aa r X i v : . [ m a t h . L O ] S e p Contents
1. Introduction 22. Preliminaries 42.1. Definitions and notations 42.2. Logical systems 73. Proofs of G¨odel’s incompleteness theorems 113.1. Introduction 113.2. Overview and modern formulation 123.2.1. Three steps towards G1 and G2 G1 G2 G1 and G2 from mathematical logic 183.3.1. Rosser’s proof 183.3.2. Recursion-theoretic proofs 193.3.3. Proofs based on Arithmetic Completeness 193.3.4. Proofs based on Kolmogorov complexity 203.3.5. Model-theoretic proofs 213.4. Proofs of G1 and G2 based on logical paradox 223.4.1. Introduction 223.4.2. Berry’s paradox 223.4.3. Unexpected Examination and Grelling-Nelson’s Paradox. 243.4.4. Yablo’s paradox 253.4.5. Beyond arithmetization 263.5. Concrete incompleteness 263.5.1. Introduction 263.5.2. Paris-Harrington and beyond 273.5.3. Harvey Friedman’s contributions 284. The limit of the applicability of G1 G1 beyond PA G1 below PA G1 via interpretability 314.3.2. The limit of G1 w.r.t. interpretation and Turing reducibility 335. The limit of the applicability of G2 G2 G2 References 47
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESSTHEOREMS
YONG CHENG
Abstract.
We give a survey of current research on G¨odel’s incom-pleteness theorems from the following three aspects: classifications ofdifferent proofs of G¨odel’s incompleteness theorems, the limit of the ap-plicability of G¨odel’s first incompleteness theorem, and the limit of theapplicability of G¨odel’s second incompleteness theorem. Introduction
G¨odel’s first and second incompleteness theorem are some of the mostimportant and profound results in the foundations of mathematics and havehad wide influence on the development of logic, philosophy, mathematics,computer science as well as other fields. Intuitively speaking, G¨odel’s in-completeness theorems express that any rich enough logical system cannotprove its own consistency , i.e. that no contradiction like 0 = 1 can be derivedwithin this system.G¨odel [46] proves his first incompleteness theorem ( G1 ) for a certain for-mal system P related to Russell-Whitehead’s Principia Mathematica basedon the simple theory of types over the natural number series and the Dedekind-Peano axioms (see [8], p.3). G¨odel announces the second incompleteness the-orem ( G2 ) in an abstract published in October 1930: no consistency proofof systems such as Principia, Zermelo-Fraenkel set theory, or the systemsinvestigated by Ackermann and von Neumann is possible by methods whichcan be formulated in these systems (see [153], p.431).G¨odel comments in a footnote of [46] that G2 is corollary of G1 (and infact a formalized version of G1 ): if T is consistent, then the consistencyof T is not provable in T where the consistency of T is formulated as thearithmetic formula which says that there exists an unprovable sentence in T . G¨odel [46] sketches a proof of G2 and promises to provide full details in Mathematics Subject Classification.
Key words and phrases.
G¨odel’s first incompleteness theorem, G¨odel’s second incom-pleteness theorem, Concrete incompleteness, Interpretation, Intensionability.First of all, I thank the lamp for my feet, and the light on my path. Some materialsof my old paper “Note on some misinterpretations of G¨odel’s incompleteness theorems”have been incorporated into this paper. I would like to thank Matthias Baaz, UlrichKohlenbach, Taishi Kurahashi, Zachiri McKenzie, Fedor Pakhomov, Michael Rathjen,Saeed Salehi, Sam Sanders and Albert Visser for their valuable comments on this work.Especially, I would like to thank Sam Sanders and Zachiri McKenzie for their proofreadingof my English writing of this paper. This paper is the research result of the Humanitiesand Social Sciences of Ministry of Education Planning Fund project “Research on G¨odel’sincompleteness theorem” (project no: 17YJA72040001). I would like to thank the fundsupport by the Humanities and Social Sciences of Ministry of Education Planning Fund. a subsequent publication. This promise is not fulfilled, and a detailed proofof G2 for first-order arithmetic only appears in a monograph by Hilbert andBernays [62]. Abstract logic-free formulations of G¨odel’s incompletenesstheorems have been given by Kleene [80] (“symmetric form”), Smullyan[124] (“representation systems”), and others. The following is a modernreformulation of G¨odel’s incompleteness theorems. Theorem 1.1 (G¨odel, [46]) . Let T be a recursively axiomatized extensionof PA . G1 If T is ω -consistent, then T is incomplete. G2 If T is consistent, then the consistency of T is not provable in T . G¨odel’s incompleteness theorems G1 and G2 are of a rather different na-ture and scope. In this paper, we will discuss different versions of G1 and G2 , from incompleteness for extensions of PA to incompleteness for systemsweaker than PA w.r.t. interpretation. We will freely use G1 and G2 to referto both G¨odel’s first and second incompleteness theorems, and their differ-ent versions. The meaning of G1 and G2 will be clear from the context inwhich we refer to them.G¨odel’s incompleteness theorems exhibit certain weaknesses and limita-tions of a given formal system. For G¨odel, his incompleteness theorems in-dicate the creative power of human reason. In Emil Post’s celebrated words:mathematical proof is an essentially creative activity (see [102], p.339). Theimpact of G¨odel’s incompleteness theorems is not confined to the communityof mathematicians and logicians; popular accounts are well-known withinthe general scientific community and beyond. G¨odel’s incompleteness theo-rems raise a number of philosophical questions concerning the nature of logicand mathematics as well as mind and machine. For the impact of G¨odel’sincompleteness theorems, Feferman said:their relevance to mathematical logic (and its offspring in thetheory of computation) is paramount; further, their philo-sophical relevance is significant, but in just what way is farfrom settled; and finally, their mathematical relevance out-side of logic is very much unsubstantiated but is the objectof ongoing, tantalizing efforts (see [35], p.434).In the literature, there are some good textbooks, and survey papers onG¨odel’s incompleteness theorems. For textbooks, we refer to [30, 102, 95,38, 121, 15, 123, 124, 55, 41]. For survey papers, we refer to [122, 8, 83, 17,138, 13, 24]. In the last twenty years, there have been a lot of advances inthe study of incompleteness. We felt that a comprehensive survey paper forthe current state-of-art of this research field is missing from the literature.The motivation of this paper is four-fold:(i) Give the reader an overview of the current state-of-art of research onincompleteness;(ii) Classify these new advances on incompleteness under some importantthemes;(iii) Propose some new questions not covered in the literature;(iv) Set the direction for the future research of incompleteness. YONG CHENG
Due to space limitations and our personal taste, it is impossible to coverall research results in the literature related to incompleteness in this survey.Therefore, we will focus on three aspects of new advances in research onincompleteness:(I) classifications of different proofs of G¨odel’s incompleteness theorems;(II) the limit of the applicability of G1 ;(III) the limit of the applicability of G2 .We think these are the most important three aspects of research on incom-pleteness and reflect the depth and breadth of the research on incomplete-ness after G¨odel. In this survey, we will focus on logical and mathematicalaspects of research on incompleteness.An important and interesting topic concerning incompleteness is missingin this paper: philosophy of G¨odel’s incompleteness theorems. For us, thewidely discussed and most important philosophical questions about G¨odel’sincompleteness theorems are: the relationship between G1 and the mecha-nism thesis, the status of G¨odel’s disjunctive thesis, and the intensionalityproblem of G2 . We leave a survey of philosophical discussions of G¨odel’sincompleteness theorems for a future philosophy paper.This paper is structured as follows. In Section 1, we introduce the moti-vation, the main content and the structure of this paper. In Section 2, welist the preliminary notions and definitions used in this paper. In Section 3,we examine different proofs of G¨odel’s incompleteness theorems and classifythese proofs based on nine criteria. In Section 4, we examine the limit ofthe applicability of G1 both for extensions of PA , and for theories weakerthan PA w.r.t. interpretation. In Section 5, we examine the limit of theapplicability of G2 , and discuss sources of indeterminacy in the formulationof the consistency statement.2. Preliminaries
Definitions and notations.
We list the definitions and notations re-quired below. These are standard and used throughout the literature.
Definition 2.1 (Basic notions) . • A language consists of an arbitrary number of relation and functionsymbols of arbitrary finite arity. For a given theory T , we use L ( T )to denote the language of T , and often equate L ( T ) with the list ofnon-logical symbols of the language. • For a formula φ in L ( T ), ‘ T ⊢ φ ’ denotes that φ is provable in T : i.e.,there is a finite sequence of formulas h φ , · · · , φ n i such that φ n = φ ,and for any 0 ≤ i ≤ n , either φ i is an axiom of T , or φ i follows fromsome φ j ( j < i ) by using one inference rule. • A theory T is consistent if no contradiction is provable in T . • We say a sentence φ is independent of T if T φ and T ¬ φ . • A theory T is incomplete if there is a sentence φ in L ( T ) which isindependent of T ; otherwise, T is complete (i.e., for any sentence φ in L ( T ), either T ⊢ φ or T ⊢ ¬ φ ). We may view nullary functions as constants, and nullary relations as propositionalvariables.
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 5
In this paper, we focus on first-order theories based on a countable lan-guage, and always assume the arithmetization of the base theory with arecursive set of non-logical symbols. For the technical details of arithme-tization, we refer to [102, 19]. Arithmetization means that any formula orfinite sequence of formulas can be coded by a natural number, called the
G¨odel number . This representation of syntax was pioneered by G¨odel.
Definition 2.2 (Basic notions following arithmetization) . • We say a set of sentences Σ is recursive if the set of G¨odel numbersof sentences in Σ is recursive. • A theory T is decidable if the set of sentences provable in T is recur-sive; otherwise it is undecidable . • A theory T is recursively axiomatizable if it has a recursive set ofaxioms (i.e. the set of G¨odel numbers of axioms of T is recursive). • A theory T is finitely axiomatizable if it has a finite set of axioms. • A theory T is locally finitely satisfiable if every finitely axiomatizedsubtheory of T has a finite model. • A theory T is recursively enumerable (r.e.) if it has a recursivelyenumerable set of axioms. • A theory T is essentially undecidable if any recursively axiomatizableconsistent extension of T in the same language is undecidable. • A theory T is essentially incomplete if any recursively axiomatizableconsistent extension of T in the same language is incomplete. • A theory T is minimal essentially undecidable if T is essentiallyundecidable, and if deleting any axiom of T , the remaining theory isno longer essentially undecidable.We denote by n the numeral representing n ∈ ω in L ( PA ). In this paper, p φ q denotes the numeral representing the G¨odel number of φ , and p φ ( ˙ x ) q denotes the numeral representing the G¨odel number of the sentence obtainedby replacing x with the value of x . Definition 2.3 (Representations, translations, and interpretations) . • A n -ary relation R ( x , · · · , x n ) on ω n is representable in T if there is aformula φ ( x , · · · , x n ) such that T ⊢ φ ( m , · · · , m n ) when R ( m , · · · , m n )holds, and T ⊢ ¬ φ ( m , · · · , m n ) when R ( m , · · · , m n ) does not hold. • We say that a total function f ( x , · · · , x n ) on ω n is representable in T if there is a formula ϕ ( x , · · · , x n , y ) such that T ⊢ ∀ y ( ϕ ( a , · · · , a n , y ) ↔ y = m ) whenever a , · · · , a n , m ∈ ω are such that f ( a , · · · , a n ) = m . • Let T be a theory in a language L ( T ), and S a theory in a language L ( S ). In its simplest form, a translation I of language L ( T ) intolanguage L ( S ) is specified by the following: – an L ( S )-formula δ I ( x ) denoting the domain of I ; – for each relation symbol R of L ( T ), as well as the equality re-lation =, an L ( S )-formula R I of the same arity; – for each function symbol F of L ( T ) of arity k , an L ( S )-formula F I of arity k + 1. The theory of completeness/incompleteness is closely related to the theory of decid-ability/undecidability (see [129]). Note that the variable x is free in the formula p φ ( ˙ x ) q but not in p φ ( x ) q . YONG CHENG • If φ is an L ( T )-formula, its I -translation φ I is an L ( S )-formula con-structed as follows: we rewrite the formula in an equivalent way sothat function symbols only occur in atomic subformulas of the form F ( x ) = y , where x i , y are variables; then we replace each such atomicformula with F I ( x, y ), we replace each atomic formula of the form R ( x ) with R I ( x ), and we restrict all quantifiers and free variablesto objects satisfying δ I . We take care to rename bound variables toavoid variable capture during the process. • A translation I of L ( T ) into L ( S ) is an interpretation of T in S if S proves the following: – for each function symbol F of L ( T ) of arity k , the formulaexpressing that F I is total on δ I : ∀ x , · · · ∀ x k − ( δ I ( x ) ∧ · · · ∧ δ I ( x k − ) → ∃ y ( δ I ( y ) ∧ F I ( x , · · · , x k − , y ))); – the I -translations of all axioms of T , and axioms of equality.The simplified picture of translations and interpretations above actuallydescribes only one-dimensional , parameter-free , and one-piece translations.For precise definitions of a multi-dimensional interpretation , an interpreta-tion with parameters , and a piece-wise interpretation , we refer to [137] [135][136] for more details.The notion of interpretation provides us with a method for comparingdifferent theories in different languages, as follows. Definition 2.4 (Interpretations II) . • A theory T is interpretable in a theory S if there exists an interpreta-tion of T in S . If T is interpretable in S , then all sentences provable(refutable) in T are mapped, by the interpretation function, to sen-tences provable (refutable) in S . • We say that a theory U weakly interprets a theory V (or V is weaklyinterpretable in U ) if V is interpretable in some consistent extensionof U in the same language (or equivalently, for some interpretation τ , the theory U + V τ is consistent). • Given theories S and T , let ‘ S ✂ T ’ denote that S is interpretable in T (or T interprets S ); let ‘ S ✁ T ’ denote that T interprets S but S does not interpret T ; we say S and T are mutually interpretable if S ✂ T and T ✂ S .Interpretability provides us with one measure of comparing strength ofdifferent theories. If theories S and T are mutually interpretable, then T and S are equally strong w.r.t. interpretation. In this paper, whenever wesay that theory S is weaker than theory T w.r.t. interpretation, this meansthat S ✁ T .A general method for establishing the undecidability of theories is de-veloped in [129]. The following theorem provides us with two methods forproving the essentially undecidability of a theory respectively via interpre-tation and representability. Theorem 2.5 (Theorem 7, Corollary 2, [129]) . URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 7 • Let T and T be two consistent theories such that T is interpretablein T . If T is essentially undecidable, then T is essentially unde-cidable. • If all recursive functions are representable in a consistent theory T ,then T is essentially undecidable. We shall also need some basic notions from recursion theory, as follows.Let φ , φ , · · · be a list of all unary computable (partial recursive) functionssuch that φ i ( j ), if it exists, can be computed from i and j . Definition 2.6 (Basic recursion theory) . • A recursively enumerable set (r.e. for short) is the domain of φ i forsome i ∈ ω , which is denoted by W i . • The notation φ i ( j ) ↑ means that the function φ i is not defined at j ,or j / ∈ W i ; and φ i ( j ) ↓ means that φ i is defined at j , or j ∈ W i .Provability logic provides us with an important tool to study the meta-mathematics of arithmetic and incompleteness. A good reference on thebasics of provability logic is [15]. Definition 2.7 (Modal logic) . The modal system K consisting of the fol-lowing axiom schemes. • All tautologies; • ✷ ( A → B ) → ( ✷ A → ✷ B );as well as two inference rules:(i) if ⊢ A and ⊢ A → B , then ⊢ B ;(ii) if ⊢ A , then ⊢ ✷ A .We denote by GL the modal system consisting of all axioms of K , all in-stances of the scheme ✷ ( ✷ A → A ) → ✷ A , and the same inference ruleswith K . We denote by GLS the modal system consisting of all theorems of GL , and all instances of the scheme ✷ A → A . However, GLS has only oneinference rule: Modus Ponens.2.2.
Logical systems.
In this section, we introduce some well-known the-ories weaker than PA w.r.t. interpretation from the literature. In Section4, we will show that these theories are essentially incomplete.Robinson Arithmetic Q is introduced in [129] by Tarski, Mostowski andRobinson as a base axiomatic theory for investigating incompleteness, andundecidability. Definition 2.8.
Robinson Arithmetic Q is defined in the language { , S , + , ×} with the following axioms: Q : ∀ x ∀ y ( S x = S y → x = y ); Q : ∀ x ( S x = ); Q : ∀ x ( x = → ∃ y ( x = S y )); Q : ∀ x ∀ y ( x + = x ); Q : ∀ x ∀ y ( x + S y = S ( x + y )); Q : ∀ x ( x × = ); Q : ∀ x ∀ y ( x × S y = x × y + x ). YONG CHENG
Robinson Arithmetic Q is very weak: it cannot even prove that additionis associative. The theory PA consists of the axioms Q - Q , Q - Q inDefinition 2.8 and the following axiom scheme of induction:( Induction ) ( φ ( ) ∧ ∀ x ( φ ( x ) → φ ( S x ))) → ∀ xφ ( x ) , where φ is a formula with at least one free variable x . Let N = h N , + , ×i denote the standard model of arithmetic.We now introduce a well-known hierarchy of L ( PA )-formulas called the arithmetical hierarchy (see [102, 55]). Definition 2.9 (Arithmetical hierarchy) . • Bounded formulas (Σ , or Π , or ∆ formula) are built from atomicformulas using only propositional connectives and bounded quanti-fiers (in the form ∀ x ≤ y or ∃ x ≤ y ). • A formula is Σ n +1 if it has the form ∃ xφ where φ is Π n . • A formula is Π n +1 if it has the form ∀ xφ where φ is Σ n . Thus,a Σ n -formula has a block of n alternating quantifiers, the first onebeing existential, and this block is followed by a bounded formula.Similarly for Π n -formulas. • A formula is ∆ n if it is equivalent to both a Σ n formula and a Π n formula.We can now formally introduce the notion of consistency in its variousguises, as well as the various fragments of Peano arithmetic PA . Definition 2.10 (Formal consistency and systems) . • A theory T is said to be ω -consistent if there is no formula ϕ ( x ) suchthat T ⊢ ∃ xϕ ( x ), and for any n ∈ ω , T ⊢ ¬ ϕ (¯ n ). • A theory T is 1 -consistent if there is no such ∆ formula ϕ ( x ). • We say a theory T is Σ -sound if for any Σ sentences φ , if T ⊢ φ ,then N | = φ . • The theory I Σ n is Q plus induction for Σ n formulas, and B Σ n +1 is I Σ plus collection for Σ n +1 formulas. • The theory I ∆ is Q plus induction for ∆ formulas. • The theory PA is the union of all I Σ n .It is well-known that the following form a strictly increasing hierarchy: I Σ , B Σ , I Σ , B Σ , · · · , I Σ n , B Σ n +1 , · · · , PA . Moreover, there are weak fragments of PA that play an important role incomputer science, namely in complexity theory ([18, 19]). These systems arebased on the following concept. Definition 2.11 (Sub-exponential functions) . • Define ω ( x ) = x | x | , and ω n +1 ( x ) = 2 ω n ( | x | ) where | x | is the lengthof the binary expression of x . • Let Ω n ≡ ( ∀ x )( ∃ y )( ω n ( x ) = y ) express that ω n ( x ) is total. The collection axiom for Σ n +1 formulas is the following principle: ( ∀ x
By [36, Proposition 2, p.299], there is a bounded formula
Exp ( x , y , z ) suchthat I Σ proves that Exp ( x , , z ) ↔ z = 1, and Exp ( x , Sy , z ) ↔ ∃ t ( Exp ( x , y , t ) ∧ z = t · x ). However, I Σ cannot prove the totality of Exp ( x , y , z ). Let exp denote the statement postulating the totality of the exponential function ∀ x ∀ y ∃ z Exp ( x , y , z ). Elementary Arithmetic ( EA ) is I ∆ + exp . Theorem 2.12 ([53, 36]) . • The theory I Σ + Ω n is interpretable in Q for any n ≥ (see [36,Theorem 3, p.304] ). • The theory I Σ + exp is not interpretable in Q . • The theory I Σ is not interpretable in I Σ + exp (see [53, Theorem1.1] , p.186). • The theory I Σ n +1 is not interpretable in B Σ n +1 (see [53, Theorem1.2] , p.186). • The theory B Σ + exp is interpretable in I Σ + exp (see [53, Theorem2.4] , p.188). • The theory B Σ + Ω n is interpretable in I Σ + Ω n for each n ≥ (see [53, Theorem 2.5] , p.189). • The theory B Σ n +1 is interpretable in I Σ n for each n ≥ (see [53,Theorem 2.6] , p.189). The theory PA − is the theory of commutative, discretely ordered semi-rings with a minimal element plus the subtraction axiom. The theory PA − has the following axioms, where the language L ( PA − ) is L ( PA ) ∪ {≤} : Definition 2.13 (The system PA − ) . • x + 0 = x ; • x + y = y + x ; • ( x + y ) + z = x + ( y + z ); • x × x ; • x × y = y × x ; • ( x × y ) × z = x × ( y × z ); • x × ( y + z ) = x × y + x × z ; • x ≤ y ∨ y ≤ x ; • ( x ≤ y ∧ y ≤ z ) → x ≤ z ; • x + 1 (cid:2) x ; • x ≤ y → ( x = y ∨ x + 1 ≤ y ); • x ≤ y → x + z ≤ y + z ; • x ≤ y → x × z ≤ y × z ; • x ≤ y → ∃ z ( x + z = y ).The theory Q + is the extension of Q in the language L ( Q + ) = L ( Q ) ∪{≤} with the following extra axioms: Definition 2.14 (The system Q + ) . The system Q + is Q plus Q : ( x + y ) + z = x + ( y + z ); Q : x × ( y + z ) = x × y + x × z ; Q : ( x × y ) × z = x × ( y × z ); Q : x + y = y + x ; Q : x × y = y × x ; Q : x ≤ y ↔ ∃ z ( x + z = y ).Andrzej Grzegorczyk considers a theory Q − in which addition and multi-plication satisfy natural reformulations of the axioms of Q but are possibly non-total functions. More exactly, the language of Q − is { , S , A, M } where A and M are ternary relations. See [36, Theorem 6, p.313]. Solovay proves that I Σ + ¬ exp is interpretable in Q (see[36, Theorem 7, p.314]). Definition 2.15 (The system Q − ) . The axioms of Q − are the axioms Q - Q of Q plus the following six axioms about A and M : A: ∀ x ∀ y ∀ z ∀ z ( A ( x, y, z ) ∧ A ( x, y, z ) → z = z ); M: ∀ x ∀ y ∀ z ∀ z ( M ( x, y, z ) ∧ M ( x, y, z ) → z = z ); G4: ∀ x A ( x, , x ); G5: ∀ x ∀ y ∀ z ( ∃ u ( A ( x, y, u ) ∧ z = S ( u )) → A ( x, S ( y ) , z )); G6: ∀ x M ( x, , G7: ∀ x ∀ y ∀ z ( ∃ u ( M ( x, y, u ) ∧ A ( u, x, z )) → M ( x, S ( y ) , z )).Samuel R. Buss [18] introduces S , a finitely axiomatizable theory, tostudy polynomial time computability. The theory S provides what isneeded for formalizing the proof of G2 in a natural and effortless way: thisprocess is actually easier in Buss’ theory than in full PA , since the restric-tions present in S prevent one from making wrong turns and inefficientchoices (see [137]).Next, we introduce adjunctive set theory AS which has a language withonly one binary relation symbol ‘ ∈ ’ satisfying AS1: ∃ x ∀ y ( y / ∈ x ). AS2: ∀ x ∀ y ∃ z ∀ u ( u ∈ z ↔ ( u = x ∨ u = y )).We now consider the theory R introduced by A. Tarski, A. Mostowski andR. Robinson in [129], and some variants of it. Definition 2.16.
Let R be the theory consisting of schemes Ax1 - Ax5 with L ( R ) = { , · · · , n, · · · , + , × , ≤} where m, n ∈ ω . Ax1: m + n = m + n ; Ax2: m × n = m × n ; Ax3: m = n if m = n ; Ax4: ∀ x ( x ≤ n → x = 0 ∨ · · · ∨ x = n ); Ax5: ∀ x ( x ≤ n ∨ n ≤ x ).As it happens, the system R contains all key properties of arithmetic forthe proof of G1 . Unlike Q , the theory R is not finitely axiomatizable. Definition 2.17 (Variations of R ) . • Let R be R without Ax5 . • Let R be the system consisting of schemes Ax1 , Ax2 , Ax3 and
Ax4 ′ where the latter is as follows Ax4 ′ : ∀ x ( x ≤ n ↔ x = 0 ∨ · · · ∨ x = n ) . • Let R be the system consisting of schemes Ax2 , Ax3 and
Ax4 ′ .The ‘concatination’ theory TC has the language { ⌢, α, β } with a binaryfunction symbol and two constants. Definition 2.18 (The system TC ) .TC1: ∀ x ∀ y ∀ z ( x ⌢ ( y ⌢ z ) = ( x ⌢ y ) ⌢ z ); TC2: ∀ x ∀ y ∀ u ∀ v ( x ⌢ y = u ⌢ v → (( x = u ∧ y = v ) ∨ ∃ w (( u = x ⌢w ∧ w ⌢ v = y ) ∨ ( x = u ⌢ w ∧ w ⌢ y = v )))); TC3: ∀ x ∀ y ( α = x ⌢ y ); TC4: ∀ x ∀ y ( β = x ⌢ y ); TC5: α = β . URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 11
We refer to [55, 120] for the definitions of primitive recursive arithmetic(
PRA ), and Weak Konig’s Lemma (
WKL ). In a nutshell, the formersystem allows us to perform ‘iteration of functions f : N → N ’, while thelatter expresses a basic compactness argument for Cantor space. Theorem 2.19 (Friedman’s conservation theorem, Theorem 2.1, [75]) . If WKL ⊢ φ , then PRA ⊢ φ for any Π sentence φ in L ( PA ) . Finally, diagonalization, in one form or other, forms the basis for the proofof G2 . The following lemma is crucial in this regard. Lemma 2.20 (The Diagonalization Lemma) . Let T be a consistent r.e. ex-tension of Q . For any formula φ ( x ) with exactly one free variable, thereexists sentence θ such that T ⊢ θ ↔ φ ( p θ q ) . Lemma 2.20 is the simplest and most often used version of the Diagonal-ization Lemma. For a generalized version of the Diagonalization Lemma,we refer to [15]. In this paper, we use the term “Diagonalization Lemma”to refer to Lemma 2.20 and some variants of the generalized version.3.
Proofs of G¨odel’s incompleteness theorems
Introduction.
In this section, we discuss different proofs of G¨odel’sincompleteness theorems in the literature, and propose nine criteria for clas-sifying them.First of all, there are no requirements on the independent sentence in G1 .In particular, such a sentence need not have any mathematical meaning.This is often the case when meta-mathematical (proof-theoretic or recursion-theoretic or model-theoretic) methods are used to construct the independentsentence. In Section 3.2-3.4, we will discuss proofs of G¨odel’s incompletenesstheorems via pure logic . In Section 3.5, we will give an overview of the“concrete incompleteness” research program which seeks to identify naturalindependent sentences with real mathematical meaning .Secondly, we say that a proof of G1 is constructive if it explicitly constructsthe independent sentence from the base theory by algorithmic means. A non-constructive proof of G1 only proves the mere existence of the independentsentence and does not show its existence algorithmically. We say that a proofof G1 for theory T has the Rosser property if the proof only assumes that T is consistent instead of assuming that T is ω -consistent or 1-consistent orΣ -sound; all these notions are introduced in Section 2.2.After G¨odel, many different proofs of G¨odel’s incompleteness theoremshave been found. These proofs can be classified using the following criteria:(C1) proof-theoretic proof;(C2) recursion-theoretic proof;(C3) model-theoretic proof;(C4) proof via arithmetization;(C5) proof via the Diagonalization Lemma;(C6) proof based on “logical paradox”;(C7) constructive proof;(C8) proof having the Rosser property; (C9) the independent sentence has natural and real mathematical content. However, these aspects are not exclusive: a proof of G1 or G2 may satisfyseveral of the above criteria.Thirdly, there are two kinds of proofs of G¨odel’s incompleteness theo-rems via pure logic: one based on logical paradox and one not based onlogical paradox. In Section 3.2, we first provide an overview of the mod-ern reformulation of proofs of G¨odel’s incompleteness theorems. We discussproofs of G¨odel’s incompleteness theorems not based on logical paradox inSection 3.3; we discuss proofs of G¨odel’s incompleteness theorems based onlogical paradox in Section 3.4.3.2. Overview and modern formulation.
In a nutshell, the three mainideas in the (modern/standard) proofs of G1 and G2 are arithmetization , representability , and self-reference , as discussed in detail in Section 3.2.1.Interesting properties of G1 and G2 are discussed in Sections 3.2.2 and 3.2.4,while the formalized notions of ‘proof’ and ‘truth’ are discussed in Sec-tion 3.2.3. Finally, we formulate a blanket caveat for the rest of this section: Unless stated otherwise, we will always assume that T is a recursivelyaxiomatizable consistent extension of Q . Other sections shall contain similar caveats and we sometimes stress these.3.2.1.
Three steps towards G1 and G2 . Intuitively speaking, G¨odel’s incom-pleteness theorems can be proved based on the following key ingredients. • Arithmetization : since G1 and G2 are theorems about propertiesof the syntax of logic, we need to somehow represent the latter, whichis 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 con-cepts, 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 thekey to proving G1 and G2 . We now discuss these three notions in detail.First of all, arithmetization has the following intuitive content: it estab-lishes a one-to-one correspondence between expressions of L ( T ) and naturalnumbers. Thus, we can translate metamathematical statements about theformal theory T into statements about natural numbers. Furthermore, fun-damental metamathematical relations can be translated in this way intocertain recursive relations, hence into relations representable in T . Conse-quently, one can speak about a formal system of arithmetic, and about itsproperties as a theory in the system itself! This is the essence of G¨odel’sidea of arithmetization, which was revolutionary at a time when computerhardware and software did not exist yet. I.e. G¨odel’s sentence is a pure logical construction (via the arithmetization of syntaxand provability predicate) and has no relevance with classic mathematics (without anycombinatorial or number-theoretic content). On the contrary, Paris-Harrington Princi-ple is an independent arithmetic sentence from classic mathematics with combinatorialcontent.
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 13
Secondly, in light of the previous, we can define certain relations on natu-ral numbers that express or represent crucial metamathematical conceptsrelated to the formal system T , like ‘proof’ and ‘consistency’. For example,modulo plenty of technical details, we can readily define a binary relationon ω expressing what it means to prove a formula in T , namely as follows: P roof T ( m, n ) if and only if n is the G¨odel number of a proof in T of theformula with G¨odel number m .Moreover, we can show that the relation P roof T ( m, n ) is recursive. Inaddition, G¨odel proves that every recursive relation is representable in PA .Next, let Proof T ( x, y ) be the formula which represents P roof T ( m, n ) in PA . From the formula
Proof T ( x, y ), we can define the ‘provability’ pred-icate Prov T ( x ) as ∃ y Proof T ( x, y ). The provability predicate Prov T ( x )satisfies the following conditions which show that formal and intuitive prov-ability have the same properties.(1) If T ⊢ ϕ , then T ⊢ Prov T ( p ϕ q );(2) T ⊢ Prov T ( p ϕ → ψ q ) → ( Prov T ( p ϕ q ) → Prov T ( p ψ q ));(3) T ⊢ Prov T ( p ϕ q ) → Prov T ( p Prov T ( p ϕ q ) q ).For the proof of G1 , G¨odel defines the G¨odel sentence G which asserts itsown unprovability in T via a self-reference construction. G¨odel shows thatif T is consistent, then T G , and if T is ω -consistent, then T ¬ G . Oneway of obtaining such a G¨odel sentence is the Diagonalization Lemma whichintuitively speaking implies that the predicate ¬ Prov T ( x ) has a fixed point ,i.e. there is a sentence θ in L ( T ) such that T ⊢ θ ↔ ¬ Prov T ( p θ q ) . Clearly, T θ while θ intuitively expresses its own unprovability, i.e. theaforementioned self-referential nature.For the proof of G2 , we first define the arithmetic sentence Con ( T ) in L ( T ) as ¬ Prov T ( p = q ) which says that for all x , x is not a code of aproof of a contradiction in T . G¨odel’s second incompleteness theorem ( G2 )states that if T is consistent, then the arithmetical formula Con ( T ), whichexpresses the consistency of T , is not provable in T . In Section 5.3, we willdiscuss some other ways of expressing the consistency of T .Finally, from the above conditions (1)-(3), one can show that T ⊢ Con ( T ) ↔ G . Thus, G2 holds: if T is consistent, then T Con ( T ). For more detailson these proofs of G1 and G2 , we refer to Chapter 2 in [102].3.2.2. Properties of G1 . In this section, we discuss some (sometimes subtle)comments on G1 .First of all, G¨odel’s proof of G1 is constructive as follows: given a consis-tent r.e. extension T of PA , the proof constructs, in an algorithmic way, atrue arithmetic sentence which is unprovable in T . In fact, one can effectivelyfind a true Π sentence G T of arithmetic such that G T is independent of T .G¨odel calls this the “incompletability or inexhaustability of mathematics”. Via arithmetization and representability, one can speak about the property of T in PA itself! Secondly, for G¨odel’s proof of G1 , only assuming that T is consistentdoes not suffice to show that G¨odel sentence is independent of T . In fact,the optimal condition to show that G¨odel sentence is independent of T is: T + Con ( T ) is consistent (see Theorems 35-36 in [64]). Thirdly, in summary, G¨odel’s proof of G1 has the following properties: • uses proof-theoretic method with arithmetization; • does not directly use the Diagonalization Lemma; • the proof formalizes the liar paradox; • the proof is constructive; • the proof does not have the Rosser property; • G¨odel’s sentence has no real mathematical content.All these characteristics of G¨odel’s proof of G1 are not necessary conditionsfor proving G1 . For example, G1 can be proved using recursion-theoreticor model-theoretic method, using the Diagonalization Lemma, using otherlogical paradoxes, using non-constructive methods, only assuming that T is consistent (i.e. having the Rosser property), and can be proved withoutarithmetization.Fourth, G1 does not tell us that any consistent theory is incomplete.In fact, there are many consistent complete first-order theories. For ex-ample, the following first-order theories are complete: the theory of denselinear orderings without endpoints ( DLO ), the theory of ordered divisiblegroups (
ODG ), the theory of algebraically closed fields of given character-istic (
ACF p ), and the theory of real closed fields ( RCF ). We refer to [32]for details of these theories. In fact, G1 only tells us that any consistentfirst-order theory containing a large enough fragment of PA (such as Q ) isincomplete: there is then a true Π sentence which is independent of theinitial theory. Turing’s work in [131] shows that any true Π -sentence ofarithmetic is provable in some transfinite iteration of PA . Feferman’s workin [34] extends Turing’s work and shows that any true sentence of arithmeticis provable in some transfinite iteration of PA .Fifth, whether a theory of arithmetic is complete depends on the languageof the theory. There are respectively recursively axiomatized complete arith-metic theories in the language of L ( , S ), L ( , S , < ) and L ( , S , <, +) (seeSection 3.1-3.2 in [30]). Containing enough information of arithmetic is es-sential for a consistent arithmetic theory to be incomplete. For example, Eu-clidean geometry is not about arithmetic but only about points, circles andlines in general; but Euclidean geometry is complete as Tarski has proved(see [130]). If the theory contains only information about the arithmeticof addition without multiplication, then it can be complete. For example,Presburger arithmetic is a complete theory of the arithmetic of addition inthe language of L ( , S , +) (see Theorem 3.2.2 in [102], p.222). Finally, con-taining the arithmetic of multiplication is not sufficient for a theory to beincomplete. For example, there exists a complete recursively axiomatizedtheory in the language of L ( , × ) (see [102], p.230). This optimal condition is much weaker than ω -consistency. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 15
Finally, it is well-known that
T h ( N , + , × ) is interpretable in T h ( Z , + , × )and T h ( Q , + , × ). Since
T h ( N , + , × ) is undecidable and has a finitely ax-iomatizable incomplete sub-theory Q , by Theorem 2.5, T h ( Z , + , × ) and T h ( Q , + , × ) are undecidable, and hence not recursively axiomatizable, butthey respectively have a finitely axiomatizable incomplete sub-theory of inte-gers and rational numbers. But T h ( R , + , × ) is decidable and recursively ax-iomatizable (even if not finitely axiomatizable). In fact, T h ( R , + , × ) = RCF (the theory of real closed field) (see [32], p.320-321). Note that this fact doesnot contradict G1 since none of N , Z and Q is definable in ( R , + , × ).3.2.3. Between truth and provability.
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 state-ments . Put another way, incompleteness does not arise at the level of Σ sentences. Indeed, we have Σ -completeness for T : for any Σ sentences φ , T ⊢ φ if and only if N | = φ . Thus, G¨odel’s sentence is a true Π sentence inthe form ∀ xφ ( x ) such that T ∀ xφ ( x ) but ‘ T ⊢ φ (¯ n )’ holds for any n ∈ ω .Secondly, define Truth = { φ ∈ L ( PA ) : N | = φ } and Prov = { φ ∈ L ( PA ) : PA ⊢ φ } , i.e. the formalized notions of ‘proof’ and ‘truth’. In thispaper, unless stated otherwise, we equate a set of sentences with the set ofG¨odel’s numbers of these sentences.Now, before G¨odel’s work, it was thought that Truth = Prov . Thus,G¨odel’s first incompleteness theorem ( G1 ) reveals the difference between thenotion of provability in PA and the notion of truth in the standard modelof arithmetic N . There are some differences between Truth and
Prov :(1)
Prov ( Truth , i.e. there is a true arithmetic sentence which is unprov-able in PA ;(2) Tarski proves that: Truth is not definable in N but Prov is definablein N ;(3) Truth is not arithmetic but
Prov is recursive enumerable.However, both
Truth and
Prov are not recursive and not representable in PA . For more details on Truth and
Prov , we refer to [102, 129].The differences between
Truth and
Prov can also be expressed in termsof arithmetical interpretations , defined as follows.
Definition 3.1 (Arithmetical interpretations) . A mapping from the set ofall modal propositional variables to the set of L ( PA )-sentences is called an arithmetical interpretation .Every arithmetical interpretation f is uniquely extended to the mapping f ∗ from the set of all modal formulas to the set of L ( T )-sentences so that f ∗ satisfies the following conditions:(1) f ∗ ( p ) = f ( p ) for each propositional variable p ;(2) f ∗ commutes with every propositional connective;(3) f ∗ ( ✷ A ) is Prov T ( p f ∗ ( A ) q ) for every modal formula A .In the following, we equate arithmetical interpretations f with their uniqueextensions f ∗ defined on the set of all modal formulas. In this way, Solovay’s The key point is: N is definable in ( Z , + , × ), and ( Q , + , × ). See chapter XVI in [32]. Arithmetical Completeness Theorems for GL and GLS characterize thedifference between
Prov and
Truth via provability logic.
Theorem 3.2 (Solovay, [127]) .Arithmetical Completeness Theorem for GL:
Let T be a Σ -soundr.e. extension of Q . For any modal formula φ in L ( GL ) , GL ⊢ φ ifand only if T ⊢ f ( φ ) for every arithmetic interpretation f . Arithmetical Completeness Theorem for GLS:
For any modal for-mula φ, GLS ⊢ φ if and only if N | = f ( φ ) for every arithmetic in-terpretation f . Finally, one can study the notion of ‘proof predicate’ as given by
Proof T ( x, y )in an abstract setting, namely as follows. Recall that T is a recursivelyaxiomatizable consistent extension of Q . We introduce general notions ofproof predicate, and provability predicate which generalize the proof pred-icate Proof T ( x, y ), and the provability predicate Prov T ( x ) defined abovein G¨odel’s proof of G1 . Definition 3.3 (Proof predicate) . We say a formula
Prf T ( x, y ) is a proofpredicate of T if it satisfies the following conditions: (1) Prf T ( x, y ) is ∆ ( PA ); (2) PA ⊢ ∀ x ( Prov T ( x ) ↔ ∃ y Prf T ( x, y ));(3) for any n ∈ ω and formula φ, N | = Proof T ( p φ q , n ) ↔ Prf T ( p φ q , n );(4) PA ⊢ ∀ x ∀ x ′ ∀ y ( Prf T ( x, y ) ∧ Prf T ( x ′ , y ) → x = x ′ ). Definition 3.4 (Provability and consistency) . We define the ‘provability’predicate Pr T ( x ) from a proof predicate Prf T ( x, y ) by ∃ y Prf T ( x, y ), andthe consistency statement Con ( T ) from a provability predicate Pr T ( x ) by ¬ Pr T ( p = q ).The items D1 - D3 below are called the Hilbert-Bernays-L¨ob derivabilityconditions . Note that D1 holds for any provability predicate Pr T ( x ). Definition 3.5 (Standard proof predicate) . We say that provability predi-cate Pr T ( x ) is standard if it satisfies D2 and D3 as follows. D1: If T ⊢ φ , then T ⊢ Pr T ( p φ q ); D2: If T ⊢ Pr T ( p φ → ϕ q ) → ( Pr T ( p φ q ) → Pr T ( p ϕ q )); D3: T ⊢ Pr T ( p φ q ) → Pr T ( p Pr T ( p φ q ) q ).We say that Prf T ( x, y ) is a standard proof predicate if the induced provabil-ity predicate from it is standard.The previous definition leads to another blanket caveat: Unless stated otherwise, we always assume that Pr T ( x ) is a standardprovability predicate, and Con ( T ) is the canonical consistency statementdefined as ¬ Pr T ( p = q ) via the standard provability predicate Pr T ( x ) . We can say that each proof predicate represents the relation “ y is the code of a proofin T of a formula with G¨odel number x ”. We say a formula φ is ∆ ( PA ) if there exists a Σ formula α such that PA ⊢ φ ↔ α ,and there exists a Π formula β such that PA ⊢ φ ↔ β . URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 17
Properties of G2 . In this section, we discuss some (sometimes subtle)comments on G2 .First of all, we examine a somewhat delicate mistake in the argumentwhich claims that, by an easy application of the compactness theorem, wecan show that for any recursive axiomatization of a consistent theory T , T can not prove its own consistency. Visser presents this argument in [140] asan interesting dialogue between Alcibiades and Socrates:Suppose a consistent theory T can prove its own consistencyunder some axiomatization. By compactness theorem, theremust be a finitely axiomatized sub-theory S of T such that S already proves the consistency of T . Since S proves theconsistency of T , it must also prove the consistency of S . So,we have a finitely axiomatized theory which proves its ownconsistency. But G2 applies to the finite axiomatization andwe have a contradiction. It follows that T can not prove itsown consistency.The mistake in this argument is: from the fact that S can prove the consis-tency of T we cannot infer that S can prove the consistency of S . Some mayargue 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 [140], we should carefully dis-tinguish three perspectives of the theory T : our external perspective, theinternal perspective of S , and the internal perspective of T . From each per-spective, the consistency of the whole theory implies the consistency of itssub-theory. From T ’s perspective, S is a sub-theory of T . But from S ’sperspective, S may not be a sub-theory of T . From the fact that T knowsthat S is a sub-theory of T , we cannot infer that S also knows that S is asub-theory of T since S is a finite sub-theory of T and may not know anyinformation that T knows, leading to the following (dramatic) conclusion: the sub-theory relation between theories is not absolute. Similarly, the notion of consistency is not absolute: from G¨odel’s proofof G2 , we cannot infer that if T is a consistent r.e. extension of Q , then Con ( T ) is independent of T . The key point is: it is not enough to show that T ¬ Con ( T ) only assuming that T is consistent. However, we can showthat Con ( T ) is independent of T assuming that T is 1-consistent. In fact,the formalized version of “if T is consistent, then Con ( T ) is independentof T ” is not provable in T . Thus, a theory may be consistent from theexternal perspective but inconsistent from the internal perspective. Theseobservations lead to the following definitions. Definition 3.6 (Reflexivity) . It is an easy fact that if T is 1-consistent and S is not a theorem of T , then Pr T ( p S q )is not a theorem of T . See [15, Theorem 4, p.97] for a modal proof in GL of this fact using the ArithmeticCompleteness Theorem for GL . For example, let S = PA + ¬ Con ( PA ). From G2 , S is consistent from the externalperspective. But since S ⊢ ¬ Con ( S ), the theory S is not consistent from the internalperspective of S . Note that PA ⊢ Pr PA ( = ) → Pr PA ( Pr PA ( = ) → = ). • A first-order theory T containing PA is said to be reflexive if T ⊢ Con ( S ) for each finite sub-theory S of T where Con ( S ) is similarlydefined as Con ( PA ); • we say the theory T is essentially reflexive if any consistent extensionof T in L ( T ) is reflexive; • Let
Con ( T ) ↾ x denote the finite consistency statement “there areno proofs of contradiction in T with ≤ x symbols”.Mostowski proves that PA is essentially reflexive (see [102, Theorem2.6.12]). In fact one can show that for every n ∈ N , I Σ n +1 ⊢ Con ( I Σ n ). For a large class of natural theories U , Pudl´ak [114] shows that the lengths ofthe shortest proofs of Con ( U ) ↾ n for n ∈ ω in the theory U itself are boundedby a polynomial in n . Pudl´ak conjectures [114] that U does not have poly-nomial proofs of the finite consistency statements Con ( U + Con ( U )) ↾ n for n ∈ ω .Finally, a big open question about G2 is: can we find a genuinely self-reference free proof of G2 ? As far as we know, at present there is no convinc-ing essentially self-reference-free proofs of either G2 or of Tarski’s Theoremof the Undefinability of Truth. In [141], Visser gives a self-reference-freeproof of G2 from Tarski’s Theorem of the Undefinability of Truth, which isa step in a program to find self-reference-free proofs of both G2 and Tarski’sTheorem (see [141]). Visser’s argument in [141] is model-theoretic and themain tool is the Interpretation Existence Lemma. Visser’s proof in [141] isnot constructive. An interesting question is then whether Visser’s argumentcan be made constructive.3.3.
Proofs of G1 and G2 from mathematical logic. In this section, wediscuss various different proofs of G1 and G2 . We mention Jech’s [65] shortproof of G2 for ZF : if ZF is consistent, then it is unprovable in ZF thatthere exists a model of ZF . Jech’s proof uses the Completeness Theorem,and also yields G2 for PA (see [65]). Other (lengthier) proofs are discussedin Sections 3.3.1-3.3.5.3.3.1. Rosser’s proof.
Rosser [116] proves a “stronger” version of G1 , calledRosser’s first incompleteness theorem, which only assumes the consistencyof T : if T is a consistent r.e. extension of Q , then T is incomplete. G¨odel’sproof of G1 assumes that T is ω -consistent. Note that ω -consistency impliesconsistency. But the converse does not hold and the notion of ω -consistencyis stronger than consistency since we can find examples of theories thatare consistent but not ω -consistent. Rosser’s proof is constructive andalgorithmically constructs the Rosser sentence that is independent of T .G¨odel’s proof of G1 uses a standard provability predicate but Rosser’s proofof G1 uses a Rosser provability predicate which is a kind of non-standard provability predicate, giving rise to the following. For a proof of this result, we refer to H´ajek and Pudl´ak [55]. We refer to [139] for more details about the Interpretation Existence Lemma. For example, assuming PA is consistent, then PA + ¬ Con ( PA ) is consistent, butnot ω -consistent. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 19
Definition 3.7.
Let T be a recursively axiomatizable consistent extension of Q , and Prf T ( x, y ) be any proof predicate of T . Define the Rosser provabilitypredicate Pr RT ( x ) to be the formula ∃ y ( Prf T ( x, y ) ∧ ∀ z ≤ y ¬ Prf T ( ˙ ¬ x, z ))where ˙ ¬ is a function symbol expressing a primitive recursive function cal-culating the code of ¬ φ from the code of φ . The fixed point of the predicate ¬ Pr RT ( x ) is called the Rosser sentence of Pr RT ( x ), i.e. a sentence θ satisfying PA ⊢ θ ↔ ¬ Pr RT ( p θ q ).In general, one can show that each Rosser sentence based on any Rosserprovability predicate of T is independent of T . In particular, this indepen-dence does not rely on the choice of the proof predicate.3.3.2. Recursion-theoretic proofs.
G¨odel’s first incompleteness theorem ( G1 )is well-known in the context of recursion theory. Recall that W e = { n ∈ ω : φ e ( n ) ↓} . Let h W e : e ∈ ω i be the list of recursive enumerable subsets of N .The following is an example of an ‘effective’ version of G1 :there exists a recursive function f such that for any e ∈ ω , if W e ⊆ Truth ,then f ( e ) is defined and f ( e ) ∈ Truth \ W e ([31]).Similarly, Avigad [2] proves G1 and G2 in terms of the undecidability of thehalting problem (see Theorem 3.1, Theorem 3.2 in [2]). Another relatedresult due to Kleene is as follows. Theorem 3.8 (Kleene’s theorem, Theorem 2.2, [119]) . For any consistentr.e. theory T that contains Q , there exists some t ∈ ω such that ϕ t ( t ) ↑ holdsbut T “ ϕ t ( t ) ↑ ” . Kleene’s proof of his theorem uses recursion theory, and is not construc-tive. Salehi and Seraji [119] show that there is a constructive proof ofKleene’s theorem, but this constructive proof does not have the Rosserproperty. Salehi and Seraji [119] comment that there could be a ‘Rosse-rian’ version of this constructive proof of Kleene’s theorem.3.3.3.
Proofs based on Arithmetic Completeness.
Hilbert and Bernays [61]present the
Arithmetic Completeness Theorem expressing that any recur-sively axiomatizable consistent theory has an arithmetically definable model.Later, Kreisel [84] and Wang [143] adapt the Arithmetic Completeness The-orem and use paradoxes to obtain undecidability results.Now, the Arithmetic Completeness Theorem is an important tool inmodel-theoretic proofs of the incompleteness theorems. For more details,we refer to [95, 71, 83]. Walter Dean [29] gives a detailed discussion onhow the Arithmetized Completeness Theorem provides a tool for obtainingformal incompleteness results from some certain paradoxes.
Theorem 3.9 (Arithmetic Completeness, Theorem 3.1, [72]) . Let T be arecursively axiomatized consistent extension of Q . There exists a formula Tr T ( x ) in L ( PA ) that defines a model of T in PA + Con ( T ) . Lemma 3.10 is a corollary of the Arithmetized Completeness Theorem,and is essential for model-theoretic proofs of the incompleteness theorems.
Lemma 3.10 ([75, 72]) . Let T be a recursively axiomatized consistent ex-tension of Q , and Tr T ( x ) is the formula as asserted in Theorem 3.9. For any model M of PA + Con ( T ) , there exists a model M of T such that forany sentence φ , M | = Tr T ( p φ q ) if and only if M | = φ . Kreisel first applies the Arithmetized Completeness Theorem to establishmodel-theoretic proofs of G2 (cf. Kreisel [86], Smory´nski [122] and Kikuchi[72]). Kikuchi-Tanaka [75], Kikuchi [72, 73] and Kotlarski [81] use the Arith-metized Completeness Theorem to give model-theoretic proofs of G2 . Forexample, Kikuchi [72] proves G2 model-theoretically via the ArithmetizedCompleteness Theorem (Lemma 3.10): if PA is consistent, then Con ( PA )is not provable in PA (see Theorem 3.4, [72]). Proofs of G2 by Kreisel [86] and Kikuchi [72] do not directly yield theformalized version of G2 . Kikuchi’s proof of G2 in [73] is not formalizablein PRA . Kikuchi and Tanaka [75] prove in
WKL that Con ( PA ) im-plies ¬ Pr PA ( p Con ( PA ) q ), since the Completeness Theorem is provable in WKL , and the key Lemma 3.10 used in Kikuchi’s proof [72] is provable in RCA . Using Theorem 2.19, Tanaka [75] proves the formalized version of G2 : PRA ⊢ Con ( PA ) → Con ( PA + ¬ Con ( PA )).One can give a simple proof of G1 via the Diagonalization Lemma (see[102]). Kotlarski [81] proves the formalized version of G1 and G2 via model-theoretic arguments (e.g. using the Arithmetized Completeness Theoremand some quickly growing functions). Kotlarski [81] proves the followingversion of G1 assuming that PA is ω -consistent, and shows that the followingsentence is provable in PA :if ∀ ϕ, x { [ ϕ ∈ ∆ ∧ ∀ y Pr PA ( ¬ ϕ ( S x , S y → ¬ Pr PA ( ∃ yϕ ( S x , y )) } , then ∃ ϕ ∈ ∆ ∃ x [ ¬ Pr PA ( ∃ yϕ ( S x , y )) ∧ ¬ Pr PA ( ¬∃ yϕ ( S x , y ))].However, it is unknown whether the method in [81] can also give anew proof of Rosser’s first incompleteness theorem. Kotlarski [81] provesthe following formalized version of G2 : PA ⊢ Con ( PA ) → Con ( PA + ¬ Con ( PA )). Later, Kotlarski [82] transforms the proof of the formalizedversion of G2 in [81] to a proof-theoretic version without the use of theArithmetized Completeness Theorem.3.3.4. Proofs based on Kolmogorov complexity.
Intuitively,
Kolmogorov com-plexity is a measure of the quantity of information in finite objects. Roughlyspeaking, the Kolmogorov complexity of a number n , denoted by K ( n ), isthe size of a program which generates n . Definition 3.11 (Kolmogorov-Chaitin Complexity, [119]) . For any naturalnumber n ∈ ω , the Kolmogorov complexity for n , denoted by K ( n ), is definedas min { i ∈ ω | ϕ i (0) ↓ = n } .If n ≤ K ( n ), then n is called random. Kolmogorov shows in 1960’s thatthe set of non-random numbers is recursively enumerable but not recursive(c.f. Odifreddi [105]). Relations between G1 and Kolmogorov complexityhave been intensively discussed in the literature (c.f. Li and Vit´anyi [94]). The idea of the proof is: assuming that PA is consistent and PA ⊢ Con ( PA ),then we get a contradiction from the fact that there is a model M of PA such that M | = Con ( PA ). The theory
RCA (Recursive Comprehension) is a subsystem of Second Order Arith-metic. For the definition of RCA , we refer to [120]. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 21
Chaitin [20] gives an information-theoretic formulation of G1 , and provesthe following weaker version of G1 in terms of Kolmogorov complexity. Theorem 3.12 (Chaitin [20, 119]) . For any consistent r.e. extension T of Q , there exists a constant c T ∈ N such that for any e ≥ c T and any w ∈ N we have T “ K ( w ) > e ” . Salehi and Seraji [119] show that we can algorithmically construct theChaitin constant c T in Theorem 3.12. I.e. for a given consistent r.e. extension T of Q , one can algorithmically construct a constant c T ∈ N such that forall e ≥ c T and all w ∈ N , we have T “ K ( w ) > e ” (see Theorem 3.4 in[119]). From Theorem 3.12, it is not clear whether “ K ( w ) > e ” holds (orwhether “ K ( w ) > e ” is independent of T ). Salehi and Seraji [119] show thatChaitin’s proof of G1 is non-constructive: there is no algorithm such thatgiven any consistent r.e. extension T of Q we can compute some w T suchthat K ( w T ) > c T holds where c T is the Chaitin constant we can compute asin Theorem 3.12 (see Theorem 3.5, [119]). If such an algorithm exists, thenfor any consistent r.e. extension T of 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 [119] also strengthen Chaitin’s Theorem 3.12 assuming T is Σ -sound: if T is a Σ -sound r.e. theory extending Q , then there existssome c T (which is computable from T ) such that for any e ≥ c T there arecofinitely many w ’s such that “ K ( w ) > e ” is independent of T (see Corollary3.7, [119]). Using a version of the Pigeonhole Principle in Q , Salehi andSeraji [119] also prove the Rosserian form of Chaitin’s Theorem: for anyconsistent r.e. extension T of Q , there is a constant c T (which is computablefrom T ) such that for any e ≥ c T there are cofinitely many w ’s such that“ K ( w ) > e ” is independent of T (see Theorem 3.9, [119]).Kikuchi [73] proves the following formalized version of G1 via Kolmogorovcomplexity for any consistent r.e. extension T of Q : there exists e ∈ ω with(1) T ⊢ Con ( T ) → ∀ x ( ¬ Pr T ( p K ( x ) > e q ));(2) T ⊢ ω - Con ( T ) → ∀ x ( e < K ( x ) → ¬ Pr T ( p K ( x ) ≤ e q )).However, this proof is not constructive. Moreover, Kikuchi [73] proves G2 via Kolmogorov complexity and the Arithmetic Completeness Theorem: if T is a consistent r.e. extension of Q , then T Con ( T ). Kikuchi’s proofof G2 in [73] cannot be formalized in PRA but can be carried out within
WKL . Thus we can also obtain a formalized version of G2 in WKL byTheorem 2.19.3.3.5. Model-theoretic proofs.
Adamowicz and Bigorajska [5] prove G2 viamodel-theoretic method using the notion of 1-closed models and existentiallyclosed models. Definition 3.13. (1) A model M of a theory T is called 1-closed (w.r.t. T ) if for any a , · · · , a n in M , any Σ formula φ and any M ′ such that M ≺ M ′ and M ′ | = T , we have: if M ′ | = φ ( a , · · · , a n ), then M | = φ ( a , · · · , a n ). In other words, we can say that M is 1-closed iffor any M ′ such that M ≺ M ′ , we have M ≺ M ′ . (2) Let K be a class of structures in the same language. A model M ∈ K is existentially closed in K if for every model N ⊇ M such that N ∈ K , we have M (cid:22) N : every existential formula with parametersfrom M which is satisfied in N is already satisfied in M .Adamowicz and Bigorajska [5] first prove G2 without the use of theArithmetized Completeness Theorem: every 1-closed model of any subthe-ory T of PA extending I ∆ + exp satisfies ¬ Con ( PA ). Then Adamow-icz and Bigorajska [5] prove the formalized version of G2 via the idea ofexistentially closed models and the Arithmetized Completeness Theorem: PA ⊢ Con ( PA ) → Con ( PA + ¬ Con ( PA )) (see Theorem 2.1, [5]). Thisis proved by showing that an arbitrary model of PA + Con ( PA ) satisfies Con ( PA + ¬ Con ( PA )).3.4. Proofs of G1 and G2 based on logical paradox. We provide asurvey of proofs of incompleteness theorems based on ‘logical paradox’.3.4.1.
Introduction.
As noted in Section 3.2.1, G¨odel’s incompleteness the-orems are closely related to paradox and self-reference. In fact, G¨odel com-ments in his famous paper [45] that “any epistemological antinomy could beused for a similar proof of the existence of undecidable propositions”.Now, the
Liar Paradox is an old and most famous paradox in modernscience. In G¨odel’s proof of G1 , we can view G¨odel’s sentence as the for-malization of the Liar Paradox. G¨odel’s sentence concerns the notion ofprovability but the liar sentence in the Liar Paradox concerns the notion oftruth in the standard model of arithmetic. There is a big difference betweenthe notion of provability and truth. G¨odel’s sentence does not lead to acontradiction as the Liar sentence does.Besides the Liar Paradox, many other paradoxes have been used to givenew proofs of incompleteness theorems: for example, Berry’s Paradox in[14, 20, 72, 77, 75, 142], Grelling-Nelson’s Paradox in [26], the UnexpectedExamination Paradox in [37, 87], and Yablo’s Paradox in [27, 76, 89, 111].We now discuss some of these paradoxes in detail.3.4.2. Berry’s paradox.
Berry’s Paradox introduced by Russell [117] is theparadox that “the least integer not nameable in fewer than nineteen syl-lables” is itself a name consisting of eighteen syllables. Informally, we saythat an expression names a natural number n if n is the unique naturalnumber satisfying the expression. Berry’s Paradox can be formalized in for-mal systems by interpreting the concept of “name” suitably. The followingis Boolos’s formulation of the concept of “name” in [14]. Definition 3.14 (Boolos [14]) . Let n ∈ ω and ϕ ( x ) be a formula with onlyone free variable x . We say that ϕ ( x ) names n if N | = ϕ ( n ) ∧ ∀ v ∀ v ( ϕ ( v ) ∧ ϕ ( v ) → v = v ).Proofs of the incompleteness theorems based on Berry’s Paradox havebeen given by Vopˇenka [142], Chaitin [20], Boolos [14], Kikuchi-Kurahashi-Sakai [77], Kikuchi [72], and Kikuchi-Tanaka [75]. In fact, Robinson first URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 23 uses Berry’s Paradox in [115] to prove Tarski’s theorem on the undefinabil-ity of truth, which anticipates the later use of Berry’s Paradox to obtainincompleteness results by Vopˇenka [142], Boolos [14] and Kikuchi [73].Boolos [14] proves a weak form of G1 in the 1980’s by formalizing Berry’sParadox in arithmetic via considering the length of formulas that namenatural numbers in the standard model of arithmetic. Using this formulationof the concept of “name”, Boolos [14] first shows that Berry’s Paradox leadsto a proof of G1 in the following form: there is no algorithm whose outputcontains all true statements of arithmetic and no false ones (i.e. the theoryof true arithmetic is not recursively axiomatizable). Barwise [6] praisesBoolos’s proof as “very lovely and the most straightforward proof of G¨odel’sincompleteness theorem that I have ever seen”. The optimal sufficient andnecessary condition for the independence of a Boolos sentence from PA isthat PA + Con ( PA ) is consistent (see [119]).Boolos’s theorem is different from G¨odel’s theorem in the following way:(1) Boolos’s theorem refers to the concept of truth but G¨odel’s theorem doesnot;(2) Boolos’s proof is not constructive, and we can prove that there is noalgorithm for computing the true but unprovable sentence;(3) Boolos’s theorem is weaker than G¨odel’s first incompleteness theorem,and hence we cannot obtain the second incompleteness theorem fromBoolos’s theorem in the standard way (see [77]).Boolos’s proof is modified by Kikuchi and Tanaka in [75, 72]. The differencebetween Kikuchi’s proof and Boolos’s proof lies in the interpretation of theword “name”. Kikuchi [72] modifies Boolos’s formulation of the concept of“name” by replacing “truth” with “provability” in the definition. Definition 3.15 (Definition 3.1, Kikuchi [72]) . Let n ∈ ω and ϕ ( x ) bea formula with only one free variable x . We say that ϕ ( x ) names n if PA ⊢ ϕ ( n ) ∧ ∀ v ∀ v ( ϕ ( v ) ∧ ϕ ( v ) → v = v ).Using this formulation of the concept of “name”, Kikuchi [72] gives aproof-theoretic proof of G1 by formalizing Berry’s paradox without the use ofthe Diagonalization Lemma. Kikuchi [72] constructs a sentence θ and showsthat if PA is consistent, then ¬ θ is not provable in PA ; if PA is ω -consistent,then θ is not provable in PA (see Theorem 2.2, [72]). Note that Kikuchi’sproof of G1 in [72] is constructive. Kikuchi and Tanaka [75] reformulateKikuchi’s proof of G1 in [72], and show in WKL that if PA + Con ( PA )is consistent, then θ is independent of PA . By Theorem 2.19, Kikuchiand Tanaka [75] prove the formalized version of G1 : PRA ⊢ Con ( PA + Con ( PA )) → ¬ Pr PA ( p θ q ) ∧ ¬ Pr PA ( p ¬ θ q ). An interesting question notcovered in [72, 75] is whether we can improve Kikuchi’s proof of G1 by onlyassuming that PA is consistent.Vopˇenka [142] proves G2 for ZF by formalizing Berry’s Paradox, viaadopting Kikuchi’s definition of the concept of “name” in [72] over mod-els of ZF : Con ( ZF ) is not provable in ZF . Vopˇenka’s proof uses the I.e. we say that ϕ ( x ) names n in ZF if ZF ⊢ ϕ ( n ) ∧ ∀ v ∀ v ( ϕ ( v ) ∧ ϕ ( v ) → v = v )where ϕ ( x ) is a formula with only one free variable x (see [142]). Completeness Theorem but does not use the Arithmetic Completeness The-orem. Kikuchi, Kurahashi and Sakai [77] show that Vopˇenka’s method canbe adapted to prove G2 for PA based on Kikuchi’s formalization of Berry’sParadox in [72] with an application of the Arithmetic Completeness Theo-rem.Proofs of G1 and G2 based on Berry’s Paradox by Vopˇenka [142], Chaitin[20], Boolos [14] and Kikuchi [72] do not use the Diagonalization Lemma.We can also prove G1 based on Berry’s Paradox using the DiagonalizationLemma. For example, Kikuchi, Kurahashi and Sakai [77] adopt Kikuchi’sdefinition of the concept of “name” in [72], and show that the independentstatement in Kikuchi’s proof in [72] can be obtained by using the Diagonal-ization Lemma.In summary, the distinctions between using and not using the Diagonal-ization Lemma, and between using and not using the Arithmetic Complete-ness Theorem are not essential for proofs of G1 and G2 based on Berry’sParadox. From the above discussions, we can characterize different proofsof G1 and G2 based on Berry’s Paradox by the method of interpreting theword “name”: Boolos [14] uses the standard model of arithmetic; Kikuchi[72] uses provability in arithmetic; Chaitin [20] and Kikuchi [73] use Kol-mogorov complexity; Kikuchi and Tanaka [75] use nonstandard models ofarithmetic; and Vopˇenka [142] uses models of ZF (see [77]).3.4.3. Unexpected Examination and Grelling-Nelson’s Paradox.
First of all,Kritchman and Raz [87] give a new proof of G2 based on Chaitin’s incom-pleteness theorem and an argument that resembles the Unexpected Exam-ination Paradox (for more details, we refer to [87]): for any consistentr.e. extension T of Q , if T is consistent, then T Con ( T ).Secondly, we say a one-place predicate is “heterological” if it does notapply to itself (e.g. “long” is heterological, since it’s not a long expression).Consider the question: is the predicate “heterological” we have just definedheterological? If “heterological” is heterological, then it isn’t heterologi-cal; and if “heterological” isn’t heterological, then it is heterological. Thiscontradiction is called Grelling-Nelson’s Paradox.Cie´sli´nski [26] presents semantic proofs of G2 for ZF and PA based onGrelling-Nelson’s Paradox. For a theory T containing ZF , Cie´sli´nski definesthe sentence HET T which says intuitively that the predicate “heterological”is itself heterological, and then shows that T HET T and T ⊢ HET T ↔ Con ( T ). Finally, Cie´sli´nski shows how to adapt the proof of G2 for ZF toa proof of G2 for PA . In fact, Cie´sli´nski [26] proves the semantic version of G2 : if T has a model, then T + ¬ Con ( T ) has a model (i.e. T Con ( T )). The Unexpected Examination Paradox is formulated as follows in [87]. The teacherannounces in class: “next week you are going to have an exam, but you will not be ableto know on which day of the week the exam is held until that day”. The exam cannot beheld on Friday, because otherwise, the night before the students will know that the examis going to be held the next day. Hence, in the same way, the exam cannot be held onThursday. In the same way, the exam cannot be held on any of the days of the week.
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 25
Yablo’s paradox.
We discuss proofs of G1 and G2 based on Yablo’sParadox in the literature. Yablo’s Paradox is an infinite version of theLiar Paradox proposed in [152]: consider an infinite sequence Y , Y , · · · ofpropositions such that each Y i asserts that Y j are false for all j > i . Differentproofs of G1 and G2 based on Yablo’s Paradox have been given by someauthors (e.g. Priest [111], Kikuchi-Kurahashi [76] and Cie´sli´nski-Urbaniak[27]).Recall that we assume by default that T is a consistent r.e. extension of Q . Priest [111] first points out that G1 can be proved by formalizing Yablo’sParadox. Priest defines a formula Y ( x ) as follows which says that for any y > x, Y ( y ) is not provable in T . Definition 3.16 (Definition 3, [89]) . A formula Y ( x ) is called a Yabloformula of T if T ⊢ ∀ x ( Y ( x ) ↔ ∀ y > x ¬ Pr T ( p Y ( ˙ y ) q )).Kurahashi [89] proves the following version of G1 , and shows that eachinstance Y ( n ) of the Yablo formula is independent of T if T is Σ -sound. Theorem 3.17 (Theorem 4, [89]) . Let Y ( x ) be a Yablo formula.(1) If T is consistent, then T Y ( n ) ;(2) If T is Σ -sound, then T ¬ Y ( n ) . Kurahashi [89] proves that T ⊢ ∀ x ( Y ( x ) ↔ Con ( T )) (see [89, Lemma6]). As a corollary, we have T ⊢ ∀ x ∀ y ( Y ( x ) ↔ Y ( y )), and G2 holds: if T is consistent, then T Con ( T ) (see [89, Corollary 8]). It is unknownwhether we can derive G1 based on Yablo’s Paradox only assuming that T is consistent. Definition 3.18 ([89]) . A formula Y R ( x ) is called a Rosser-type Yabloformula of Prf T ( x, y ) if PA ⊢ ∀ x ( Y R ( x ) ↔ ∀ y > x ¬ Pr RT ( x )( p Y R ( ˙ y ) q )).Theorem 3.19 shows that the Rosser-type Yablo formula is independentof any Σ -sound theory T . Theorem 3.19 (Theorem 10, [89]) . Let
Prf T ( x, y ) be any standard proofpredicate of T , and Y R ( x ) be any Rosser-type Yablo formula of Prf T ( x, y ) .Given n ∈ ω , if T is consistent, then T Y R ( n ) ; if T is Σ -sound, then T ¬ Y R ( n ) . The independence of Y R ( n ) for T which is not Σ -sound is discussed in[89]. For a consistent but not Σ -sound theory, the situation of Rosser-typeYablo formulas is quite different from that of Rosser sentences. Kurahashi[89] shows that for any consistent but not Σ -sound theory, the independenceof each instance of a Rosser-type Yablo formula depends on the choice ofstandard proof predicates (see Theorem 12 and Theorem 25 in [89]). Kura-hashi [89] shows that for any consistent but not Σ -sound theory T , thereis a standard proof predicate of T such that each instance Y R ( n ) of theRosser-type Yablo formula Y R ( x ) based on this proof predicate is provablein T for any n ∈ ω . Moreover, Kurahashi [89] constructs a standard proofpredicate of T and a Rosser-type Yablo formula Y R ( x ) based on this proofpredicate such that each instance of Y R ( x ) is independent of T . Proofs ofthese results use the technique of Guaspari and Solovay in [52]. Cie´sli´nski and Urbaniak [27] conjecture that any two distinct instances Y R ( m ) and Y R ( n ) of a Rosser-type Yablo formula Y R ( x ) based on a stan-dard proof predicate are not provably equivalent. Leach-Krouse [88] con-structs a standard proof predicate, and a Rosser-type Yablo formula Y R ( x )based on this proof predicate such that T ⊢ ∀ x ∀ y ( Y R ( x ) ↔ Y R ( y )).Kurahashi [89] constructs a counterexample to Cie´sli´nski and Urbaniak’sconjecture: a standard proof predicate, and a Rosser-type Yablo formula Y R ( x ) based on this proof predicate such that ∀ x ∀ y ( Y R ( x ) ↔ Y R ( y )) isnot provable in T (Corollary 20, [89]). Thus the provability of the sentence ∀ x ∀ y ( Y R ( x ) ↔ Y R ( y )) also depends on the choice of standard proof pred-icates (see Corollary 20-21, [89]). Proofs of these results by Kurahashi andLeach-Krouse also use the technique of Guaspari and Solovay in [52]. Aninteresting open question is: whether there is a standard proof predicatesuch that Y R ( n ) and Y R ( n + 1) are not provably equivalent for some n ∈ ω (see [89]).3.4.5. Beyond arithmetization.
All the proofs of G1 we have discussed usearithmetization. Andrzej Grzegorczyk proposes the theory TC in [49] as apossible alternative theory for studying incompleteness and undecidability,and shows that TC is essentially incomplete and mutually interpretable with Q without arithmetization.Now, in PA we have numbers that can be added or multiplied; while in TC , one has strings (or texts) that can be concatenated. In G¨odel’s proof,the only use of numbers is coding of syntactical objects. The motivations foraccepting strings rather than numbers as the basic notion are as follows: onmetamathematical level, the notion of computability can be defined withoutreference to numbers; for Grzegorczyk, dealing with texts is philosophicallybetter justified since intellectual activities like reasoning, communicatingor even computing involve working with texts not with numbers (see [49]).Thus, it is natural to define notions like undecidability directly in terms oftexts instead of natural numbers. Grzegorczyk only proves the incomplete-ness of TC in [49]. Later, Grzegorczyk and Zdanowski [50] prove that TC is essentially incomplete.3.5. Concrete incompleteness.
Introduction.
All proofs of G¨odel’s incompleteness theorems we havediscussed above make use of meta-mathematical or logical methods, and theindependent sentence constructed has a clear meta-mathematical or logicalflavour which is devoid of real mathematical content. To be blunt, froma purely mathematical point of view, G¨odel’s sentence is artificial and notmathematically interesting. G¨odel’s sentence is constructed not by reflectingabout arithmetical properties of natural numbers, but by reflecting aboutan axiomatic system in which those properties are formalized (see [63]). Anatural question is then: can we find true sentences not provable in PA withreal mathematical content? The research program concrete incompleteness is the search for natural independent sentences with real mathematical con-tent. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 27
This program has received a lot of attention because despite G¨odel’s in-completeness theorems, one can still cherish the hope that all natural andmathematically interesting sentences about natural numbers are provable orrefutable in PA , and that elementary arithmetic is complete w.r.t. naturaland mathematically interesting sentences. However, after G¨odel, many nat-ural independent sentences with real mathematical content have been found.These independent sentences have a clear mathematical flavor, and do notrefer to the arithmetization of syntax and provability.In this section, we provide an overview of the research on concrete incom-pleteness. The survey paper [16] provides a good overview on the state-of-the-art up to Autumn 2006. For more detailed discussions about concreteincompleteness, we refer to Cheng [22], Bovykin [16] and Friedman [41].3.5.2. Paris-Harrington and beyond.
Paris and Harrington [108] proposethe first mathematically natural statement independent of PA : the Paris-Harrington Principle PH which generalizes the finite Ramsey theorem . G¨odel’ssentence is a pure logical construction (via the arithmetization of syntaxand 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 clas-sic mathematics with combinatorial content as we will show. We refer to [63]for more discussions about the distinction between mathematical arithmeticsentences and meta-mathematical arithmetic sentences.For set X and n ∈ ω , let [ X ] n be the set of all n -elements subset of X .We identify n with { , · · · , n − } . Definition 3.20 (Paris-Harrington Principle ( PH ), [108]) . For all m, n, c ∈ ω , there is N ∈ ω such that for all f : [ N ] m → c , we have: ( ∃ H ⊆ N )( | H | ≥ n ∧ H is homogeneous for f ∧ | H | > min( H )). Theorem 3.21 (Paris-Harrington, [108]) . The principle PH is true but notprovable in PA . Now, PH has a clear combinatorial flavor, and is of the form ∀ x ∃ yψ ( x, y )where ψ is a ∆ formula. It can be shown that for any given natural number n , PA ⊢ ∃ yψ ( n, y ), i.e. all particular instances of PH are provable in PA .Following PH , many other mathematically natural statements indepen-dent of PA with combinatorial or number-theoretic content have been for-mulated: the Kanamori-McAloon principle [70], the Kirby-Paris sentence[109], the Hercules-Hydra game [109], the Worm principle [9, 58], the flip-ping principle [79], the arboreal statement [97], the kiralic and regal prin-ciples [28], and the Pudl´ak’s principle [112, 54] (see [16], p.40). In fact, allthese principles are equivalent to PH (see [16], p.40).An interesting and amazing fact is: all the above mathematically nat-ural principles are in fact provably equivalent in PA to a certain meta-mathematical sentence. Consider the following reflection principle for Σ sentences: for any Σ sentence φ in L ( PA ), if φ is provable in PA , then φ is true. Using the arithmetization of syntax, one can write this princi-ple as a sentence of L ( PA ), and denote it by Rfn Σ ( PA ) (see [102, p.301]).McAloon has shown that PA ⊢ PH ↔ Rfn Σ ( PA ) (see [102], p.301), and similar equivalences can be established for the other independent princi-ples mentioned above. Equivalently, all these principles are equivalent toso-called 1-consistency of PA (see [8, p.36], [9, p.3] and [102, p.301]).The above phenomenon indicates that the difference between mathemat-ical and meta-mathematical statements is perhaps not as huge as we mighthave expected. Moreover, the above principles are provable in fragmentsof Second-Order Arithmetic and are more complex than G¨odel’s sentence:G¨odel’s sentence is equivalent to Con ( PA ) in PA ; but all these principlesare not only independent of PA but also independent of PA + Con ( PA )(see [8, p.36] and [102, p.301]).3.5.3. Harvey Friedman’s contributions.
Incompleteness would not be com-plete without mentioning the work of Harvey Friedman who is a centralfigure in research on the foundations of mathematics after G¨odel. He hasmade many important contributions to concrete mathematical incomplete-ness. The following quote is telltale:the long range impact and significance of ongoing investiga-tions in the foundations of mathematics is going to dependgreatly on the extent to which the Incompleteness Phenom-ena touches normal concrete mathematics (see [41], p.7).In the following, we give a brief introduction to H. Friedman’s work on con-crete mathematical incompleteness. In his early work, H. Friedman exam-ines how one uses large cardinals in an essential and natural way in numbertheory, as follows.the quest for a simple meaningful finite mathematical the-orem that can only be proved by going beyond the usualaxioms for mathematics has been a goal in the foundationsof mathematics since G¨odel’s incompleteness theorems (see[40], p.805).H. Friedman shows in [39, 40] that there are many mathematically naturalcombinatorial statements in L ( PA ) that are neither provable nor refutablein ZFC or ZFC + large cardinals. H. Friedman’s more recent monograph[41] is a comprehensive study of concrete mathematical incompleteness. H.Friedman studies concrete mathematical incompleteness over different sys-tems, ranging from weak subsystems of PA to higher-order arithmetic and ZFC . H. Friedman lists many concrete mathematical statements in L ( PA )that are independent of subsystems of PA , or stronger theories like higher-order arithmetic and set theory.The theories RCA (Recursive Comprehension), WKL (Weak Konig’sLemma), ACA (Arithmetical Comprehension), ATR (Arithmetic Trans-finite Recursion) and Π - CA (Π -Comprehension) are the most famousfive subsystems of Second-Order Arithmetic ( SOA ), and are called the ‘BigFive’. For the definition of
SOA and the ‘Big Five’, we refer to [120].To give the reader a better sense of H. Friedman’s work, we list somesections dealing with concrete mathematical incompleteness in [41]. • Section 0.5 on Incompleteness in Exponential Function Arithmetic;
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 29 • Section 0.6 on Incompleteness in Primitive Recursive Arithmetic,Single Quantifier Arithmetic,
RCA , and WKL ; • Section 0.7 on Incompleteness in Nested Multiply Recursive Arith-metic and Two Quantifier Arithmetic; • Section 0.8 on Incompleteness in Peano Arithmetic and
ACA ; • Section 0.9 on Incompleteness in Predicative Analysis and
ATR ; • Section 0.10 on Incompleteness in Iterated Inductive Definitions andΠ - CA ; • Section 0.11 on Incompleteness in second-order arithmetic and
ZFC − ; • Section 0.12 on Incompleteness in Russell Type Theory and ZermeloSet Theory; • Section 0.13 on Incompleteness in
ZFC using Borel Functions; • Section 0.14 on Incompleteness in
ZFC using Discrete Structures.H. Friedman [41] provides us with examples of concrete mathematicaltheorems not provable in subsystems of Second-Order Arithmetic strongerthan PA , and a number of concrete mathematical statements provable inThird-Order Arithmetic but not provable in Second-Order Arithmetic.Related to Friedman’s work, Cheng [22, 25] gives an example of concretemathematical theorems based on Harrington’s principle which is isolatedfrom the proof of the Harrington’s Theorem (the determinacy of Σ gamesimplies the existence of zero sharp), and shows that this concrete theoremsaying that Harrington’s principle implies the existence of zero sharp is ex-pressible in Second-Order Arithmetic, not provable in Second-Order Arith-metic or Third-Order Arithmetic, but provable in Fourth-Order Arithmetic(i.e. the minimal system in higher-order arithmetic to prove this concretetheorem is Fourth-Order Arithmetic).Many other examples of concrete mathematical incompleteness, and thediscussion of this subject in 1970s-1980s can be found in the four volumes[126, 125, 106, 11]. Weiermann’s work in [144]-[149] provides us with moreexamples of naturally mathematical independent sentences. We refer to [41]for new advances in Boolean Relation Theory and for more examples ofconcrete mathematical incompleteness.4. The limit of the applicability of G1 Introduction.
In this section, we discuss the limit of the applicabilityof G1 based on the following two questions. • To what extent does G1 apply to extensions of PA ? • To what extent does G1 apply to theories weaker than PA w.r.t. in-terpretation?We list some generalizations of G1 needed below. Let Γ denote either Σ n or Π n for some n ≥
1, and Γ d denote either Π n or Σ n . We say a sentence ϕ is Γ-conservative over theory T if for any Γ sentence ψ , T ⊢ ψ whenever T + ϕ ⊢ ψ . Fact 4.1 (Guaspari [51]) . Let T be a consistent r.e. extension of Q . Thenthere is a Γ d sentence φ such that φ is Γ-conservative over T and T φ . ZFC − denotes ZFC with the Power Set Axiom deleted and Collection instead ofReplacement. If T ⊢ ¬ φ , then φ is not Γ-conservative over T because T is consistent.Thus, we can view Fact 4.1 as an extension of Rosser’s first incompletenesstheorem. Solovay improves this fact and shows that there is a Γ d sentence φ such that φ is Γ-conservative over T , ¬ φ is Γ d -conservative over T , but φ is independent of T . Fact 4.2 (Mostowski [100]) . Let { T n : n ∈ ω } be an r.e. sequence of consis-tent theories extending Q . Then there is a Π sentence φ such that for any n ∈ ω , T n φ , and T n ¬ φ .4.2. Generalizations of G1 beyond PA. We study generalization of G1 for extensions of PA w.r.t. interpretation. We know that G1 applies to allconsistent r.e. extensions of PA . A natural question is then: whether G1 can be extended to non-r.e. arithmetically definable extensions of PA .Kikuchi [74] and Salehi-Seraji [118] make contributions to generalize G¨odel-Rosser’s first incompleteness theorem to non-r.e. arithmetically definableextensions of PA . Definition 4.3 ([74]) . Let T be a consistent extension of Q .(1) T is Σ n -definable if there is a Σ n formula φ ( x ) such that n is the G¨odelnumber of some sentence of T if and only if N | = φ ( n ). (2) T is Σ n -sound if for all Σ n sentences φ , T ⊢ φ implies N | = φ ; T is soundif T is Σ n -sound for any n ∈ ω .(3) T is Σ n -consistent if for all Σ n formulas φ with φ = ∃ xθ ( x ) and θ ∈ Π n − ,if T ⊢ ¬ θ ( n ) for all n ∈ ω , then T φ .(4) T is Π n -decisive if for all Π n sentences φ , either T ⊢ φ or T ⊢ ¬ φ holds.From G1 , we have: if T is a Σ -definable and Σ -sound extension of Q ,then T is not Π -decisive. Kikuchi [74] and Salehi-Seraji [118] generalize G1 to arithmetically definable theories via the notion of “Σ n -sound”. Theorem 4.4 (Theorem 4.8 [74], Theorem 2.5 [118]) . If T is a Σ n +1 -definable and Σ n -sound extension of Q , then T is not Π n +1 -decisive. Salehi and Seraji [118] point out that Theorem 4.4 has a constructiveproof: given a Σ n +1 -definable and Σ n -sound extension T of Q , one caneffectively construct a Π n +1 sentence which is independent of T . The opti-mality of Theorem 4.4 is shown by Salehi and Seraji in [118]: there existsa Σ n − -sound and Σ n +1 -definable complete extension of Q for any n ≥ G1 to arithmetically definable theoriesvia the notion of “Σ n -consistent”. Theorem 4.5 (Theorem 4.9 [74], Theorem 4.3 [118]) . If T is a Σ n +1 -definable and Σ n -consistent extension of Q , then T is not Π n +1 -decisive. Theorem 4.5 is also optimal: the complete Σ n − -sound and Σ n +1 -definabletheory constructed in the proof of Theorem 2.6 in [118] is also Σ n − -consistentsince if a theory is Σ n -sound, then it is Σ n -consistent. The proof of Theo-rem 4.5 cannot be constructive as the following theorem shows. Recall that N is the standard model of arithmetic. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 31
Theorem 4.6 (Non-constructivity of Σ n -consistency incompleteness, The-orem 4.4, [118]) . For n ≥ , there is no (partial) recursive function f (evenwith the oracle n ) such that if m codes (the G¨odel code) a Σ n +1 -formulawhich defines an Σ n -consistent extension T of Q , then f ( m ) halts and codesa Π n +1 sentence which is independent of T . In a summary, G1 can be generalized to the incompleteness of Σ n +1 -definable and Σ n -sound extensions of Q constructively; and to the incom-pleteness of Σ n +1 -definable and Σ n -consistent extensions of Q non-constructively(when n > Generalizations of G1 below PA. We study generalizations of G1 for theories weaker than PA w.r.t. interpretation.4.3.1. Generalizations of G1 via interpretability. We show that G1 can begeneralized to theories weaker than PA via interpretability. Indeed, thereexists a weak recursively axiomatizable consistent subtheory T of PA suchthat each recursively axiomatizable theory S in which T is interpretable isincomplete (see [129]). To generalize this fact further, we propose a newnotion “ G1 holds for T ”, as follows. Definition 4.7.
Let T be a consistent r.e. theory. We say G1 holds for T iffor any recursively axiomatizable consistent theory S , if T is interpretablein S , then S is incomplete.First of all, for a consistent r.e. theory T , it is not hard to show that thefollowings are equivalent (see [23]):(1) G1 holds for T ;(2) T is essentially incomplete;(3) T is essentially undecidable.It is well-known that G1 holds for many weaker theories than PA w.r.t. in-terpretation (e.g. Robinson arithmetic Q ).Secondly, we mention theories weaker than PA w.r.t. interpretation forwhich G1 holds. We first review some essentially undecidable theories weakerthan PA w.r.t. interpretation in the literature (i.e. G1 holds for these the-ories). For the definition of theory Q , I Σ n , B Σ n , PA − , Q + , Q − , S , AS , EA , PRA , R , R , R and R , we refer to Section 2.Robinson shows that any consistent r.e. theory that interprets Q is un-decidable, and hence Q is essentially undecidable. The fact that Q is es-sentially undecidable is very useful and can be used to prove the essentiallyundecidability of other theories via Theorem 2.5. Since Q is finitely axioma-tized, it follows that any theory that weakly interprets Q is also undecidable.The Lindenbaum algebras of all r.e. theories that interpret Q are recur-sively isomorphic (see Pour-El and Kripke [110]). In fact, Q is minimalessentially undecidable in the sense that if deleting any axiom of Q , thenthe remaining theory is not essentially undecidable and has a complete de-cidable extension (see [129, Theorem 11, p.62]). Salehi and Seraji [118] remark that there indeed exists some 0 n +1 -(total) recursivefunction f such that if m codes a Σ n +1 -formula defining an Σ n -consistent extension T of Q , then f ( m ) halts and codes a Π n +1 sentence independent of T . Thirdly, Nelson [103] embarks on a program of investigating how muchmathematics can be interpreted in Robinson’s Arithmetic Q : what can beinterpreted in Q , and what cannot be interpreted in Q . In fact, Q rep-resents a rich degree of interpretability since a lot of stronger theories areinterpretable in it as we will show in the following passages. For example,using Solovay’s method of shortening cuts (see [52]), one can show that Q interprets fairly strong theories like I ∆ + Ω on a definable cut.Fourth, we discuss some prominent fragments of PA extending Q fromthe literature. As a corollary of Theorem 2.12, we have: • The theories Q , I Σ , I Σ +Ω , · · · , I Σ +Ω n , · · · , B Σ , B Σ +Ω , · · · , B Σ + Ω n , · · · are all mutually interpretable; • I Σ + exp and B Σ + exp are mutually interpretable; • For n ≥ I Σ n and B Σ n +1 are mutually interpretable; • Q ✁ I Σ + exp ✁ I Σ ✁ I Σ ✁ · · · ✁ I Σ n ✁ · · · ✁ PA .Since any consistent r.e. theory which interprets Q is essentially undecidable, G1 holds for all these fragments of PA extending Q .Fifth, we discuss some weak theories mutually interpretable with Q in theliterature. It is interesting to compare Q with its bigger brother PA − . From[137], PA − is interpretable in Q , and hence Q is mutually interpretable with PA − . The theory Q + is interpretable in Q (see Theorem 1 in [36], p.296),and thus mutually interpretable with Q . A. Grzegorczyk asks whether Q − is essentially undecidable. ˇSvejdar [128] provids a positive answer to Grze-gorczyk’s original question by showing that Q is interpretable in Q − usingthe Solovay’s method of shortening cuts. Thus Q − is essentially undecidableand mutually interpretable with Q .Sixth, by [36], I Σ is interpretable in S , and S is interpretable in Q .Hence S is essentially undecidable and mutually interpretable with Q . Thetheory AS interprets Robinson’s Arithmetic Q , and hence is essentiallyundecidable. Nelson [103] shows that AS is interpretable in Q . Thus, AS is mutually interpretable with Q .Seventh, Grzegorczyk and Zdanowski [50] formulate but leave unansweredan interesting problem: are TC and Q mutually interpretable? M. Ganea[43] provs that Q is interpretable in TC using the detour via Q − (i.e. firstshow that Q − is interpretable in TC ; since Q is interpretable in Q − , thenwe have Q is interpretable in TC ). Sterken and Visser [134] give a proof ofthe interpretability of Q in TC not using Q − . Note that TC is easily inter-pretable in the bounded arithmetic I Σ . Thus, TC is mutually interpretablewith Q .Note that R ✁ Q since Q is not interpretable in R (if Q is interpretable in R , then Q is interpretable in some finite fragment of R ; however R is locallyfinitely satisfiable and any model of Q is infinite). Visser [136] provides uswith a unique characterization of R . Theorem 4.8 (Visser, Theorem 6, [136]) . For any consistent r.e. theory T , T is interpretable in R if and only if T is locally finitely satisfiable. In fact, if T is locally finitely satisfiable, then T is interpretable in R via a one-pieceone-dimensional parameter-free interpretation. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 33
Since relational Σ sentences have the finite model property, by Theo-rem 4.8, any consistent theory axiomatized by a recursive set of Σ sentencesin a finite relational language is interpretable in R . Since all recursive func-tions are representable in R (see [129, theorem 6], p.56), as a corollary ofTheorem 2.5, R is essentially undecidable. Cobham shows that R has astronger property than essential undecidability. Vaught gives a proof ofCobham’s theorem 4.9 via existential interpretation in [132]. Theorem 4.9 (Cobham, [132]) . Any consistent r.e. theory that weaklyinterprets R is undecidable. Eighth, we discuss some variants of R in the same language as L ( R ) = { , · · · , n, · · · , + , × , ≤} . The theory R is no longer essentially undecidablein the same language as R . In fact, whether R is essentially undecidabledepends on the language of R : if L ( R ) = { , S , + , × , ≤} with ≤ defined interms of +, then R is essentially undecidable (Cobham first observed that R is interpretable in R in the same language { , S , + , ×} , and hence R is essentially undecidable (see [132] and [68])). The theory R is essentiallyundecidable since R is interpretable in R (see [68], p.62).However R is not minimal essentially undecidable. From [68], R is in-terpretable in R , and hence R is essentially undecidable. The theory R is minimal essentially undecidable in the sense that if we delete anyaxiom scheme of R , then the remaining system is not essentially undecid-able. By essentially the same argument as in [137], we can show that anyconsistent r.e. theory that weakly interprets R is undecidable.Kojiro Higuchi and Yoshihiro Horihata introduce the theory of concate-nation WTC − ǫ , which is a weak subtheory of Grzegorczyk’s theory TC ,and show that WTC − ǫ is minimal essentially undecidable and WTC − ǫ ismutually interpretable with R (see [60]).In summary, we have the following pictures:(1) Theories PA − , Q + , Q − , TC , AS , S and Q are all mutually interpretable,and hence G1 holds for them;(2) Theories R , R , R and WTC − ǫ are mutually interpretable, and hence G1 holds for them;(3) R ✁ Q ✁ EA ✁ PRA ✁ PA .4.3.2. The limit of G1 w.r.t. interpretation and Turing reducibility. We dis-cuss the limit of G1 for theories weaker than PA w.r.t. interpretation,i.e. finding a theory with minimal degree of interpretation for which G1 holds.First of all, a natural question is: is Q the weakest finitely axiomatizedessentially undecidable theory w.r.t. interpretation such that R ✁ Q ? The The theory R has a decidable complete extension given by the theory of reals with ≤ as the empty relation on reals. Another way to show that R is essentially undecidable is to prove that all recursivefunctions are representable in R . If we delete
Ax2 , then the theory of natural numbers with x × y defined as x + y isa complete decidable extension; if we delete Ax3 , then the theory of models with only oneelement is a complete decidable extension; if we delete
Ax4 ′ , then the theory of reals is acomplete decidable extension. following theorem tells us that the answer is no: for any finitely axioma-tized subtheory A of Q that extends R , we can find a finitely axiomatizedsubtheory B of A such that B extends R and B does not interpret A . Theorem 4.10 (Visser, Theorem 2, [137]) . Suppose A is a finitely axiom-atized consistent theory and R ⊆ A . Then there is a finitely axiomatizedtheory B such that R ⊆ B ⊆ A and B ✁ A . Define X = { S : R ✂ S ✁ Q and S is finitely axiomatized } . Theorem 4.10shows that the structure h X, ✁ i is not well-founded. Theorem 4.11 (Visser, Theorem 12, [137]) . Suppose A and B are finitelyaxiomatized theories that weakly interpret Q . Then there are finitely axiom-atized theories A ⊇ A and B ⊇ B such that A and B are incomparable (i.e. A B and B A ). Theorem 4.11 shows that there are incomparable theories extending Q w.r.t. interpretation.Up to now, we do not have an example of essentially undecidable the-ory that is weaker than R w.r.t. interpretation. To this end, we introduceJeˇ r ´ a bek’s theory Rep
PRF . Let
PRF denote the sets of all partial recursivefunctions. The language L ( Rep
PRF ) consists of constant symbols n for each n ∈ ω , and function symbols f of appropriate arity for each partial recursivefunction f . Definition 4.12 (The system
Rep
PRF ) . The theory
Rep
PRF has axioms: • n = m for n = m ∈ ω ; • f ( n , · · · , n k − ) = m for each k -ary partial recursive function f suchthat f ( n , · · · , n k − ) = m where n , · · · , n k − , m ∈ ω .The theory Rep
PRF is essentially undecidable since all recursive functionsare representable in it. Since
Rep
PRF is locally finitely satisfiable, by The-orem 4.8,
Rep
PRF ✂ R . Jeˇ r ´ a bek [67] proves that R is not interpretable in Rep
PRF . Thus
Rep
PRF ✁ R .Cheng [23] provides more examples of a theory S such that G1 holds for S and S ✁ R , and shows that we can find many theories T such that G1 holdsfor T and T ✁ R based on Jeˇ r ´ a bek’s work [67] which uses model theory. Theorem 4.13 (Cheng, [23]) . For any recursively inseparable pair h A, B i ,there is a r.e. theory U h A,B i such that G1 holds for U h A,B i , and U h A,B i ✁ R . Define D = { S : S ✁ R and G1 holds for theory S } . Theorem 4.13shows that we could find many witnesses for D . Naturally, we could ask thefollowing questions: Question 4.14. (1) Is h D , ✁ i well-founded?(2) Are any two elements of h D , ✁ i comparable?(3) Does there exist a minimal theory w.r.t. interpretation such that G1 holds for it?We conjectured the following answers to these questions: h D , ✁ i is notwell founded, h D , ✁ i has incomparable elements, and there is no minimaltheory w.r.t. interpretation for which G1 holds. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 35
Finally, let R be the structure of the r.e. degrees with the ordering ≤ T induced by Turing reducibility with the least element and the greatestelement ′ . We have discussed the limit of applicability of G1 w.r.t. inter-pretation. A natural question is: what is the limit of applicability of G1 w.r.t. Turing reducibility. Let D = { S : S < T R and G1 holds for theory S } where S < T R stands for S ≤ T R but R (cid:2) T S .Cheng [23] shows that for any Turing degree < d < ′ , there is a theory U such that G1 holds for U , U < T R , and U has Turing degree d . Asa corollary of this result and known results about the degree structure of hR , < T i in recursion theory, we can answer above questions for the structure h D , < T i : Theorem 4.15 (Cheng, [23]) . (1) h D , < T i is not well-founded;(2) h D , < T i has incomparable elements;(3) There is no minimal theory w.r.t. Turing reducibility such that G1 holdsfor it. Moreover, Cheng [23] shows that for any Turing degree < d < ′ , thereis a theory U such that G1 holds for U , U ✂ R , and U has Turing degree d . Thus, examining the limit of applicability of G1 w.r.t. interpretation ismuch harder than that w.r.t. Turing reducibility. The structure of h D , ✁ i isa deep and interesting open question for future research.5. The limit of the applicability of G2 Introduction.
In our view, G2 is fundamentally different from G1 .In fact, both mathematically and philosophically, G2 is more problematicthan G1 for the following reason. On one hand, in the case of G1 , we canconstruct a natural independent sentence with real mathematical content without referring to arithmetization and provability predicates. On the otherhand, the meaning of G2 strongly depends on how we exactly formulate theconsistency statement.Similar to [56], we call a result intensional if it depends on (the detailsof) the representation used. Thus, G1 can be called extensional (that is,non-intensional), while G2 is (highly) intensional. We refer to Section 5.3for more discussion on the intensionality of G2 . In this section, we discussthe limit of applicability of G2 : under what conditions G2 holds, and underwhat conditions G2 fails. In Section 5.2, we discuss generalizations of G2 .5.2. Some generalizations of G2 . After G¨odel, generalizations of G2 arethe subject of extensive studies. We know that G2 holds for any consistentr.e. extension of PA . However, it is not true that G2 holds for any extensionof PA . For example, Karl-Georg Niebergall [104] shows that the theory( PA + RFN ( PA )) ∩ ( PA + all true Π sentences) can prove its own canonicalconsistency sentence. Similarly to G1 , one can generalise G2 to arithmetically definable non-r.e. extensions of PA . Kikuchi and Kurahashi [74] reformulate G2 as: if S is a For the definition of
RFN ( PA ), we refer to [95]: RFN ( PA ) = {∀ x ((Γ( x ) ∧ Pr PA ( x )) → Tr Γ ( x )) : Γ arbitrary } . Σ -definable and consistent extension of PA , then for any Σ definition σ ( u )of S , S Con σ ( S ) (see fact 5.1 [74]). Kikuchi and Kurahashi [74] generalize G2 to arithmetically definable non-r.e. extensions of PA and prove that if S is a Σ n +1 -definable and Σ n -sound extension of PA , then there exists aΣ n +1 definition σ ( u ) of some axiomatization of T h ( S ) such that Con σ ( S ) isindependent of S . This corollary shows that the witness for the generalizedversion of G1 can be provided by the appropriate consistency statement.Chao and Seraji [21] give another generalization of G2 to arithmeticallydefinable non-r.e. extensions of PA : for each n ∈ ω , any Σ n +1 -definableand Σ n -sound extension of PA cannot prove its own Σ n -soundness (see [21,Theorem 2]). The optimality of this generalization is shown in [21]: thereis a Σ n +1 -definable and Σ n − -sound extension of PA that proves its ownΣ n − -soundness for n > T be a consistent r.e. extension of Q . Kreisel [85] shows that ¬ Con ( T )is Π -conservative over T which is a generalization of G2 . We can alsogeneralize G2 via the notion of standard provability predicate. Theorem 5.1.
Let T be any consistent r.e. extension of Q . If Pr T ( x ) is astandard provability predicate, then T Con ( T ) . Lev Beklemishev and Daniyar Shamkanov [10] prove that in an abstractsetting that presupposes the presence of G¨odel’s fixed point (instead of di-rectly constructing it, as in the case of formal arithmetic), the Hilbert-Bernays-L¨ob conditions implies G2 even with fairly minimal conditions onthe underlying logic. The following two theorems, due to Feferman andVisser, generalize G2 in terms of the notion of interpretation. Theorem 5.2 (Feferman’s theorem on the interpretability of inconsistency,[33]) . If T is a consistent r.e. extension of Q , then T + ¬ Con ( T ) is inter-pretable in T . Theorem 5.3 (Pudl´ak, [113, 55]) . There is no consistent r.e. theory S such that ( Q + Con ( S )) ✂ S . As a corollary of Theorem 5.3, for any consistent r.e. theory S that in-terprets Q , G2 holds for S : S Con ( S ). The Arithmetic CompletenessTheorem tells us that S ✂ ( Q + Con ( S )) (see [135] for the details). As acorollary, we have the following version of G2 which highlights the inter-pretability power of consistency statements. Corollary 5.4.
For any consistent r.e. theory S , we have S ✁ ( Q + Con ( S )) . Definition 5.5.
Let T be a consistent extension of Q . A formula I ( x ) withone free variable (understood as a number variable) is a definable cut in T (in short, a T -cut) if(1) T ⊢ I ( );(2) T ⊢ ∀ x ( I ( x ) → I ( x + 1));(3) T ⊢ ∀ x ∀ y ( y < x ∧ I ( x ) → I ( y )). Instead of Robinson’s Arithmetic Q , we can as well have taken S , or PA − , or I ∆ + Ω . Moreover, instead of an arithmetical theory we can have employed a stringtheory like Grzegorczyk’s theory TC or adjunctive set theory AS . All these theories arethe same in the sense that they are mutually interpretable (see [138]). URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 37
Definition 5.6.
Let T ⊇ I Σ , let J be a T -cut and let τ be a Σ exp -definitionof T . (1) Pr Iτ ( x ) is the formula ∃ y ( I ( y ) ∧ Proof Iτ ( x, y )) (saying that there is a τ -proof of x in I ).(2) Con Iτ is the formula ¬∃ y ( I ( y ) ∧ Proof Iτ ( = , y )).The following theorem generalizes G2 to definable cuts. Theorem 5.7 (Theorem 3.11, [55]) . Let T ⊇ I Σ , let J be a T -cut and τ a Σ exp -definition of T . Then T Con Iτ . Next, consider a theory U and an interpretation N of the Tarski-Mostowski-Robinson theory R in U . A U -predicate △ is an L -predicate for U, N if itsatisfies the following L¨ob conditions. We write △ A for △ ( p A q ), where p A q is the numeral of the G¨odel number of A and we interpret the numbers via N . The G¨odel numbering is supposed to be fixed and standard. Definition 5.8 (L¨ob conditions) .L1: ⊢ A ⇒ ⊢ △ A ; L2: △ A, △ ( A → B ) ⊢ △ B ; L3: △ A ⊢ △△ A . Proposition 5.9 (L¨ob’s theorem, Theorem 3.3.2, [138]) . Suppose that U isa theory, N is an interpretation of the theory R in U , and △ is a U -predicatethat is an L -predicate for U, N . Then:(1) For all U -sentences A we have: if U ⊢ △ A → A , then U ⊢ A ;(2) For all U -sentences A we have: U ⊢ △ ( △ A → A ) → △ A . As a corollary of Proposition 5.9, we formulate a general version of G2 which does not mention the notion of provability predicate. Theorem 5.10 (Visser, [138]) . For all consistent theories U and all inter-pretations N of R in U and all L -predicates △ for U, N , we have U ¬△ ⊥ . The intensionality of G2 . In this section, we discuss the intension-ality of G2 which reveals the limit of the applicability of G2 .5.3.1. Introduction.
For a consistent theory T , we say that G2 holds for T ifthe consistency of T is not provable in T . However, this definition is vague,and whether G2 holds for T depends on how we formulate the consistencystatement. We refer to this phenomenon as the intensionality of G2 . In fact, G2 is essentially different from G1 due to the intensionality of G2 : “whether G2 holds for the base theory” depends on how we formulate the consistencystatement in the first place.Both mathematically and philosophically, G2 is more problematic than G1 . In the case of G1 , we are mainly interested in the fact that some sentenceis independent of PA . We make no claim to the effect that that sentence“really” expresses what we would express by saying “ PA cannot prove thissentence”. But in the case of G2 , we are also interested in the content We extend the language L ( PA ) by a new unary function symbol 2 x for the x -th powerof two. The extended language is denoted L ( exp ). A formula is Σ exp if it results fromatomic formulas of L ( exp ) by iterated application of logical connectives and boundedquantifiers of the form ( ∀ x ≤ y ) or ( ∃ x ≤ y ) (see [55]). of the consistency statement. We can say that G1 is extensional in thesense that we can construct a concrete independent mathematical statementwithout referring to arithmetization and provability predicate. However, G2 is intensional and “whether G2 holds for T ” depends on varied factors as wewill discuss.In this section, unless stated otherwise, we assume the following: • T is a consistent r.e. extension of Q ; • the canonical arithmetic formula to express the consistency of thebase theory T is Con ( T ) , ¬ Pr T ( = ); • the canonical numbering we use is G¨odel’s numbering; • the provability predicate we use is standard; • the formula representing the set of axioms is Σ .The intensionality of G¨odel sentence and the consistency statement hasbeen widely discussed in the literature (e.g. Halbach-Visser [56, 57], Visser[135]). Halbach and Visser examine the sources of intensionality in the con-struction of self-referential sentences of arithmetic in [56, 57], and argue thatcorresponding to the three stages of the construction of self-referential sen-tences of arithmetic, there are at least three sources of intensionality: coding,expressing a property and self-reference. The three sources of intensionalityare not independent of each other, and a choice made at an earlier stage willhave influences on the availability of choices at a later stage. Visser [135]locates three sources of indeterminacy in the formalization of a consistencystatement for a theory T :(I) the choice of a proof system;(II) the choice of a way of numbering;(III) the choice of a specific formula representing the set of axioms of T .In summary, the intensional nature ultimately traces back to the variousparameter choices that one has to make in arithmetizing the provabilitypredicate. That is the source of both the intensional nature of the G¨odelsentence and the consistency sentence.Based on this and other works in the literature, we argue that “whether G2 holds for the base theory” depends on the following factors:(1) the choice of the provability predicate (Section 5.3.2);(2) the choice of the formula expressing consistency (Section 5.3.3);(3) the choice of the base theory (Section 5.3.4);(4) the choice of the numbering (Section 5.3.5);(5) the choice of the formula representing the set of axioms (Section 5.3.6).These factors are not independent, and a choice made at an earlier stagemay have effects on the choices available at a later stage. In the following,unless stated otherwise, when we discuss how G2 depends on one factor,we always assume that other factors are fixed, and only the factor we arediscussing is varied. For example, Visser [135] rests on fixed choices for (1)-(2) and (4)-(5) but varies the choice of (3); Grabmayr [47] rests on fixedchoices for (1)-(3) and (5) but varies the choice of (4); Feferman [33] restson fixed choices for (1)-(4) but varies the choice of (5). URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 39
The choice of provability predicate.
In this section, we show that“whether G2 holds for the base theory” depends on the choice of the prov-ability predicate we use.As Visser argues in [138], being a consistency statement is not an absoluteconcept but a role w.r.t. a choice of provability predicate (see Visser [138]).From Theorem 5.1, G2 holds for standard provability predicates. However, G2 may fail for non-standard provability predicates.Mostowski [101] gives an example of a Σ provability predicate for which G2 fails. Let Pr MT ( x ) be the Σ formula “ ∃ y ( Prf T ( x, y ) ∧ ¬ Prf T ( p = q , y ))” where Prf T ( x, y ) is a ∆ formula saying that “ y is a proof of x ”.Then ¬ Pr MT ( p = 0 q ) is trivially provable in PA . We know that G2 holds forprovability predicates satisfying D1 - D3 . Since the formula Pr MT ( x ) satisfies D1 and D3 , it does not satisfy D2 . Mostowski’s example [101] shows that G2 may fail for Σ provability predicates satisfying D1 and D3 .One important non-standard provability predicate is Rosser provabilitypredicate Pr RT ( x ) introduced by Rosser [116] to improve G¨odel’s first in-completeness theorem. Recall that we have defined the Rosser provabilitypredicate in Definition 3.7. The consistency statement Con R ( T ) based on aRosser provability predicate Pr RT ( x ) is naturally defined as ¬ Pr RT ( p = q ).It is an easy fact that for any sentence φ and Rosser provability predicate Pr RT ( x ), if T ⊢ ¬ φ , then T ⊢ ¬ Pr RT ( p φ q ) (see [78, Proposition 2.1]). As acorollary, the consistency statement Con R ( T ) based on Rosser provabilitypredicate Pr RT ( x ) is provable in T . In this sense, we can say that G2 fails forthe consistency statement constructed from Rosser provability predicates.We can construct different Rosser provability predicates with varied prop-erties. We know that each Rosser provability predicate Pr RT ( x ) does notsatisfy at least one of conditions D2 and D3 . Guaspari and Solovay [52]establish a very powerful method of constructing a new proof predicate withrequired properties from a given proof predicate by reordering nonstandardproofs. Applying this tool, Guaspari and Solovay [52] construct a Rosserprovability predicate for which both D2 and D3 fail. Arai [1] constructsa Rosser provability predicate with condition D2 , and a Rosser provabilitypredicate with condition D3 .Slow provability, introduced by S.D. Friedman, M. Rathjen and A. Weier-mann [42], is another notion of nonstandard provability for PA in the lit-erature. The slow consistency statement Con ∗ ( PA ) asserts that a contra-diction is not slow provable in PA (for the definition of Con ∗ ( PA ), werefer to [42]). In fact, G2 holds for slow provability: Friedman, Rathjen andWeiermann show that PA Con ∗ ( PA ) (see [42, Proposition 3.3]). More-over, Friedman, Rathjen and Weiermann [42] show that PA + Con ∗ ( PA ) Con ( PA ) (see [42, Theorem 3.10]), and the logical strength of the theory PA + Con ∗ ( PA ) lies strictly between PA and PA + Con ( PA ): PA PA + Con ∗ ( PA ) PA + Con ( PA ). Henk and Pakhomov [59] study threevariants of slow provability, and show that the associated consistency state-ment of each of these notions of provability yields a theory that lies strictlybetween PA and PA + Con ( PA ) in terms of logical strength. The choice of the formula expressing consistency.
We show that “whether G2 holds for the base theory” depends on the choice of the arithmetic for-mula used to express consistency. In the literature, an arithmetic formulais usually used to express the consistency statement. Artemov [3] arguesthat in Hilbert’s consistency program, the original formulation of consis-tency “no sequence of formulas is a derivation of a contradiction” is aboutfinite sequences of formulas, not about arithmetization, proof codes, andinternalized quantifiers.The canonical consistency statement, the arithmetical formula Con ( PA ),says that for all x , x is not a code of a proof of a contradiction in PA . Ina nonstandard model of PA , the universal quantifier “for all x ” ranges overboth standard and nonstandard numbers, and hence Con ( PA ) expresses theconsistency of both standard and nonstandard proof codes (see [3]). Thus, Con ( PA ) is stronger than the original formulation of consistency whichonly talks about sequences of formulas and such sequences have only stan-dard codes. Hence, Artemov [3] concludes that G2 , saying that PA cannotprove Con ( PA ), does not actually exclude finitary consistency proofs of theoriginal formulation of consistency (see [3]).Artemov shows that the original formulation of consistency admits a di-rect proof in informal arithmetic and this proof is formalizable in PA (see[3]). Artemov’s work establishes the consistency of PA by finitary meansand vindicates Hilbert’s consistency program to some extent.In the following, we use a single arithmetic sentence to express the con-sistency statement. Among consistency statements defined via arithmetiza-tion, there are three candidates of arithmetic formulas to express consistencyas follows:(1) Con ( T ) , ∀ x ( Fml ( x ) ∧ Pr T ( x ) → ¬ Pr T ( ˙ ¬ x )); (2) Con ( T ) , ¬ Pr T ( p = q );(3) Con ( T ) , ∃ x ( Fml ( x ) ∧ ¬ Pr T ( x )).G¨odel originally formulates G2 with the consistency statement Con ( T ):if T is consistent, then T Con ( T ). In the literature, Con ( T ) is thewidely used canonical consistency statement. Note that Con ( T ) implies Con ( T ), and Con ( T ) implies Con ( T ). However the converse implicationsdo not hold in general (see [92]). Kurahashi [92] proposes different sets ofderivability conditions (local version, uniform version and global version),and examines whether they are sufficient to show the unprovability of theseconsistency statements (e.g. Con ( T ) , Con ( T ) and Con ( T )). HB1: If T ⊢ φ → ϕ , then T ⊢ Pr T ( p φ q ) → Pr T ( p ϕ q ). HB2: T ⊢ Pr T ( p ¬ φ ( x ) q ) → Pr T ( p ¬ φ ( ˙ x ) q ). HB3: T ⊢ f ( x ) = 0 → Pr T ( p f ( ˙ x ) = 0 q ) for every primitive recursiveterm f ( x ). Informal arithmetic is the theory of informal elementary number theory containingrecursive identities of addition and multiplication as well as the induction principle. Theformal arithmetic PA is just the conventional formalization of the informal arithmetic(see [3]). Fml ( x ) is the formula which represents the relation that x is a code of a formula. URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 41
HB1 - HB3 is called the Hilbert-Bernays derivability condition. If a prov-ability predicate Pr T ( x ) satisfies HB1 - HB3 , then T Con ( T ) (see [62]).Kurahashi [93] constructes two Rosser provability predicates satisfying theHilbert-Bernays derivability condition. Thus, the Hilbert-Bernays derivabil-ity condition is not sufficient to prove that T Con ( T ).L¨ob [96] proves that if Pr T ( x ) satisfies the Hilbert-Bernays-L¨ob deriv-ability condition D1 - D3 (see Definition 3.5), then L¨ob’s theorem holds: forany sentence φ , if T ⊢ Pr T ( p φ q ) → φ , then T ⊢ φ . It is well-known thatL¨ob’s theorem implies G2 : T Con ( T ) (see [102]). Thus, if a provabil-ity predicate Pr T ( x ) satisfies D1 - D3 , then T Con ( T ). Kurahashi [92,Proposition 4.11] constructes a provability predicate Pr T ( x ) with conditions D1 - D3 , but T ⊢ Con ( T ). Thus, the Hilbert-Bernays-L¨ob derivability con-dition D1 - D3 is not sufficient to prove that T Con ( T ).Montagna [98] proves that if a provability predicate Pr T ( x ) satisfies thefollowing two conditions, then T Con ( T ):(1) T ⊢ ∀ x (“x is a logical axiom” → Pr T ( x ));(2) T ⊢ ∀ x ∀ y ( Fml ( x ) ∧ Fml ( y ) → ( Pr T ( x → y ) → ( Pr T ( x ) → Pr T ( y )))).5.3.4. The choice of base theory.
We show that “whether G2 holds for thebase theory” depends on the base theory we choose. A foundational ques-tion about G2 is: how much information about arithmetic is required for theproof of G2 . If the base system does not contain enough information aboutarithmetic, then G2 may fail. The widely used notion of consistency is con-sistency in proof systems with cut elimination. However, notions like cutfreeconsistency, Herbrand consistency, tableaux consistency, and restricted con-sistency for different base theories behave differently (see [135]). We do haveproof systems that prove their own cutfree consistency: for example, finitelyaxiomatized sequential theories prove their own cut-free consistency on adefinable cut (see [140], p.25).A natural question is: whether G2 can be generalized to base systemsweaker than PA w.r.t. interpretation. As a corollary of Theorem 5.3, wehave Q Con ( Q ) and hence Q Con ( Q ). Bezboruah and Shepherdson[12] define the consistency of Q as the sentence Con ( Q ) , and prove that G2 holds for Q : Q Con ( Q ). However, the method used by Bezboruahand Shepherdson in [12] is quite different from Theorem 5.3. Bezboruah-Shepherdson’s proof depends on some specific assumptions about the coding,does not easily generalize to stronger theories, and tells us nothing aboutthe question whether Q can prove its consistency on some definable cut (see[137]). The next question is: whether G2 holds for other theories weakerthan Q w.r.t. interpretation (e.g. R ). In a forthcoming paper, we will showthat G2 holds for R via the canonical consistency statement. However, wecan find weak theories mutually interpretable with R for which G2 fails.Willard [151] explores the generality and boundary-case exceptions of G2 over some base theories. Willard constructs examples of r.e. arithmeticaltheories that cannot prove the totality of their successor functions but can This sentence says that for any x , if x is the code of a formula φ and φ is provablein Q , then ¬ φ is not provable in Q . prove their own canonical consistencies (see [150], [151]). However, the the-ories Willard constructs are not completely natural since some axioms areconstructed using G¨odel’s Diagonalization Lemma. Pakhomov [107] con-structs a more natural example of this kind. Pakhomov [107] defines atheory H <ω , and shows that it proves its own canonical consistency. UnlikeWillard’s theories, H <ω isn’t an arithmetical theory but a theory formulatedin the language of set theory with an additional unary function. From [107], H <ω and R are mutually interpretable. Hence, the theory H <ω can be re-garded as the set-theoretic analogue of R from the interpretability theoreticpoint of view.From Theorem 5.3, G2 holds for any consistent r.e. theory interpreting Q . However, it is not true that G2 holds for any consistent r.e. theoryinterpreting R since H <ω interprets R , but G2 fails for H <ω . We know thatif S ✂ T and G1 holds for S , then G1 holds for T . However, it is not truethat if S ✂ T and G2 holds for S , then G2 holds for T since R ✂ H <ω , G2 holds for R but G2 fails for H <ω . This shows the difference between Q and R , and the difference between G1 and G2 .One way to eliminate the intensionality of G2 is to uniquely characterizethe consistency statement. In [135], Visser proposes the interesting ques-tion of a coordinate-free formulation of G2 and a unique characterization ofthe consistency statement. Visser [135] shows that consistency for finitelyaxiomatized sequential theories can be uniquely characterized modulo EA -provable equivalence (see [135], p.543). But characterizing the consistency ofinfinitely axiomatized r.e. theories is more delicate and a big open problemin the current research on the intensionality of G2 .After G¨odel, Gentzen constructs a theory T ∗ (primitive recursive arith-metic with the additional principle of quantifier-free transfinite inductionup to the ordinal ǫ ) , and proves the consistency of PA in T ∗ . Gentzen’stheory T ∗ contains Q but does not contain PA since T ∗ does not provethe ordinary mathematical induction for all formulas. By the ArithmetizedCompleteness Theorem, Q + Con ( PA ) interprets PA . Since Gentzen’s the-ory T ∗ contains Q and T ∗ ⊢ Con ( PA ), Gentzen’s theory T ∗ interprets PA . By Pudl´ak’s result that no consistent r.e. extension T of Q can in-terpret Q + Con ( T ), PA does not interpret Gentzen’s theory T ∗ . Thus PA ✁ T ∗ . Gentzen’s work has opened a productive new direction in prooftheory: finding the means necessary to prove the consistency of a giventheory. More powerful subsystems of Second-Order Arithmetic have beengiven consistency proofs by Gaisi Takeuti and others, and theories that havebeen proved consistent by these methods are quite strong and include mostordinary mathematics.5.3.5. The choice of numbering.
We show that “whether G2 holds for thebase theory” depends on the choice of the numbering encoding the language.For the influence of different numberings on G2 , we refer to [47]. Anyinjective function γ from a set of L ( PA )-expressions to ω qualifies as anumbering. G¨odel’s numbering is a special kind of numberings under whichthe G¨odel number of the set of axioms of PA is recursive. In fact, G2 is ǫ is the first ordinal α such that ω α = α . URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 43 sensitive to the way of numberings. Let γ be a numbering and p ϕ γ q denote γ ( ϕ ), i.e., the standard numeral of the γ -code of ϕ . Definition 5.11 (Relativized L¨ob conditions) . A formula Pr γT ( x ) is said tosatisfy L¨ob’s conditions relative to γ for the base theory T if for all L ( PA )-sentences ϕ and ψ we have that: D1 ∗ : If T ⊢ ϕ , then PA ⊢ Pr γT ( p ϕ γ q ); D2 ∗ : T ⊢ Pr γT ( p ( ϕ → ψ ) γ q ) → ( Pr γT ( p ϕ γ q ) → Pr γT ( p ψ γ q )); D3 ∗ : T ⊢ Pr γT ( p ϕ γ q ) → Pr γT ( p ( Pr γT ( p ϕ γ q )) γ q );Grabmayr [47] examines different criteria of acceptability, and proves theinvariance of G2 with regard to acceptable numberings (for the definition ofacceptable numberings, we refer to [47]). Theorem 5.12 (Invariance of G2 under acceptable numberings, Theorem4.8, [47]) . Let γ be an acceptable numbering and T be a consistent r.e. ex-tension of Q . If Pr γT ( x ) satisfies L¨ob’s conditions D1 ∗ - D3 ∗ relative to γ for T , then T ¬ Pr γT ( p ( = ) γ q ) . Theorem 5.12 shows that G2 holds for acceptable numberings. But G2 may fail for non-acceptable numberings. Grabmayr [47] gives some examplesof deviant numberings γ such that G2 fails w.r.t. γ : T ⊢ Pr γT ( p ( = ) γ q ). Definition 5.13.
We say that α ( x ) is a numeration of T if for any n , wehave PA ⊢ α ( n ) if and only if n is the G¨odel number of some φ ∈ T .5.3.6. The choice of the formula representing the set of axioms.
We showthat “whether G2 holds for T ” depends on the way the axioms of T arerepresented.First of all, Definition 5.14 gives a more general definition of provabilitypredicate and consistency statement for T w.r.t. the numeration of T . Definition 5.14.
Let T be any consistent r.e. extension of Q and α ( x ) bea formula in L ( T ).(1) Define the formula Prf α ( x, y ) saying “ y is the G¨odel number of a proof ofthe formula with G¨odel number x from the set of all sentences satisfying α ( x )”.(2) Define the provability predicate Pr α ( x ) of α ( x ) as ∃ y Prf α ( x, y ) and theconsistency statement Con α ( T ) as ¬ Pr α ( p = q ).For each formula α ( x ), we have: D2 ′ PA ⊢ Pr α ( p ϕ → ψ q ) → ( Pr α ( p ϕ q ) → Pr α ( p ψ q )) . If α ( x ) is a numeration of T , then Pr α ( x ) satisfies the following properties(see [90, Fact 2.2]): D1 ′ : If T ⊢ ϕ , then PA ⊢ Pr α ( p ϕ q ); D3 ′ : If ϕ is Σ , then PA ⊢ ϕ → Pr α ( p ϕ q ).Now we give a new reformulation of G2 via numerations. Theorem 5.15.
Let T be any consistent r.e. extension of Q . If α ( x ) is any Σ numeration of T , then T Con α ( T ) . In fact, G2 holds for any Σ numeration of T , but fails for some Π numeration of T . Feferman [33] constructs a Π numeration π ( x ) of T suchthat G2 fails, i.e. Con π ( T ) , ¬ Pr π ( p = q ) is provable in T . Feferman’sconstruction keeps the proof predicate and its numbering fixed but variesthe formula representing the set of axioms. Notice that Feferman’s predicatesatisfies D1 and D2 , but does not satisfy D3 . Feferman’s example showsthat G2 may fail for provability predicates satisfying D1 and D2 .Generally, Feferman [33] shows that if T is a Σ -definable extension of Q , then there is a Π definition τ ( u ) of T such that T ⊢ Con τ ( T ). Insummary, G2 is not coordinate-free (it is dependent on numerations of PA ).An important question is how to formulate G2 in a general way such that itis coordinate-free (independent of numerations of T ).The properties of the provability predicate are intensional and depend onthe numeration of the theory. I.e., under different numerations of T , theprovability predicate may have different properties. It may happen that T has two numerations α ( x ) and β ( x ) such that Con α ( T ) is not equivalentto Con β ( T ). For example, under G¨odel’s recursive numeration τ ( x ) andFeferman’s Π numeration π ( x ) of T , the corresponding consistency state-ment Con τ ( T ) and Con π ( T ) are not equivalent. But PA does not knowthis fact, i.e. PA ¬ ( Con τ ( T ) ↔ Con π ( T )) since PA ¬ Con τ ( T ).Generally, Kikuchi and Kurahashi prove in [74, Corollary 5.11] that if T is Σ n +1 -definable and not Σ n -sound, then there are Σ n +1 definitions σ ( x )and σ ( x ) of T such that T ⊢ Con σ ( T ) and T ⊢ ¬ Con σ ( T ).Provability logic is an important tool for the study of incompleteness andmeta-mathematics of arithmetic. The origins of provability logic (e.g. Henkin’sproblem, the isolation of derivability conditions, L¨ob’s theorem) are allclosely tied to G¨odel’s incompleteness theorems historically. In this sense, wecan say that G¨odel’s incompleteness theorems play a unifying role betweenfirst order arithmetic and provability logic.Provability logic is the logic of properties of provability predicates. Notethat G2 is very sensitive to the properties of the provability predicate used inits formulation. Provability logic provides us with a new viewpoint and animportant tool that can be used to understand incompleteness. Provabilitylogic based on different provability predicates reveals the intensionality ofprovability predicates which is one source of the intensionality of G2 .Let T be a consistent r.e. extension of Q , and τ ( u ) be any numerationof T . Recall that an arithmetical interpretation f is a mapping from theset of all modal propositional variables to the set of L ( T )-sentences. Everyarithmetical interpretation f is uniquely extended to the mapping f τ fromthe set of all modal formulas to the set of L ( T )-sentences such that f τ satisfies the following conditions: • f τ ( p ) is f ( p ) for each propositional variable p ; • f τ ( ⊥ ) is = ; • f τ commutes with every propositional connective; • f τ ( ✷ A ) is Pr τ ( p f τ ( A ) q ) for every modal formula A . URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 45
Provability logic provides us with a new way of examining the intensionalityof provability predicates. Under different numerations of T , the provabil-ity predicate may have different properties, and hence may correspond todifferent modal principles under different arithmetical interpretations. Definition 5.16.
Given a numeration τ ( u ) of T , the provability logic PL τ ( T )of τ ( u ) is defined to be the set of modal formulas A such that T ⊢ f τ ( A ) forall arithmetical interpretations f .Note that the provability logic PL τ ( T ) of a Σ n numeration τ ( x ) of T isa normal modal logic. A natural and interesting question is: which nor-mal modal logic can be realized as a provability logic PL τ ( T ) of some Σ n numeration τ ( x ) of T ? An interesting research program is to classify theprovability logic PL α ( T ) according to the numeration α ( x ) of T . We firstdiscuss Σ numerations of T . Theorem 5.17 (Generalized Solovay’s Arithmetical Completeness Theo-rem, Theorem 2.5, [90]) . Let T be any consistent r.e. extension of Q . If T is Σ -sound, then for any Σ numeration α ( x ) of T , the provability logic PL α ( T ) is precisely GL . Moreover, Visser [133] examines all possible provability logics for Σ nu-merations of Σ -unsound theories. To state Visser’s result, we need somedefinitions. Definition 5.18 (Definition 3.5-3.6, [91]) . We define the sequence { Con nτ : n ∈ ω } recursively as follows:(1) Con τ is = ;(2) Con n +1 τ is ¬ Pr τ ( p ¬ Con nτ q ).The height of τ ( u ) is the least natural number n such that T ⊢ ¬ Con nτ ifsuch an n exists. If not, the height of τ ( u ) is ∞ .For Σ -unsound theories, Visser proves that PL τ ( T ) is determined bythe height of the numeration τ ( u ). Visser [133, Theorem 3.7] shows that theheight of τ ( u ) is ∞ if and only if PL ( τ ) = GL ; and the height of τ ( u ) is n ifand only if PL ( τ ) = GL + ✷ n ⊥ . Beklemishev [7, Lemma 7] shows that if T is Σ -unsound, then the height of Σ numerations of T can take any valuesexcept 0.Let U be any consistent theory of arithmetic. Based on the previouswork by Artemov, Visser and Japaridze, Beklemishev [7] proves that for Σ numeration τ of U , PL τ ( U ) coincides with one of the logics GL α , D β , S β and GL − β where α and β are subsets of ω and β is cofinite (for definitionsof GL α , D β , S β and GL − β , we refer to [7, 4]).Feferman [33] constructs a Π numeration π ( x ) of T such that the con-sistency statement Con π ( T ) defined via Pr π ( x ) is provable in T . Thus,the provability logic PL π ( T ) of Pr π ( x ) contains the formula ¬ ✷ ⊥ , and isdifferent from GL . However, the exact axiomatization of the provabilitylogic PL π ( T ) under Feferman’s numeration π ( x ) is not known. Kurahashi[90] proves that for any recursively axiomatized consistent extension T of PA , there exists a Σ numeration α ( x ) of T such that the provability logic PL α ( T ) is the modal system K . As a corollary, the modal principles com-monly contained in every provability logic PL α ( T ) of T is just K .It is often thought that a provability predicate satisfies D1 - D3 if andonly if G2 holds (i.e. for the induced consistency statement Con ( T ) fromthe provability predicate, T Con ( T )). But this is not true. From Defi-nition 5.14, conditions D1 - D2 hold for any numeration of T . Whether theprovability predicate satisfies condition D3 depends on the numeration of T .For any Σ -numeration α ( x ) of T , D3 holds for Pr α ( x ). From Kurahashi[90], there is a Σ -numeration α ( x ) of T such that the provability logic forthat numeration is precisely K . Since K ¬ (cid:3) ⊥ , as a corollary, G2 holdsfor T , i.e. Con α ( T ) defined as ¬ Pr α ( p = q ) is not provable in T . Butthe L¨ob condition D3 does not hold since K (cid:3) A → (cid:3)(cid:3) A . This gives usan example of a Σ numeration α ( x ) of T such that D3 does not hold for Pr α ( x ) but G2 holds for T . Thus, G2 may hold for a provability predicatewhich does not satisfy the L¨ob condition D3 .Moreover, Kurahashi [91] proves that for each n ≥
2, there exists a Σ numeration τ ( x ) of T such that the provability logic PL τ ( T ) is just themodal logic K + (cid:3) ( (cid:3) n p → p ) → (cid:3) p . Hence there are infinitely many normalmodal logics that are provability logics for some Σ numeration of T . A goodquestion from Kurahashi [91] is: for n ≥
2, is the class of provability logics PL τ ( T ) for Σ n numerations τ ( x ) of T the same as the class of provabilitylogics PL τ ( T ) for Σ n +1 numerations τ ( x ) of T ? However, this question isstill open as far as we know. Define that KD = K + ¬ ✷ ⊥ . A natural andinteresting question, which is also open as far as we know, is: can we find anumeration τ ( x ) of T such that PL τ ( T ) = KD ?In summary, G2 is intensional with respect to the following parameters:the formalization of consistency, the base theory, the method of numbering,the choice of a provability predicate, and the representation of the set ofaxioms. Current research on incompleteness reveals that G2 is a deep andprofound theorem both mathematically and philosophically in the founda-tions of mathematics, and there is a lot more to be explored about theintensionality of G2 . 6. Conclusion
We conclude this paper with some personal comments. To the author,the research on concrete incompleteness is very deep and important.After G¨odel, people have found many different proofs of incompletenesstheorems via pure logic, and many concrete independent statements withreal mathematical contents. As Harvey Friedman comments, the researchon concrete mathematical incompleteness shows how the IncompletenessPhenomena touches normal concrete mathematics, and reveals the impactand significance of the foundations of mathematics.Harvey Friedman’s research project on concrete incompleteness plans toshow that we will be able to find, in just about any subject of mathemat-ics, many natural looking statements that are independent of
ZFC . Harvey
URRENT RESEARCH ON G ¨ODEL’S INCOMPLETENESS THEOREMS 47
Friedman’s work is very profound and promising, and will reveal that incom-pleteness is everywhere in mathematics, which, if it is true, may be one of themost important discoveries after G¨odel in the foundations of mathematics.
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