Truth, Disjunction, and Induction

By a well-known result of Kotlarski, Krajewski, and Lachlan (1981), first-order Peano arithmetic $PA$ can be conservatively extended to the theory $CT^{-}[PA]$ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This results motivates the general question of determining natural axioms concerning the truth predicate that can be added to $CT^{-}[PA]$ while maintaining conservativity over $PA$. Our main result shows that conservativity fails even for the extension of $CT^{-}[PA]$ obtained by the seemingly weak axiom of disjunctive correctness $DC$ that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, $CT^{-}[PA] + DC$ implies $Con(PA)$. Our main result states that the theory $CT^{-}[PA] + DC$ coincides with the theory $CT_0[PA]$ obtained by adding $\Delta_0$-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cie\'sli\'nski (2010). For our proof we develop a new general form of Visser's theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (L\"ob's version of) G\"odel's second incompleteness theorem, rather than by using the Visser-Yablo paradox, as in Visser's original proof (1989).


INTRODUCTION
By a theorem of Krajewski, Kotlarski, and Lachlan [12], every countable recursively saturated model M of PA (Peano Arithmetic) carries a 'full satisfaction class', i.e., there is a subset S of the universe M of M that 'decides' the truth/falsity of each sentence of arithmetic in the sense of M -even sentences of nonstandard length -while obeying the usual recursive clauses guiding the behavior of a Tarskian satisfaction predicate. This remarkable theorem implies that theory 1. Theorem. CT − [I∆ 0 + Exp] + DC and CT 0 [PA] axiomatize the same first order theory.
In the above theorem, CT − [I∆ 0 + Exp] is the weakening of CT − [PA] obtained by replacing the 'base theory' PA with its fragment consisting of Robinson's arithmetic Q, along with the scheme for ∆ 0 -induction and the totality of the exponential function; and DC (disjunctive correctness) is the statement asserting that a disjunction of arithmetical sentences of arbitrary finite length is true (in the sense of T) iff one of the disjuncts is true. Coupled with Le lyk's aforementioned result [15], Theorem  The plan of the paper is as follows: in Section 2 we review preliminary definitions and results, including more precise versions of those definitions and results mentioned in this introduction. In Section 3 we establish the veracity of the principle IC (Inductive Correctness, often referred to in the literature as "internal induction") within CT − [I∆ 0 + Exp] + DC. This is demonstrated by first establishing a new general form of Visser's theorem [19] on nonexistence of infinite descending chains of truth definitions with the help of (Löb's version of) Gödel's second incompleteness theorem instead of the Visser-Yablo paradox. In Section 4 we show that CT 0 [PA] is a subtheory of CT − [PA] + DC + IC; thus Sections 3 and 4 together constitute the proof of the hard direction of Theorem 1 since it is routine to verify that both DC and IC are theorems of CT 0 [PA]. We close the paper with some open problems in Section 5.
Historical Note. The concept of disjunctive correctness first appeared in the work of Krajewski [13, p.133], who called it "∨-completeness"; the current terminology was coined in a working paper of Enayat and Visser that was privately circulated in 2011, only a fragment [6] of which has been published so far. The working paper included the claim that CT − [PA] + DC is conservative over PA, but the proof outline presented in the paper was found in 2013 to contain a significant gap by Cieśliński and his (then) doctoral students Le lyk and Wcis lo. On the other hand, in 2012 Enayat found a proof of CT 0 [PA] within CT − [I∆ 0 + Exp] + DC + IC; his proof was only privately circulated, and later was presented in the doctoral dissertation of Le lyk [15]. This proof forms the content of Section 4 of this paper. In light of these developments, and the wellknown conservativity of CT − [PA] + IC over PA (see Theorem 2.3), the question of conservativity of CT − [PA] + DC over PA came to prominence amongst truth theory experts [5, p.226], and has been unsuccessfully attacked by a number of researchers since 2013, until Pakhomov established IC within CT − [I∆ 0 + Exp] + DC as in Section 3 of this paper, which, coupled with Enayat's aforementioned result, yields Theorem 1 and exhibits the unexpected arithmetical strength of DC.

PRELIMINARIES
2.1. Definition. (a) L A is the usual language of first order arithmetic {+, ·, S(x), <, 0}. To simplify matters, we will assume that the logical constants of first order logic consist only of ¬ (negation), ∨ (disjunction), and ∃ (existential quantification); in particular ∀ (universal quantification) as well as ∧ (conjunction) and other propositional connectives are treated here as derived notions.
(b) Given a language L ⊇ L A , an L-formula ϕ is said to be a ∆ 0 (L)-formula if all the quantifiers of ϕ are bounded by L-terms, i.e., they are of the form ∃x ≤ t, or of the form ∀x ≤ t, where t is a term of L not involving x. Given a predicate U(x), L A+U is the language L A ∪ {U(x)}.
(c) Given a language L ⊇ L A , I∆ 0 (L) is the scheme of induction over natural numbers for ∆ 0 (L)formulae. We shall omit the reference to L if L = L A , e.g., a ∆ 0 -formula is a ∆ 0 (L A )-formula; and we shall use I∆ 0 (U) to abbreviate I∆ 0 (L A+U ).
(d) I∆ 0 + Exp is the fragment of Peano arithmetic obtained by strengthening Robinson's arithmetic Q with I∆ 0 and with the sentence Exp that expresses the totality of the exponential function y = 2 x . It is well-known that Exp can be written as ∀x∃yExp(x, y), where Exp(x, y) is a ∆ 0 -predicate which, provably in I∆ 0 , satisfies the familiar algebraic laws governing the graph of the exponential function, cf. (e) Sent A (x) is the L A -formula that expresses "x is the Gödel-number of a formula of L A with no free variables", and Form n A (x) is the L A -formula that expresses "x is the Gödel-number of a formula of L A with precisely n free variables". We use Sent A and Form n A to refer to the corresponding definable classes of Gödel-numbers of L A -formulae.
(f ) Given a (base) theory B whose language is L A and which extends I∆ 0 + Exp, CT − [B] is the theory obtained by strengthening B by adding the sentences tarski 0 through tarski 4 described below, where we use the following conventions: τ 1 and τ 2 vary over Gödel-numbers of closed L A -terms, τ • i is the value of the term coded by τ i , ϕ and ψ range over Gödel-numbers of L Asentences, v ranges over variables, γ(v) ranges over Form 1 A , and x is the numeral representing the value of x.
(h) DC (disjunctive correctness) is the L A+T -sentence asserting that T commutes with disjunctions of arbitrary length, i.e., DC asserts that for all numbers s and for all sequences ϕ i : i < s from Sent A , the following equivalence holds: where for definiteness i<s ϕ i is defined 3 by recursion: (i) We will employ the abbreviation • Note that the commutativity of T with negation implies that DC and CC are equivalent.
(j) IC (inductive correctness 4 ) is the L A+T -sentence asserting that each L A -instance of induction over natural numbers is true, i.e., IC asserts that for all unary L A -formulae ψ = ψ(x), T( Ind ψ ) holds, where Ind ψ is the following L A -sentence that asserts that ψ is inductive: The B = PA case of Theorem 2.2 below, and its elaboration Theorem 2.3, were first established in the work of Krajewski, Kotlarski, and Lachlan [12] for B = PA, where PA is formulated in a relational language, and 'domain constants' play the role of numerals. As mentioned in the introduction to this paper, their result was couched in model theoretic terms involving the notions of recursive saturation and satisfaction classes, but it is well-known that their formulation is equivalent to appropriately formulated conservatity assertions (the key ingredients of this equivalence are the following facts: Every consistent theory in a countable language has a recursively saturated model, and countable recursively saturated models are resplendent). Later Kaye [10] developed the theory of satisfaction classes over models of PA in languages incorporating function symbols; his work was extended by Engström [7] to truth classes over models of PA in functional languages. 5 More recently, newer and more informative proofs of Theorems 2.2 and 2.3 have been found in the joint work of Visser and Enayat [6] (with base theories that support a modicum of coding, and which are formulated in a relational language), and by Leigh [14] (for functional base theories that support a modicum of coding). As verified by Cieśliński [5,Ch. 7], the Visser-Enayat model theoretic methodology can be extended so as to accommodate functional languages.
The direction (a) ⇒ (b) of Theorem 2.4 below is due to Kotlarski [11]; the other direction is due to Le lyk [15].

Theorem. (Kotlarski-Le lyk)
The following theories are deductively equivalent: The direction (a) ⇒ (b) of Theorem 2.5 below is due to Cieśliński [3], who refined Kotlarski's proof of the direction (a) ⇒ (b) of Theorem 2.4; the other direction involves a routine induction.

DISJUNCTIVE CORRECTNESS IMPLIES INDUCTIVE CORRECTNESS
3.1. Definition. ITB (iterated truth biconditionals) is a theory formulated in two-sorted first order logic. The first sort x, y, z . . . of ITB is for the 'natural numbers'. The second sort α, β, γ, . . . is for the indices of truth definition. The language L ITB of ITB is obtained by augmenting the language L A of arithmetic with two binary predicates: α ≺ β and T(α, x), but we shall write T(α, x) as T α (x) to display the indexicality of α. The axioms of ITB come in three groups. The first group consists of the axioms of Q (Robinson arithmetic); the second group stipulates that ≺ is a strict partial order; and the third group consists of the following biconditionals B ϕ : where ϕ ranges over all L ITB -sentences, and for each index variable α, ϕ ≺α denotes the relativization of ϕ to the cone of indices below α, i.e. the formula obtained by replacing all the quantifiers of the form ∀β (∃β) with ∀β ≺ α (∃β ≺ α), and if there is a bounded instance of α we make the appropriate renaming. Clearly ϕ ≺α = ϕ if ϕ is a purely arithmetical formula.
• Note that we take the theory ITB over the variant of many-sorted logic that allows domains of some sorts to be empty.
• We will use the following convention to lighten the notation: The notation ϕ for the Gödel number of a formula ϕ will be generally used, but the corner-notation will be omitted when ϕ appears inside of a truth predicate T, or inside an indexed version of T.
The proof of the following theorem was inspired by the recent James Walsh proof [18] of nonexistence of infinite recursive provably descending chains of sentences with respect to < Conorder.
We will verify that θ(x) satisfies the following Hilbert-Bernays-Löb derivability conditions in DTB; in what follows ϕ and ψ range over all sentences of the language of DTB: It is easy to see that for every axiom A of DTB we have DTB ⊢ A ≺α . The assumption embodied in DTB that the set of indices is nonempty and has no minimal element is invoked only at this step of the proof, in verifying that DTB ⊢ A ≺α for A : = ∀γ∃β(β ≺ γ) ∧ ∃γ(γ = γ).
Routine considerations show that HBL-1 holds based on an easy induction on the number of steps of the proof of ϕ within DTB.
For a given ϕ and ψ, HBL-2 follows directly from the biconditional axioms B ϕ→ψ , B ϕ , and B ψ of ITB.
Finally, HBL-3 holds since: On the other hand, the formula ¬θ( 0 = 1 ) is provable in ITB, and therefore in DTB. So by Löb's version 6 of Gödel's second incompleteness theorem, DTB is inconsistent.

Lemma.
CT − [I∆ 0 + Exp] + DC proves IC. 6 Löb's paper [17], in which the venerable 'Löb's Theorem' was proved, is responsible for the now common standard textbook framework for the presentation of 'abstract' form of Gödel's second incompleteness theorem: If T is a theory extending Q that supports a unary predicate θ(x) satisfiying condition HBL-1, HBL2, and HBL-3, and γ is a T -provable fixed point of ¬θ (i.e., the equivalence of γ and ¬θ( γ ) is provable in T ), then either ¬∃x θ(x) is not provable in T , or T is inconsistent. See, e.g., [1,Ch. 18], for the presentation of such a general form of Gödel's second incompleteness theorem.
Proof. By Theorem 3.2 we can fix an inconsistent finite subtheory DTB − of DTB. Suppose DTB − contains only biconditionals for the formulae ϕ 0 , . . . , ϕ k−1 . We will use ITB − to denote the subtheory of ITB whose only biconditional axioms are {B ϕ i : i < k} .
For the rest of the proof we will reason in CT − [I∆ 0 + Exp] + DC. In order to prove IC we assume for a contradiction that some arithmetical ψ(x) is not inductive in the sense of T, i.e., we have: By induction on natural numbers n we define interpretations ι n of ITB − . Note that each interpretation is just a finite sequence of formulae and thus could be easily represented in arithmetic. The interpretation of arithmetic in each ι n is the identity interpretation, but the domain of indices of truth definition ι n is given by the formula D (n) (x): For all n the relation ≺ is interpreted by <. The formula T α (x) is interpreted by the formula IT (n) (y, x), where y corresponds to α, and x corresponds to itself: where ι m (ϕ i ) is the ι m -translation of the sentence ϕ i . It is easy to see that this definition could be carried out in I∆ 0 + Exp.
Let us now prove that the translations given by ι n are indeed the desired interpretations inside T, i.e., we need to prove that for all n and axioms A of ITB − we have T(ι n (A)). Clearly it is the case for all the axioms of Q and the axioms of partial order for ≺. Now let us show that for any s < k : ( ) T(ι n (∀α(T α (ϕ s ) ↔ ϕ ≺α s ))).
By compositional axioms, we just need to show that for all u such that u < n and T(¬ψ(u)) we have: Now by compositional axioms and DC (in the form of CC, as explained in part (i) of Definition 2.1) our task can be reduced to proving the equivalence: In order to prove this we will show by induction on subformulae θ of ϕ s that for the universal closure θ of θ: Note that since ϕ s is a fixed formula with finitely many subformulae, actually this external induction will be formalizable in CT − [I∆ 0 + Exp]+ DC despite the fact that it lacks the induction axiom for the appropriate class of formulae. The only non-trivial case here is the case when θ is T α (x): T (ι u (∀α∀xT α (x))) ↔ T(ι n (∀α ≺ u ∀x(T α (x)))).
Hence we just need to show that for all p < u such that T(¬ψ(p)), and for all o, the following pair of formulae (whose only formal difference is in the bound for indices of the second conjunction) are equivalent: But since p < u < n, we trivially use DC (in the form of CC) to show that the formulae are indeed equivalent. Thus we conclude that ( ) holds.
Choose some n such that T(¬ψ(n)); this is possible since we assumed that induction fails for ψ(x) in the sense of T. It is easy to see that ι n actually is an interpretation of DTB − inside T. Now we could follow the proof of inconsistency of DTB − inside T to derive a contradiction, thereby completing the proof of IC. Proof. This is an immediate consequence of Lemma 3.3, and the provability of Tarski biconditionals in CT − [I∆ 0 + Exp].

DISJUNCTIVE CORRECTNESS + INDUCTIVE CORRECTNESS IMPLIES ∆ 0 (T)-INDUCTION
In this section we shall prove that I∆ 0 (T) is provable in CT − [PA] + DC + IC, which, coupled with Lemma 3.3 completes the proof of the nontrivial direction of Theorem 1. We begin with a key definition:

Definition.
In what follows ∈ Ack is "Ackermann's epsilon", i.e., x ∈ Ack y is the arithmetical formula that expresses "the x-th bit of the binary expansion of y is a 1".
The following lemma shows that over I∆ 0 + Exp the scheme I∆ 0 (U) is equivalent to the single sentence 'U is piecewise coded'. The lemma is folklore; we present the proof for the sake of completeness.
(a −→ b): Assume I∆ 0 (U). Given u, let w be the Ackermann-code for the set of predecessors of u. Clearly w = i<u 2 i = 2 u − 1, and w is an upper bound for any w ′ that codes a subset of the predecessors of u. Consider the ∆ 0 -formula: A simple induction using I∆ 0 (U) shows that ∀u δ(u) holds.
(b → a): A straightforward induction on the complexity of ∆ 0 (U)-formulae shows that if U is piecewise coded and δ(U, x 0 , ..., x n−1 ) is a ∆ 0 (U)-formula, then PC δ holds. We could then trivially deduce the least number principle for ∆ 0 (U)-formula, which is of course equivalent to I∆ 0 (U).
On the other hand, ∀u ψ(u) is the conclusion of the formula Ind ψ (asserting the inductive property of ψ) given by IC. So (2) follows directly from IC and the easily verified facts T(ψ(0)) and T(∀u (ψ(u) → ψ(u + 1))).

4.3.1.
Remark. Lemma 4.3 can be readily strengthened to a more general result whose proof we leave to the reader: the ω-interpretation that can be informally described by the motto "sets are T-extensions of unary arithmetical formulae" satisfies ACA 0 , provably in CT − [PA] + IC. Indeed, it is a theorem of CT − [PA] that IC is equivalent to the veracity of ACA 0 within this interpretation.

Lemma. CT
Proof. Reason in CT − [I∆ 0 + Exp] + DC + IC. By Lemma 4.2, it suffices to show that T is piecewise coded. Let ϕ i : i < u be the sequence of arithmetical sentences such that ϕ i is the sentence with Gödel-number i if there is such a sentence, and otherwise ϕ i is the sentence 0 = 1.
We wish to show that {i < u : T(ϕ i )} is coded. Towards this goal, consider the unary formula θ(x) ∈ Form A given by: (→) Suppose T(ϕ i ) for some i < u. Then T ((i = i) ∧ ϕ i ) , and hence by DC we haveT(θ(i)).
(←) Suppose T(θ(i)) for some i < u. Then by DC, there is some j < u such that T (i = j) ∧ ϕ j . So T(ϕ i ) holds since T commutes with conjunction and T(i = j) holds iff i = j.

CLOSING REMARKS AND OPEN QUESTIONS
5.1. Question. Is the generalization of Theorem 1 in which CT − is weakened to CS − (where S stands for satisfaction) true?
• The notion CS − [B] is defined in [6] for base theories B formulated in relational languages, using the notation B FS (FS for "full satisfaction"); and in [5,Ch. 7] for functional languages. We expect that an examination of the proofs in Sections 3 and 4 would show that this question has a positive answer.
• In the above, S 1 2 is Buss's well-known arithmetical theory whose provable recursive functions are precisely the functions computable in polynomial time, as in [2]. For the above question to make sense, part (f) of Definition 1.1 should be adjusted so as to accommodate the fact that the language of S 1 2 extends L A . In the proof of Lemma 3.3, most likely it is possible to use some tricks with effective formulae in order to modify the definition of ι n in such a way that their sizes will be polynomial. But in order for the construction to work we will also need to ensure that DC is still enough to show that ι n are indeed interpretations inside the truth predicate.

Question. Is there a fixed-point free proof of Lemma 3.3?
• The proof of Lemma 3.3 is based on Theorem 3.2, whose proof implicitly relies (in the very last step) on the existence of a fixed point for the formula ¬θ(x).