consistency statement. We refer to this phenomenon as the intensionality of $\sf G2$. Both mathematically and philosophically, $\sf G2$ is more problematic than $\sf G1$. The difference between $\sf G1$ and $\sf G2$ is that in the case of $\sf G1$ we are mainly interested in the fact that it shows that some sentence is undecidable if $\mathbf{PA}$ is $\omega$-consistent. We make no claim to the effect that that sentence “really” expresses what we would express by saying “$\mathbf{PA}$ cannot prove this sentence”.282828I would also like to thank the referee for pointing out this difference between $\sf G1$ and $\sf G2$. But in the case of $\sf G2$ we are also interested in the content of the statement.

The status of $\sf G2$ is essentially different from $\sf G1$ due to the intensionality of $\sf G2$. We can say that ${\sf G1}$ is extensional in the sense that we can construct a concrete independent mathematical statement without referring to arithmetization and provability predicate. However, ${\sf G2}$ is intensional and “whether ${\sf G2}$ holds for $T$” depends on varied factors as we will discuss.

In the following, we give a very brief discussion of the intensionality of $\sf G2$ (we refer to [8] for more details). In this section, unless otherwise stated, we make the following assumptions:

1. (1)

The theory $T$ is a recursively axiomatized consistent extension of $\mathbf{Q}$; 2. (2)

The canonical arithmetic formula to express the consistency of $T$ is $\mathbf{Con}(T)\triangleq\neg\mathbf{Pr}_{T}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$; 3. (3)

The canonical numbering we use is Gödel’s numbering; 4. (4)

The provability predicate we use is standard; 5. (5)

The formula numerating the axiom set of $T$ is $\Sigma^{0}_{1}$.

Based on works in the literature, we argue that “whether $\sf G2$ holds for $T$” depends on the following factors:

1. (1)

the choice of the base theory $T$; 2. (2)

the choice of a provability predicate; 3. (3)

the choice of an arithmetic formula to express consistency; 4. (4)

the choice of a numbering; 5. (5)

the choice of a specific formula numerating the axiom set of $T$.

These factors are not independent of each other, and a choice made at an earlier stage may have influences on the choices made at a later stage. In the following, when we discuss how $\sf G2$ depends on one factor, we always assume that other factors are fixed as in the default assumptions we make and only the factor we are discussing is varied. For example, Visser [59] rests on fixed choices for (1) and (3)-(5) but varies the choice of (2); Grabmayr [19] rests on fixed choices for (1)-(2) and (4)-(5) but varies the choice of (3); Feferman [11] rests on fixed choices for (1)-(4) but varies the choice of (5).

In the following, we give a brief discussion of how $\sf G2$ depends on the above five factors. For more discussions of these factors, we refer to [8].

“Whether $\sf G2$ holds for $T$” depends on the choice of the base theory. An foundational question about $\sf G2$ is: how much of information about arithmetic is required for the proof of $\sf G2$. If the base theory does not contain enough information about arithmetic, then $\sf G2$ may fail in the sense that the consistency statement is provable in the base theory. Willard [63] explored the generality and boundary-case exceptions of $\sf G2$ under some base theories. Willard constructed examples of recursively enumerable arithmetical theories that couldn’t prove the totality of successor function but could prove their own canonical consistency (see [62, 63]). Pakhomov [42] defined a theory $H_{<\omega}$ and showed that it proves its own canonical consistency. Unlike Willard’s theories, $H_{<\omega}$ isn’t an arithmetical theory but a theory formulated in the language of set theory with an additional unary function.

“Whether $\sf G2$ holds for $T$” depends on the definition of provability predicate. Recall that $T$ is a recursively axiomatizable consistent extension of $\mathbf{Q}$. Being a consistency statement is not an absolute concept but a role w.r.t.  a choice of the provability predicate. Note that $\sf G2$ holds for any standard provability predicate in the sense that if provability predicate $\mathbf{Pr}_{T}(x)$ is standard, then $T\nvdash\neg\mathbf{Pr}_{T}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$. However, $\sf G2$ may fail for some non-standard provability predicates. Rosser provability predicate is an important kind of non-standard provability predicate in the study of meta-mathematics of arithmetic. Define the Rosser provability predicate $\mathbf{Pr}^{R}_{T}(x)$ as the formula $\exists y(\mathbf{Prf}_{T}(x,y)\wedge\forall z\leq y\neg\mathbf{Prf}_{T}(\dot{\neg}(x),z))$.292929$\dot{\neg}$ is a function symbol expressing a primitive recursive function calculating the code of $\neg\phi$ from the code of $\phi$. Define the consistency statement $\mathbf{Con}^{R}(T)$ via Rosser provability predicate as $\neg\mathbf{Pr}^{R}_{T}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$. Then $\sf G2$ fails for Rosser provability predicate: $T\vdash\mathbf{Con}^{R}(T)$.

“Whether $\sf G2$ holds for $T$” depends on the choice of arithmetic formulas to express consistency. We have different ways to express the consistency of $T$. The canonical arithmetic formula to express the consistency of $T$ is $\mathbf{Con}(T)\triangleq\neg\mathbf{Pr}_{T}(\ulcorner\mathbf{0}\neq\mathbf{0}\urcorner)$. Another way to express the consistency of $T$ is $\mathbf{Con}^{0}(T)\triangleq\forall x(\mathbf{Fml}(x)\wedge\mathbf{Pr}_{T}(x)\rightarrow\neg\mathbf{Pr}_{T}(\dot{\neg}x))$.303030$\mathbf{Fml}(x)$ is the formula which represents the relation that $x$ is a code of a formula.

Kurahashi [31] constructed a Rosser provability predicate such that $\sf G2$ holds for the consistency statement formulated via $\mathbf{Con}^{0}(T)$ (i.e.  the consistency statement formulated via $\mathbf{Con}^{0}(T)$ and Rosser provability predicate is not provable in $T$), but $\sf G2$ fails for the consistency statement formulated via $\mathbf{Con}(T)$ (i.e.  the consistency statement formulated via $\mathbf{Con}(T)$ and Rosser provability predicate is provable in $T$).

“Whether $\sf G2$ holds for $T$” depends on the choice of numberings. Any injective function $\gamma$ from a set of $L(\mathbf{PA})$-expressions to $\omega$ qualifies as a numbering. Gödel’s numbering is a special kind of numberings under which the Gödel number of the set of axioms of $\mathbf{PA}$ is recursive. Grabmayr [19] showed that $\sf G2$ holds for acceptable numberings; But $\sf G2$ fails for some non-acceptable numberings.313131For the definition of acceptable numberings, we refer to Grabmayr [19].

Finally, “Whether $\sf G2$ holds for $T$” depends on the numeration of $T$. As a generalization, $\sf G2$ holds for any $\Sigma^{0}_{1}$ numeration of $T$: if $\alpha(x)$ is a $\Sigma^{0}_{1}$ numeration of $T$, then $T\nvdash\mathbf{Con}_{\alpha}(T)$. However, $\sf G2$ fails for some $\Pi^{0}_{1}$ numerations of $T$. For example, Feferman [11] constructed a $\Pi^{0}_{1}$ numeration $\tau(u)$ of $T$ such that $\sf G2$ fails under this numeration: $T\vdash\mathbf{Con}_{\tau}(T)$.