Isaacson’s thesis on arithmetical truth

Abstract

Isaacson’s thesis claims that Peano arithmetic is complete with respect to arithmetical truth as defined by Isaacson. According to this thesis, the incompleteness phenomenon revealed in Gödel’s incompleteness theorems does not pertain to arithmetical incompleteness. In our analysis, we discuss both the advantages and disadvantages of Isaacson’s thesis. Additionally, we propose seven case examples that may pose potential challenges to Isaacson’s thesis. To effectively defend Isaacson’s thesis, one must address these case examples. We conclude that either Isaacson’s notion of arithmetical truth is not right, or that it is difficult to defend Isaacson’s thesis.

1 Introduction

Isaacson’s thesis claims that Peano arithmetic ($\textsf{PA}$) is complete with respect to arithmetical truth: $\textsf{PA}$proves all and only arithmetical truths as defined by Isaacson (Isaacson, 1996, p. 222). That is, a sentence in the language of arithmetic is arithmetical as defined by Isaacson if and only if it is provable in $\textsf{PA}$. Within the arithmetically expressible truths of mathematics, Isaacson distinguishes between those truths that are purely arithmetical and those that are essentially higher-order. For Isaacson, there should be a boundary between these two kinds of truths (Isaacson, 1996, p. 221). Isaacson’s thesis asserts that the theory $\textsf{PA}$ is precisely the boundary of these two kinds of truths.

Isaacson’s thesis, introduced by Isaacson in 1987/1996 and 1992, has been discussed in the literature  (Smith, 2008; Tatton-Brown, 2018; Dopico, 2024). One reason why Isaacson’s thesis merits further study is its relevance to the debates surrounding the Implicit Commitment Thesis (ICT) in contemporary philosophy of mathematics (e.g., Brauer, 2023; Nicolai and Piazza, 2019). A key notion in Isaacson’s thesis is “arithmetical truth”. Notably, Isaacson’s conception of arithmetical truth differs from the standard definition in logic.¹ According to Isaacson (1996), a statement is considered arithmetical if its truth can be directly discerned from our understanding of the structure of natural numbers or from other truths expressed in the language of arithmetic that are themselves arithmetical.²

Isaacson adopts a classical view that equates the mathematical truth of sentences in the language of arithmetic with satisfaction in the standard model of arithmetic. “Being arithmetical” is a property that applies only to mathematically true statements expressible in the language of arithmetic. Isaacson’s notion of arithmetical truth specifies what kinds of mathematical truths, expressible in the language of arithmetic, are considered arithmetical. In this paper, whenever we say that a sentence in the language of arithmetic is arithmetical in Isaacson’s sense, we mean that its truth is directly perceivable based on our articulation of our grasp of the structure of the natural numbers or directly perceivable from truths in the language of arithmetic that are themselves arithmetical.

In this paper, we analyze Isaacson’s thesis, examine how we could defend it, and discuss its advantages and disadvantages. We propose seven case examples that may pose potential challenges to Isaacson’s thesis. Defenders of Isaacson’s thesis should address these examples. We conclude that either Isaacson’s notion of arithmetical truth is not right, or that it is difficult to defend Isaacson’s thesis.

The structure of this paper is as follows. In Sect. 2, we provide an overview of Gödel’s incompleteness theorems and some examples of independent sentences of $\textsf{PA}$ from the research on concrete incompleteness. In Sect. 3, we analyze Isaacson’s thesis. In Sect. 4, we examine what we should provide for defending Isaacson’s thesis. We propose seven case examples that may constitute potential challenges to Isaacson’s thesis. In Sect. 5, we summarize and conclude the paper.

2 Preliminaries

Isaacson (1996) discusses two kinds of independent sentences of $\textsf{PA}$: Gödel sentences and the consistency statement of $\textsf{PA}$ from Gödel’s incompleteness theorems, as well as examples of independent sentences of $\textsf{PA}$ from the research on concrete incompleteness such as the Paris-Harrington sentence. In this section, to help readers better understand Isaacson’s thesis, we provide an overview of these two kinds of independent sentences of $\textsf{PA}$.

The three key ingredients in Gödel’s proof of the incompleteness theorems are arithmetization, representability, and self-reference. Arithmetization establishes a correspondence between expressions in the language of $\textsf{PA}$ and natural numbers. Through arithmetization, one can translate metamathematical statements about arithmetic as a formal system into statements about natural numbers; moreover, fundamental metamathematical relations can be translated into certain recursive relations on natural numbers (Murawski, 1999). Gödel proves that every recursive relation is representable in $\textsf{PA}$. That is, through representability, we can represent these recursive relations in $\textsf{PA}$. Consequently, one can speak about the properties of $\textsf{PA}$ within $\textsf{PA}$ itself! This is the essence of Gödel’s idea of arithmetization and representability. For details on arithmetization and representability, we refer to Murawski (1999).

Let $\textsf{Prf}(x,y)$ be the formula that represents in $\textsf{PA}$ the recursive relation $Prf(m,n)$ which says that $n$ is the Gödel number of a proof in $\textsf{PA}$ of the formula with Gödel number $m$. From the formula $\textsf{Prf}(x,y)$, we can define the provability predicate $\textsf{Pr}(x)$ as $\exists y \textsf{Prf}(x,y)$. In this paper, we use $\overline{n}$ to denote the numeral of $n\in\omega$, and use $\ulcorner\phi\urcorner$ to denote the numeral of the Gödel number of $\phi$ for a given formula $\phi$.

For the proof of the first incompleteness theorem ($\textsf{G1}$), Gödel constructs a Gödel sentence $\textsf{G}$ in the language of arithmetic, stating that $\textsf{G}$ itself is not provable in $\textsf{PA}$, and shows that if $\textsf{PA}$ is consistent, then $\textsf{PA}\nvdash \textsf{G}$; thus, $\textsf{G}$ is a mathematically true sentence. For the proof of the second incompleteness theorem (G2), we first define the sentence $\textsf{Con(PA)}$ as $\neg \textsf{Pr}(\ulcorner 0\neq 0\urcorner)$ in the language of arithmetic. One can show that $\textsf{PA}\vdash \textsf{Con(PA)}\leftrightarrow \textsf{G}$. Thus, $\textsf{G2}$ holds: if $\textsf{PA}$ is consistent, then $\textsf{PA}\nvdash \textsf{Con(PA)}$. For more details on the proofs of $\textsf{G1}$ and $\textsf{G2}$, we refer to Murawski (1999) and Smorynski (Smoryński, 1977).

Now we provide an overview of examples of independent sentences of $\textsf{PA}$ from the research on concrete incompleteness. In Gödel’s proof, the independent sentence constructed has a clear meta-mathematical flavor that is devoid of concrete mathematical content. Gödel’s sentence is a purely logical construction via the arithmetization of syntax and the provability predicate, and has no relevance to classical mathematics (e.g., it lacks any number-theoretic content). From a purely mathematical point of view, Gödel’s sentence is artificial and not mathematically interesting.

After Gödel, a natural question arises: can we find mathematically true sentences that are not provable in $\textsf{PA}$ and have concrete mathematical content? The research program aimed at finding “natural” independent sentences with concrete mathematical content is referred to as concrete incompleteness in the literature. This program has garnered significant attention after Gödel because, despite Gödel’s incompleteness theorems, one can still hope that all mathematically interesting sentences about natural numbers are provable or refutable in $\textsf{PA}$ (i.e., $\textsf{PA}$ is complete with respect to mathematically interesting sentences).

However, after Gödel, researchers have discovered many independent sentences of $\textsf{PA}$, such as the Paris-Harrington sentence, the Kanamori-McAloon principle, the Goodstein sequence, the Hercules-Hydra game, the Worm principle, and the kiralic and regal principles.³ The formulation of these sentences does not refer to the arithmetization of syntax or provability predicates. These independent sentences arguably possess a clear mathematical flavor with concrete mathematical content. An interesting and remarkable discovery in the literature is that many mathematically concrete independent sentences of $\textsf{PA}$ from concrete incompleteness (all examples listed above) are, in fact, provably equivalent over $\textsf{PA}$ to the $\Sigma^0_1$-reflection principle of $\textsf{PA}$.⁴

3 An analysis of Isaacson’s thesis

In this section, we analyze Isaacson’s thesis from the following aspects: Isaacson’s notion of arithmetical truth, the nature of independent sentences of $\textsf{PA}$, Isaacson’s notion of coding, and Isaacson’s notion of “higher order concept”.

3.1 Isaacson’s notion of arithmetical truth

In this subsection, we examine Isaacson’s notion of arithmetical truth and argue that: (1) Isaacson’s notion of arithmetical truth is vague; (2) Isaacson holds a restrictive notion of arithmetical truth; (3) Isaacson’s notion of arithmetical truth is confined to the standard structure of arithmetic.

3.1.1 Isaacson’s notion of arithmetical truth is vague

Recall that whenever we say that a sentence in the language of arithmetic is arithmetical in Isaacson’s sense, it means that its truth is directly perceivable based on our articulation of our grasp of the structure of natural numbers or directly perceivable from truths in the language of arithmetic that are themselves arithmetical.

According to Isaacson’s thesis, the set of sentences in the language of arithmetic that are arithmetical in Isaacson’s sense is the same as the set of all theorems of $\textsf{PA}$. However, the former set is not formally well-defined. For Isaacson, what counts as arithmetical in Isaacson’s sense has to do with the way in which we are able to directly perceive a statement’s truth (Isaacson, 1996). Isaacson’s notion of arithmetical truth is not a precise formal notion; as a conceptual and epistemic notion, it is vague in several respects.

Some terminologies in Isaacson’s definition of arithmetical truth are ambiguous. What does “directly perceivable” mean? What is the difference between “directly perceivable” and “indirectly perceivable”? Who is the agent of perceiving? Is it perceived by a single individual or by a community of people?⁵ What does “grasp” mean? What is the difference between perceiving and grasping?

The notion of direct perceivability is crucial in Isaacson’s conception of arithmetical truth. According to Isaacson’s thesis, the truths that we can directly perceive based on the purely arithmetical content of the structure of natural numbers are precisely the theorems of Peano arithmetic. It is important to distinguish between the meaning of direct perceivability and how it applies to mathematical statements.⁶ For the application to mathematical statements, the notion of direct perceivability applies to very concrete objects. Unfortunately, the meaning of direct perceivability remains vague in Isaacson (1996). This vagueness consequently leaves Isaacson’s notion of arithmetical truth equally unclear.

3.1.2 A restrictive notion of arithmetical truth

The mainstream conception of arithmetical truth equates it with satisfiability in the standard structure of arithmetic (which we call the classical view). According to the classical view, any truth in the standard model of arithmetic is viewed as arithmetical, regardless of whether there is a way to directly perceive its truth in purely arithmetical terms.

According to Isaacson’s thesis, his notion of arithmetical truth is restrictive. Isaacson’s thesis claims that the set of arithmetical truths equates to the set of theorems of $\textsf{PA}$. According to Isaacson’s thesis, true sentences in the language of arithmetic that are unprovable in $\textsf{PA}$ are not arithmetical. For Isaacson, an arithmetical truth can be perceived as true directly from the purely arithmetical content of a conceptual analysis of the structure of natural numbers. He distinguishes between purely arithmetical truths (truths directly perceived only via purely arithmetical content) and higher-order truths (truths perceived via real higher-order concepts). Isaacson claims that “the system $\textsf{PA}$ arises as constituting the purely arithmetical content of our full understanding of the concept of natural number, where that understanding is implicitly and inherently higher-order” (Isaacson, 1996, p. 209).

Now, we compare Isaacson’s notion of arithmetical truth with Dummett and Gödel’s view. Dummett (1963) defines a concept to be indefinitely extensible if it is not possible to delineate the range of objects to which the concept applies: i.e., “a concept is indefinitely extensible if any attempt to delineate the extension of the concept leads to an instance of the concept not so delineated”  (Shapiro, 2000, p. 196). Dummett argues that the incompleteness theorems show that the notion of arithmetical truth is indefinitely extensible  (Shapiro, 2000, p. 196). Let $T$ be any effective procedure for enumerating arithmetical truths. An application of the incompleteness theorems yields an arithmetical truth $\phi$ not enumerated by $T$. So $T$ fails to characterize arithmetical truth  (Shapiro, 2000, p. 197).

For Gödel, who is a realist, determining properties of mathematical objects such as natural numbers requires going beyond ‘intuition’ and developing powerful mathematical theories  (Shapiro, 2000, p. 208). Gödel wrote: “it has turned out that $\cdots$ the solution of arithmetical problems requires the use of assumptions essentially transcending arithmetic” (Gödel 1944, p. 449).

Isaacson’s notion of arithmetical truth is not indefinitely extensible. According to Isaacson’s thesis, a sentence in the language of arithmetic is arithmetical if and only if it is provable in $\textsf{PA}$. The limit of arithmetical truth in Isaacson’s sense is just the limit of provability in $\textsf{PA}$. The set of true sentences in the standard model of arithmetic is not recursively enumerable. Since the set of theorems of $\textsf{PA}$ is recursively enumerable, according to Isaacson’s thesis, the set of arithmetical truths in his sense is also recursively enumerable, and any true sentence not included in the enumeration of this set of arithmetical truths is not arithmetical in Isaacson’s sense. Moreover, unlike in Gödel’s view, Isaacson’s notion of arithmetical truth is not a notion that requires going beyond the resources of arithmetic.⁷

In summary, Isaacson holds a distinctive view of arithmetical truth: a restrictive notion that is not indefinitely extensible.

3.1.3 Isaacson’s notion of arithmetical truth is confined to the standard structure of arithmetic

Our conceptual analysis of natural numbers is inevitably related to specific structures of natural numbers. Different structures may embody distinct arithmetical contents. For any countable ordinal $\alpha$, we can naturally induce a well-ordering $< _\alpha$ on $\omega$ of order type $\alpha$, such that $(\alpha,\in)$ is isomorphic to (or can be coded by) the structure $(\omega, < _\alpha)$. Since there are $\aleph_1$ (the first uncountable cardinal) many countable ordinals, there are $\aleph_1$ distinct structures of natural numbers, each with its own unique order type. Isaacson (1996) states that any attempt to establish the mathematical truth of the finite Kruskal theorem “must proceed from essentially set-theoretic principles which are beyond anything required to establish the structure of the natural numbers, a well-ordering of order type $\omega$” (Isaacson, 1996, p. 220). In this context, “the structure of natural numbers” specifically refers to a structure of order type $\omega$. Since the standard structure of arithmetic has order type $\omega$, we can conclude that “the structure of natural numbers” refers to the standard structure of arithmetic in the sense of order isomorphism. It is important to note that Isaacson’s concept of arithmetical truth is linked to the structure of natural numbers. Therefore, we can assert that Isaacson’s notion of arithmetical truth is confined to the standard structure of arithmetic of order type $\omega$.

Consider the transfinite induction principle $TI(\alpha)$ (a sentence in the language of arithmetic coding transfinite induction of order type $\alpha$). According to Isaacson’s thesis, $TI(\alpha)$ for any $\omega\leq\alpha < \epsilon_0$ is arithmetical in his sense, since, as a classic result from ordinal analysis, $TI(\alpha)$ for $\omega\leq\alpha < \epsilon_0$ is provable in $\textsf{PA}$. Since we can perceive the truths of $TI(\alpha)$ for all $\omega\leq\alpha < \epsilon_0$, our perception should refer to structures over $\omega$ of any countable order type less than $\epsilon_0$. However, Isaacson’s notion of arithmetical truth is confined to the standard structure of arithmetic of order type $\omega$.

By the completeness theorem of first-order logic, a sentence $A$ in the language of arithmetic is provable in $\textsf{PA}$ if and only if $A$ is satisfied in any model of $\textsf{PA}$. We know that $\textsf{PA}$ is not categorical and has many (in fact, continuum many) non-standard models in the language of arithmetic.⁸ Thus, provability in $\textsf{PA}$ implicitly refers to all structures of $\textsf{PA}$ in the language of arithmetic. If Isaacson’s thesis holds, then his notion of arithmetical truth should also refer to all structures of $\textsf{PA}$. Yet, Isaacson’s notion of arithmetical truth remains confined to the standard structure of arithmetic.

3.2 The nature of independent sentences of $\textsf{PA}$

One motivation of Isaacson’s work (1996) comes from how to view the incompleteness phenomenon reflected in Gödel’s incompleteness theorems and the nature of those true but unprovable sentences of $\textsf{PA}$ that have been discovered in the literature.

Gödel’s incompleteness theorems tell us that $\textsf{PA}$ is incomplete. However, according to Isaacson’s thesis, Gödel’s incompleteness theorems are not about arithmetical incompleteness, and $\textsf{PA}$ is complete with respect to arithmetical truth in Isaacson’s sense. For Isaacson, any independent sentence of $\textsf{PA}$ is not arithmetical in Isaacson’s sense. He believes that “$\textsf{PA}$ indeed occupies an intrinsic, conceptually well-defined region of arithmetical truth”, and $\textsf{PA}$ “consists of those truths which can be perceived as true directly from the purely arithmetical content of a categorical conceptual analysis of the notion of natural number” (Isaacson, 1996, p. 203).

Now we discuss Isaacson’s view on the nature of independent sentences of $\textsf{PA}$. Isaacson (1996) discussed two kinds of independent sentences of $\textsf{PA}$. One type consists of sentences formulated via arithmetization, such as Gödel sentences and the consistency statement $\textsf{Con(PA)}$; the other type includes independent sentences from the research on concrete incompleteness, whose formulations do not use arithmetization. Examples of the second kind of independent sentences discussed by Isaacson (1996) include the Paris-Harrington sentence, the Goodstein sentence, and Friedman’s finitization of Kruskal’s Theorem. Isaacson argues that his thesis is tested and supported by these examples.

What is Isaacson’s view on the nature of independent sentences of $\textsf{PA}$ as revealed in Gödel’s incompleteness theorems, such as Gödel sentences and $\textsf{Con(PA)}$? According to Isaacson’s thesis, Gödel sentences and $\textsf{Con(PA)}$ are not arithmetical in his sense. However, Isaacson does not directly explain why there is no way by which the truths of Gödel sentences and $\textsf{Con(PA)}$ can be directly perceived in purely arithmetical terms. Instead, Isaacson (1996) emphasizes that Gödel sentences and $\textsf{Con(PA)}$ do not directly address properties of natural numbers but rather properties of the formal system in which they are formalized via coding.⁹

What is Isaacson’s view on the nature of independent sentences of $\textsf{PA}$ from the research on concrete incompleteness? Isaacson agrees that independent sentences from the research on concrete incompleteness are much more genuinely and purely mathematical than Gödel sentences (Isaacson, 1996, p. 215). However, Isaacson claims that all independent sentences of $\textsf{PA}$ from concrete incompleteness have a special status and are not arithmetical in his sense.

Consider the Paris-Harrington sentence ($\textsf{PH}$), introduced by Paris and Harrington (1977). This sentence generalizes the finite Ramsey theorem.¹⁰ The statement $\textsf{PH}$ is true in the standard model of arithmetic. Isaacson (1996) does not deny that the Paris-Harrington sentence is mathematical, but he does not consider it arithmetical in his sense. Paris and Harrington (1977) proved that $\textsf{PH}$ is provably equivalent over $\textsf{PA}$ to the $\Sigma^0_1$-reflection principle of $\textsf{PA}$. Isaacson interprets this result as “revealing something of the implicit (hidden) higher-order content of the Paris-Harrington sentence”: “the Paris-Harrington sentence expresses a strong reflective property about the whole formalization of arithmetic, and is thereby implicitly higher-order, and so non-arithmetical” (Isaacson, 1996, pp. 218–219). For Isaacson, since most independent sentences of $\textsf{PA}$ from concrete incompleteness (all examples listed in Sect. 2) are equivalent to the $\Sigma^0_1$-reflection principle of $\textsf{PA}$, all these examples are not arithmetical in his sense.

In summary, Isaacson’s thesis advocates for a new perspective on the nature of true but unprovable sentences of $\textsf{PA}$ (Isaacson, 1996, p. 203). From the classical view of arithmetical truth, any true sentence in the language of arithmetic is considered arithmetical, regardless of whether it is a theorem of $\textsf{PA}$ or an unprovable sentence. However, Isaacson argues that the status of Gödel sentences and the Paris-Harrington sentence is fundamentally different from that of theorems of $\textsf{PA}$. If our notion of arithmetical truth implies that some independent sentences of $\textsf{PA}$ are arithmetical while others are not, we must explain why this distinction exists. From Isaacson’s perspective, any true-but-unprovable sentence of $\textsf{PA}$ is not arithmetical in his sense, and independent sentences of $\textsf{PA}$ arising from concrete incompleteness share the same status as Gödel sentences: they are all non-arithmetical in Isaacson’s sense.

3.3 Isaacson’s notion of coding

In this section, we examine Isaacson’s notion of coding, which is crucial for understanding his thesis. Regarding the nature of truths expressible in the language of arithmetic that are not provable in $\textsf{PA}$, Isaacson wrote: “there is no way by which their truth can be (directly) perceived in purely arithmetical terms” and they “contain hidden higher-order concepts, or infinitary concepts, where what is hidden is revealed by the recognition of the phenomenon of coding” (Isaacson, 1996, p. 203). Isaacson views “arithmetically expressible independent truths as exemplifying the expressive richness of finite structures, perceivable through the phenomenon of coding” (Isaacson, 1996, p. 222).

We first distinguish between coding and definability (or representability). Definability (representability) concerns how to use a formula in the language of a formal system to express a property of a set. For example, the set of prime numbers is definable in the language of arithmetic. Coding, on the other hand, involves defining an injective mapping from a set to the set of natural numbers. In logic, definability is always related to a formal system, but coding is not necessarily so.

The notion of coding is popular in the literature, but the same term may refer to different phenomena in the practices of coding in mathematics and logic. In this paper, we distinguish three kinds of coding based on the method used.

The first kind of coding comes from pure mathematics (excluding mathematical logic). Generally speaking, we can regard any injective mapping from a set in pure mathematics (excluding sets from mathematical logic, such as sets of ordinals or cardinals, or sets of formulas of some formal system) to the set of natural numbers as a way of coding in this first kind. We take an example: coding a finite sequence of natural numbers by a natural number. There are varied ways to do this and all of them belong to the first kind of coding, which are standard techniques in number theory.

The second kind of coding employs methods from mathematical logic but does not refer to a formal system. For example, via the Cantor’s normal form theorem, we can code any ordinal below $\epsilon_0$ by a natural number.¹¹ This method uses set theory (some logical methods) but does not refer to a formal system.

The third kind of coding refers to the arithmetization of a formal system. Arithmetization codes the symbols (i.e., the elements of the signature of the language and the logical constants) of a formal system by natural numbers. Moreover, we can code the formulas in the language of a formal system by natural numbers using some coding method from the first kind. To distinguish this kind of coding from the others, we usually refer to it as “arithmetization”, which is a powerful tool in the meta-mathematics of arithmetic and an important technique in the proof of Gödel’s incompleteness theorems.

In summary, the common characteristic of the three kinds of coding is that they all involve defining an injective mapping from a set to the set of natural numbers. For the first kind of coding, the mapping is defined from a set in pure mathematics, excluding mathematical logic, and does not refer to a formal system. For the second kind of coding, the mapping is defined from a set in mathematical logic but does not refer to a formal system. For the third kind of coding, the mapping refers to a formal system and is defined from a set of symbols in the language of that formal system.

What does “coding” mean in Isaacson’s view? It seems that in some places in Isaacson (1996), the term “coding” refers to arithmetization (the third kind of coding), while in other places, it may refer to combinations of the three kinds of coding. For example, Isaacson wrote: “the phenomenon of coding reveals fixed links between two situations or facts, one in the structure of arithmetic, the other in the realm of syntax of a formal system. These facts, and the link between them, are revealed by the description of the coding” (Isaacson, 1996, p. 214). Here, “coding” refers to the third kind: arithmetization. In another instance, Isaacson wrote: “coding pulls the ostensibly higher-order truth into the arithmetical” (Isaacson, 1996, p. 221). Here, the meaning of “coding” may encompass any of the three kinds we have discussed: to pull the ostensibly higher-order truth into the arithmetical, we may use combinations of the three kinds of coding. The coding that pulls the ostensibly higher-order truth into the arithmetical may involve some formal system, thus invoking the third kind of coding, arithmetization. To formalize a mathematical statement in $\textsf{PA}$, we may also use the first or second kind of coding.

Moreover, we should distinguish between the use of coding in the formulation of a statement and its use in the proof of a statement. For example, the Paris-Harrington theorem states that the Paris-Harrington sentence is not provable in $\textsf{PA}$. To express the finite version of the Ramsey theorem in the language of $\textsf{PA}$, we use the first or second kind of coding to express the Paris-Harrington sentence. For the formulation of the Paris-Harrington sentence, we do not use arithmetization. However, the proof of the Paris-Harrington theorem may use arithmetization.

The Paris-Harrington theorem has various proofs in the literature; some use arithmetization while others may not. There are two general methods for proving the unprovability of the Paris-Harrington sentence in $\textsf{PA}$: model-theoretical proofs via indicators and model theory of non-standard models of $\textsf{PA}$, and proof-theoretic proofs via the classification of provably total recursive functions of $\textsf{PA}$. A model-theoretical proof may use arithmetization (e.g., using the definability of the satisfaction relation for $\Delta_0$ formulas or partial truth predicates  (Katz & Reimann, 2018). However, a proof-theoretic proof using the classification of provably total recursive functions of $\textsf{PA}$ could be arithmetization-free, i.e., without the use of arithmetization of syntax, even if it may use some first or second kind of coding  (Weiermann, 2006).

Regarding the role of coding, Isaacson states that “the relationship of coding constitutes a rigid link between the arithmetical and the higher-order truths” (Isaacson, 1996, p. 221). For Isaacson, the effect of coding is two-sided. On the one hand, some true sentences in the language of arithmetic may have hidden higher-order concepts via coding and are not arithmetical in his sense, such as Gödel sentences and the Paris-Harrington sentence. In these cases, coding “pulls the ostensibly arithmetical truth up into the higher-order” (Isaacson, 1996, p. 221). On the other hand, in a case such as $TI(\omega^{\omega})$, an arithmetical sentence coding transfinite induction of order type $\omega^{\omega}$, coding pulls the ostensibly higher-order truth into the arithmetical and allows for a proof of a statement in strictly number-theoretic terms (Isaacson, 1996, p. 221). Thus, a statement containing concepts that are seemingly higher-order may actually be arithmetical in Isaacson’s sense. In summary, “coding is simply a device that, in most cases, allows us to discern whether a seemingly higher-order truth is arithmetical after all, or whether a seemingly arithmetical truth is higher-order”  (Dopico, 2024, pp. 16–17).

3.4 Isaacson’s notion of “higher order concept”

In this section, we examine Isaacson’s notion of “higher order concept”. Isaacson’s distinctions between finitary concepts and higher order concepts are clarified further in Isaacson (1992). Examples of higher-order concepts in Isaacson (1996) include arbitrary subsets, infinitary notions in the sense of presupposing an infinite totality, the notion of well-ordering, the consistency of a formal system, provability in a formal system, and truth (Isaacson, 1996, pp. 210, 216; Horsten, 2001, p. 181).

The notion of “higher order concept” is crucial for understanding Isaacson’s thesis. He argues that “we can only arrive at our full understanding of the concept of natural number on the basis of some higher-order understanding” (Isaacson, 1996, p. 209). Isaacson distinguishes between seemingly higher-order and genuinely higher-order concepts. Sometimes, a statement may contain seemingly higher-order concepts that are not genuinely higher-order and may, in fact, be arithmetical in Isaacson’s sense. According to Isaacson (1996), containing genuinely higher-order concepts is a sufficient condition for being non-arithmetical in his sense. It appears that for Isaacson, containing genuinely higher-order concepts is also a necessary condition for being non-arithmetical: if a true sentence in the language of arithmetic is non-arithmetical in his sense, then there is no way to directly perceive its truth in purely arithmetical terms, and perceiving its truth must appeal to some genuinely higher-order concept.

If our perception of the truth of a sentence in the language of arithmetic does not appeal solely to purely arithmetical content, it must appeal to some genuinely higher-order concept. Conversely, if our perception of the truth of a sentence does not invoke any genuinely higher-order concept, it only appeals to the purely arithmetical content of the structure of natural numbers. According to Isaacson’s thesis, the boundary between purely arithmetical truths and higher order truths is precisely $\textsf{PA}$: if a sentence in the language of arithmetic is provable in $\textsf{PA}$, then it belongs to the collection of purely arithmetical truths in Isaacson’s sense; if a true sentence in the language of arithmetic is not provable in $\textsf{PA}$, then its truth cannot be directly perceived based on purely arithmetical content, and our perception of its truth must appeal to some genuinely higher-order concept.

We conclude our analysis of Isaacson’s thesis with several comments on its advantages. Firstly, Isaacson introduces a new notion of arithmetical truth, proposing a restrictive definition that contrasts with the views of many scholars who consider arithmetical truth to be indefinitely extensible. Isaacson’s notion of arithmetical truth is philosophically meaningful, even if it is vague in some respects. Secondly, Isaacson’s thesis offers a fresh perspective on understanding the incompleteness phenomenon and the nature of $\textsf{PA}$. According to Isaacson’s thesis, the incompleteness phenomenon revealed in Gödel’s incompleteness theorems and the research on concrete incompleteness is not fundamentally about arithmetical incompleteness. Isaacson argues that $\textsf{PA}$ encapsulates our intuitions about the purely arithmetical content of our understanding of the structure of natural numbers. According to Isaacson, PA is complete with respect to his notion of arithmetical truth, even though it is deductively incomplete. While this is a strong claim that may be challenging to defend, it represents a philosophically meaningful viewpoint.

4 What we should provide for defending Isaacson’s thesis

In this section, we examine what we should provide (or explain) to defend Isaacson’s thesis. To support Isaacson’s thesis, we should justify the following two claims:

• Claim A: If a sentence in the language of arithmetic is provable in $\textsf{PA}$, then it is arithmetical in Isaacson’s sense.

• Claim B: If a true sentence in the language of arithmetic is unprovable in $\textsf{PA}$, then it is not arithmetical in Isaacson’s sense.

Arguing for a sentence’s status as arithmetical or not in Isaacson’s sense is the primary way to justify Isaacson’s thesis.¹² If we want to argue that a sentence in the language of arithmetic is arithmetical in Isaacson’s sense, we need to show that its truth is directly perceivable based on our grasp of the structure of natural numbers or directly perceivable from truths in the language of arithmetic that are themselves arithmetical. Conversely, if we want to argue that a true sentence in the language of arithmetic is not arithmetical, we need to demonstrate that its truth is neither directly perceivable based on our grasp of the structure of natural numbers nor directly perceivable from truths in the language of arithmetic that are themselves arithmetical.

We will argue that justifying Claim A and Claim B according to Isaacson’s notion of arithmetical truth is not an easy task. In the following, we propose seven case examples that may constitute potential challenges to Isaacson’s thesis. To defend Isaacson’s thesis, we should respond to these examples. If proponents of Isaacson’s thesis cannot adequately explain these examples, they may pose real challenges to Isaacson’s thesis.

4.1 Example 1: transfinite induction principles

Isaacson (1996) regarded the transfinite induction principle of order type $\alpha$ for $\omega\leq\alpha < \epsilon_0$, denoted by $TI(\alpha)$, as a possible challenge to his thesis.¹³ These possible challenges are not addressed in Isaacson (1996). It is a classic result in proof theory that $TI(\epsilon_0)$ is not provable in $\textsf{PA}$, while for any $\alpha < \epsilon_0, TI(\alpha)$ is provable in $\textsf{PA}$: $\textsf{PA}$ can prove that any ordinal less than $\epsilon_0$ is well ordered. To defend Claim A, we should show that $TI(\alpha)$ for $\omega\leq\alpha < \epsilon_0$ is arithmetical in Isaacson’s senese. One main contribution of Dopico (2024) is to address this question.

Dopico (2024) defends Isaacson’s thesis and argues that $TI(\alpha)$ for $\alpha < \epsilon_0$ is arithmetical based on the following two points: (1) the $\textsf{PA}$-proof of transfinite induction for infinite ordinals below $\epsilon_0$ employs exclusively resources from finite mathematics; and (2) ordinals below $\epsilon_0$ are finitary objects despite being infinite. Dopico (2024) asserts that “the standard proof in $\textsf{PA}$ of transfinite induction claims $TI(\alpha)$ for $\alpha < \epsilon_0$ involves no higher-order notions and are based solely on arithmetical truths”  (Dopico, 2024, p. 14). He correctly points out that “the way $\textsf{PA}$ deals with ordered sets of order-type less than $\epsilon_0$ does not go beyond the strictly finite, and they are somehow tractable in a finitary way”  (Dopico, 2024, p. 14).

To express ordinals in the language of arithmetic, an ordinal notation is needed, i.e., a way to code ordinals less than $\epsilon_0$ by natural numbers. This can be done in various ways, and one standard way is via Cantor’s normal form theorem: any ordinal $0 < \alpha < \epsilon_0$ can be written in an unique way as

$$\alpha = {\omega ^{{\alpha _1}}} \cdot {k_1} + \cdots + {\omega ^{{\alpha _n}}} \cdot {k_n},$$

where $k_1, \cdots, k_n$ are positive integers and $\alpha > \alpha_1 > \alpha_2 > \cdots > \alpha_n$. Based on this observation, infinite ordinals $\omega \leq \alpha < \epsilon_0$ can be coded (using the second kind of coding) as finite objects and are not inherently higher-order.

We agree that ordinals below $\epsilon_0$ are essentially finitary objects and are not inherently higher order. Dopico (2024) has explained why $TI(\alpha)$ for $\alpha < \epsilon_0$ is arithmetical in Isaacson’s sense. However, to defend Isaacson’s thesis, Dopico should also explain why $TI(\epsilon_0)$ is not arithmetical in Isaacson’s sense. We make two comments here.

Firstly, not only ordinals less than $\epsilon_0$ but also any ordinal $\leq\Gamma_0$ can be coded by natural numbers, where $\Gamma_0$ is a countable ordinal much larger than $\epsilon_0$ and the smallest impredicative ordinal  (Pohlers, 2009).¹⁴ In particular, $\epsilon_0$ is an infinite countable ordinal that can be encoded in terms of finite objects of some structure over $\omega$, and hence is finitary. If we assume that an ordinal $\alpha$ being finitary is a sufficient condition for $TI(\alpha)$ being arithmetical in Isaacson’s sense, then for any ordinal $\alpha \leq \Gamma_0$, $TI(\alpha)$ is arithmetical in Isaacson’s sense, and especially, $TI(\epsilon_0)$ would also be arithmetical in Isaacson’s sense. However, since $TI(\epsilon_0)$ is not provable in $\textsf{PA}$, if Isaacson’s thesis holds, then $TI(\epsilon_0)$ cannot be arithmetical in Isaacson’s sense, leading to a contradiction. Thus, if Isaacson’s thesis holds, an ordinal $\alpha$ being finitary is not a sufficient condition for $TI(\alpha)$ being arithmetical in Isaacson’s sense.

Dopico (2024) did not assert that ordinals below $\epsilon_0$ being finitary is what allows for the existence of a proof in $\textsf{PA}$ that only employs resources from finite mathematics.¹⁵ Instead, Dopico (2024) stated: “Indeed, what facilitates the nested induction is the fact that ordinals up to $\epsilon_0$ are capable of being treated as finite objects, as is revealed by their specific Cantor normal form”  (Dopico, 2024, p. 17). For Dopico, the ability to express ordinals below $\epsilon_0$ through Cantor normal form is what allows for Gentzen jumps that only employ resources from finite mathematics  (Dopico, 2024, p. 15). We know that ordinals below $\Gamma_0$ can be coded by natural numbers via Veblen normal form.¹⁶ It remains to be explained why only ordinals that can be expressed via Cantor normal form allow for Gentzen’s jumps, why ordinals below $\Gamma_0$ coded via Veblen normal form do not permit a proof of $TI(\alpha)$ for $\alpha < \Gamma_0$ in $\textsf{PA}$, and why a proof of $TI(\alpha)$ for $\epsilon_0 \leq \alpha < \Gamma_0$ cannot solely employ resources from finite mathematics.

Secondly, another issue concerns the influence of the choice of an ordinal notation system on the provability of transfinite induction principles in $\textsf{PA}$. When we code an infinite countable ordinal by a natural number, we adopt some kind of ordinal notation system. It is a well-known open question what constitutes a “natural” ordinal notation system in proof theory. Under different ordinal notation systems, the coded sentence in the language of arithmetic may exhibit distinct properties.

There is a common agreement that the standard ordinal notation system for ordinals below $\epsilon_0$ constitutes a natural/good well-ordering of $\omega$, and under this ordinal notation system, $TI(\alpha)$ for $\alpha < \epsilon_0$ is indeed provable in $\textsf{PA}$. However, for some other ordinal notation systems, $TI(\alpha)$ for $\alpha < \epsilon_0$ may not be provable in $\textsf{PA}$. For example, Kreisel devised a definition of a primitive recursive well-ordering of order type $\omega$ such that $\textsf{PA}$ does not prove the transfinite induction principle of order type $\omega$  (Kreisel, 1968, pp. 333–334). We denote Kreisel’s example of the induction principle of order type $\omega$ by $KTI(\omega)$. To defend Isaacson’s thesis, we need to explain why, under Kreisel’s ordinal notation system, $KTI(\omega)$ is not arithmetical in Isaacson’s sense: more specifically, why under Kreisel’s definition, the truth of $KTI(\omega)$ cannot be directly perceived in purely arithmetical terms.

4.2 Example 2: Fermat’s last theorem

We know that Wiles proved Fermat’s Last Theorem. According to the classical view of arithmetical truth, Fermat’s Last Theorem is considered an arithmetical truth, a view that most mathematicians agree upon. However, according to Isaacson’s thesis, whether Fermat’s Last Theorem is an arithmetical truth depends on whether it is provable in $\textsf{PA}$. As far as we know, the question of whether Fermat’s Last Theorem is provable in $\textsf{PA}$ remains open. For more discussions about the prospect of proving Fermat’s Last Theorem in $\textsf{PA}$ or even in weak fragments of $\textsf{PA}$, we refer to McLarty (2010, 2024).

Wiles’ proof employs cohomology of sheaves on certain Grothendieck topologies, which requires the existence of an uncountable Grothendieck universe  (McLarty, 2010). This fact indicates that Wiles’ proof of Fermat’s Last Theorem utilizes some higher-order concepts in Isaacson’s sense, and thus his proof is not formalizable in $\textsf{PA}$. However, from this fact alone, we cannot conclude that perceiving the truth of Fermat’s Last Theorem necessitates going beyond the purely arithmetical content of the structure of natural numbers, as it is possible that a fundamentally new proof of Fermat’s Last Theorem could be found within $\textsf{PA}$ in the future. To defend Isaacson’s thesis based on Fermat’s Last Theorem, we must consider two cases as follows.

Suppose Fermat’s Last Theorem is provable in $\textsf{PA}$. In this case, according to Isaacson’s thesis, Fermat’s Last Theorem is an arithmetical truth in Isaacson’s sense. To defend Isaacson’s thesis, we need to justify that the truth of Fermat’s Last Theorem is directly perceivable based on our grasp of the standard structure of natural numbers or directly perceivable from truths in the language of arithmetic that are themselves arithmetical. Given the complexity of any proof of Fermat’s Last Theorem, we can imagine that finding a deduction of Fermat’s Last Theorem from the axioms of $\textsf{PA}$ is not an easy task.

Suppose Fermat’s Last Theorem is not provable in $\textsf{PA}$. In this scenario, according to Isaacson’s thesis, Fermat’s Last Theorem is not arithmetical in Isaacson’s sense. To defend Isaacson’s thesis, we need to argue that the truth of Fermat’s Last Theorem is neither directly perceivable based on our grasp of the standard structure of natural numbers nor directly perceivable from truths in the language of arithmetic that are themselves arithmetical. This, too, is a challenging task.

It is possible that we may never know whether Fermat’s Last Theorem is provable in $\textsf{PA}$. If that is the case, it would mean we would never know whether Fermat’s Last Theorem is an arithmetical truth in Isaacson’s sense, assuming Isaacson’s thesis holds.

In summary, even if most mathematicians agree that Fermat’s Last Theorem is an arithmetical truth, it remains an open question for logicians whether it is provable in $\textsf{PA}$, and for some philosophers of mathematics whether it qualifies as an arithmetical truth in Isaacson’s sense. Fermat’s Last Theorem is either provable in $\textsf{PA}$ or unprovable in $\textsf{PA}$. In either case, to defend Isaacson’s thesis, there is something we must explain.

4.3 Example 3: consistency statements

It is a philosophically open question what counts as a provability predicate and what counts as a consistency statement. In this paper, we assume that $\textsf{Con(PA)} =\neg \textsf{Pr}(\ulcorner 0=1\urcorner)$ is the standard consistency statement formulated based on Gödel coding, along with the standard provability predicate $\textsf{Pr}(x)$, as in Murawski (1999). According to Isaacson’s thesis, by Gödel’s second incompleteness theorem, $\textsf{Con(PA)}$ is not arithmetical in Isaacson’s sense. There are various ways of expressing consistency. The provability of consistency statements of $\textsf{PA}$ is related to the intensionality of these statements, which refers to the phenomenon that whether a consistency statement is provable in $\textsf{PA}$ depends on various factors.

Firstly, whether a consistency statement is provable in $\textsf{PA}$ depends on the base theory to which this consistency statement refers. We know that $\textsf{PA}$ is reflective in the sense that it proves the consistency of any of its finite sub-theories. To justify Claim A, we need to show that for any finite sub-theory $T$ of $\textsf{PA}$, $\textsf{Con}(T)$ is arithmetical in Isaacson’s sense. Dopico (2024) claims that the explanation why $TI(\alpha)$ for $\omega\leq\alpha < \epsilon_0$ is arithmetical in Isaacson’s sense also explains why $\textsf{Con(T)}$ is arithmetical in Isaacson’s sense for a sub-theory $T$ of $\textsf{PA}$. The key argument from Dopico (2024) is: “the form $\textsf{Con(T)}$, where $T$ is a theory of arithmetic weaker than $\textsf{PA}$, can be proven equivalent to a transfinite induction claim up to a certain ordinal below $\epsilon_0$, over a subsystem of $\textsf{PA}$ proof-theoretically weaker than $T$ itself”  (Dopico, 2024, p. 13). However, we did not find that this claim has been proven in the literature, and we do not believe it is correct. Indeed in ordinal analysis, the proof-theoretic ordinals of many sub-theories of $\textsf{PA}$ are known. Assuming that the proof-theoretic ordinal of a sub-theory $T$ of $\textsf{PA}$ is $\alpha$, this only means that $\textsf{Con(T)}$ is provable from $\textsf{PRA}+ TI(\alpha)$; generally, this does not imply that $\textsf{Con(T)}$ is equivalent with $TI(\alpha)$. For example, $\textsf{Con(PA)}$ is not equivalent to $TI(\alpha)$ for any ordinal $\alpha$, since $TI(\alpha)$ for $\alpha < \epsilon_0$ is provable in $\textsf{PA}$, while $\textsf{Con(PA)}$ is not provable in $\textsf{PA}$, and $TI(\epsilon_0)$ is equivalent to the $1$-consistency of $\textsf{PA}$, which is strictly stronger than $\textsf{Con(PA)}$.¹⁷

Secondly, whether a consistency statement is provable in $\textsf{PA}$ depends on the provability predicate we use. The standard consistency statement is formulated via a standard provability predicate. If a consistency statement is formulated using other provability predicates (e.g., non-standard provability predicates), it may be provable in $\textsf{PA}$. For example, let $\textsf{Pr}^{\textsf{R}}\textsf{(x)}$ be a Rosser provability predicate and $\textsf{Con}^{\textsf{R}}\textsf{(PA)}=\neg \textsf{Pr}^{\textsf{R}}\textsf{(0=1)}$ be the corresponding consistency statement based on the Rosser provability predicate $\textsf{Pr}^{\textsf{R}}\textsf{(x)}$.¹⁸ It is a well known fact that $\textsf{Con}^{\textsf{R}}\textsf{(PA)}$ is provable in $\textsf{PA}$ (Murawski, 1999). To defend Isaacson’s thesis, we need to explain why $\textsf{Con}^{\textsf{R}}\textsf{(PA)}$ is arithmetical in Isaacson’s sense.

Thirdly, whether a consistency statement is provable in $\textsf{PA}$ depends on the method of arithmetization we use. Under some unusual method of arithmetization, the corresponding consistency statement may be provable in $\textsf{PA}$  (Grabmayr, 2021). In this case, to defend Isaacson’s thesis, we need to explain why, under this unusual method of arithmetization, the corresponding consistency statement is arithmetical in Isaacson’s sense.

Fourthly, whether a consistency statement is provable in $\textsf{PA}$ depends on how we express the notion of consistency. Artemov (2024) argues that in Hilbert’s consistency program, the original formulation of consistency is “no sequence of formulas is a derivation of a contradiction”, which pertains to finite sequences of formulas, not to arithmetization, proof codes, and internalized quantifiers. The standard consistency statement $\textsf{Con(PA)}$ says that for any $x$, $x$ is not a code of a proof of a contradiction in $\textsf{PA}$. In a nonstandard model of $\textsf{PA}$, the universal quantifier “for all $x$” ranges over both standard and nonstandard numbers, and hence $\textsf{Con(PA)}$ expresses the consistency of both standard and nonstandard proofs  (Artemov, 2024). Thus, $\textsf{Con(PA)}$ is stronger than the original formulation of consistency, which only addresses sequences of formulas with standard codes. We can view the original formulation of consistency as a local notion and the standard consistency statement $\textsf{Con(PA)}$ as a global notion.

Artemov (2024) argued that Gödel’s second incompleteness theorem does not actually exclude finitary consistency proofs of the original formulation of consistency. He shows that the original formulation of consistency admits a direct proof that is formalizable in $\textsf{PA}$  (Artemov, 2024). This fact does not contradict Gödel’s second incompleteness theorem since Gödel’s theorem only shows that the global notion of consistency $\textsf{Con(PA)}$ is not provable in $\textsf{PA}$, while Artemov shows that the local notion of consistency is provable in $\textsf{PA}$. To defend Isaacson’s thesis, we need to explain why this local notion of consistency is arithmetical in Isaacson’s sense, while the global notion of consistency is not.

Fifthly, whether a consistency statement is provable in $\textsf{PA}$ depends on how we numerate the axioms of $\textsf{PA}$. We say a formula $\phi(x)$ with one free variable is a numeration of a set $A\subseteq \omega$ in $\textsf{PA}$ if for any $n\in \omega, n\in A$ iff $\textsf{PA}\vdash \phi(\overline{n})$. We say $\phi(x)$ is a $\Sigma^0_1$ ($\Pi^0_1$) numeration if $\phi(x)$ is $\Sigma^0_1$ ($\Pi^0_1$). Feferman (1960) shows that under any $\Sigma^0_1$ numeration of the axioms of $\textsf{PA}$, the corresponding consistency statement formulated via a standard way of coding and a standard provability predicate is not provable in $\textsf{PA}$; however, for some $\Pi^0_1$ numeration of the axioms of $\textsf{PA}$, the corresponding consistency statement is provable in $\textsf{PA}$. To defend Isaacson’s thesis, we need to explain why, under some $\Pi^0_1$ numerations of the axioms of $\textsf{PA}$, the corresponding consistency statement is arithmetical in Isaacson’s sense; while under any $\Sigma^0_1$ numeration of the axioms of $\textsf{PA}$, the corresponding consistency statement is not arithmetical in Isaacson’s sense.

4.4 Example 4: provable meta-mathematical sentences including non-arithmetical sub-sentences

In this paper, we define a sentence as meta-mathematical if its formulation employs arithmetization. Isaacson (1996) claims that some meta-mathematical sentences (such as the Gödel sentence for $\textsf{PA}$, the consistency statement $\textsf{Con(PA)}$, and the $\Sigma^0_1$-reflection principle of $\textsf{PA}$) are not arithmetical in Isaacson’s sense. We always assume that $\textsf{Pr}(x)$ is a standard provability predicate. Since the formulation of the standard provability predicate $\textsf{Pr}(x)$ uses arithmetization, any sentence that includes the standard provability predicate $\textsf{Pr}(x)$ is a meta-mathematical sentence.

Let us consider meta-mathematical sentences provable in $\textsf{PA}$, which include some meta-mathematical sub-sentence that is not arithmetical in Isaacson’s sense. Here, we exclude those meta-mathematical sentences that are logically entailed by a provable sentence in $\textsf{PA}$ without including a sub-sentence that is not arithmetical in Isaacson’s sense. For example, we exclude $0=0 \vee \textsf{Con(PA)}$, which is logically entailed by $0=0$.¹⁹ To defend Isaacson’s thesis, we need to show meta-mathematical sentences provable in $\textsf{PA}$ with the above properties are arithmetical in Isaacson’s sense. There are many such meta-mathematical sentences, and we will provide some typical examples here.

Firstly, if $A$ is a $\Sigma^0_1$ sentence in the language of arithmetic, then $A\rightarrow \textsf{Pr}(\ulcorner A\urcorner)$ is provable in $\textsf{PA}$. To justify that Claim A holds, we need to show that $A\rightarrow \textsf{Pr}(\ulcorner A\urcorner)$ is arithmetical in Isaacson’s sense for any $\Sigma^0_1$ sentence $A$. In particular, we need to show that $\neg \textsf{Con(PA)}\rightarrow \textsf{Pr}(\ulcorner \neg \textsf{Con(PA)}\urcorner)$ is arithmetical in Isaacson’s sense, even though $\textsf{Con(PA)}$ is not arithmetical in Isaacson’s sense.

Secondly, the formalized version of the first incompleteness theorem is provable in $\textsf{PA}$: $\textsf{PA}\vdash (\textsf{Con(PA)} \rightarrow \neg \textsf{Pr}(\ulcorner \textsf{G}\urcorner)) \wedge (1$-$\textsf{Con(PA)}\rightarrow \neg \textsf{Pr}(\ulcorner\neg \textsf{G}\urcorner))$, where $\textsf{G}$ is a Gödel sentence and $1$-$\textsf{Con(PA)}$ is the formalization of the notion of the $1$-consistency of $\textsf{PA}$. To justify that Claim A holds, we need to show that $\textsf{Con(PA)} \rightarrow \neg \textsf{Pr}(\ulcorner \textsf{G}\urcorner)$ and $1$-$\textsf{Con(PA)}\rightarrow \neg \textsf{Pr}(\ulcorner\neg \textsf{G}\urcorner)$ are arithmetical in Isaacson’s sense, even though $\textsf{Con(PA)}$ and $1$-$\textsf{Con(PA)}$ are not arithmetical in Isaacson’s sense.

Thirdly, the formalized version of the second incompleteness theorem is provable in $\textsf{PA}$: $\textsf{PA}\vdash \textsf{Con(PA)}\rightarrow \neg \textsf{Pr}(\ulcorner \textsf{Con(PA)}\urcorner)$. To justify that Claim A holds, we need to show that $\textsf{Con(PA)}\rightarrow \neg \textsf{Pr}(\ulcorner \textsf{Con(PA)}\urcorner)$ is arithmetical in Isaacson’s sense, even though $\textsf{Con(PA)}$ is not arithmetical in Isaacson’s sense.

Since there are many such examples of meta-mathematical sentences provable in $\textsf{PA}$ that include some sub-sentence which is non-arithmetical in Isaacson’s sense, justifying that they are, in fact, arithmetical in Isaacson’s sense may be an infinite task. However, it may be possible to formulate a general argument.

4.5 Example 5: phase transitions

Research on concrete incompleteness reveals the phenomenon of phase transitions from provability to unprovability of a statement in $\textsf{PA}$ by varying a threshold parameter. Weiermann (2007) wrote: “Phase transition is a type of behavior wherein small changes of a parameter of a system cause dramatic shifts in some globally observed behavior of the system, such shifts being usually marked by a sharp ‘threshold point’”.

The idea of phase transition is roughly as follows. Let $A_f$ be a statement in the language of arithmetic with a number-theoretic function parameter $f$. Assuming that the following conditions hold: (1) for any $f$, $A_f$ is always true; (2) if $f$ is very slow growing, then $\textsf{PA}\vdash A_f$; if $f$ grows reasonably fast, then $\textsf{PA}\nvdash A_f$; (3) if $\textsf{PA}\vdash A_f$ and $g$ is eventually dominated by $f$, then $\textsf{PA}\vdash A_g$; then there will be a phase transition threshold point. Determining the threshold point of phase transition provides valuable information about what makes a true statement unprovable in $\textsf{ PA}$  (Weiermann, 2009).

Weiermann’s research program on classifying phase transitions for independence results of $\textsf{PA}$ is deep and foundational. In the literature, phase transitions of concrete incompleteness of $\textsf{PA}$ have been studied case by case for different examples of concrete incompleteness of $\textsf{PA}$. For distinct examples of concrete incompleteness, behaviors of phase transitions are quite different. For more results on phase transitions of independence results, see Weiermann (2009).

We take one classic example of phase transition here. Given $f: \omega\rightarrow \omega$, let $\textsf{FKT}_f$ denote a finite version of Kruskal’s Theorem via a parameter $f$ which is expressible in $\textsf{PA}$ via coding.²⁰ For any primitive recursive real number $r$, $f_r$ is a number theoretic function and $\textsf{FKT}_{f_r}$ is a sentence in the language of arithmetic  (Weiermann, 2003). All these sentences $\textsf{FKT}_{f_r}$ for varied primitive recursive real numbers $r$ are true in the standard model of arithmetic  (Weiermann, 2003). Weiermann (2003) defines a primitive recursive constant real number $c$ and shows that: for any primitive recursive real number $r$, if $r \leq c$, then $\textsf{PA}\vdash \textsf{FKT}_{f_r}$; and if $r > c$, then $\textsf{PA}\nvdash \textsf{FKT}_{f_r}$.

The phenomenon of phase transitions from concrete incompleteness characterizes the boundary between provability and unprovability of $\textsf{PA}$ in a precise way. The research on phase transitions provides us with a new perspective for examining the limits of incompleteness (or the limits of provability) of $\textsf{PA}$.

Assuming that Isaacson’s thesis holds, the threshold points of provability in $\textsf{PA}$ correspond to the threshold points of arithmetical truth in Isaacson’s sense. For example, the primitive recursive real number $c$ is a threshold point of arithmetical truth for the sentences $\textsf{FKT}_{f_r}$ with a primitive recursive parameter $r$: if $r \leq c$, then $\textsf{FKT}_{f_r}$ is arithmetical in Isaacson’s sense, and if $r > c$, then $\textsf{FKT}_{f_r}$ is not arithmetical in Isaacson’s sense. To defend Isaacson’s thesis, we should explain why $\textsf{FKT}_{f_r}$ is arithmetical for $r \leq c$, but not arithmetical for $r > c$ according to Isaacson’s notion of arithmetical truth. Specifically, we need to explain why the truth of $\textsf{FKT}_{f_r}$ is directly perceivable in purely arithmetical terms for any primitive recursive real number $r \leq c$, and why the truth of $\textsf{FKT}_{f_r}$ is not directly perceivable in purely arithmetical terms for any primitive recursive real number $r > c$.

For distinct examples of concrete incompleteness, the corresponding phase transition threshold points are different. To defend Isaacson’s thesis, we need to explain how these threshold points of provability of $\textsf{PA}$ could be the threshold points of arithmetical truth in Isaacson’s sense. Isaacson did not discuss possible threshold points of arithmetical truth in his sense. It seems challenging to explain the threshold points of arithmetical truth using Isaacson’s epistemic notion of arithmetical truth. Thus, the phenomenon of phase transitions may pose a significant challenge to Isaacson’s thesis.

4.6 Example 6: undecidable sentences via Diophantine equations

At the Königsberg meeting Gödel attended in 1930, von Neumann asked Gödel whether number-theoretical undecidable propositions could also be constructed. Gödel replied, “Of course, undecidable propositions about integers could be so constructed, but they would contain concepts quite different from those occurring in number theory like addition and multiplication”  (Wang, 1981). However, “shortly afterward Gödel, to his own astonishment, succeeded in turning the undecidable proposition into a polynomial form preceded by quantifiers (over natural numbers)”  (Wang, 1981).

A Diophantine equation is one of the form $f(\overrightarrow{x})=0$, where $f$ is a polynomial with integer coefficients (this can be rewritten as $P_1(\overrightarrow{x})=P_2(\overrightarrow{x})$, where $P_i$ are polynomials over $\omega\setminus \{0\}$). A set $X\subseteq \omega$ is Diophantine if and only if there exists a polynomial $p(x, y_1, \cdots, y_n)$ with integer coefficients such that $x \in X$ iff there are $y_i\in \omega (1\leq i\leq n)$ such that $p(x, y_1, \cdots, y_n) = 0$. The $\textsf{MRDP}$ theorem states that for any $X\subseteq \omega$, $X$ is recursively enumerable if and only if $X$ is Diophantine  (Davis, 1973). The proof of the $\textsf{MRDP}$ theorem is completely constructive in the sense that for any $n$-ary recursive enumerable relation $P$ with an index $e$,²¹ we can effectively find a polynomial $f$ and a number $m$ such that $P(\overrightarrow{x})$ iff $\exists y_1 \cdots\exists y_m [f(\overrightarrow{x},y_1 \cdots, y_m)=0]$.

The following form of the first incompleteness theorem due to Gödel follows from the $\textsf{MRDP}$ theorem: we can explicitly construct a Diophantine equation $t_1(\overrightarrow{x})=t_2(\overrightarrow{x})$ which has no solution in natural numbers, but $\textsf{PA}\nvdash \forall \overrightarrow{x} (t_1(\overrightarrow{x}) \neq t_2(\overrightarrow{x}))$. Since $t_1(\overrightarrow{x})=t_2(\overrightarrow{x})$ is a Diophantine equation, it is reasonable to view $\forall \overrightarrow{x} (t_1(\overrightarrow{x}) \neq t_2(\overrightarrow{x}))$ as an arithmetical truth, as it is true in the standard model of arithmetic and directly discusses properties of natural numbers. To defend Isaacson’s thesis, we should explain why $\forall \overrightarrow{x} (t_1(\overrightarrow{x}) \neq t_2(\overrightarrow{x}))$ is not arithmetical in Isaacson’s sense.

We say that a formula is Diophantine if it consists of a block of existential quantifiers followed by a Diophantine equation. From the $\textsf{MRDP}$ theorem, we know that for any $\Sigma^0_1$ formula $\psi(x)$ in the language of arithmetic, we can effectively find a Diophantine formula $\delta(x)$ such that $\textsf{PA} \vdash \forall x (\psi(x) \leftrightarrow \delta(x))$. For the $\Pi^0_1$ sentence $\textsf{Con}(\textsf{PA})$, we can effectively find a polynomial $f$ and a number $m$ such that $\textsf{PA}\vdash \textsf{Con(PA)} \leftrightarrow \neg \exists y_1 \cdots\exists y_m (f(y_1 \cdots, y_m)=0)$.

Let us consider the following argument showing that Isaacson’s thesis is false: (1) since $f(y_1 \cdots, y_m)=0$ is a Diophantine equation talking about basic operations of natural numbers, $\neg \exists y_1 \cdots\exists y_m (f(y_1 \cdots, y_m)=0))$ is arithmetical; (2) since $\textsf{Con(PA)}$ is equivalent with an arithmetical sentence, $\textsf{Con(PA)}$ is arithmetical. According to Isaacson’s thesis, $\textsf{Con(PA)}$ is not arithmetical, which leads to a contradiction. Thus, Isaacson’s thesis is false.

There are two possible objections to this argument. The first objection concerns step (1): even if a Diophantine equation discusses basic operations of natural numbers, to perceive the truth of $\neg \exists {y_1} \cdots \exists {y_m}(f({y_1} \cdots {y_m}) = 0)$, we cannot rely solely on the purely arithmetical content of the structure of natural numbers; some implicitly higher-order concepts are needed. To address this objection and defend Isaacson’s thesis, one should explain why, for such a simple statement discussing basic operations (addition and multiplication) of natural numbers, its truth is neither directly perceivable based on our grasp of the structure of natural numbers nor directly perceivable from truths in the language of arithmetic that are themselves arithmetical.

The second objection concerns step (2): some may argue that even if $\textsf{Con(PA)}$ is equivalent to an arithmetical sentence in Isaacson’s sense, we cannot conclude that $\textsf{Con(PA)}$ is also arithmetical in Isaacson’s sense, since two equivalent sentences may not share the same properties. However, Isaacson (1996) uses a similar argument to assert that the Paris-Harrington sentence is not arithmetical in Isaacson’s sense: since the Paris-Harrington sentence is equivalent to the $\Sigma^0_1$ reflection principle of $\textsf{PA}$, which is not arithmetical in Isaacson’s sense, the Paris-Harrington sentence is also not arithmetical in Isaacson’s sense. If Isaacson’s argument holds, we can reasonably argue that since $\textsf{Con(PA)}$ is equivalent to an arithmetical sentence in Isaacson’s sense, $\textsf{Con(PA)}$ is also arithmetical in Isaacson’s sense.

4.7 Example 7: Independent sentences of PA from concrete incompleteness

Isaacson argued that all examples of independent sentences of $\textsf{PA}$ from concrete incompleteness discussed in Isaacson (1996) are not arithmetical in Isaacson’s sense. The two main reasons from Isaacson (1996) are: (1) these sentences are provably equivalent over $\textsf{PA}$ to the $\Sigma^0_1$ reflection principle of $\textsf{PA}$, which is not arithmetical in Isaacson’s sense; (2) perceiving the truth of these sentences appeals to hidden higher-order concepts via the technique of coding.

For (1), Isaacson argued that the $\Sigma^0_1$ reflection principle of $\textsf{PA}$ is not about properties of natural numbers but about the reflecting property of the base system, and thus it is not arithmetical in Isaacson’s sense (Isaacson, 1996, pp. 218–219). Even if the $\Sigma^0_1$ reflection principle of $\textsf{PA}$ is not arithmetical in Isaacson’s sense, the following argument is not convincing for us: since the Paris-Harrington sentence is provably equivalent over $\textsf{PA}$ to the $\Sigma^0_1$ reflection principle of $\textsf{PA}$, the Paris-Harrington sentence is also not arithmetical in Isaacson’s sense. This argument relies on the assumption that if two independent sentences $A$ and $B$ in the language of arithmetic are provably equivalent in $\textsf{PA}$, then for any property $P$, $A$ has property $P$ iff $B$ has property $P$. This reasoning is unconvincing to us since two equivalent sentences may have different properties. We agree that for definable properties in the language of arithmetic, equivalent sentences have the same given definable property: more formally, if $P$ is a property defined by a formula $\phi$ and $A$ and $B$ are provably equivalent in $\textsf{PA}$, then if $\phi(A)$ holds in $\textsf{PA}$, then $\phi(B)$ holds in $\textsf{PA}$, and vice versa.

However, Isaacson failed to formulate his notion of arithmetical truth as a definable property in the language of arithmetic. For undefinable properties in the language of arithmetic, two equivalent sentences may have different properties. Even if the Paris-Harrington sentence is equivalent over $\textsf{PA}$ to the $\Sigma^0_1$ reflection principle of $\textsf{PA}$, we cannot assert that these two sentences have no differences in any aspect. At least literally, the formulation of the Paris-Harrington sentence does not use arithmetization, while the formulation of the $\Sigma^0_1$ reflection principle of $\textsf{PA}$ does.

For (2), we have explained that the use of coding is not a sufficient condition for a sentence to be non-arithmetical in Isaacson’s sense. The effect of coding depends on the types of coding we use and how coding is applied. The formulation of the $\Sigma^0_1$ reflection principle of $\textsf{PA}$ uses arithmetization, the third kind of coding. However, the formulation of the Paris-Harrington sentence does not use arithmetization, even if expressing the finite version of the Ramsey theorem in $\textsf{PA}$ requires the first or second kind of coding. We have distinguished coding in the formulation of a statement from coding in the proof of a statement in Sect. 3.3. Some model theoretical proofs of the Paris-Harrington theorem may use arithmetization (e.g., see Katz and Reimann (2018)). However, a proof-theoretic proof of the Paris-Harrington theorem using the classification of provably total recursive functions of $\textsf{PA}$ could be arithmetization-free, even if it may use some first or second kind of coding  (Weiermann, 2006).

All independent sentences of $\textsf{PA}$ from concrete incompleteness discussed in this paper are $\Pi^0_2$ sentences in the form $\forall x\exists y A(x,y)$, where $A(x,y)$ is a bounded formula.²² All these $\Pi^0_2$ sentences are true in the standard model of arithmetic. Given such a $\Pi^0_2$ sentence $\forall x\exists y A(x,y)$, for any $x\in\omega$, there exists a unique least $y\in\omega$ such that $A(x,y)$ holds in the standard model of arithmetic. Naturally, we can define a total recursive function $f$ such that $f(m)$ is the least $n$ such that $\mathbb{N}\models A(m,n)$. However, all these $\Pi^0_2$ sentences $\forall x\exists y A(x,y)$ are not provable in $\textsf{PA}$; that is, the induced function $f$ is not provably total in $\textsf{PA}$. Even if for any $x\in\omega$, there exists $y\in\omega$ such that $y=f(x)$ holds in the standard model of arithmetic, such a $y$ must be so large that there is no way of proving in $\textsf{PA}$ that it exists. The function $f$ grows too fast for us to show that it is provably total in $\textsf{PA}$. However, for any such a $\Pi^0_2$ sentence $\forall x\exists y A(x,y)$, even if it is not provable in $\textsf{PA}$, each of its instances is provable in $\textsf{PA}$; that is, for any $n\in\omega$, $\exists y A(\overline{n},y)$ is provable in $\textsf{PA}$.²³

If Isaacson’s thesis holds, then for any such a $\Pi^0_2$ sentence $\forall x\exists y A(x,y)$ from concrete incompleteness, we have: (1) $\forall x\exists y A(x,y)$ is not arithmetical in Isaacson’s sense; (2) for any $n\in\omega$, $\exists y A(\overline{n},y)$ is arithmetical in Isaacson’s sense. Note that (1) and (2) directly follow from Isaacson’s thesis. To defend Isaacson’s thesis, one should explain (1) and (2) according to Isaacson’s notion of arithmetical truth: why we can directly perceive the truth of $\exists y A(\overline{n},y)$ for each $n\in\omega$ based solely on the purely arithmetical content of the structure of natural numbers, but cannot directly perceive the truth of $\forall x \exists y A(x,y)$ based solely on the purely arithmetical content of the structure of natural numbers.

5 Conclusion

In this paper, we first analyze Isaacson’s notion of arithmetical truth and his thesis, and then discuss the advantages and disadvantages of Isaacson’s thesis. Isaacson’s thesis is a philosophical statement that has not yet been sufficiently justified; in this sense, it remains a conjecture. Justifying Claim A and Claim B in Sect. 4 is a substantial undertaking and not easy for us.

Isaacson (1996) did not provide sufficient arguments to defend his thesis. An interesting phenomenon in Isaacson (1996) is that, despite proposing an informal definition of arithmetical truth, he rarely directly argued for his conclusions based on this definition when asserting that some true-but-unprovable sentences of $\textsf{PA}$ are not arithmetical in Isaacson’s sense. Instead, he often resorted to other criteria to justify his conclusions, such as discussing the properties of formal systems rather than the properties of natural numbers, the use of coding, and the presence of hidden higher-order concepts.

Due to the vagueness in Isaacson’s notion of arithmetical truth, it is challenging to determine whether a sentence is arithmetical or non-arithmetical based on his definition. For us, the first step toward clarifying the claim that “there is no way truths in the language of arithmetic which are unprovable in $\textsf{PA}$ can be directly perceived in purely arithmetical terms” is to clarify the notions of “direct perceivability” and “purely arithmetical terms”.

In Sects. 4.1 to 4.7, we propose seven case examples that may pose potential challenges to Isaacson’s thesis and discuss the issues that need to be addressed in order to defend Isaacson’s thesis. Advocates of Isaacson’s thesis should carefully consider these case examples when formulating their arguments. If they are unable to adequately respond to the challenges outlined in Sects. 4.1 to 4.7, then these examples may indeed represent significant challenges to Isaacson’s thesis. Therefore, to effectively defend Isaacson’s thesis, it is essential to address all of these case examples, which will require considerable effort.

Although there are numerous potential challenges to Isaacson’s thesis and it has not yet been thoroughly justified, we cannot conclude that Isaacson’s thesis is entirely wrong. The conclusion we can reasonably draw is that either Isaacson’s notion of arithmetical truth is not right, or that defending Isaacson’s thesis is inherently difficult.