3.2.1 Three steps towards G1 and G2

Intuitively speaking, Gödel’s incompleteness theorems can be proved based on the following key ingredients.

• • Arithmetization: since G1 and G2 are theorems about properties of the syntax of logic, we need to somehow represent the latter, which is done via a coding scheme called arithmetization.

• • Representations: the notion of ‘proof’ and related concepts in G1 and G2 are then expressed (‘represented’) via arithmetization.

• • Self-reference: given a representation of ‘proof’ and related concepts, one can write down formal statements that intuitively express ‘self-referential’ things like ‘this sentence does not have a proof’.

As we will see, the intuitively speaking ‘self-referential’ statements are the key to proving G1 and G2. We now discuss these three notions in detail.

First of all, arithmetization has the following intuitive content: it establishes a one-to-one correspondence between expressions of $L(T)$ and natural numbers. Thus, we can translate metamathematical statements about the formal theory T into statements about natural numbers. Furthermore, fundamental metamathematical relations can be translated in this way into certain recursive relations, hence into relations representable in T. Consequently, one can speak about a formal system of arithmetic, and about its properties as a theory in the system itself (see [Reference Murawski97])! This is the essence of Gödel’s idea of arithmetization, which was revolutionary at a time when computer hardware and software did not exist yet.

Secondly, in light of the previous, we can define certain relations on natural numbers that express or represent crucial metamathematical concepts related to the formal system T, like ‘proof’ and ‘consistency’. For example, modulo plenty of technical details, we can readily define a binary relation on $\omega ^2$ expressing what it means to prove a formula in T, namely as follows:

Moreover, we can show that the relation $\textit{Proof}_T(m,n)$ is recursive. In addition, Gödel proves that every recursive relation is representable in ${\mathbf {PA}}$ .

Next, let $\mathbf {Proof}_T(x,y)$ be the formula which represents $\textit {Proof}_T(m,n)$ in $\mathbf {PA}$ .Footnote ⁶ From the formula $\mathbf {Proof}_T(x,y)$ , we can define the ‘provability’ predicate $\mathbf {Prov}_T(x)$ as $\exists y \mathbf {Proof}_T(x,y)$ . The provability predicate $\mathbf {Prov}_T(x)$ satisfies the following conditions which show that formal and intuitive provability have the same properties.

1. (1) If $T \vdash \varphi $ , then $T \vdash \mathbf {Prov}_T(\ulcorner \varphi \urcorner )$ ;

2. (2) $T \vdash \mathbf {Prov}_T(\ulcorner \varphi \rightarrow \psi \urcorner )\rightarrow (\mathbf {Prov}_T(\ulcorner \varphi \urcorner )\rightarrow \mathbf {Prov}_T(\ulcorner \psi \urcorner ))$ ;

3. (3) $T\vdash \mathbf {Prov}_T(\ulcorner \varphi \urcorner ) \rightarrow \mathbf {Prov}_T(\ulcorner \mathbf {Prov}_T(\ulcorner \varphi \urcorner )\urcorner )$ .

For the proof of $\textsf {G1}$ , Gödel defines the Gödel sentence $\mathbf {G}$ which asserts its own unprovability in T via a self-reference construction. Gödel shows that if T is consistent, then $T\not\vdash \mathbf {G}$ , and if T is $\omega $ -consistent, then $T\not\vdash \neg \mathbf {G}$ . One way of obtaining such a Gödel sentence is the Diagnolisation Lemma which intuitively speaking implies that the predicate $\neg \mathbf {Prov}_T(x)$ has a fixed point, i.e., there is a sentence $\theta $ in $L(T)$ such that

$$ \begin{align*} T \vdash \theta \leftrightarrow\neg \mathbf{Prov}_T(\ulcorner \theta \urcorner). \end{align*} $$

Clearly, $T\not\vdash \theta $ while $\theta $ intuitively expresses its own unprovability, i.e., the aforementioned self-referential nature.

For the proof of $\textsf {G2}$ , we first define the arithmetic sentence $\mathbf {Con}(T)$ in $L(T)$ as $\neg \mathbf {Prov}_T(\ulcorner \mathbf {0}\neq \mathbf {0}\urcorner )$ which says that for all x, x is not a code of a proof of a contradiction in T. Gödel’s second incompleteness theorem ( $\textsf {G2}$ ) states that if T is consistent, then the arithmetical formula $\mathbf {Con}(T)$ , which expresses the consistency of T, is not provable in T. In Section 5.3, we will discuss some other ways of expressing the consistency of T.

Finally, from the above conditions (1)–(3), one can show that $T\vdash \mathbf {Con}(T)\leftrightarrow \mathbf {G}$ . Thus, $\textsf {G2}$ holds: if T is consistent, then $T\not\vdash \mathbf {Con}(T)$ . For more details on these proofs of $\textsf {G1}$ and $\textsf {G2}$ , we refer to Chapter 2 in [Reference Murawski97].

3.2.2 Properties of $\mathsf {G1}$

In this section, we discuss some (sometimes subtle) comments on $\textsf {G1}$ .

First of all, Gödel’s proof of $\textsf {G1}$ is constructive as follows: given a consistent r.e. extension T of $\mathbf {PA}$ , the proof constructs, in an algorithmic way, a true arithmetic sentence which is unprovable in T. In fact, one can effectively find a true $\Pi ^0_1$ sentence $G_T$ of arithmetic such that $G_T$ is independent of T. Gödel calls this the “incompletability or inexhaustability of mathematics”.

Secondly, for Gödel’s proof of $\textsf {G1}$ , only assuming that T is consistent does not suffice to show that Gödel sentence is independent of T. In fact, the optimal condition to show that Gödel sentence is independent of T is: $T+\mathbf {Con}(T)$ is consistent (see Theorems 35 and 36 in [Reference Isaacson, DeVidi, Hallett and Clark62]).Footnote ⁷

Thirdly, in summary, Gödel’s proof of $\textsf {G1}$ has the following properties:

• • uses proof-theoretic method with arithmetization;

• • does not directly use the Diagnolisation Lemma;

• • the proof formalizes the liar paradox;

• • the proof is constructive;

• • the proof does not have the Rosser property;

• • Gödel’s sentence has no real mathematical content.

All these characteristics of Gödel’s proof of $\textsf {G1}$ are not necessary conditions for proving $\textsf {G1}$ . For example, $\textsf {G1}$ can be proved using recursion-theoretic or model-theoretic method, using the Diagnolisation Lemma, using other logical paradoxes, using non-constructive methods, only assuming that T is consistent (i.e., having the Rosser property), and can be proved without arithmetization.

Fourth, $\textsf {G1}$ does not tell us that any consistent theory is incomplete. In fact, there are many consistent complete first-order theories. For example, the following first-order theories are complete: the theory of dense linear orderings without endpoints ( $\mathbf {DLO}$ ), the theory of ordered divisible groups ( $\mathbf {ODG}$ ), the theory of algebraically closed fields of given characteristic ( $\mathbf {ACF_p}$ ), and the theory of real closed fields ( $\mathbf {RCF}$ ). We refer to [Reference Enderton and Szczerba31] for details of these theories. In fact, $\textsf {G1}$ only tells us that any consistent first-order theory containing a large enough fragment of $\mathbf {PA}$ (such as $\mathbf {Q}$ ) is incomplete: there is then a true $\Pi ^0_1$ sentence which is independent of the initial theory. Turing’s work in [Reference Turing126] shows that any true $\Pi ^0_1$ -sentence of arithmetic is provable in some transfinite iteration of ${\mathbf {PA}}$ . Feferman’s work in [Reference Feferman33] extends Turing’s work and shows that any true sentence of arithmetic is provable in some transfinite iteration of $\mathbf {PA}$ .

Fifth, whether a theory of arithmetic is complete depends on the language of the theory. There are respectively recursively axiomatized complete arithmetic theories in the language of $\, L(\mathbf {0}, \mathbf {S})$ , $L(\mathbf {0}, \mathbf {S}, <)$ , and $L(\mathbf {0}, \mathbf {S}, <, +)$ (see Sections 3.1 and 3.2 in [Reference Enderton29]). Containing enough information of arithmetic is essential for a consistent arithmetic theory to be incomplete. For example, Euclidean geometry is not about arithmetic but only about points, circles, and lines in general; but Euclidean geometry is complete as Tarski has proved (see [Reference Tarski and Givant124]). If the theory contains only information about the arithmetic of addition without multiplication, then it can be complete. For example, Presburger arithmetic is a complete theory of the arithmetic of addition in the language of $L(\mathbf {0}, \mathbf {S}, +)$ (see [Reference Murawski97, Theorem 3.2.2, p. 222]). Finally, containing the arithmetic of multiplication is not sufficient for a theory to be incomplete. For example, there exists a complete recursively axiomatized theory in the language of $L(\mathbf {0}, \times )$ (see [Reference Murawski97, p. 230]).

Finally, it is well-known that $Th(\mathbb {N}, +, \times )$ is interpretable in $Th(\mathbb {Z}, +, \times )$ and $Th(\mathbb {Q}, +, \times )$ .Footnote ⁸ Since $Th(\mathbb {N}, +, \times )$ is undecidable and has a finitely axiomatizable incomplete sub-theory $\mathbf {Q}$ , by Theorem 2.6, $Th(\mathbb {Z}, +, \times )$ and $Th(\mathbb {Q}, +, \times )$ are undecidable, and hence not recursively axiomatizable, but they respectively have a finitely axiomatizable incomplete sub-theory of integers and rational numbers. But $Th(\mathbb {R}, +, \times )$ is decidable and recursively axiomatizable (even if not finitely axiomatizable). In fact, $Th(\mathbb {R}, +, \times )=\mathbf {RCF}$ (the theory of real closed field) (see [Reference Enderton and Szczerba31, p. 320–321]). Note that this fact does not contradict $\textsf {G1}$ since none of $\mathbb {N}, \mathbb {Z}$ , and $\mathbb {Q}$ is definable in $(\mathbb {R}, +, \times )$ .

3.2.3 Between truth and provability

In this paper, unless stated otherwise, we equate a set of sentences with the set of Gödel’s numbers of these sentences. We discuss the formalized notions of ‘truth’ and ‘proof’, and how they relate to incompleteness.

First of all, truth and provability are the same for purely existential statements. Put another way, incompleteness does not arise at the level of $\Sigma ^0_1$ sentences. Indeed, we have $\Sigma ^0_1$ -completeness for T: for any $\Sigma ^0_1$ sentences $\phi $ , $T\vdash \phi $ if and only if $\mathfrak {N}\models \phi $ . Thus, Gödel’s sentence is a true $\Pi ^0_1$ sentence in the form $\forall x \phi (x)$ such that $T\nvdash \forall x \phi (x)$ but ‘ $T\vdash \phi (\bar {n})$ ’ holds for any $n\in \omega $ .

Secondly, the properties of $\textbf {Truth}$ are essentially different from that of $\textbf {Prov}$ . Before Gödel’s work, it was thought that $\textbf {Truth}=\textbf {Prov}$ . Thus, Gödel’s first incompleteness theorem( $\textsf {G1}$ ) reveals the difference between the notion of provability in ${\mathbf {PA}}$ and the notion of truth in the standard model of arithmetic $\mathfrak {N}$ . There are some differences between $\textbf {Truth}$ and $\textbf {Prov}$ :

• • $\textbf {Prov}\subsetneq \textbf {Truth}$ , i.e., there is a true arithmetic sentence which is unprovable in $\mathbf {PA}$ ;

• • Tarski proves that: $\textbf {Truth}$ is not definable in $\mathfrak {N}$ but $\textbf {Prov}$ is definable in $\mathfrak {N}$ ;

• • $\textbf {Truth}$ is not arithmetic but $\textbf {Prov}$ is recursive enumerable.

However, both $\textbf {Truth}$ and $\textbf {Prov}$ are not recursive and not representable in $\mathbf {PA}$ . For more details on $\textbf {Truth}$ and $\textbf {Prov}$ , we refer to [Reference Murawski97, Reference Tarski, Mostowski and Robinson125].

Thirdly, the differences between $\textbf {Truth}$ and $\textbf {Prov}$ can also be expressed in terms of arithmetical interpretations, defined as follows.

Every arithmetical interpretation f is uniquely extended to the mapping $f^{\ast }$ from the set of all modal formulas to the set of $L(T)$ -sentences so that $f^{\ast }$ satisfies the following conditions:

• • $f^{\ast }(p)=f(p)$ for each propositional variable p;

• • $f^{\ast }$ commutes with every propositional connective;

• • $f^{\ast }(\boxed{} A)$ is $\mathbf {Prov}_T(\ulcorner f^{\ast }(A)\urcorner )$ for every modal formula A.

In the following, we equate arithmetical interpretations f with their unique extensions $f^{\ast }$ defined on the set of all modal formulas. In this way, Solovay’s Arithmetical Completeness Theorems for $\mathbf {GL}$ and $\mathbf {GLS}$ characterize the difference between $\textbf {Prov}$ and $\textbf {Truth}$ via provability logic.

Finally, one can study the notion of ‘proof predicate’ as given by $\mathbf {Proof}_T(x, y)$ in an abstract setting, namely as follows. Recall that T is a recursively axiomatizable consistent extension of $\mathbf {Q}$ . We introduce general notions of proof predicate and provability predicate which generalize the proof predicate $\mathbf {Proof}_T(x, y)$ and the provability predicate $\mathbf {Prov}_T(x)$ defined above in Gödel’s proof of $\textsf {G1}$ .

The items $\mathbf {D1}$ – $\mathbf {D3}$ below are called the Hilbert–Bernays–Löb derivability conditions. Note that $\mathbf {D1}$ holds for any provability predicate $\mathbf {Pr}_T(x)$ .

The previous definition leads to another blanket caveat:

Unless stated otherwise, we always assume that $\mathbf {Pr}_T(x)$ is a standard provability predicate, and $\mathbf {Con}(T)$ is the canonical consistency statement defined as $\neg \mathbf {Pr}_T(\ulcorner \mathbf {0}\neq \mathbf {0}\urcorner )$ via the standard provability predicate $\mathbf {Pr}_T(x)$ .

3.2.4 Properties of $\textsf {G2}$

In this section, we discuss some (sometimes subtle) comments on $\textsf {G2}$ .

First of all, we examine a somewhat delicate mistake in the argument which claims that, by an easy application of the compactness theorem, we can show that for any recursive axiomatization of a consistent theory T, T cannot prove its own consistency. Visser presents this argument in [Reference Visser136] as an interesting dialogue between Alcibiades and Socrates:

The mistake in this argument is: from the fact that S can prove the consistency of T we cannot infer that S can prove the consistency of S. Some may argue that since S is a sub-theory of T and S can prove the consistency of T, then of course S can prove the consistency of S.

However, as Visser correctly points out in [Reference Visser136], we should carefully distinguish three perspectives of the theory T: our external perspective, the internal perspective of S, and the internal perspective of T. From each perspective, the consistency of the whole theory implies the consistency of its sub-theory. From T’s perspective, S is a sub-theory of T. But from S’s perspective, S may not be a sub-theory of T. From the fact that T knows that S is a sub-theory of T, we cannot infer that S also knows that S is a sub-theory of T since S is a finite sub-theory of T and may not know any information that T knows, leading to the following (dramatic) conclusion:

Similarly, the notion of consistency is not absolute. For example, let $S=\mathbf {PA}+ \neg \mathbf {Con}(\mathbf {PA})$ . From $\textsf {G2}$ , S is consistent from the external perspective. But since $S\vdash \neg \mathbf {Con}(S)$ , the theory S is not consistent from the internal perspective of S. Note that $\mathbf {PA}\vdash \mathbf {Pr}_{\mathbf {PA}}(\mathbf {0}\neq \mathbf {0})\rightarrow \mathbf {Pr}_{\mathbf {PA}}(\mathbf {Pr}_{\mathbf {PA}}(\mathbf {0}\neq \mathbf {0})\rightarrow \mathbf {0}\neq \mathbf {0})$ . Thus, a theory may be consistent from the external perspective but inconsistent from the internal perspective.

From Gödel’s proof of $\textsf {G2}$ , we cannot infer that if T is a consistent r.e. extension of $\mathbf {Q}$ , then $\mathbf {Con}(T)$ is independent of T. The key point is: it is not enough to show that $T\nvdash \neg \mathbf {Con}(T)$ only assuming that T is consistent. However, we can show that $\mathbf {Con}(T)$ is independent of T assuming that T is 1-consistent.Footnote ¹¹ In fact, the formalized version of “if T is consistent, then $\mathbf {Con}(T)$ is independent of T” is not provable in T.Footnote ¹²

Mostowski proves that $\mathbf {PA}$ is essentially reflexive (see [Reference Murawski97, Theorem 2.6.12]). In fact one can show that for every $n \in \mathbb {N}$ , $I\Sigma ^0_{n+1}\vdash \mathbf {Con}(I\Sigma ^0_{n})$ .Footnote ¹³ For a large class of natural theories U, Pudlák [Reference Pudlák109] shows that the lengths of the shortest proofs of $\mathbf {Con}(U)\!\!\upharpoonright n$ for $n\in \omega $ in the theory U itself are bounded by a polynomial in n. Pudlák conjectures [Reference Pudlák109] that U does not have polynomial proofs of the finite consistency statements $\mathbf {Con}(U + \mathbf {Con}(U))\upharpoonright n$ for $n\in \omega $ .

Finally, a big open question about $\textsf {G2}$ is: can we find a genuinely self-reference free proof of $\textsf {G2}$ ? As far as we know, at present there is no convincing essentially self-reference-free proofs of either $\textsf {G2}$ or of Tarski’s Theorem of the Undefinability of Truth. In [Reference Visser135], Visser gives a self-reference-free proof of $\textsf {G2}$ from Tarski’s Theorem of the Undefinability of Truth, which is a step in a program to find self-reference-free proofs of both $\textsf {G2}$ and Tarski’s Theorem (see [Reference Visser135]). Visser’s argument in [Reference Visser135] is model-theoretic and the main tool is the Interpretation Existence Lemma.Footnote ¹⁴ Visser’s proof in [Reference Visser135] is not constructive. An interesting question is then whether Visser’s argument can be made constructive.

3.3.1 Rosser’s proof

Rosser [Reference Rosser111] proves a “stronger” version of $\textsf {G1}$ , called Rosser’s first incompleteness theorem, which only assumes the consistency of T: if T is a consistent r.e. extension of $\mathbf {Q}$ , then T is incomplete. Gödel’s proof of $\textsf {G1}$ assumes that T is $\omega $ -consistent. Note that $\omega $ -consistency implies consistency. But the converse does not hold and the notion of $\omega $ -consistency is stronger than consistency since we can find examples of theories that are consistent but not $\omega $ -consistent.Footnote ¹⁵ Rosser’s proof is constructive and algorithmically constructs the Rosser sentence that is independent of T. Gödel’s proof of $\textsf {G1}$ uses a standard provability predicate but Rosser’s proof of $\textsf {G1}$ uses a Rosser provability predicate which is a kind of non-standard provability predicate, giving rise to the following.

In general, one can show that each Rosser sentence based on any Rosser provability predicate of T is independent of T. In particular, this independence does not rely on the choice of the proof predicate.

3.3.2 Recursion-theoretic proofs

Gödel’s first incompleteness theorem ( $\textsf {G1}$ ) is well-known in the context of recursion theory. Recall that $W_e=\{n\in \omega : \phi _e(n)\!\!\downarrow \}$ . Let $\langle W_e: e\in \omega \rangle $ be the list of recursive enumerable subsets of $\mathbb {N}$ . The following is an example of an ‘effective’ version of $\textsf {G1}$ (see [Reference Enderton30]):

Similarly, Avigad [Reference Avigad4] proves $\textsf {G1}$ and $\textsf {G2}$ in terms of the undecidability of the halting problem (see Theorems 3.1 and 3.2 in [Reference Avigad4]). Another related result due to Kleene is as follows.

Kleene’s proof of his theorem uses recursion theory, and is not constructive. Salehi and Seraji [Reference Salehi and Seraji114] show that there is a constructive proof of Kleene’s theorem, but this constructive proof does not have the Rosser property. Salehi and Seraji [Reference Salehi and Seraji114] comment that there could be a ‘Rosserian’ version of this constructive proof of Kleene’s theorem.

3.3.3 Proofs based on Arithmetic Completeness

Hilbert and Bernays [Reference Hilbert and Bernays60] present the Arithmetic Completeness Theorem expressing that any recursively axiomatizable consistent theory has an arithmetically definable model. Later, Kreisel [Reference Kreisel80] and Wang [Reference Wang138] adapt the Arithmetic Completeness Theorem and use paradoxes to obtain undecidability results.

Now, the Arithmetic Completeness Theorem is an important tool in model-theoretic proofs of the incompleteness theorems. For more details, we refer to [Reference Kaye and Kotlarski67, Reference Kotlarski79, Reference Lindström91]. Walter Dean [Reference Dean28] gives a detailed discussion on how the Arithmetized Completeness Theorem provides a tool for obtaining formal incompleteness results from some certain paradoxes.

Lemma 3.11 is a corollary of the Arithmetized Completeness Theorem, and is essential for model-theoretic proofs of the incompleteness theorems.

Kreisel first applies the Arithmetized Completeness Theorem to establish model-theoretic proofs of $\textsf {G2}$ (cf. Kreisel [Reference Kreisel82], Smoryński [Reference Smoryński and Barwise119], and Kikuchi [Reference Kikuchi68]). Kikuchi–Tanaka [Reference Kikuchi and Tanaka74], Kikuchi [Reference Kikuchi68, Reference Kikuchi69], and Kotlarski [Reference Kotlarski77] use the Arithmetized Completeness Theorem to give model-theoretic proofs of $\textsf {G2}$ . For example, Kikuchi [Reference Kikuchi68] proves $\textsf {G2}$ model theoretically via the Arithmetized Completeness Theorem (Lemma 3.11): if $\mathbf {PA}$ is consistent, then $\mathbf {Con}(\mathbf {PA})$ is not provable in $\mathbf {PA}$ (see [Reference Kikuchi68, Theorem 3.4]).Footnote ¹⁶

Proofs of $\textsf {G2}$ by Kreisel [Reference Kreisel82] and Kikuchi [Reference Kikuchi68] do not directly yield the formalized version of $\textsf {G2}$ . Kikuchi’s proof of $\textsf {G2}$ in [Reference Kikuchi69] is not formalizable in $\mathbf {PRA}$ . Kikuchi and Tanaka [Reference Kikuchi and Tanaka74] prove in $\mathbf {WKL_0}$ that $\mathbf {Con}(\mathbf {PA})$ implies $\neg \mathbf {Pr}_{\mathbf {PA}}(\ulcorner \mathbf {Con}(\mathbf {PA})\urcorner )$ , since the Completeness Theorem is provable in $\mathbf {WKL_0}$ , and the key Lemma 3.11 used in Kikuchi’s proof [Reference Kikuchi68] is provable in $\mathbf {RCA_0}$ .Footnote ¹⁷ Using Theorem 2.22, Tanaka [Reference Kikuchi and Tanaka74] proves the formalized version of $\textsf {G2}$ : $\mathbf {PRA}\vdash \mathbf {Con}(\mathbf {PA})\rightarrow \mathbf {Con}(\mathbf {PA}+\neg \mathbf {Con}(\mathbf {PA}))$ .

One can give a simple proof of $\textsf {G1}$ via the Diagnolisation Lemma (see [Reference Murawski97]). Kotlarski [Reference Kotlarski77] proves the formalized version of $\textsf {G1}$ and $\textsf {G2}$ via model-theoretic arguments (e.g., using the Arithmetized Completeness Theorem and some quickly growing functions). Kotlarski [Reference Kotlarski77] proves the following version of $\textsf {G1}$ assuming that $\mathbf {PA}$ is $\omega $ -consistent, and shows that the following sentence is provable in $\mathbf {PA}$ :

However, it is unknown whether the method in [Reference Kotlarski77] can also give a new proof of Rosser’s first incompleteness theorem. Kotlarski [Reference Kotlarski77] proves the following formalized version of $\textsf {G2}$ : $\mathbf {PA}\vdash \mathbf {Con(\mathbf {PA})} \rightarrow \mathbf {Con}(\mathbf {PA}+\neg \mathbf {Con}(\mathbf {PA}))$ . Later, Kotlarski [Reference Kotlarski78] transforms the proof of the formalized version of $\textsf {G2}$ in [Reference Kotlarski77] to a proof-theoretic version without the use of the Arithmetized Completeness Theorem.

3.3.4 Proofs based on Kolmogorov complexity

Intuitively, Kolmogorov complexity is a measure of the quantity of information in finite objects. Roughly speaking, the Kolmogorov complexity of a number n, denoted by $K(n)$ , is the size of a program which generates n.

If $n \leq K(n)$ , then n is called random. Kolmogorov shows in 1960s that the set of non-random numbers is recursively enumerable but not recursive (cf. Odifreddi [Reference Odifreddi100]). Relations between $\textsf {G1}$ and Kolmogorov complexity have been intensively discussed in the literature (cf. Li and Vitányi [Reference Li, Vitányi and van Leeuwen90]). Chaitin [Reference Chaitin19] gives an information-theoretic formulation of $\textsf {G1}$ , and proves the following weaker version of $\textsf {G1}$ in terms of Kolmogorov complexity.

Salehi and Seraji [Reference Salehi and Seraji114] show that we can algorithmically construct the Chaitin constant $c_T$ in Theorem 3.13. I.e., for a given consistent r.e. extension T of $\mathbf {Q}$ , one can algorithmically construct a constant $c_T\in \mathbb {N}$ such that for all $e \geq c_T$ and all $w \in \mathbb {N}$ , we have (see Theorem 3.4 in [Reference Salehi and Seraji114]). From Theorem 3.13, it is not clear whether “ $K(w)> e$ ” holds (or whether “ $K(w)> e$ ” is independent of T). Salehi and Seraji [Reference Salehi and Seraji114] show that Chaitin’s proof of $\textsf {G1}$ is non-constructive: there is no algorithm such that given any consistent r.e. extension T of $\mathbf {Q}$ we can compute some $w_T$ such that $K(w_T)> c_T$ holds where $c_T$ is the Chaitin constant we can compute as in Theorem 3.13 (see [Reference Salehi and Seraji114, Theorem 3.5]). If such an algorithm exists, then for any consistent r.e. extension T of $\mathbf {Q}$ , we can compute some $c_T$ and $w_T$ such that $K(w_T)> c_T$ is true but unprovable in T.

Salehi and Seraji [Reference Salehi and Seraji114] also strengthen Chaitin’s Theorem 3.13 assuming T is $\Sigma ^0_1$ -sound: if T is a $\Sigma ^0_1$ -sound r.e. theory extending $\mathbf {Q}$ , then there exists some $c_T$ (which is computable from T) such that for any $e \geq c_T$ there are cofinitely many w’s such that “ $K(w)> e$ ” is independent of T (see [Reference Salehi and Seraji114, Corollary 3.7]). Using a version of the Pigeonhole Principle in $\mathbf {Q}$ , Salehi and Seraji [Reference Salehi and Seraji114] also prove the Rosserian form of Chaitin’s Theorem: for any consistent r.e. extension T of $\mathbf {Q}$ , there is a constant $c_T$ (which is computable from T) such that for any $e \geq c_T$ there are cofinitely many w’s such that “ $K(w)> e$ ” is independent of T (see [Reference Salehi and Seraji114, Theorem 3.9]).

Kikuchi [Reference Kikuchi69] proves the following formalized version of $\textsf {G1}$ via Kolmogorov complexity for any consistent r.e. extension T of $\mathbf {PA}$ : there exists $e \in \omega $ with:

• • $T \vdash \mathbf {Con}(T) \rightarrow \forall x(\neg \mathbf {Pr}_T(\ulcorner K (x)>e\urcorner ))$ ;

• • $T \vdash \omega $ - $\mathbf {Con}(T) \rightarrow \forall x(e < K(x) \rightarrow \neg \mathbf {Pr}_T(\ulcorner K (x) \leq e\urcorner ))$ .

However, this proof is not constructive. Moreover, Kikuchi [Reference Kikuchi69] proves $\textsf {G2}$ via Kolmogorov complexity and the Arithmetic Completeness Theorem: if T is a consistent r.e. extension of $\mathbf {PA}$ , then $T\nvdash \mathbf {Con}(T)$ . Kikuchi’s proof of $\textsf {G2}$ in [Reference Kikuchi69] cannot be formalized in $\mathbf {PRA}$ but can be carried out within $\mathbf {WKL_0}$ . Thus we can also obtain a formalized version of $\textsf {G2}$ in $\mathbf {WKL_0}$ by Theorem 2.22.

3.3.5 Model-theoretic proofs

Adamowicz and Bigorajska [Reference Adamowicz and Bigorajska1] prove $\textsf {G2}$ via model-theoretic method using the notion of 1-closed models and existentially closed models.

Adamowicz and Bigorajska [Reference Adamowicz and Bigorajska1] first prove $\textsf {G2}$ without the use of the Arithmetized Completeness Theorem: every 1-closed model of any subtheory T of $\mathbf {PA}$ extending $I\Delta _0 + \mathbf {exp}$ satisfies $\neg \mathbf {Con}(\mathbf {PA})$ . Then Adamowicz and Bigorajska [Reference Adamowicz and Bigorajska1] prove the formalized version of $\textsf {G2}$ via the idea of existentially closed models and the Arithmetized Completeness Theorem: $\mathbf {PA} \vdash \mathbf {Con}(\mathbf {PA}) \rightarrow \mathbf {Con}(\mathbf {PA} + \neg \mathbf {Con}(\mathbf {PA}))$ (see [Reference Adamowicz and Bigorajska1, Theorem 2.1]). This is proved by showing that an arbitrary model of $\mathbf {PA} + \mathbf {Con}(\mathbf {PA})$ satisfies $\mathbf {Con}(\mathbf {PA} + \neg \mathbf {Con(PA)})$ .

3.4.2 Berry’s Paradox

Berry’s Paradox introduced by Russell [Reference Russell112] is the paradox that “the least integer not nameable in fewer than 19 syllables” is itself a name consisting of 18 syllables. Informally, we say that an expression names a natural number n if n is the unique natural number satisfying the expression. Berry’s Paradox can be formalized in formal systems by interpreting the concept of “name” suitably. The following is Boolos’s formulation of the concept of “name” in [Reference Boolos13].

Proofs of the incompleteness theorems based on Berry’s Paradox have been given by Vopěnka [Reference Vopěnka137], Chaitin [Reference Chaitin19], Boolos [Reference Boolos13], Kikuchi–Kurahashi–Sakai [Reference Kikuchi, Kurahashi and Sakai73], Kikuchi [Reference Kikuchi68], and Kikuchi–Tanaka [Reference Kikuchi and Tanaka74]. In fact, Robinson first uses Berry’s Paradox in [Reference Robinson110] to prove Tarski’s theorem on the undefinability of truth, which anticipates the later use of Berry’s Paradox to obtain incompleteness results by Vopěnka [Reference Vopěnka137], Boolos [Reference Boolos13], and Kikuchi [Reference Kikuchi69].

Boolos [Reference Boolos13] proves a weak form of $\textsf {G1}$ in the 1980s by formalizing Berry’s Paradox in arithmetic via considering the length of formulas that name natural numbers in the standard model of arithmetic. Using this formulation of the concept of “name”, Boolos [Reference Boolos13] first shows that Berry’s Paradox leads to a proof of $\textsf {G1}$ in the following form: there is no algorithm whose output contains all true statements of arithmetic and no false ones (i.e., the theory of true arithmetic is not recursively axiomatizable). Barwise praises Boolos’s proof as “very lovely and the most straightforward proof of Gödel’s incompleteness theorem that I have ever seen”. The optimal sufficient and necessary condition for the independence of a Boolos sentence from $\mathbf {PA}$ is that $\mathbf {PA}+\mathbf {Con}(\mathbf {PA})$ is consistent (see [Reference Salehi and Seraji114]).

Boolos’s theorem is different from Gödel’s theorem in the following way:

• • Boolos’s theorem refers to the concept of truth but Gödel’s theorem does not;

• • Boolos’s proof is not constructive, and we can prove that there is no algorithm for computing the true but unprovable sentence;

• • Boolos’s theorem is weaker than Gödel’s first incompleteness theorem, and hence we cannot obtain the second incompleteness theorem from Boolos’s theorem in the standard way (see [Reference Kikuchi, Kurahashi and Sakai73]).

Boolos’s proof is modified by Kikuchi and Tanaka in [Reference Kikuchi68, Reference Kikuchi and Tanaka74]. The difference between Kikuchi’s proof and Boolos’s proof lies in the interpretation of the word “name”. Kikuchi [Reference Kikuchi68] modifies Boolos’s formulation of the concept of “name” by replacing “truth” with “provability” in the definition.

Using this formulation of the concept of “name”, Kikuchi [Reference Kikuchi68] gives a proof-theoretic proof of $\textsf {G1}$ by formalizing Berry’s Paradox without the use of the Diagnolisation Lemma. Kikuchi [Reference Kikuchi68] constructs a sentence $\theta $ and shows that if $\mathbf {PA}$ is consistent, then $\neg \theta $ is not provable in $\mathbf {PA}$ ; if $\mathbf {PA}$ is $\omega $ -consistent, then $\theta $ is not provable in $\mathbf {PA}$ (see [Reference Kikuchi68, Theorem 2.2]). Note that Kikuchi’s proof of $\textsf {G1}$ in [Reference Kikuchi68] is constructive. Kikuchi and Tanaka [Reference Kikuchi and Tanaka74] reformulate Kikuchi’s proof of $\textsf {G1}$ in [Reference Kikuchi68], and show in $\mathbf {WKL_0}$ that if $\mathbf {PA}+\mathbf {Con}(\mathbf {PA})$ is consistent, then $\theta $ is independent of $\mathbf {PA}$ . By Theorem 2.22, Kikuchi and Tanaka [Reference Kikuchi and Tanaka74] prove the formalized version of $\textsf {G1}$ : $\mathbf {PRA}\vdash \mathbf {Con}(\mathbf {PA}+\mathbf {Con}(\mathbf {PA}))\rightarrow \neg \mathbf {Pr}_{\mathbf {PA}}(\ulcorner \theta \urcorner )\wedge \neg \mathbf {Pr}_{\mathbf {PA}}(\ulcorner \neg \theta \urcorner )$ . An interesting question not covered in [Reference Kikuchi68, Reference Kikuchi and Tanaka74] is whether we can improve Kikuchi’s proof of $\textsf {G1}$ by only assuming that $\mathbf {PA}$ is consistent.

Vopěnka [Reference Vopěnka137] proves $\textsf {G2}$ for $\mathbf {ZF}$ by formalizing Berry’s Paradox, via adopting Kikuchi’s definition of the concept of “name” in [Reference Kikuchi68] over models of $\mathbf {ZF}$ Footnote ¹⁸ : $\mathbf {Con(ZF)}$ is not provable in $\mathbf {ZF}$ . Vopěnka’s proof uses the Completeness Theorem but does not use the Arithmetic Completeness Theorem. Kikuchi, Kurahashi, and Sakai [Reference Kikuchi, Kurahashi and Sakai73] show that Vopěnka’s method can be adapted to prove $\textsf {G2}$ for $\mathbf {PA}$ based on Kikuchi’s formalization of Berry’s Paradox in [Reference Kikuchi68] with an application of the Arithmetic Completeness Theorem.

Proofs of $\textsf {G1}$ and $\textsf {G2}$ based on Berry’s Paradox by Vopěnka [Reference Vopěnka137], Chaitin [Reference Chaitin19], Boolos [Reference Boolos13], and Kikuchi [Reference Kikuchi68] do not use the Diagnolisation Lemma. We can also prove $\textsf {G1}$ based on Berry’s Paradox using the Diagnolisation Lemma. For example, Kikuchi, Kurahashi, and Sakai [Reference Kikuchi, Kurahashi and Sakai73] adopt Kikuchi’s definition of the concept of “name” in [Reference Kikuchi68], and show that the independent statement in Kikuchi’s proof in [Reference Kikuchi68] can be obtained by using the Diagnolisation Lemma.

In summary, the distinctions between using and not using the Diagnolisation Lemma, and between using and not using the Arithmetic Completeness Theorem are not essential for proofs of $\textsf { G1}$ and $\textsf {G2}$ based on Berry’s Paradox. From the above discussions, we can characterize different proofs of $\textsf {G1}$ and $\textsf {G2}$ based on Berry’s Paradox by the method of interpreting the word “name”: Boolos [Reference Boolos13] uses the standard model of arithmetic; Kikuchi [Reference Kikuchi68] uses provability in arithmetic; Chaitin [Reference Chaitin19] and Kikuchi [Reference Kikuchi69] use Kolmogorov complexity; Kikuchi and Tanaka [Reference Kikuchi and Tanaka74] use nonstandard models of arithmetic; and Vopěnka [Reference Vopěnka137] uses models of $\mathbf {ZF}$ (see [Reference Kikuchi, Kurahashi and Sakai73]).

3.4.3 Unexpected Examination and Grelling–Nelson’s Paradox

First of all, Kritchman and Raz [Reference Kritchman and Raz83] give a new proof of $\textsf {G2}$ based on Chaitin’s incompleteness theorem and an argument that resembles the Unexpected Examination ParadoxFootnote ¹⁹ (for more details, we refer to [Reference Kritchman and Raz83]): for any consistent r.e. extension T of $\mathbf {PA}$ , if T is consistent, then $T\nvdash \mathbf {Con}(T)$ .

Secondly, we say a one-place predicate is “heterological” if it does not apply to itself (e.g., “long” is heterological, since it’s not a long expression). Consider the question: is the predicate “heterological” we have just defined heterological? If “heterological” is heterological, then it isn’t heterological; and if “heterological” isn’t heterological, then it is heterological. This contradiction is called Grelling–Nelson’s Paradox.

Cieśliński [Reference Cieśliński25] presents semantic proofs of $\textsf {G2}$ for $\mathbf {ZF}$ and $\mathbf {PA}$ based on Grelling–Nelson’s Paradox. For a theory T containing $\mathbf {ZF}$ , Cieśliński defines the sentence $\mathbf {HET_T}$ which says intuitively that the predicate “heterological” is itself heterological, and then shows that $T\nvdash \mathbf {HET_T}$ and $T\vdash \mathbf {HET_T}\leftrightarrow \mathbf {Con}(T)$ . Finally, Cieśliński shows how to adapt the proof of $\textsf {G2}$ for $\mathbf {ZF}$ to a proof of $\textsf {G2}$ for $\mathbf {PA}$ . In fact, Cieśliński [Reference Cieśliński25] proves the semantic version of $\textsf {G2}$ : if T has a model, then $T + \neg \mathbf {Con}(T)$ has a model (i.e., $T\nvdash \mathbf {Con}(T)$ ).

3.4.4 Yablo’s Paradox

We discuss proofs of $\textsf {G1}$ and $\textsf {G2}$ based on Yablo’s Paradox in the literature. Yablo’s Paradox is an infinite version of the Liar Paradox proposed in [Reference Yablo146]: consider an infinite sequence $Y_1, Y_2, \ldots $ of propositions such that each $Y_i$ asserts that $Y_j$ are false for all $j> i$ . Different proofs of $\textsf {G1}$ and $\textsf {G2}$ based on Yablo’s Paradox have been given by some authors (e.g., Priest [Reference Priest106], Cieśliński–Urbaniak [Reference Cieśliński and Urbaniak26], and Kikuchi–Kurahashi [Reference Kikuchi and Kurahashi70]).

Recall that we assume by default that T is a consistent r.e. extension of $\mathbf {Q}$ . Priest [Reference Priest106] first points out that $\textsf { G1}$ can be proved by formalizing Yablo’s Paradox. Priest defines a formula $Y(x)$ as follows which says that for any $y> x, Y (y)$ is not provable in T.

Cieśliński and Urbaniak originally prove the following version of $\textsf {G1}$ , and show that each instance $Y(\overline {n})$ of the Yablo formula is independent of T if T is $\Sigma ^0_1$ -sound (or 1-consistent).

Cieśliński and Urbaniak originally prove that $T \vdash \forall x(Y(x)\leftrightarrow \mathbf {Con}(T))$ (see [Reference Cieśliński and Urbaniak26, Theorems 21 and 22]). As a corollary, we have $T \vdash \forall x\forall y(Y(x)\leftrightarrow Y(y))$ , and $\textsf {G2}$ holds: if T is consistent, then $T\nvdash \mathbf {Con}(T)$ .

Theorem 3.20 shows that the Rosser-type Yablo formula is independent of any $\Sigma ^0_1$ -sound theory T.

The independence of $Y^R(\overline {n})$ for T which is not $\Sigma ^0_1$ -sound is discussed in [Reference Kurahashi85]. For a consistent but not $\Sigma ^0_1$ -sound theory, the situation of Rosser-type Yablo formulas is quite different from that of Rosser sentences. Kurahashi [Reference Kurahashi85] shows that for any consistent but not $\Sigma ^0_1$ -sound theory, the independence of each instance of a Rosser-type Yablo formula depends on the choice of standard proof predicates (see Theorems 12 and 25 in [Reference Kurahashi85]). Kurahashi [Reference Kurahashi85] shows that for any consistent but not $\Sigma ^0_1$ -sound theory T, there is a standard proof predicate of T such that each instance $Y^R(\overline {n})$ of the Rosser-type Yablo formula $Y^R(x)$ based on this proof predicate is provable in T for any $n \in \omega $ . Moreover, Kurahashi [Reference Kurahashi85] constructs a standard proof predicate of T and a Rosser-type Yablo formula $Y^R(x)$ based on this proof predicate such that each instance of $Y^R(x)$ is independent of T. Proofs of these results use the technique of Guaspari and Solovay in [Reference Guaspari and Solovay50].

Cieśliński and Urbaniak [Reference Cieśliński and Urbaniak26] conjecture that any two distinct instances $Y^R(\overline {m})$ and $Y^R(\overline {n})$ of a Rosser-type Yablo formula $Y^R(x)$ based on a standard proof predicate are not provably equivalent. Leach–Krouse [Reference Leach-Krouse84] and Kurahashi [Reference Kurahashi85] construct a standard proof predicate, and a Rosser-type Yablo formula $Y^R(x)$ based on this proof predicate such that $T\vdash \forall x\forall y (Y^R(x) \leftrightarrow Y^R(y))$ (see [Reference Leach-Krouse84, Theorem 9] and [Reference Kurahashi85, Corollary 21]).

Kurahashi [Reference Kurahashi85] constructs a partial counterexample to Cieśliński and Urbaniak’s conjecture: a standard proof predicate, and a Rosser-type Yablo formula $Y^R(x)$ based on this proof predicate such that $\forall x\forall y (Y^R(x) \leftrightarrow Y^R(y))$ is not provable in T ([Reference Kurahashi85, Corollary 20]). Thus the provability of the sentence $\forall x\forall y (Y^R(x) \leftrightarrow Y^R(y))$ also depends on the choice of standard proof predicates (see [Reference Kurahashi85, Corollaries 20 and 21]). Proofs of these results by Leach-Krouse and Kurahashi also use the technique of Guaspari and Solovay in [Reference Guaspari and Solovay50]. An interesting open question is: whether there is a standard proof predicate such that $Y^R(\overline {n})$ and $Y^R(\overline {n+1})$ are not provably equivalent for some $n \in \omega $ (see [Reference Kurahashi85])?

3.4.5 Beyond arithmetization

All the proofs of $\textsf {G1}$ we have discussed use arithmetization. Andrzej Grzegorczyk proposes the theory $\mathbf {TC}$ in as a possible alternative theory for studying incompleteness and undecidability, and shows that $\mathbf {TC}$ is essentially incomplete and mutually interpretable with $\mathbf {Q}$ without arithmetization.

Now, in $\mathbf {PA}$ we have numbers that can be added or multiplied; while in $\mathbf {TC}$ , one has strings (or texts) that can be concatenated. In Gödel’s proof, the only use of numbers is coding of syntactical objects. The motivations for accepting strings rather than numbers as the basic notion are as follows: on metamathematical level, the notion of computability can be defined without reference to numbers; for Grzegorczyk, dealing with texts is philosophically better justified since intellectual activities like reasoning, communicating, or even computing involve working with texts not with numbers. Thus, it is natural to define notions like undecidability directly in terms of texts instead of natural numbers. Grzegorczyk only proves the incompleteness of $\mathbf {TC}$ in. Later, Grzegorczyk and Zdanowski [Reference Grzegorczyk, Zdanowski, Ehrenfeucht, Marek and Srebrny48] prove that $\mathbf {TC}$ is essentially incomplete.

3.5.2 Paris–Harrington and beyond

Paris and Harrington [Reference Paris, Harrington and Barwise103] propose the first mathematically natural statement independent of $\mathbf {PA}$ : the Paris–Harrington Principle ( $\textsf {PH}$ ) which generalizes the finite Ramsey theorem. Gödel’s sentence is a pure logical construction (via the arithmetization of syntax and provability predicate) and has no relevance with classic mathematics (without any combinatorial or number-theoretic content). On the contrary, Paris–Harrington Principle is an independent arithmetic sentence from classic mathematics with combinatorial content as we will show. We refer to [Reference Isaacson, Barwise, Kaplan, Keisler, Suppes and Troelstra61] for more discussions about the distinction between mathematical arithmetic sentences and meta-mathematical arithmetic sentences.

Now, $\textsf {PH}$ has a clear combinatorial flavor, and is of the form $\forall x \exists y\psi (x, y)$ where $\psi $ is a $\Delta ^0_0$ formula. It can be shown that for any given natural number n, $\mathbf {PA} \vdash \exists y\psi (\overline {n}, y)$ , i.e., all particular instances of $\textsf {PH}$ are provable in $\mathbf {PA}$ .

Following $\textsf {PH}$ , many other mathematically natural statements independent of $\mathbf {PA}$ with combinatorial or number-theoretic content have been formulated: the Kanamori–McAloon principle, the Kirby–Paris sentence [Reference Paris and Kirby104], the Hercules–Hydra game [Reference Paris and Kirby104], the Worm principle [Reference Beklemishev7, Reference Hamano and Okada56], the flipping principle [Reference Kirby75], the arboreal statement [Reference Mills93], the kiralic and regal principles [Reference Clote and Mcaloon27], and the Pudlák’s principle [Reference Hájek and Paris52, Reference Pudlák107] (see [Reference Bovykin15, p. 40]). In fact, all these principles are equivalent to $\textsf {PH}$ (see [Reference Bovykin15, p.40]).

An interesting and amazing fact is: all the above mathematically natural principles are in fact provably equivalent in $\mathbf {PA}$ to a certain meta-mathematical sentence. Consider the following reflection principle for $\Sigma ^0_1$ sentences: for any $\Sigma ^0_1$ sentence $\phi $ in $L(\mathbf {PA})$ , if $\phi $ is provable in $\mathbf {PA}$ , then $\phi $ is true. Using the arithmetization of syntax, one can write this principle as a sentence of $L({\mathbf {PA}})$ , and denote it by $\textsf {Rfn}_{\Sigma ^0_1}(\mathbf {PA})$ (see [Reference Murawski97, p. 301]). McAloon has shown that $\mathbf {PA} \vdash \textsf {PH}\leftrightarrow \textsf {Rfn}_{\Sigma ^0_1}(\mathbf {PA})$ (see [Reference Murawski97, p. 301]), and similar equivalences can be established for the other independent principles mentioned above. Equivalently, all these principles are equivalent to so-called 1-consistency of $\mathbf {PA}$ (see [Reference Beklemishev8, p. 36], [Reference Beklemishev7, p. 3], and [Reference Murawski97, p. 301]).

The above phenomenon indicates that the difference between mathematical and meta-mathematical statements is perhaps not as huge as we might have expected. Moreover, the above principles are provable in fragments of Second-Order Arithmetic and are more complex than Gödel’s sentence: Gödel’s sentence is equivalent to $\mathbf {Con}(\mathbf {PA})$ in $\mathbf {PA}$ ; but all these principles are not only independent of $\mathbf {PA}$ but also independent of $\mathbf {PA}+ \mathbf {Con}(\mathbf {PA})$ (see [Reference Beklemishev8, p. 36] and [Reference Murawski97, p. 301]).

3.5.3 Harvey Friedman’s contributions

Incompleteness would not be complete without mentioning the work of Harvey Friedman who is a central figure in research on the foundations of mathematics after Gödel. He has made many important contributions to concrete mathematical incompleteness. The following quote is telltale:

In the following, we give a brief introduction to H. Friedman’s work on concrete mathematical incompleteness. In his early work, H. Friedman examines how one uses large cardinals in an essential and natural way in number theory, as follows.

H. Friedman shows in [Reference Friedman38, Reference Friedman39] that there are many mathematically natural combinatorial statements in $L(\mathbf {PA})$ that are neither provable nor refutable in $\mathbf {ZFC}$ or $\mathbf {ZFC}$ + large cardinals. H. Friedman’s more recent monograph [Reference Friedman40] is a comprehensive study of concrete mathematical incompleteness. H. Friedman studies concrete mathematical incompleteness over different systems, ranging from weak subsystems of $\mathbf {PA}$ to higher-order arithmetic and $\mathbf {ZFC}$ . H. Friedman lists many concrete mathematical statements in $L(\mathbf {PA})$ that are independent of subsystems of $\mathbf {PA}$ , or stronger theories like higher-order arithmetic and set theory.

The theories $\mathbf {RCA_0}$ (Recursive Comprehension), $\mathbf {WKL_0}$ (Weak Konig’s Lemma), $\mathbf {ACA_0}$ (Arithmetical Comprehension), $\mathbf {ATR_0}$ (Arithmetic Transfinite Recursion), and $\Pi ^1_1$ - $\mathbf {CA_0}$ ( $\Pi ^1_1$ -Comprehension) are the most famous five subsystems of Second-Order Arithmetic ( $\mathbf {SOA}$ ), and are called the ‘Big Five’. For the definition of $\mathbf {SOA}$ and the ‘Big Five’, we refer to [Reference Simpson117].

To give the reader a better sense of H. Friedman’s work, we list some sections dealing with concrete mathematical incompleteness in [Reference Friedman40].

• • Section 0.5 on Incompleteness in Exponential Function Arithmetic.

• • Section 0.6 on Incompleteness in Primitive Recursive Arithmetic, Single Quantifier Arithmetic, $\mathbf {RCA_0}$ , and $\mathbf {WKL_0}$ .

• • Section 0.7 on Incompleteness in Nested Multiply Recursive Arithmetic and Two Quantifier Arithmetic.

• • Section 0.8 on Incompleteness in Peano Arithmetic and $\mathbf {ACA_0}$ .

• • Section 0.9 on Incompleteness in Predicative Analysis and $\mathbf {ATR_0}$ .

• • Section 0.10 on Incompleteness in Iterated Inductive Definitions and $\Pi ^1_1$ - $\mathbf {CA_0}$ .

• • Section 0.11 on Incompleteness in Second-Order Arithmetic and $\mathbf {ZFC}^{-}$ .Footnote ²⁰

• • Section 0.12 on Incompleteness in Russell Type Theory and Zermelo Set Theory.

• • Section 0.13 on Incompleteness in $\mathbf {ZFC}$ using Borel Functions.

• • Section 0.14 on Incompleteness in $\mathbf {ZFC}$ using Discrete Structures.

H. Friedman [Reference Friedman40] provides us with examples of concrete mathematical theorems not provable in subsystems of Second-Order Arithmetic stronger than $\mathbf {PA}$ , and a number of concrete mathematical statements provable in Third-Order Arithmetic but not provable in Second-Order Arithmetic.

Related to Friedman’s work, Cheng [Reference Cheng21, Reference Cheng and Schindler24] gives an example of concrete mathematical theorems based on Harrington’s principle which is isolated from the proof of the Harrington’s Theorem (the determinacy of $\Sigma ^1_1$ games implies the existence of zero sharp), and shows that this concrete theorem saying that Harrington’s principle implies the existence of zero sharp is expressible in Second-Order Arithmetic, not provable in Second-Order Arithmetic or Third-Order Arithmetic, but provable in Fourth-Order Arithmetic (i.e., the minimal system in higher-order arithmetic to prove this concrete theorem is Fourth-Order Arithmetic).

Many other examples of concrete mathematical incompleteness, and the discussion of this subject in 1970s–1980s can be found in the four volumes [Reference Berline, Mcaloon and Ressayre10, Reference Pacholski and Wierzejewski101, Reference Simpson115, Reference Simpson116]. Weiermann’s work in [Reference Weiermann139–Reference Weiermann143] provides us with more examples of naturally mathematical independent sentences. We refer to [Reference Friedman40] for new advances in Boolean Relation Theory and for more examples of concrete mathematical incompleteness.

4.3.1 Generalizations of $\textsf {G1}$ via interpretability

We show that $\textsf {G1}$ can be generalized to theories weaker than $\mathbf {PA}$ via interpretability. Indeed, there exists a weak recursively axiomatizable consistent subtheory T of $\mathbf {PA}$ such that each recursively axiomatizable theory S in which T is interpretable is incomplete (see [Reference Tarski, Mostowski and Robinson125]). To generalize this fact further, we propose a new notion “ $\textsf {G1}$ holds for T”, as follows.

First of all, for a consistent r.e. theory T, it is not hard to show that the followings are equivalent (see [Reference Cheng22]):

• • $\textsf {G1}$ holds for T.

• • T is essentially incomplete.

• • T is essentially undecidable.

It is well-known that $\textsf {G1}$ holds for many weaker theories than $\mathbf {PA}$ w.r.t. interpretation (e.g., Robinson Arithmetic $\mathbf {Q}$ ).

Secondly, we mention theories weaker than $\mathbf {PA}$ w.r.t. interpretation for which $\textsf {G1}$ holds. We first review some essentially undecidable theories weaker than $\mathbf {PA}$ w.r.t. interpretation from the literature (i.e., $\textsf {G1}$ holds for these theories). For the definition of theory $\mathbf {Q}$ , $I\Sigma _n$ , $B\Sigma _n$ , $\mathbf {PA}^{-}$ , $\mathbf {Q}^+$ , $\mathbf {Q}^{-}$ , $\mathbf {S^1_2}$ , $\mathbf {AS}$ , $\mathbf {EA}$ , $\mathbf {PRA}$ , $\mathbf {R}$ , $\mathbf {R}_0$ , $\mathbf {R}_1$ , and $\mathbf {R}_2$ , we refer to Section 2.