Truth, disjunction, and induction

Abstract

By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic PA{{\mathsf {P}}}{{\mathsf {A}}} can be conservatively extended to the theory CT−[PA]{{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[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 result motivates the general question of determining natural axioms concerning the truth predicate that can be added to CT−[PA]{{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]} while maintaining conservativity over PA {{\mathsf {P}}}{{\mathsf {A}}}. Our main result shows that conservativity fails even for the extension of CT−[PA]{{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]} obtained by the seemingly weak axiom of disjunctive correctness DC{{\mathsf {D}}}{{\mathsf {C}}} that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, CT−[PA]+DC{{\mathsf {C}}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC} implies Con(PA)\mathsf {Con}(\mathsf {PA}). Our main result states that the theory CT−[PA]+DC{\mathsf {C}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC} coincides with the theory CT0[PA]{\mathsf {C}}{\mathsf {T}}_{0}\mathsf {[PA]} obtained by adding Δ0 \Delta _{0}-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (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 (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).

Full text

Archive for Mathematical Logic (2019) 58:753–766
https://doi.org/10.1007/s00153-018-0657-9
Mathematical Logic
Truth, disjunction, and induction
Ali Enayat1·Fedor Pakhomov2
Received: 25 June 2018 / Accepted: 17 December 2018 / Published online: 4 February 2019
© The Author(s) 2019
Abstract
By a well-known result of Kotlarski et al. (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
result 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 CT0[PA] obtained by adding 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 (Löb’s version of) Gödel’s second incompleteness theorem, rather than
by using the Visser–Yablo paradox, as in Visser’s original proof (1989).
Keywords Axiomatic truth ·Compositional theory of truth ·Conservativity
Mathematics Subject Classification 03F30
The work of Fedor Pakhomov is supported by the Russian Science Foundation under Grant 16-11-10252
and performed at Steklov Mathematical Institute of Russian Academy of Sciences. F. Pakhomov was
selected as one of the Young Russian Mathematics award winners, and he would like to thank its sponsors
and jury; his work on this particular paper was funded from another source. Sections 2 and 4 of this paper
were contributed by A. Enayat; Section 3 was contributed by F. Pakhomov; Sections 1 and 5 were
contributed jointly by the authors. The authors are grateful to the anonymous referees for their meticulous
and valuable feedback.
BAli Enayat
ali.enayat@gu.se
Extended author information available on the last page of the article
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754 A. Enayat, F. Pakhomov
1 Introduction
By a theorem of Krajewski, Kotlarski, and Lachlan [12], every countable recursively
saturated model Mof PA (Peano Arithmetic) carries a ‘full satisfaction class’, i.e.,
there is a subset Sof the universe Mof Mthat ‘decides’ the truth/falsity of each sen-
tence 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 CT−[PA] (compositional truth
over PA with induction only for the language LAof arithmetic) is conservative over
PA, i.e., if an LA-sentence ϕis provable in CT−[PA], then ϕis already provable in PA.
New proofs of this conservativity result were given by Visser and Enayat [5]using
basic model theoretic ideas, and by Leigh [14] using proof theoretic tools; these new
proofs make it clear that in the Krajewski–Kotlarski–Lachlan theorem the theory PA
can be replaced by any ‘base’ theory that supports a modicum of coding machinery
for handling elementary syntax.
On the other hand, it is well-known [9, Thm. 8.39 and Cor. 8.40] that the consistency
of PA (and much more) is readily provable in the stronger theory CT[PA], which is the
result of strengthening CT−[PA] with the scheme of induction over natural numbers
for all LA+T-formulae, where LA+T:= LA∪{T(x)}.1Indeed, it is straightforward to
demonstrate the consistency of PA within the subsystem CT1[PA] of CT[PA], where
CTn[PA] is the subtheory of CT[PA] with the scheme of induction over natural numbers
limited to LA+T-formulae that are at most of complexity n[16,Thm.2.8].
The discussion above leaves open whether CT0[PA] is conservative over PA.Kot-
larski [11] established that CT0[PA] is a subtheory of CT−[PA] +Ref(PA), where
Ref(PA) is the LA+T-sentence stating that “every first order consequence of PA is
true”. Recently Łełyk [15] demonstrated that the converse also holds, which imme-
diately implies that CT0[PA] is not conservative over PA since Con(PA) is readily
provable in CT−[PA] +Ref(PA).2Kotlarski’s aforementioned theorem was refined by
Cie´sli´nski [3] who proved that CT−[PA]+“Tis closed under propositional proofs” and
CT−[PA] +Ref(PA) axiomatize the same theory. The main result of this paper, in turn,
refines Cie´sli´nski’s result by demonstrating:
Theorem 1 CT−[I0+Exp] +DC and CT0[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 func-
tion; and DC (disjunctive correctness) is the statement asserting that a disjunction of
1CT[PA] dwarfs PA in arithmetical strength: By a classical theorem (discovered by a number of researchers,
including Feferman and Takeuti, and explained in [9]) the arithmetical consequences of CT[PA] are the same
as the arithmetical consequences of ACA, the subsystem of second order arithmetic obtained by adding the
full scheme of induction over natural numbers (in the language of second order arithmetic) to the well-known
subsystem ACA0of second order arithmetic.
2This result was first claimed by Kotlarski [11], but his proof outline of Ref(PA) within CT0[PA] was found
to contain a serious gap in 2011 by Heck and Visser; this gap cast doubt over the veracity of Kotlarski’s
claim until the issue was resolved by Łełyk in his doctoral dissertation [15]. Łełyk’s work was preceded by
the discovery of an elegant proof of the nonconservativity of CT0[PA] over PA by Wcisło and Łełyk [16].
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Truth, disjunction, and induction 755
arithmetical sentences of arbitrary finite length is true (in the sense of T) iff one of the
disjuncts is true. Coupled with Łełyk’s aforementioned result [15], Theorem 1shows
that CT−[I0+Exp] +DC,CT−[PA] +DC, and CT0[PA] +Ref(PA) are axiomatizations
of the same theory.
The plan of the paper is as follows: in Sect.2we review preliminary definitions and
results, including more precise versions of those definitions and results mentioned
in this introduction. In Sect. 3we establish the veracity of the principle IC (Induc-
tive 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 [20] on nonexistence of infinite descending chains of truth def-
initions with the help of (Löb’s version of) Gödel’s second incompleteness theorem
instead of the Visser–Yablo paradox. In Sect. 4we show that CT0[PA] is a subtheory
of CT−[PA] +DC +IC; thus Sects. 3and 4together constitute the proof of the hard
direction of Theorem 1since it is routine to verify that both DC and IC are theorems
of CT0[PA]. We close the paper with some open problems in Sect. 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 [5] 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 Cie´sli ´nski and his
(then) doctoral students Łełyk and Wcisło. On the other hand, in 2012 Enayat found
a proof of CT0[PA] within CT−[I0+Exp] +DC +IC; his proof was only privately
circulated, and later was presented in the doctoral dissertation of Łełyk [15]. This
proof forms the content of Sect. 4of this paper. In light of these developments, and the
well-known 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 [4, p.226], and had been unsuccessfully attacked by a number of researchers
since 2013, until Pakhomov established IC within CT−[I0+Exp] +DC as in Sect. 3
of this paper, which, coupled with Enayat’s aforementioned result, yields Theorem 1
and exhibits the unexpected arithmetical strength of DC .
2 Preliminaries
Definition 2.1 (a) LAis 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⊇LA,anL-formula ϕis said to be a 0(L)-formula if all the
quantifiers of ϕare bounded by L-terms, i.e., they are of the form ∃x≤t,orof
the form ∀x≤t, where tis a term of Lnot involving x. Given a predicate U(x),
LA+Uis the language LA∪{U(x)}.
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756 A. Enayat, F. Pakhomov
(c) Given a language L⊇LA,I0(L)is the scheme of induction over natural numbers
for 0(L)-formulae. We shall omit the reference to Lif L=LA, e.g., a 0-formula
is a 0(LA)-formula; and we shall use I0(U)to abbreviate I0(LA+U).
(d) I0+Exp is the fragment of Peano arithmetic obtained by strengthening Robin-
son’s arithmetic Qwith I0and with the sentence Exp that expresses the totality
of the exponential function y=2x. 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,sat-
isfies the familiar algebraic laws governing the graph of the exponential function,
cf. [8, Sec. V3(c)].
(e) SentA(x)is the LA-formula that expresses “xis the Gödel-number of a formula
of LAwith no free variables”, and Formn
A(x)is the LA-formula that expresses “x
is the Gödel-number of a formula of LAwith precisely nfree variables”. We use
SentAand Formn
Ato refer to the corresponding definable classes of Gödel-numbers
of LA-formulae.
(f) Given a (base) theory Bwhose language is LAand which extends I0+Exp,
CT−[B]is the theory obtained by strengthening Bby adding the sentences tarski0
through tarski4described below, where we use the following conventions: τ1and
τ2vary over Gödel-numbers of closed LA-terms, τ◦
iis the value of the term coded
by τi,ϕand ψrange over Gödel-numbers of LA-sentences, vranges over variables,
γ(v)ranges over Form1
A, and xis the numeral representing the value of x.
tarski0:= ∀x(T(x)→SentA(x)).
tarski1:= ∀τ1,τ
1T(τ1=τ2)↔τ◦
1=τ◦
2.
tarski2:= ∀ϕ((T(¬ϕ) ↔¬T(ϕ)).
tarski3:= ∀ϕ, ψ (T(ϕ ∨ψ) ↔(T(ϕ) ∨T(ψ ))).
tarski4:= ∀v∀γ(v)(T(∃vγ(v)
)↔∃xT(γ (x)).
(g) CT0[B]:=CT−[B]∪I0(T).
(h) DC (disjunctive correctness) is the LA+T-sentence asserting that Tcommutes with
disjunctions of arbitrary length, i.e., DC asserts that for all numbers sand for all
sequences ϕi:i<sfrom SentA, the following equivalence holds:
T
i<s
ϕi↔∃i<sT(ϕi),
where for definiteness i<sϕiis defined3by recursion: i<0ϕi:= ϕ0and
i<t+1ϕi:= i<tϕi∨ϕt.
(i) We will employ the abbreviation i<sϕifor ¬i<s¬ϕi, and CC (conjunctive
correctness) for the LA+T-sentence
T
i<s
ϕi↔∀i<sT(ϕi).
3One can also formulate disjunctive correctness in a stronger way by asserting that all disjunctions (and
not just the ones that are grouped to the left) are well-behaved with respect to T. But the current frugal form,
as shown by Theorem 1, ends up implying the seemingly stronger form since it is easy to show in CT0that
the two forms are equivalent.
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Truth, disjunction, and induction 757
•Note that the commutativity of Twith negation implies that DC and CC are
equivalent.
(j) IC (inductive correctness4)istheLA+T-sentence asserting that each LA-instance
of induction over natural numbers is true, i.e., IC asserts that for all unary LA-
formulae ψ=ψ(x),T(Indψ)holds, where Indψis the following LA-sentence
that asserts that ψis inductive:
ψ(0)→(∀x(ψ(x)→ψ(x+1))→∀xψ(x)).
The B=PA case of Theorem 2.2 below, and its elaboration Theorem 2.3,were
first established in the work of Krajewski et al. [12]forB=PA, where PA is for-
mulated 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 func-
tion symbols; his work was extended by Engström [6] to truth classes over models
of PA in functional languages.5More recently, newer and more informative proofs
of Theorems 2.2 and 2.3 have been found in the joint work of Visser and Enayat [5]
(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 Cie´sli ´nski [4, Ch. 7], the Visser-Enayat model
theoretic methodology can be extended so as to accommodate functional languages.
Theorem 2.2 CT−[B] is conservative over Bfor every arithmetical base theory B
extending I0+Exp.
Theorem 2.3 CT−[PA] +IC is conservative over PA.
The direction (a)⇒(b)of Theorem 2.4 below is due to Kotlarski [11]; the other
direction is due to Łełyk [15].
Theorem 2.4 (Kotlarski–Łełyk) The following theories are deductively equivalent:
(a) CT−[PA]+Ref(PA).
(b) CT0[PA].
The direction (a)⇒(b)of Theorem 2.5 below is due to Cie´sli ´nski [3], who refined
Kotlarski’s proof of the direction (a)⇒(b)of Theorem 2.4; the other direction
involves a routine induction.
Theorem 2.5 (Cie´sli´nski) The following theories are deductively equivalent:
(a) CT−[PA] +“Tis closed under propositional proofs”.
(b) CT0[PA].
4This condition has been referred to as Int (internal induction) in the literature.
5The subtle distinction between satisfaction classes and truth classes, and their close relationship, is
explained in [5]and[4, Ch. 7].
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758 A. Enayat, F. Pakhomov
3 Disjunctive correctness implies inductive correctness
Definition 3.1 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 LITB of ITB
is obtained by augmenting the language LAof arithmetic with two binary predicates:
α≺βand T(α, x), but we shall write T(α, x)as Tα(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 consists of a single axiom asserting that ≺
is a transitive relation; and the third group consists of the following biconditionals Bϕ:
Bϕ:= ∀αTα(ϕ)↔ϕ≺α,
where ϕranges over all LITB -sentences, and for each index variable α,ϕ≺αdenotes
the relativization of ϕto 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 αwe make the appropriate renaming. Clearly ϕ≺α=ϕif 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.
•Although we haven’t required ≺to be irreflexive, we treat it as an irreflexive
relation; in particular we define a minimal element αto be an element such that
∀β¬(β ≺α). The reason is that Theorem 3.2 below implies that ITB proves the
irreflexivity of ≺. Alternatively one could show that ITB proves irreflexivity of ≺
by observing that the existence of a model of ITB with a reflexive point contradicts
Tarksi’s undefinability of truth theorem.
•We will use the following convention to lighten the notation: The notation ϕfor
the Gödel number of a formula ϕwill be generally used, but the corner-notation
will be omitted when ϕappears inside of a truth predicate T,orinsideanindexed
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 <Con-order. We note that Theorem 3.2 is similar to a result by Flumini
and Sato [7], which states that the second order principle asserting the existence of
iteration of 0
1-comprehension over a preorder ≺implies that ≺is well-founded.
Theorem 3.2 ITB +∃α(α =α) proves the existence of a ≺-minimal element. Equiv-
alently, the following theory DTB (descending truth biconditionals) is inconsistent:
DTB := ITB +∀α∃β(β ≺α) +∃α(α =α).
Proof We prove the inconsistency of DTB by Löb’s version of Gödel’s second
incompleteness theorem6: We exhibit a formula θ(x)that satisfies the HBL (Hilbert–
Bernays–Löb) conditions for a provability predicate over the theory DTB. This enables
6Löb’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
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Truth, disjunction, and induction 759
us to justify the inconsistency of DTB by showing that DTB proves the “consistency”
sentence ¬θ(0=1).
Consider the formula θ(x):
θ(x):= ∀α(Tα(x)).
We will verify that θ(x)satisfies the HBL conditions listed below provably in DTB;in
what follows ϕand ψrange over all sentences of the language of DTB:
HBL-1. DTB ϕ⇒ DTB θ(ϕ).
HBL-2. DTB θ(ϕ→ψ)→(θ(ϕ)→θ(ψ)).
HBL-3. DTB θ(ϕ)→θ(θ(ϕ)).
Since we have biconditionals, in order to prove HBL-1 it is enough to show that for
each sentence ϕ,ifDTB ϕthen DTB ∀αϕ
≺α. The latter is the case since for any
model Mof DTB and index ain M, the theory DTB holds in the model M≺athat is
the restriction of Mto all indices ≺a.
For a given ϕand ψ, HBL-2 follows directly from the biconditional axioms Bϕ→ψ,
Bϕ, and Bψof ITB.
Finally, HBL-3 holds since:
ITB θ(θ(ϕ))←→ ∀α∀β((β ≺α) →Tβ(ϕ)).
On the other hand, the formula ∀α¬Tα(0=1)is provable in ITB, hence the formula
¬θ(0=1)is provable in ITB, and therefore in ITB+∃α(α =α). So by Löb’s version
of Gödel’s second incompleteness theorem, DTB is inconsistent. 
Lemma 3.3 CT−[I0+Exp]+DC proves IC.
Proof By Theorem 3.2 we can fix an inconsistent finite subtheory DTB−of DTB.
Suppose DTB−contains only biconditionals for the formulae ϕ0,...,ϕ
k−1.We
will use ITB−to denote the subtheory of ITB whose only biconditional axioms are
Bϕi: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:
¬T(ψ(0)→(∀x(ψ(x)→ψ(x+1))→∀xψ(x))) .
Within CT−[I0+Exp]+DC we use induction on nto define translations ιnfrom
the language of ITB to the language of first-order arithmetic such that from the point
of view of Tall of them will be interpretations of ITB−, i.e., we will have T(ιn(ϕ)),
for all axioms ϕof ITB−. We will arrive at a contradiction by showing that ¬T(ψ(n))
Footnote 6 continued
theorem: If Tis a consistent theory extending Qthat supports a unary predicate θ(x)satisfying conditions
HBL-2, HBL-2, and HBL-3, then Tdoesn’t prove θ(0=1), i.e., the consistency sentence corresponding
to θ(the intended meaning of θ(x)is “the sentence with Gödel number xis provable in T”). See, e.g., [1,
Ch. 18], for the presentation of such a general form of Gödel’s second incompleteness theorem.
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760 A. Enayat, F. Pakhomov
implies that ιnis an interpretation of DTB−; thus it will be necessary to consider nthat
are non-standard from external point of view.
Note that each translation ιnconsists of finitely many formulae, giving the interpre-
tation of the domains and all symbols of the signature of ITB, and thus could be easily
represented by a number. The interpretation of arithmetic in each ιnis the identity
interpretation, but the domain of indices of truth definition ιnis given by the following
formula D(n)(x):
x<n∧¬ψ(x).
For all nthe relation ≺is interpreted by <. The formula Tα(x)is interpreted by the
formula IT(n)(y,x), where ycorresponds to α, and xcorresponds to itself:

i<k
(x=ϕi→
m<n
((y=m∧¬ψ(m)) →ιm(ϕi))),
where ιm(ϕi)is the ιm-translation of the sentence ϕi. 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 ιnare indeed the desired interpre-
tations inside T, i.e., we need to prove that for all nand axioms Aof ITB−we have
T(ιn(A)). Clearly it is the case for all the axioms of Qand the axioms of partial order
for ≺. Now let us show that for any s<k:
T(ιn(∀α(Tα(ϕs)↔ϕ≺α
s))). ()
By compositional axioms, we just need to show that for all usuch that u<nand
T(¬ψ(u)) we have:
T
i<kϕs=ϕi→
m<nu=m∧¬ψ(m)→ιm(ϕi)↔Tιnϕ≺u
s.
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:
T(ιu(ϕs)) ↔Tιnϕ≺u
s.
In order to prove this we will show by induction on subformulae θof ϕsthat for the
universal closure θof θ:
Tιu(θ↔Tιn(θ ≺u).
Note that since ϕsis 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 θis Tα(x):
T(ιu(∀α∀xTα(x))) ↔T(ιn(∀α≺u∀x(Tα(x)))).
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Truth, disjunction, and induction 761
Hence we just need to show that for all p<usuch 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:
T
i<ko=ϕi→
m<up=m∧¬ψ(m)→ιm(ϕi),
T
i<ko=ϕi→
m<np=m∧¬ψ(m)→ιm(ϕi).
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 nsuch 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 ιnactually is an interpretation of
DTB−inside T. We externally fix some proof of inconsistency from axioms of DTB−
and follow it inside Tto derive a contradiction, thereby completing the proof of IC .
Corollary 3.4 CT−[I0+Exp]+DC proves PA, and therefore CT−[I0+Exp]+DC
and CT−[PA] +DC axiomatize the same theory.
Proof This is an immediate consequence of Lemma 3.3, and the provability of Tarski
bi-conditionals in CT−[I0+Exp].
Remark 3.5 Note that Theorem 3.2 could be regarded as a strengthening of Tarski’s
undefinability of truth theorem: Tarski’s theorem essentially states that there could be
no hierarchy of truth definitions whose set of indices contains a reflexive point. To the
best of the authors’ knowledge there is no known proof of Tarski’s theorem that avoids
the use of any kind of diagonalization constructions.7In a preprint of this paper we
raised a question of whether Lemma 3.3, which we proved using Theorem 3.2, could
be proved more directly without the use of diagonalization. Later we noticed that it
is possible to replace the use of Theorem 3.2 in the proof of Lemma 3.3 with a result
by Flumini and Sato [7, Thm. 1], which has a rather simple proof that in our opinion
could be regarded as diagonalization-free. Since this other proof is of methodological
interest, we will provide its general outline.
The results of Flumini and Sato in [7] are fairly general and are applicable to various
second-order systems. To keep our presentation compact we will just formulate a direct
corollary of [7, Thm. 1] that will be relevant to us. The only axioms of our base system
of second-order arithmetic Bwill be those of I0+Exp. We denote as Ind the usual
second-order induction principle:
∀X(0∈X∧∀x(x∈X→x+1∈X)→∀xx∈X).
For a set Xand number awe denote as (X)athe set {b|a,b∈X}and for binary
relation ≺we denote as X≺athe set {a,b|a,b∈ Xand a≺a}. For a second
7We refer to the introduction of Visser’s paper [21] for a discussion of the role of diagonalization in the
proofs of Gödel’s second incompleteness theorem and Tarski’s truth undefinability theorem.
123
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762 A. Enayat, F. Pakhomov
order formula ϕ(X,y)and binary relation ≺we could naturally write the formula
Hier(ϕ, ≺,H)that expresses that His a hierarchy along ≺produced by ϕ, i.e., His
such that ∀x((H)x={y|ϕ(H≺x,y)}). The usual arithmetic transfinite recursion
principle states that hierarchies exist for any well-ordering ≺and formula ϕwithout
second-order quantifiers. The corollary of [7, Thm. 1] that will be relevant to us is
that over Bthere is an arithmetic (moreover 0
1)formulaϕ(X,y)(with additional
variables) such that the axiom Ind is implied by the existence of ϕ-hierarchies along
arbitrarily large proper initial segments of natural numbers. Formally this principle is
∀z∃HHier(ϕ, <z,H), where <zis the restriction of the usual order <on naturals
to the numbers below z.
Now consider the ω-interpretation of the language of second-order arithmetic in
CT−[I0+Exp], where the range of sets consists of all the sets {n|T(ϕ(n))}. It is easy
to see that IC is equivalent to the validity of Ind in this interpretation.8So in view of the
above in order to show that DC implies Ind it will be enough to prove that DC implies
that the universal closure of ∀z∃HHier(ϕ, <z,H)holds in this interpretation. To
achieve the latter we could just directly construct the formula defining the relevant
hierarchy and then use DC to verify that it indeed has the desired property.
4 Disjunctive correctness + inductive correctness implies
10(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 4.1 In what follows ∈Ack is “Ackermann’s epsilon”, i.e., x∈Ack yis the
arithmetical formula that expresses “the x-th bit of the binary expansion of yis a 1”.
(a) For a unary predicate U(x),theLA+Usentence PCU(read as “Uis piece-wise
coded”) is the following sentence:
∀u∃y∀x[(U(x)∧x<u)↔x∈Ack y].
(b) More generally, given an n-ary LA+U-formula ϕ(U,x0,...,xn−1),PCϕis the
following LA+U-sentence:
∀u∃y∀x0,...∀xn−1(ϕ(U,x0,...,xn−1)∧(x0<u∧ ···∧xn−1<u)
↔xi:i<n∈Ack y,
where xi:i<nis a canonical code for the ordered n-tuple (x0,...,xn−1).
The following lemma shows that over I0+Exp the scheme I0(U)is equivalent
to the single sentence ‘Uis piecewise coded’.The lemma is folklore; we present the
proof for the sake of completeness.
8See Remark 4.3.1 for a sharper formulation of this equivalence.
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Truth, disjunction, and induction 763
Lemma 4.2 The following are equivalent over I0+Exp:
(a) I0(U).
(b) PCU.
Proof We will reason in I0+Exp. Recall that both x=2yand x∈Ack yhave
0-definitions within I0[8,Ch.V].
(a−→ b): Assume I0(U).Givenu,letwbe the Ackermann-code for the set of
predecessors of u(i.e., ∀x(x<u↔x∈Ack w)). Clearly w=i<u2i=2u−1,
and wis an upper bound for any wthat codes a subset of the predecessors of u.Let
δ(u,w)be the 0-formula below:
(w =2u)→[∃y<w∀x<u((U(x)∧x<u)↔x∈Ack y)].
A simple induction on u(where wis treated as parameter) using I0(U)shows that
∀u∀wδ(u,w) holds, which completes the proof that PCUholds.
(b→a): A straightforward induction on the complexity of 0(U)-formulae shows
that:
(∗)If Uis piecewise coded and δ(U,x0,...,xn−1)is a 0(U)-formula, then PCδholds.
The base case of the induction is clearly equivalent to the assumption that Uis piece-
wise coded. What allows the inductive steps to be smoothly carried out is that, provably
in I10+Exp, ∈Ack obeys many familiar axioms of set theory, as verified in [8,Ch.I,
Thm. 1.39]. As an example, in the existential case of the induction, we suppose that
δ(U,x0,...,xn−1)is a 0(U)-formula such that PCδholds, and then establish PCδ,
where δ=∃x0<t(x0,...,xn−1)δ(U,x0,...,xn−1)for some term t.Todoso,let
us fix any number uand demonstrate that there is an ∈Ack-set s, where:
s={
xi:0<i<n|x1,...,xn−1<uand δ(U,x1,...,xn−1)}.
Let vbe a number such that the value t(x0,...,xn−1)≤vfor x0,...,xn−1<u.By
PCδwe have the following ∈Ack-set s:
s={
xi:i<n|x0,...,xn−1<v and δ(U,x0,...,xn−1)}.
Using 0-Separation we construct the set sas:
{xi:0<i<n|x1,...,xn−1<uand ∃x0<t(x0,...,xn−1)xi:i<n∈Ack s)}.
With (∗)at our disposal, we could trivially deduce the least number principle for
0(U)-formula, which is of course equivalent to I0(U).
•In the next lemma and its proof, Code(c,ϕ,u)denotes the ternary LA+T-formula
∀xx<u∧Tϕ(x)↔x∈Ack c,
and PC(ϕ) denotes the LA+T-formula ∀u∃cCode(c,ϕ,u).
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764 A. Enayat, F. Pakhomov
Lemma 4.3 CT−[I0+Exp]+IC proves ∀ϕ( Form1
A(ϕ) →PC(ϕ)).
Proof We reason in CT−[I0+Exp]+IC. Given ϕ(x)in Form 1
A, we need to show:
(1) ∀u∃cCode(c,ϕ,u).
By the compositional properties of T, (1) is equivalent to:
(2) T(∀uψ(u)), where ψ(u):= ∃c(∀x<uϕ(x)↔x∈Ack c).
On the other hand, ∀uψ(u)is the conclusion of the formula Indψ(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))).
Remark 4.3.1 Lemma 4.3 can be readily strengthened to a more general result whose
proof we leave to the reader: CT−[I0+Exp]+IC verifies ACA0for the ω-interpretation
of the language of second-order arithmetic, where the range of sets consists of all the
sets {n|T(ϕ(n))}, where ϕis any first-order formula. Moreover, it is a theorem of
CT−[I0+Exp]that IC is equivalent to the veracity of ACA0within this interpretation.
Lemma 4.4 CT−[I0+Exp]+DC +IC I0(T).
Proof Reason in CT−[I0+Exp]+DC +IC. By Lemma 4.2, it suffices to show that
Tis piecewise coded. Let ϕi:i<ube the sequence of arithmetical sentences such
that ϕiis the sentence with Gödel-number iif there is such a sentence, and otherwise
ϕiis the sentence 0 =1. We wish to show that {i<u:T(ϕi)}is coded. Towards this
goal, consider the unary formula θ(x)∈FormAgiven by:
θ(x):= 
i<u(x=i)∧ϕi.
Claim (∗)∀i<uT(ϕi)↔T(θ(i)).
(→)Suppose T(ϕi)for some i<u. Then T(i=i)∧ϕi, and hence by DC we have
T(θ(i)).
(←)Suppose T(θ(i)) for some i<u. Then by DC, there is some j<usuch that
T(i=j)∧ϕj.SoT(ϕi)holds since Tcommutes with conjunction and T(i=j)
holds iff i=j.
By coupling Claim (∗)together with Lemma 4.3, we can conclude that i<u:Tθ(i)
is coded. 
5 Closing remarks and open questions
Question 5.1 Is the generalization of Theorem 1in which CT−is weakened to CS−
(where Sstands for satisfaction) true?
•The notion CS−[B]is defined in [5] for base theories Bformulated in relational
languages, using the notation BFS (FS for “full satisfaction”); and in [4, Ch. 7] for
functional languages. We expect that an examination of the proofs in Sects. 3and
4would show that this question has a positive answer.
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Truth, disjunction, and induction 765
Question 5.2 Is IC provable in CT−[S1
2]+DC?
•In the above, S1
2is 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 S1
2extends LA. In the proof of
Lemma 3.3, most likely it is possible to use some tricks with effective formulae
(see [19, Section 3]) in order to modify the definition of ιnin 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 ιnare indeed interpretations
inside the truth predicate.
Question 5.3 Let DCElim be the ‘half’ of DC that asserts: if a finite disjunction is true,
then at least one of the disjuncts is also true. Is CT−[PA +DCElim]conservative over
PA?
•DC can be written as the conjunction of two implications DCElim and DCIntro,
where DCIntro is the converse of DCElim (i.e., DCIntro asserts: a disjunction is true,
whenever at least one of its disjuncts is true). Recent joint of work of Wcisło,
Łełyk and Enayat (to appear) shows that DCIntro can be conservatively added to
CT−[PA] +IC +∀xTruen(x)→T(x):n<ω
.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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Affiliations
Ali Enayat1·Fedor Pakhomov2
Fedor Pakhomov
pakhfn@mi.ras.ru
1Department of Philosophy, Linguistics, and the Theory of Science, University of Gothenburg,
Gothenburg, Sweden
2Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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