The Logic of Turing Progressions

V


Introduction
After Gödel's Incompleteness Theorems [10], we know that any consistent arithmetical theory, sufficiently rich, is incomplete. Moreover, by the Second Incompleteness Theorem, we know that such a theory cannot prove its own consistency.
In other words, given a consistent arithmetical theory T containing, say elementary arithmetic EA, we have that T ⊢ Con(T ) and therefore, extending T by the consistency assertion Con(T ), we get a stronger theory. If we started with a sound theory T , the new system T + Con(T ) is affected once more by Gödel's Incompleteness Theorems so it cannot prove its own consistency. Hence we can extend it by adding the corresponding consistency assertion obtaining a new stronger, but again incomplete system.
In 1936 Turing started working on his PhD dissertation under the direction of Alonzo Church. The results of this work were published in 1939 under the title Systems of logic based on ordinals [15]. Here, Turing introduced what are now known as Turing progressions which are hierarchies of theories that arise by iterating this process of adding consistency statements to a base theory, even to transfinite levels. Turing progressions are widely used in proof theory and can serve the purpose of gauging the proof-theoretic strength of mathematical theories that contain arithmetic.
It is known that the standard propositional provability logic GL containing a modality to model the standard formalized provability predicate can be used to denote any finite Turing progression. The logic GLP was introduced by Japaridze in [12] and contained a range of modalities [n], one for each n < ω, that can be interpreted as a sequence of ever increasing provability predicates.
In [4] it is shown that transfinite Turing progressions up to the ordinal ε 0 can be approximated using the polymodal provability logic GLP. In particular, [4] showed how an ordinal analysis of Peano Arithmetic can be performed mainly within GLP.
On the one hand, GLP has many good algebraic properties and is very expressible in that it constitutes for an alternative ordinal notation system. On the other hand, the link from GLP to Turing progressions is only by approximating the progression so that applications of GLP require many technical results. In this paper a first investigation is undertaken to see if there are logics which are still expressible and have good algebraic properties but which can be used to directly denote Turing progressions rather than just approximating them. In this sense, our logic is inspired by what is called Reflection Calculus RC ( [7], [6], [8]). Like with RC, we shall focus on strictly positive formulas which are formulas that do not contain disjunction nor implication.
The arrangement of the paper is the following: Section 2 introduces some basic arithemtical definitions such as n-consistency or ∆ 0 -presented theories. In this Section we shall also discuss the notion of Smooth Turing progressions and how they are presented within arithmetic.
In Section 3 we introduce a propositional modal language using ordinal modalities, which are ordered pairs n, α where α ∈ ε 0 and n ∈ ω. We will also establish a way to interpret our modal formulae in (possibly infinitely axiomatized) theories that contain a sufficient amount of arithmetic. This way, the intended reading of n, α ϕ will be the n-consistency α times iterated over the arithmetical interpretation of ϕ.
In Section 4 we will introduce the system TC (for Turing Calculus). The derivable objects of this system are sequents, i.e., expressions of the form ϕ ⊢ ψ whose intended interpretation is to express the entailment between the theories denoted by the modal formulas ϕ and ψ.
In Section 5, we will see how to fix the interpretation for these sequents within EA + and we prove the Soundness of TC with respect to this interpretation. Finally, in Section 6 we will explore the relations between the closed fragment of RC and the finite fragment of TC. elimination can be formalized. Here, EA + is Robinson's arithmetic Q together with induction for bounded formulas. Note that EA + in our formulation has a function symbol for the super-exponential function x → 2 x x defined by 2 x 0 := x and 2 x y+1 := 2 2 x y . To be more precise, we will consider ∆ 0 -presented theories 1 , that are, theories whose set of Gödel numbers can be defined on the natural numbers by a bounded formula. Note that since supexp is in our language, our notion of ∆ 0 -presentable does not coincide with the more standard one from the literature where one considers theories whose set of axioms are definable via an elementary formula. Again, via Craig's trick, we think that this difference is not too important.

Arithmetical preliminaries
As usual, by T (x) we denote ∃y Prf T (y, x); an arithmetical Σ 1 -formula expressing that there is y such that y codes a proof in T of a formula with code x. By n we denote the numeral S(S(. . . S(0) . . .)) (n times). We will use just n when it cannot be mislead with a variable. Often, we write T (ϕ(ẋ)) instead of T ( ϕ(ẋ) ), where ϕ(ẋ 1 , . . . ,ẋ j ) denotes the map sending n 1 , . . . , n k to the Gödel number ϕ(n 1 , . . . , n j ) .
Definition 2.1 (n-consistency). A theory U is called n-consistent if U together with the set of all true Π n -sentences is consistent.
The n-consistency of the theory U can equivalently be expressed (see e.g. [5]) by the arithmetical formula: where x ∈ Π n+1 expresses that x is the Gödel number of a Π n+1 sentence, and Tr n+1 is the standard Π n+1 -truth definition for Π n+1 -formulas. Notice that the arithmetical complexity of Con n (U ) is Π n+1 .

Definition 2.2 (Presented theories).
A theory U is ∆ 0 -presented iff there is a bounded arithmetical formula σ(x) that defines the sets of Gödel numbers coding the axioms of U in the standard model of arithmetic. By ǫ(x) we denote the canonical numeration of EA + .

Smooth Turing Progressions
Turing progressions are hierarchies of theories such that given an initial theory T , we can construct a transfinite sequence of extensions of T by iteratedly adding n-consistency statements. These progressions can be defined according to the following conditions below. Let us assume that we have fixed some recursive limit ordinal Λ together with a natural ordinal notation system for Λ that is elementary represented.
Here we will consider Smooth Turing progressions, studied by Beklemishev in among others [5] and [1]. We give a slightly different presentation.
Suppose we are given some elementary well-ordering (D, ≺) and an initial theory T . The conditions T1-T3 can be reformulated by the unique following clause: given such a family of theories (T α n ) α∈D n<ω we have that they can be represented within EA + through some arithmetical formula numerating their axioms. This way, consider the elementary formula τ σ(z) n (x, y) where x is a variable for an ordinal α ∈ D, y stands for the coding of some arithmetical formula and σ(z) is an elementary formula enumerating the axioms of some base theory. Hence, roughly speaking, the formula tells us that the formula coded by y is an axiom of T n α where the initial theory is numerated by the elementary formula σ(z). We shall often refrain from distinguishing an ordinal and its code, and likewise for other objects that are coded, like formulas and proofs.
We say that τ σ(z) n (α, x) enumerates the α-th theory of a progression based on iteration of consistency along (D, ≺) with base σ(z) if: The existence of such τ σ(z) n (α, x) is guaranteed by the fixed point theorem.

Strictly Positive Signature
As we have pointed out, we shall work with a positive propositional modal signature consisting of one constant symbol ⊤, one logical connective ∧ and a set of modal connectives M := { n, α : n < ω and α < ε 0 }, named ordinal modalities The set of formulas in this language is defined as follows: we denote the smallest set such that: Lε 0 we can find some special formulas, named Ordinal Worms -OWs-, that are defined as follows: The set of OWs denoted by W On is inductively defined as: Also, for any n < ω we define F succ <n ⊂ F <n ⊂ Form 0 Lε 0 as follows: iii) if ϕ ∈ F <n and m < n, then m, β ϕ ∈ F <n .

Arithmetical Interpretation for Modal Formulas
Let us introduce now the arithmetical interpretation of our modal formulae in terms of the τ -formulae previously presented. Notice that the conjunction of modal formulas is intended to mean the union of theories. Hence, we map the interpretation of modal conjunctions to the disjunction of the respective interpretations.

Note that since Form 0
Lε 0 has no propositional variables, we can identify a modal formula with its arithmetical interpretation unambiguously.
For the sake of clarity, and since we are working in the close fragment, we will use the following notation: given ϕ ∈ Form 0 Lε 0 by Th ϕ we denote Th σ where ϕ * (x) = σ(x), following Definition 3.4. If ϕ * (x) = ǫ(x) we use just EA + instead of Th ǫ . Also if ϕ := n, α ψ we write (Th ψ ) α n for the theory of ( n, α ψ) * .

Turing Calculus
In the previous Section we have defined an arithmetical interpretation that allows us to denote Turing progressions by modal formulas in Form 0 Lε 0 . In this section we introduce the logic TC in this modal language whose main goal is to express valid relations that hold between Turing progressions. It is worth mentioning the special character of Axioms 5 and 8 since both axioms are modal formulations of principles related to Schmerl's Fine Structure Theorem, also known as Schmerl's formula (see [14] and [3]). In order to simplify notation, here we shall also introduce the so-called hyperexponential functions e n : On → On, as studied in [9], where On denotes the class of ordinals, e 0 is the identity function, e 1 : α → −1 + ω α and e n+m = e n • e m . Together with the hyper-exponential functions, we also introduce the IT-function, i : (N × Ord) k → Ord, defined as follows: i) i((n, γ)) = e n (γ); ii) i((n 0 , γ 0 ), . . . , (n j+1 , γ j+1 )) = e nj+1 ( i((n 0 , γ 0 ), . . . , (n j , γ j )) ) · (1 + γ j+1 ).
By convention we take that for any n, n, 0 ϕ is just ϕ. Also, instead of saying that A ⊢ B is provable in TC, we shall often just write A ⊢ B. Let us, by way of exercise, prove a monotonicity property of TC.  Some other useful properties that we shall use without further mention are expressed in the next Proposition. 4. m + n, α ϕ ⊢ m, α ϕ; 5. m + n, α ϕ ≡ m + n, α m, β ϕ for β < e n (α) and n > 0.

FTP-Interpretation
In this section we shall prove that the logic TC adequately describes graded Turing progressions. First we shall make the link between our logic and graded Turing progressions more precise by introducing the Formalized Turing Progression (FTP) Interpretation.

FTP-interpretation
The Formalized Turing Progression Interpretation (FTP), is the representation of the entailment between the corresponding first order theories within EA + , via the Π 2 -sentence expressing such derivability. Thus given ϕ, ψ ∈ Form 0 Lε 0 the intended interpretation of ϕ ⊢ ψ is the arithmetical statement: Before proving Soundness we would like to introduce some useful tools that will be used along our proof.

Soundness
Proof. By induction on the length of a TC proof of ϕ ⊢ ψ. It is easy to see that the first rule preserves validity. Recall that the arithmetical interpretation of a conjunction is the union of the arithmetical interpretation of the conjuncts.
The first three axioms are easily seen to be arithmetically valid. In particular, Axiom 3 expresses the provably monotone property for Smooth Turing progressions. For a proof of Soundness of Axiom 4 see Lemma 2.6 in [2]. Validity of Axiom 6 follows from Proposition 5.4 Item 2. The remaining axioms and rules are separately proven to be sound in the next subsection.

Comparing Reflection Calculus and Turing Progressions Calculus
In this Section, we will discuss some relations between the finite formulas of TC and the closed fragment of RC. Here, by finite formulas we mean the formulas in Form 0 Lε 0 such that every element in the ordinal modality is less than ω. The set of these formulas, denoted by Form 0 Lω , is inductively defined as follows: Lω , then m, n ϕ ∈ Form 0 Lω for m, n < ω.
As expected, OWs in Form 0 Lω behave better that OWs using transfinite ordinals. Some of these properties are listed below: Lω satisfying the following: Proof. This can be checked by induction on the length of A. Base case is trivial. 2. for all k, 0 ≤ k ≤ j, n k = l k+(i−j) and m k = v k+(i−j) . Notice that previous Proposition does not hold in general for A, B ∈ W On . For example, consider 0, ω + 1 ⊤ and 1, 1 ⊤. Neither 0, ω + 1 ⊤ ⊢ 1, 1 ⊤ nor 1, 1 ⊤ ⊢ 0, ω + 1 ⊤ holds. 2. B is of the form n 0 , m 0 . . . n j , m j ⊤ where n i < n i+1 for 0 ≤ i < j.
In [7], Beklemishev showed the arithmetical completeness of RC with respect to a natural arithmetical interpretation where modal formulas are translated into arithmetical formulas defining the set of Gödel numbers that code the Axioms of arithmetical theories extending PA. More precisely, ⊤ is translated into the formula numerating the Axioms of PA, ε P A , the conjunction is translated into the disjunction of the translations and n ϕ is translated into the arithmetical formula Con n ((ϕ) * ) that defines the set of Gödel numbers of PA together with the Gödel number of the formula Con n ((ϕ) * ) where (ϕ) * is the arithmetical interpretation of ϕ. The completeness theorem states the following, given ϕ, ψ ∈ Form RC : ϕ ⊢ ψ iff PA ⊢ ∀x ( ψ * (x) → ϕ * (x)).