SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS

Abstract

The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every countable recursively saturated model of PA\mathsf {PA} (Peano arithmetic) carries a full satisfaction class. This result implies that the compositional theory of truth over PA\mathsf {PA} commonly known as CT−[PA]\mathsf {CT}^{-}[\mathsf {PA}] is conservative over PA\mathsf {PA} . In contrast, Pakhomov and Enayat (2019) showed that the addition of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true iff one of its disjuncts is true) to CT−[PA]\mathsf {CT}^{-}[\mathsf {PA}] axiomatizes the theory of truth CT0[PA]\mathsf {CT}_{0}[\mathsf {PA}] that was shown by Wcisło and Łełyk (2017) to be nonconservative over PA\mathsf {PA} . The main result of this paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full satisfaction classes over arbitrary countable recursively saturated models of PA\mathsf {PA} that satisfy arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its approximations.

Full text

The Review of Symbolic Logic,Page1of 18
SATISFACTION CLASSES WITH APPROXIMATE
DISJUNCTIVE CORRECTNESS
ALI ENAYAT
Department of Philosophy, Linguistics, and Theory of Science,
University of Gothenburg
Abstract. The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every
countable recursively saturated model of PA (Peano arithmetic) carries a full satisfaction class.
This result implies that the compositional theory of truth over PA commonly known as CT–[PA]
is conservative over PA. In contrast, Pakhomov and Enayat (2019) showed that the addition
of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true
iff one of its disjuncts is true) to CT–[PA] axiomatizes the theory of truth CT0[PA] that was
shown by Wcisło and Łełyk (2017) to be nonconservative over PA. The main result of this
paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full
satisfaction classes over arbitrary countable recursively saturated models of PA that satisfy
arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–
Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its
approximations.
§1. Introduction. Intuitively speaking, a binary relation Son a model Mof PA
(Peano arithmetic) is said to be a satisfaction class if S‘behaves like’ the usual
Tarskian satisfaction relation SatMon M, but in sharp contrast to the usual Tarskian
satisfaction relation on M,ifMis nonstandard, then Sis required to decide the ‘truth’
of at least some nonstandard ‘formulae’ of M. This notion was brought to prominence
in the 1970s and 1980s, thanks to the fine efforts of a number of logicians, including
(in alphabetical order) Barwise, Kotlarski, Krajewski, Lachlan, Murawski, Ratajczyk,
Kossak, Schlipf, Schmerl, Smith, and Wilmers. A notable precursor is Robinson whose
landmark 1963 paper [28] probed the subtle obstacles in the development of a coherent
Tarski-style semantic framework for nonstandard formulae.
The flurry of activity in the 1970s and 1980s revealed two distinct ‘flavors’ of
satisfaction classes: full satisfaction classes and inductive satisfaction classes. A full
satisfaction class on a model Mof arithmetic is required to decide the ‘truth’ of every
arithmetic ‘formula’ in M(including the nonstandard ones, if Mis nonstandard)
while obeying the usual Tarskian recursive clauses that relate the truth of a formula to
the truth of its components. In contrast, an inductive satisfaction class Son a model
Mis required to satisfy the following two properties: (1) Sobeys Tarski’s recursive
clauses for (at least) all standard arithmetical formulae; and (2) the expansion (M,S)
Received: November 22, 2023.
2020 Mathematics Subject Classification: Primary 03C62, 03F30, Secondary 03F25.
Key words and phrases: Peano arithmetic, axiomatic theories of truth, conservativity, disjunctive
correctness.
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic.
1doi:10.1017/S1755020324000182
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

2ALI ENAYAT
of Msatisfies the scheme of induction in the language obtained by augmenting the
arithmetical language with a new predicate symbol representing S. Kaye’s textbook
[18] covers the basics of both types of satisfaction classes. It is known that if Mis
a model of PA that carries a satisfaction class that is either full or inductive, then
Mis recursively saturated. For countable models, the converse also holds, i.e., every
countable recursively saturated model Mof PA carries a full satisfaction class S1,as
well as an inductive satisfaction class S2; but the existence of a satisfaction class on
Mthat is both full and inductive implies that the formal consistency of PA holds in
M(and much more) and thus by G¨
odel’s second incompleteness theorem not every
countable recursively saturated model of PA carries a satisfaction class that is both full
and inductive. The monograph [20] by Kossak and Schmerl includes a more advanced
treatment of inductive satisfaction classes, but it does not contain material on full
satisfaction classes. This can be explained by the fact that inductive satisfaction classes
have proved to be indispensable in the model theory of arithmetic, but full satisfaction
classes have not played a comparable role. To the best of the author’s knowledge,
the only known prominent application of full satisfaction classes over nonstandard
models to the model theory of PA is due to Smith [29], who employed them to calibrate
the logical complexity of the notions of recursive saturation and resplendence in the
context of models of PA (as respectively Σ1
1and Δ1
2).
However, full satisfaction classes have captured the imagination of philosophical
logicians since they are intimately linked with the grand topic of axiomatic theories of
truth, and in particular they shed light on the vibrant debate concerning the deflationist
conception of truth, especially in connection with the so-called conservativeness
argument. This philosophical interest has galvanized the subject of satisfaction classes
and has led to a new wave of technical advances and questions over the past decade.
The monographs by Halbach [16] and Cie´
sli´
nski [4] provide an overview of the
philosophical motivations as well as minutiae of the technical preliminaries.
There is also a notable methodological asymmetry between inductive satisfaction
classes and full satisfaction classes: given an arbitrary model Mof PA it is routine
to construct an elementary extension of Mthat carries an inductive satisfaction class
with the help of the standard tools of the trade (the ingredients are definable partial
satisfaction classes, and compactness). However, an air of mystery has surrounded
full satisfaction classes ever since the original tour de force Kotlarski–Krajewski–
Lachlan construction [23] that was based on the ad hoc technology of M-logic (an
infinitary logical system based on a nonstandard model M). The Kotlarski–Krajewski–
Lachlan construction implies that the axiomatic theory of truth commonly known
as CT–[PA]=CT–+PA is conservative over PA, i.e., if an arithmetical sentence ϕis
provable in CT–[PA], then ϕis already provable in PA.HereCT–is a finitely axiomatized
theory formulated in the language of arithmetic augmented with a truth predicate T
(the minus superscript indicates that no instances of the induction scheme mentioning
Thave been added to the theory). Decades later, a new versatile model-theoretic
method of constructing full satisfaction classes was presented by Visser and the author
[12]; this new method has been refined in various directions, e.g., as in Cie´
sli´
nski’s
monograph [4], and in the joint work of Łełyk and Wcisło with the author [9]. The
conservativity of CT–[PA]overPA has also been established by proof theoretic methods
by Leigh [25], and more recently by Cie´
sli´
nski [5].
The conservativity of CT–[PA]overPA together with G ¨
odel’s second incompleteness
theorem implies that the sentence Con(PA) expressing the consistency of PA is not
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 3
provable in CT–[PA].However, it is well-known that the formal 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 formulae in extended language (i.e., the language obtained by extending the
language of PA with a truth predicate). Indeed, it is straightforward to demonstrate
the consistency of PA within the subsystem CT1[PA]ofCT[PA], where CTn[PA]isthe
subtheory of CT[PA] with the scheme of induction over natural numbers limited to
formulae in the extended language that are at most of complexity Σn[32, theorem 2.8].
However, the case of CT0[PA] has taken considerable effort to analyze. Kotlarski [22]
established that CT0[PA ] is a subtheory of CT–[PA]+Ref(PA), where +Ref(PA)isthe
sentence in the extended language stating that “every first-order consequence of PA is
true”. Łełyk [26] demonstrated that the converse also holds, which immediately implies
that CT0[PA] is not conservative over PA since the formal consistency of PA is readily
provable in CT–[PA]+Ref (PA).1Kotlarski’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. Later Cie´
sli´
nski’s result was refined
by Pakhomov and the author [10] by demonstrating that CT–[IΔ0+Exp]+DC and
CT0[PA] axiomatize the same first-order theory, where DC (disjunctive correctness)
is the axiom that states that a disjunction is true iff one of its disjuncts is true. This
result in particular shows that CT–[IΔ0+Exp]+DC and CT–[PA]+DC axiomatize
the same theory.
The recent work of Cie´
sli´
nski, Łełyk, and Wcisło [7] refined the aforementioned work
of Pakhomov and the author by showing that CT–[PA]+DCout is an axiomatization
of CT0[PA], where DCout is the ‘half’ of DC that says every true disjunction has
a true disjunct. In summary, the arithmetical strength of CT–[PA] augmented with
seemingly innocuous axioms such as “truth is closed propositional proofs” or even “If
a disjunction is true, then it has a true disjunct” goes beyond PA.2The philosophical
ramifications of the nonconservativity of CT–[PA]+DC has been explored by Fujimoto
[14], whose work shows that the nonconservativity of CT–[PA]+DC over PA introduces
a new twist to the conservativity argument in relation to the deflationist conception of
truth.
In this paper we present two new constructions of extensional satisfaction classes
over models of PA, such satisfaction classes are inter-definable with their less famous
siblings known as ‘truth classes’ (see Proposition 2.14). The specific features of the
truth classes constructed in this paper can be summarized as follows.
The first construction (at work in the proofs of Theorem 3.3 and 3.4) employs basic
results in the model theory of PA to show that given a countable recursively saturated
model Mof PA, there are arbitrarily large cuts Iin Mwith the property that there is
atruthclassTon Mthat is compositional for the collection of sentences that have at
most ioccurrences of closed terms for some iin I, and it satisfies disjunctive correctness
1This result was first claimed by Kotlarski [22], 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 [26]. Łełyk’s work was preceded by the discovery of an elegant proof of the
nonconservativity of CT0[PA]overPA by Wcisło and Łełyk [32].
2The proof in [7] uses the version of CT–that includes the so-called regularity axiom; it is not
clear if the main result of [7] holds without the inclusion of the regularity axiom. In contrast,
the formulation in [10]ofCT–does not include the regularity axiom.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

4ALI ENAYAT
for such sentences. Note that the truth classes constructed in Theorem 3.3 and 3.4 are
not full, and as indicated in Question 3.7 it is not clear whether they can be extended
to full satisfaction classes.
The second construction (at work in the proof of Theorem 4.5) is far more involved
than the first one since it uses a vast array of technical results to elaborate the
construction of full truth classes in the joint paper [12] of Visser and the author to show
that given a countable recursively saturated model Mof PA, there are arbitrarily large
cuts Iin Msuch that there is a full truth class Ton Mthat is disjunctively correct for
disjunctions whose number of disjuncts is in I. This shows that in the aforementioned
result of Pakhomov and the author the assumption of disjunctive correctness cannot
be replaced with any of its approximations. As indicated in Remark 4.7, Theorem
4.5 implies the conservativity of an axiomatic theory of truth over PA that uses
a parameterized family of truth predicate Tx(where xvaries over the domain of
discourse) such that Txsatisfies CT–and Txis disjunctively correct for disjunctions
whose number of disjuncts is at most x.
We should draw attention to the recent joint work [1] of Athar Abdul-Quader and
Mateusz Łełyk who also explore approximations of disjunctive correctness, but their
results are complementary to those obtained in this paper since their focus is on a
different set of problems (see also Remark 4.6).
The main idea of the first construction was discovered by the author in 2009; the
same idea was earlier hit upon in 1980 by Smory´
nski in a letter to Jim Schmerl; for more
detail see Section 6 of [8] (which is a preliminary version of the present paper). The
protoform of the second construction appeared in a privately circulated manuscript
[11] written in collaboration with Albert Visser, and is included here with his kind
permission.
§2. Preliminaries. In this section we present the relevant notations, conventions,
definitions, and results that are needed in the subsequent sections.
2.1. Models of arithmetic.
Definition 2.1. The language of Peano Arithmetic, LPA ,is{+,·,S,0}.We use the
convention of writing M,M0,N, etc. to (respectively) denote the universes of discourse
of structures M,M0,N,etc. In what follows Mand Nare models of PA.
(a) PA (Peano arithmetic) is the result of adding the scheme of induction for all
LPA -formulae to the finitely axiomatizable theory known as (Robinson’s) Q.
(b) Σ0=Π
0=Δ
0= the collection of LPA -formulae all of whose quantifiers are
bounded by L-terms (i.e., they are of the form ∃x≤t, or of the form ∀x≤t,
where tis an LPA -term not involving x). More generally, Σn+1 consists of
formulae of the form ∃x0... ∃xk–1 ϕ,whereϕ∈Πn; and Πn+1 consists of
formulae of the form ∀x0... ∀xk–1 ϕ,whereϕ∈Σn(with the convention that
k= 0 corresponds to an empty block of quantifiers).
(c) If p∈Mand ϕ(x.y)isanLPA -formula, where xis an n-tuple of
variables,ϕ(x,p)M:= {m∈Mn:M|=ϕ(m, p}.ForX⊆Mn,Xis M-
definable if X=ϕ(x,p)Mfor some LPA -formula ϕ(x.y) and for some
parameter pin M.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 5
(d) A subset Xof Mis M-finite (orM-coded)ifX=cEfor some c∈M,where
cE={m∈M:M|=m∈Ack c},and m∈Ack cis shorthand for the LPA -
formula expressing “the mth bit of the binary expansion of cis 1”.
(e) We identify the longest well-founded initial segment of models of PA with the
ordinal . In this context, when Mis nonstandard, we refer to members of 
as standard elements, and we refer to as the standard cut of M.
(f) A subset Iof Mis a cut of Mif Iis an initial segment of Mwith no last
element.
Both constructions presented in this paper employ known isomorphism results about
countable recursively saturated models of arithmetic. The following classical result can
be found in Kaye’s monograph [18].
Theorem 2.2. Suppose Mand Nare countable recursively saturated models of PA.M
and Nare isomorphic iff SSy(M) = SSy(N)and Th(M)=Th(N).
In order to state the next isomorphism theorem we need to recall the following
definitions:
Definition 2.3. Suppose Iis a proper cut of M.I is -codedfrombelow(above)inM
iff for some c∈M, {(c)n:n∈}is cofinal in I(downward cofinal in M\I);here(c)n
is the exponent of the nth prime in the prime factorization of cwithin M.
Definition 2.4. Suppose Iis a cut of a model Mof PA .SSyI(M)is the collection
of subsets of Ithat are coded in M. More precisely, SSyI(M)consists of sets of the
form cE∩I, as cranges in M,wherecE={x∈M:M|=x∈Ack c}and ∈Ack is the
Ackermann membership defined by: x∈Ack ciff the xth digit of the binary expansion of
cis 1.
Theorem 2.5 (Kossak–Kotlarski [19, theorem 2.1, corollary 2.3]). Suppose Mand N
are countable recursively saturated models of PA with Th(M)=Th(N), and furthermore
suppose that Iis a cut shared by Mand Nsuch that SSyI(M) = SSyI(N)and Iis not
-coded from above in Mor in N.ThenMand Nare isomorphic over I, i.e., there is
an isomorphism fbetween Mand Nsuch that f(i)=ifor each i∈I.
Remark 2.6. Suppose Mis a nonstandard model of PA.
(a) Thanks to overspill, a cut Iof Mcannot be both -coded from above and
-coded from below in M. In particular, if cis a nonstandard element of
M, then the cut determined by finite powers of cis not -coded from above
in M. Thus there are arbitrarily large cuts in Mthat are not -coded from
above in M.
(b) It is easy to see that if Iis a strong cut of M, then Iis not -coded from above.
(c) Suppose Mis countable and recursively saturated. There are arbitrarily large
strong cuts Jof Msuch that J≺Mand J∼
=M. One way of seeing this is as
follows: by the resplendence property of M,Mcarries an inductive satisfaction
class S. Next, use the Phillips–Gaifman refinement of the MacDowell–
Specker theorem to build a countable conservative elementary end extension
(M∗,S∗)of(M,S). Note that ACA0holds in M,SSyM(M∗))since M∗
is a conservative extension of Mand therefore by [20, theorem 7.3.2] Mis
strong in M∗.Also observe that M∗is recursively saturated since M∗carries
an inductive satisfaction class. By part (a), given any cin Mthere is a cut
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

6ALI ENAYAT
Iof Mthat includes csuch that Iis not -coded from above in M. Since
M∗is an end extension of M,Iis not -coded from above in M∗and
SSyI(M) = SSyI(M∗).Therefore by Theorem 2.5 there is an isomorphism
f:M∗→Mthat is the identity on I. So if we let J=f(M), it is evident that
Jis a strong cut of Mthat includes c,J≺M,and J∼
=M.
2.2. Satisfaction and truth classes. Truth and satisfaction are often used inter-
changeably in mathematical logic, and this conflation is also at work when it comes to
the terms ‘truth class’ and ‘satisfaction class’. In this subsection we provide the precise
definitions of each. As we shall see in Proposition 2.14, a truth class is essentially an
extensional satisfaction class.3
Definition 2.7. We will use the following abbreviations relating to the arithmetization
of syntax; note that all the formulae in the list below are LPA -formulae.
(a) Form(x) is the formula expressing “xis (the code of) an LPA -formula”.
(b) Sent(x) is the conjunction of Form(x) and the formula expressing “xhas no
free variables”.
(c) Var(x) is the formula expressing “xis (the code of) a variable”.
(d) Asn(α) is the formula expressing “αis the code of an assignment”, where an
assignment here simply refers to a function whose domain consists of a (finite)
set of variables.
(e) y∈FV(x) is the formula expressing “Form(x)andyis a free variable of x”.
(f) y∈Dom(α) is the formula expressing “the domain of αincludes y”.
(g) Asn(α, x) is the following formula expressing “αis an assignment for x”:
Form(x)∧Asn(α)∧∀yy∈Dom(α)↔y∈FV(x)).
(h) For assignments αand α,α⊇αexpresses “the domain of αextends the
domain of αand α(v)=α(v) for all vin the domain of α”.
(i) xyis the formula expressing “xis the code of an immediate subformula
of the LPA -formula coded by y”, i.e., xyabbreviates the conjunction of
y∈Form and the following disjunction:
(y=¬x)∨∃z((y=x∨z)∨(y=z∨x)) ∨∃v∈Var (y=∃vx
).
(j) Given an LPA -term sand an assignment αwhose domain includes the free
variables of s,val(s, α )isthePA-definable function that outputs the value of s
when its free variables are replaced by the values specified by α.
Definition 2.8. The theory CS–(F)defined below is formulated in an expansion of LPA
by adding a fresh binary predicate S(x, y)(denoting satisfaction) and a fresh unary
predicate F(denoting a specified collection of formulae). The binary/unary distinction
is of course not an essential one since PA has access to a definable pairing function.
3For the subtle differences between satisfaction classes and truth classes see [4,6]. Historically
speaking, Krajewski [24] employed the framework of satisfaction classes overvarious theories
formulated in relational languages, however, the later series of papers [23,29,30] all used
the framework of truth classes over PA formulated in a relational language, augmented with
‘domain constants’ (which is the approach taken in [12]). Later, Kaye [18] developed the
theory of satisfaction classes over models of PA in languages incorporating function symbols;
Kaye’s work was extended by Engstr ¨
om [13] to truth classes over models of PA in functional
languages.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 7
However, the binary/unary distinction at the conceptual level marks the key difference
between the concepts of satisfaction and truth.
(a) CS–(F)is the conjunction of the universal generalizations of the formulae
tarski0(S,F) through tarski4(S,F) described below with the proviso that in what
follows αand αrange over assignments, and sand trange over LPA-terms
(that might have free variables). It is helpful to bear in mind that the axioms
of CS–(F) collectively express “Fis a subset of arithmetical formulae that is
closed under immediate subformulae; each member of Sis an ordered pair of
the form (x, α), where xis in Fand αis an assignment for x;andSsatisfies
Tarski’s compositional clauses for a satisfaction predicate”.
•tarski0(S,F):=F(x)→Form(x)∧yx∧F(x)→F(y)∧
S(x, α)→F(x)∧α∈Asn(x)).
•tarski1(S,F):=F(x)∧x=s=t∧α∈Asn(x)→S(x, α)↔val(s, α )=
val(t, α).
•tarski2(S,F):=F(x)∧(x=¬y)∧α∈Asn(x)→S(x, α)↔¬S(y, α).
•tarski3(S,F):=F(x)∧(x=y1∨y2)∧α∈Asn(x)→
S(x, α)↔Sy1,α FV(y1))∨Sy2,α FV(y2)).
•tarski4(S,F):=F(x)∧(x=∃vy)∧α∈Asn(x))→S(x, α)↔∃α⊇
αS(y, α).
(b) CS–is the theory whose axioms are obtained by substituting the predicate F(x)
by the LPA -formula Form(x) in the axioms of CS–(F). Thus the axioms on CS–
are formulated in the language obtained by adding Sto LPA (with no mention
of F).
Definition 2.9. Let M|=PA, and suppose F⊆FormM={m∈M:M|=Form(m)}.
(a) A subset Sof M2is said to be an F-satisfaction class on Mif (M,F,S)|=
CS–(F), here the interpretation of Fis Fand the interpretation of Sis S. S is a
satisfaction class on Mif Sis an F-satisfaction class for some F.
(b) An F-satisfaction class Sis extensional if for all ϕ0and ϕ1in F,M|=(ϕ0,α
0)∼
(ϕ1,α
1) implies (ϕ0,α
0)∈Siff (ϕ1,α
1)∈S, where (ϕ0,α
0)∼(ϕ1,α
1)means
that after substituting numerals in ϕ0in accordance with α0we obtain precisely
the same formula obtained by substituting numerals in ϕ1in accordance
with α1.
(c) A subset Tof Mis said to be a full satisfaction class on Mif (M,S)|=CS–
(equivalently: if (M,F,S)|=CT–(F)) for F=FormM.
Definition 2.10. The theory CT–(F)defined below is formulated in an expansion of
LPA by adding a fresh unary predicate T(x)(denoting satisfaction) and a fresh unary
predicate F(denoting a specified collection of formulae).
(a) CT–(F)is the conjunction of the universal generalizations of the formulae
tarski0(S,F) through tarski4(S,F).In what follows we use following conven-
tions: FSent(x) expresses “xis an LPA -sentence obtained by substituting closed
terms of LPA for every free variable of a formula in F”; yxexpresses “yis an
immediate subformula of x” as in Definition 2.7(i); sand trange over closed
terms of LPA ;s◦denotes the value of the closed term s, i.e., s◦=val(α, ∅)
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

8ALI ENAYAT
(where val is as in Definition 2.7(j) and ∅is the empty assignment); and
its variants (,
1,2) range over elements of FSent;ϕ(v) ranges over LPA -
formulae; F≤1(ϕ(v)) expresses “F(ϕ)andϕhas at most one free variable v”;
and ϕ[x/v] is (the code of ) the formula obtained by substituting all occurrences
of the variable vin ϕwith the numeral representing x.
•tarski0(T,F):=T(x)→FSent(x)∧yx∧F(x)→F(y).
•tarski1(T,F):=T(s=t)↔s◦=t◦.
•tarski2(T,F):=FSent()∧=¬)→T()↔¬T()).
•tarski3(T,F):=FSent()∧=1∨2→T()↔T(1)∨T(2)).
•tarski4(T,F):=F≤1(ϕ(v)) →T(∃vϕ)↔∃xT(ϕ[x/v]).
(b) CT–is the theory whose axioms are obtained by substituting the predicate F(x)
by the LPA -formula Form(x) in the axioms of CT–(F). Thus the axioms on CT–
are formulated in the language obtained by adding Tto LPA (with no mention
of F).
Definition 2.11. Let M|=PA, and suppose F⊆FormM, and Fis closed under direct
subformulae of M. Recall that FSent(M,F )consists of m∈Msuch that (M,F)satisfies
“mis an LPA -sentence obtained by substituting closed terms of LPA for the free variables
of a formula in F”.
(a) A subset Tof Mis an F-truth class on Mif (M,F,T)|=CT–(F), here the
interpretation of Fis Fand the interpretation of Tis T. T is a truth class on M
if Tis an F-truth class for some F.
(b) A subset Tof Mis a full truth class on Mif (M,T)|=CT–; equivalently: if
(M,F,T)|=CT–(F)forF=FormM.
(c) An F-truth class Ton Mis F-disjunctively correct, if whenever ∈FSentMand
{ϕi:i<m}is an M-coded subset of SentM(for some possibly nonstandard
m∈M), the following holds:
M|==
i<m
ϕi=⇒(∈T⇔∃i<mϕ
i∈T),()
where any grouping of {ϕi:i<m}canbeusedinMfor forming the
disjunction .
(d) Given a cut Iof M,anF-truth class Ton Mis I-disjunctively correct if Tis
disjunctively correct for disjunctions whose number of disjuncts is a member
of I; more precisely, if () holds whenever ∈FSentMand m∈I.
(e) Let ValMbe the set of theorems of first-order logic as computed in M.Given
an F-truth class Ton M,Tis F-deductively correct if whenever ∈T,∈
FSentM,and(→)∈ValM, then ∈T.
(f) Given a cut Iof Mand a truth class Ton M,Tis I-deductively correct if Tis
F-deductively correct for F=FormM∩I.
Remark 2.12. Let Mbe a nonstandard model of PA, and SatMbe the usual Tarskian
truth predicate on M.SatMinduces an -truth class TrueMon M,whereis the
standard cut of M. More specifically, suppose ϕ(x1,...,x
k)is a standard k-ary formula,
and t1,...,t
kare (codes of ) closed LPA -terms in the sense of M(thus t1,...,t
kneed not
be standard ). Since Mis a model of PA , the closed-term-evaluation function t→ t◦is
M-definable, and therefore there are elements m1,...,m
kin Msuch that M|=t◦
i=mi
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 9
for 1≤i≤k. Thus True Mcan be defined as follows: ϕ(t1/x1,...,t
k/xk)∈True Miff
(ϕ, α)∈SatM,whereαis the assignment given by xi→ mi.It is evident that TrueMis
-disjunctively and -deductively correct.
Remark 2.13. It is well-known [15, sec. V.5] that for each nonzero n∈there is
a unary Σn-formula Tru eΣnthat serves as a universal Σn-predicate within PA (indeed
IΔ0+Expsuffices for this purpose). Thus, if M|=PA , and Fn=the set of Σn-formulae
as computed in M,thenTrue M
Σnis an Fn-truth class that is Fn-disjunctively and Fn-
deductively correct. Note that if Mis nonstandard, Fnincludes nonstandard formulae.
The following proposition codifies the inter-definability of truth classes and
extensional satisfaction classes. The close relationship between extensional satisfaction
classes and truth classes was first made explicit in [12]; for further elaborations see [4,
Chapter 7] and [31]. In what follows num is the PA-definable function m→num m,where
mis the numeral for m∈M, and ϕ(num ◦α)istheLPA -sentence obtained by replacing
each occurrence of a free variable xof ϕwith m,whereα(x)=m.
Proposition 2.14. Suppose M|=PA ,Tis an F-truth class on M, and Sis an
extensional F-satisfaction class on M.
(a) S(T)is an extensional F-satisfaction class on M,whereS(T)is defined as the
collection of ordered pairs(ϕ, α)such that ϕ(num ◦α)∈T.
(b) T(S)is an F-truth class onM,whereT(S)is defined as the collection of ϕsuch
that (ϕ, ∅)∈S(where ∅is the empty assignment).
(c) S(T(S)) = S,and T(S(T)) = T.
The following result plays a key role in this paper; for a modern exposition see [15,
theorem I.4.33].
Mostowski’s Reflection Theorem 2.15. For each n∈, the consistency of the set of
sentences in True Σnis provable in PA . In particular, PA proves the consistency of each of
its finitely axiomatizable subtheories.
Remark 2.16. Mostowski’s Reflection Theorem, together with the Arithmetized
Completeness Theorem allow us to start with any model Mof PA and build an end
extension Nof Mthat satisfies PA, and moreover Nis strongly interpretable in Min
the sense that Nis interpretable in Mand there is an M-definable F-truth class TNon
Nfor F=FormM.To see this, consider the arithmetical formula (i):=Con(PAi),
where PA iis the set of axioms of PA whose code is at most i. Mostowski’s Reflection
Theorem implies that M|=(n)for each standard n. Hence by overspill M|=(e)for
some nonstandard element eof M. By invoking the Arithmetized Completeness Theorem
(as in [18, theorem 13.13]) within Mwe can conclude that there is a model Nof PA whose
elementary diagram is M-definable; and moreover, by a straightforward internal recursion
within Mthere is an M-definable embedding jof Monto an initial segment of N. Thus
by identifying Mwith its image under j, we can assume without loss of generality that Nis
an end extension of M. Note that the Arithmetized Completeness Theorem applied within
Myields an M-definable predicate T0such that (N,T
0)satisfies the slightly weaker form
of CT–(F)for F=FormMin which the closed terms are limited to those in M.However,
thanks to the availability of the closed-term-evaluation function in N, the simple ‘trick’
used in Remark 2.12 can be used to obtain an M-definable predicate Tsuch that (N,F,T)
satisfies CT–(F).Note that since Nis an end extension of M,Tru eM
Σ1⊆True M
Σ1, and
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

10 ALI ENAYAT
therefore if sis a closed arithmetical term in the sense of M,thenvalM(s)=val
N(s).
Since T0is the Tarskian satisfaction class of the structure Nas computed in the model M
of PA,T0as well as Thave many regularity properties of Tarskian satisfaction classes; in
particular Tis both M-deductively and M-deductively closed. As shown in (Stage 1 of )
the proof of Theorem 3.3,ifMis a countable recursively saturated model of PA,wecan
additionally require that there is an isomorphism f:N→M. Therefore by defining the
cut Iof Mas f(M), the image f(T)of Tis under fis an F∩I-satisfaction class on M
such that f(T)is both I-disjunctively and I-deductively correct.
§3. The first construction. Suppose Iis a cut of a countable recursively saturated
model Mof PA. In Definition 3.1 we define a subset FrugM
Iof FormM, and then in
Theorem 3.3 we build an F-truth class Ton Mfor F=FrugM
(i.e., for I=, where
is the standard cut of M) such that Tis F-disjunctively correct (in the sense of
Definition 2.11(c)). In Theorem 3.4 we extend this result to arbitrarily large cuts Iof
M. Theorem 3.3 and 3.4 could have been packaged as a single result, but in the interest
of offering the reader a clear intuition of the mechanism of the first construction we
opted for the current format; as a result we provide the full details of the proof of
Theorem 3.3 and only offer a proof outline for Theorem 3.4.
Definition 3.1. Aformulaϕis frugal if ϕhas no occurrence of a closed term. We use
Frug(ϕ)for the LPA -formula that expresses “ϕis a frugal LPA-formula”. Given M|=PA,
and m∈M,wedefine:
FrugM
≤m:= ϕ∈M:M|=Frug(ϕ)∧“ϕhas at most mdistinct free variables”)}.
For a cut Iof M,FrugM
I:= 
i∈I
FrugM
≤i.
Remark 3.2. The above definitions make it evident that for any cut Iof Mthat is closed
under addition, FrugM
Iis closed under Boolean connectives, existential quantification, and
immediate subformulae (here we are assuming the coding of formulae is done in a standard
way that ensures that the code of every immediate subformulae of a given formula ϕis
less than the code of ϕ). In particular, if Tis an F-truth class on Mfor F=FrugM
Iand 
is a sentence in Mthat is obtained from replacing all of the free variables of a formula in
Fwith closed terms, i.e., ∈FSent(M,F ),thenbytarski2(T,F)either ∈Tor ¬∈T
(but not both). Also note that in this context the sentences ∈FSent(M,F )are precisely
those sentences ∈SentMsuch that for some i∈Ithe number of distinct closed terms
that occur in is i(as computed in M).
Theorem 3.3. Suppose Mis a countable recursively saturated model of PA, and let
F=FrugM
. There is an F-truth class Ton Mthat is F-disjunctively correct.
Proof. The proof has two stages.
STAGE 1. In this stage, starting with a countable recursively saturated model Mof PA,
we use a variant of the construction outlined in Remark 2.16 to build an appropriate
end extension Nof Mwith two key properties: Mand Nare isomorphic, and yet
there is an F-truth class TNon Nthat is definable in M,whereF=FormM. To build
such an N, we first observe that by recursive saturation we can find an element cin M
that codes Th(M) by realizing the recursive type Σ(v), where
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 11
Σ(v):={ϕ↔(ϕ∈Ack v):ϕis sentence of LPA },
where ϕis the G ¨
odel number for ϕ,∈Ack is “Ackermann’s ∈”, as in Definition 2.1(d).
Next, let ϕn:n∈be a recursive enumeration of LPA -sentences, and consider the
arithmetical formula (i) defined below:
(i):=Con({ϕj:j<i∧ϕj∈Ack c}),
where Con(X) expresses the formal consistency of X. It is easy to see, using Mostowski’s
Reflection Theorem and our choice of cthat M|=(n) for each standard n. Hence
by overspill M|=(d) for some nonstandard element dof M. By invoking the
Arithmetized Completeness Theorem [18, theorem 13.13] within M, with a reasoning
similar to that in Remark 2.16 we can conclude that there is a model Nof Twith the
following properties:
(1) Th(M)=Th(N).
(2) There is an M-definable TNsuch that (N,FormM,T)|=CT–(F).
(3) Nis recursively saturated.
(4) There is an M-definable embedding jof Mas an initial segment of N.
(5) SSy(M) = SSy(N).
(6) There is an isomorphism f:M→N.
Note that (3) is a consequence of (2) and a routine overspill argument, as in
Proposition 15.4 of [18]; (5) is consequence of (4) since standard systems are preserved
by end extension; and (6) follows from (1) and (5) by Theorem 2.2.
STAGE 2. In this stage we construct the desired F-truth class Ton Mfor F=FrugM
.
Suppose ϕ(x1,x
2,...,x
k)∈F,wherek∈, and let t1,t
2,...,t
kbe elements of
ClTermM(closed terms in the sense of M). Note that ϕas well as t1,t
2,...,t
k
are allowed to be nonstandard. Using the truth class TNas in (2) of Stage 1,
together with the isomorphism fas (6) of Stage 1, we define Tto consist of
ϕ(t1,t
2, ... , tk)∈FSent(M,F )such that
ϕ(f(m1),f(m2),...,f(mk)) ∈TN,
where m1,m
2,...,m
kare elements of Msuch that M|=mi=t◦
ifor 1 ≤i≤k. In other
words,
ϕ(t1,t
2,...,t
k)∈Tiff ϕ(f(t◦
1),f(t◦
2),...,f(t◦
k))) ∈TN.
In the above we simply write t◦for the element of Msuch that M|=m=t◦. As noted
in the last sentence of Remark 2.16, since t∈M, t◦as-computed-in-Mcoincides with
t◦as-computed-in N.
We first verify that Tis an F-truth class on M. By the definition of T,the
axiom tarski0(F)=∀xT(x)→x∈FSentclearly holds in (M,F,T). Recall that
tarski1(F,T) asserts the equivalence of T(s=t)ands◦=t◦for all closed terms sand
t.Toseethat(M,F,T)|= tarski1(F) we argue as follows:
(s=t)∈Tiff f(s◦) = f(t◦)∈TN,
[by the definition of T]
f(s◦) = f(t◦)∈TNiff f(s◦)=f(t◦),
[since (x)◦=xand (N,F,T
N)|= tarski0(F)]
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

12 ALI ENAYAT
f(s◦)=f(t◦)iffs◦=t◦.
[since fis injective].
Next we verify that (M,F,T) satisfies tarski2(F,T), which stipulates that Tcommutes
with negation. Suppose ϕ(t1,t
2,...,t
k)=¬(t1,t
2,...,t
k)∈FSent(M,F ).Then we
have:
¬(t1,t
2,...,t
k)∈Tiff ¬(f(t◦
1),f(t◦
2),...,f(t◦
k)) ∈TN,
[by the definition of T]
¬(f(t◦
1),f(t◦
2),...,f(t◦
k)) ∈TNiff (f(t◦
1),f(t◦
2), ... , f(t◦
k)) /∈TN,
[since (N,F,T
N)|= tarski2(F)]
(f(t◦
1),f(t◦
2), ... , f(t◦
k)) /∈TNiff (t1,t
2,...,t
k)/∈T.
[by the definition of T]
An argument similar to the above shows that Tcommutes with disjunction, thus
tarski3(F,T) holds in (M,F,T); we leave the proof for the reader.
The axiom tarski4(F) stipulates that Tcommutes with existential quantification. For
this purpose suppose ϕ(t1,t
2,...,t
k)=∃v(x, t1,...,t
k)∈FSent(M,F ).Then we have:
∃v(x, t1, ... , tk)∈Tiff ∃v(v, f(t◦
1),f(t◦
2),...,f(t◦
k)) ∈TN,
[by the definition of T]
∃v(v, f (t◦
1),f(t◦
2),...,f(t◦
k)) ∈TNiff ∃b∈N(b,f(t◦
1),f(t◦
2),...,f(t◦
k)) ∈TN,
[since (N,F
0,T
N)|= tarski4(F)]
∃b∈N(b,f(t◦
1),f(t◦
2), ... , f(t◦
k)) ∈TNiff ∃c∈Mf(c)=b, (c,t
1,...,t
k)∈T.
[by the definition of T, together with surjectivity of f]
This concludes our verification that Tis an F-satisfaction class on M.
To see that Tis F-disjunctively correct suppose =
i<m
ϕi∈FSent(M,F )=SentM

(where mis a possibly nonstandard element of M). Note that each ϕican be written
as ϕi(t1,...,t
k) for some k∈with the understanding that the closed terms occurring
in ϕiare among t1,...,t
k.Then we have:

i<m
ϕi(t1,...,t
k)∈Tiff 
i<m
ϕi(f(t◦
1),f(t◦
2),...,f(t◦
k))∈TN,
[by the definition of T]

i<m
ϕi(f(t◦
1),f(t◦
2),...,f(t◦
k))∈TNiff ∃i<mϕ
i(f(t◦
1),f(t◦
2),...,f(t◦
k)) ∈TN,
[since TNis F-disjunctively correct and ∈FSent(M,F)],
∃i<mϕ
i(f(t◦
1),f(t◦
2), ... , f(t◦
k)) ∈TNiff ∃i<m
i<m
ϕi(t1,...,t
k)∈T.
[by the definition of T]
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 13
Theorem 3.4. Suppose Mis a countable recursively saturated model of PA. There are
arbitrarily large cuts I≺Msuch that I∼
=Mwith the property that there is an F-truth
class Ton Mfor F=FrugM
Isuch that Tis F-disjunctively correct.
Proof outline. Proceed as in Stage 1 of the proof of Theorem 3.3 to get hold of
the model N, but in Stage 2 invoke Remark 2.6(c) and use Theorem 2.5 instead of
Theorem 2.2 to get hold of an isomorphism fbetween Mand Nthat pointwise fixes
a prescribed cut of M. Note that by Remark 2.6(c) there are arbitrarily large strong
cuts Iin Msuch that I≺Mand I∼
=M, and by Remark 2.6(b) such cuts are not
-coded from above in M.
Remark 3.5. The proofs of Theorem 3.3 and 3.4 show that these two results can be
strengthened by requiring the truth predicate Tto satisfy further desirable properties such
as alphabetic correctness, which stipulates that Tis invariant under the renaming of bound
variables, and generalized term-extensionality, which stipulates that Tis invariant under
replacing of terms with the same value. As shown by Łełyk and Wcisło [27, theorem 23]
the theory obtained by the addition of axioms stipulating alphabetic correctness and
generalized term extensionality to CT–[PA]remains conservative over PA.
Remark 3.6. In answer to a question of the author, Lawrence Wong noted that Wciłso’s
proof of Lachlan’s theorem (as presented in [21]) shows that if Mis a model of PA that
has an expansion to a FormM
-truth class, then Mis recursively saturated.
Question 3.7. Does every countable recursively saturated model Mof PA carry a full
truth class Tthat is FormM
-disjunctively correct? More specifically, are the truth classes
constructed in Theorem 3.3 and 3.4 extendable to full truth classes?
§4. The second construction. The second construction in this paper employs an
arithmetized form of the main construction in [12] as in Theorem 4.1. Our method
of proof of Theorem 4.1 is through an arithmetization of the construction of a truth
class satisfying of CT–[PA],presented both in [9,21], which refine the model-theoretic
construction given in [12]forPA formulated in a relational language. As we shall
see, the compactness and elementary chain argument used in the model-theoretic
conservativity proof of CT–[PA]overPA canbeprovedinthefragmentIΣ
2of PA
with the help of the so-called Low Basis Theorem of Recursion Theory.4Note that
the results of this section that appear before Theorem 4.5 can be also established
by the arithmetization method presented in [9], or by taking advantage of the main
proof-theoretic result of [25].
Theorem 4.1 (Joint with Albert Visser). Suppose PA Con(B),where Bis some
recursively axiomatizable theory extending IΔ0+Exp.ThenPA strongly interprets
CT–[B], i.e., there are definitions 1and 2within PA such that for every model M
of PA,M
1=(M∗,T)|=CT–[B]and M
2is the elementary diagram of (M∗,T)as
viewed from M.
Proof. Before starting the proof, we need to review some key definitions and ideas
from Recursion Theory. In what follows Xand Yare subsets of .
4Indeed, using the technology of LL1-sets of [15, theorem 4.2.7.1, p. 104], the main argument
presented in this section can be carried out in IΣ1.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

14 ALI ENAYAT
•Xis low-Δ2,ifXis Δ2,andX≤T0(where ≤Tdenotes Turing reducibility,
and Zis the Turing-jump of Z).
•More generally, Yis low-Δ2in the oracle X,ifYis Δ2in the oracle X,and
Y≤TX.
By classical recursion theory, we have:
(1) X≤TYiff Xis Δ2in the oracle Y. Next, observe that if X≤T0and
Y≤TX, then Y≤T0, hence Yis low-Δ2by (1). Therefore:
(2) If Xis low-Δ2,andYis low-Δ2in the oracle X, then Yis low-Δ2.
The classical ‘Low Basis Theorem’ of Jockusch and Soare [17] asserts that every
infinite finitely branching recursive tree has an infinite branch Bsuch that Bis low-Δ2.
Moreover, it is known that the Low Basis Theorem is provable in IΔ0+BΣ
2(this is
due to Clote, whose argument is presented in [15, chap. I, sec. 3(c)]). In particular,
IΔ0+BΣ
2can prove that every countable consistent Δ1-set of first-order sentences
has a low-Δ2completion. Since the Henkin construction of a model of a prescribed
complete theory can already be performed in IΣ1, this shows that
(3) IΔ0+BΣ
2proves that every consistent Δ1set of first-order sentences has a
model Msuch that the satisfaction predicate SatMof Mis low-Δ2.
We are now ready to establish Theorem 4.1. We argue model-theoretically for ease
of exposition, and we will work with satisfaction classes instead of truth classes.
•From this point on, we assume the reader is familiar with [12] and follow the
notation therein.
Let M|=PA ,andBbe as in the assumption of the theorem. By (3) there is a
model Nof Bthat is strongly interpretable in Msuch that SatNis low-Δ2. Using
a truth-predicate for Δ2-predicates we can execute the construction of Lemma 3.1 of
[12] (the same idea can be applied to the counterpart of Lemma 3.1 in the proof
of the conservativity of [9,21]); the key idea is that if Sat(N,S)is low-Δ2, then so is
Th+(N), where Th+(N) is the theory defined as in [12, Lemma 3.1] involving extra
predicates added to the language of arithmetic whose consistency is established by
building a partial satisfaction predicate ‘by hand’ by examining an arbitrarily chosen
finite configuration of N-formulae. Therefore there is some elementary extension N1
of N, and some FormN-satisfaction class S1⊇Ssuch that Sat(N1,S1)is also low-Δ2,
thanks to (2) and the Low Basis Theorem. This shows that the construction
(N,S)→ (N1,S
1)
can be uniformly described in IΣ2+ Con(B). This allows one to obtain an M-definable
increasing sequence of structures (Mi,S
i):i∈M(as in the proof of Theorem 3.3
of [12]) with the additional feature that each (Mi,S
i)is strongly interpretable in M.
Consider the limit model:
(N,S)=
i∈M
(Mi,S
i).
(N,S)is clearly interpretable in M, and additionally (thanks to the elementary chains
theorem applied within M) the reduct Nof (N,S)is strongly interpretable in M, thus
M“knows” that Nis a model of B.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 15
However, there is no reason to expect that the expansion (N,S)is strongly
interpretable in M. To circumvent this problem, we appeal to a trick found by Łełyk
(first introduced in [9]) to take advantage of (N,S)in order to show that Msatisfies
Con(CT–[B]), thereby concluding the existence of a model of CT–[B] that is strongly
interpretable in M. The key idea is to resort to (1) the classical fact of proof theory
that any deduction in first order logic can be replaced by another deduction with same
conclusion but which has the subformula property,5(2) the fact that CT–[B]isthe
result of augmenting Bwith only finitely many truth axioms, and (3) the fact that M
has full access to the elementary diagram of N.The veracity of Con(CT–[B]) within
Mthen allows us to invoke the completeness theorem of first-order logic within Mto
get hold of a model of CT–[B] that is strongly interpretable in M.
Corollary 4.2. PA strongly interprets CT–[PA ].
Proof. Let Mbe an arbitrary model of PA. Within M,ifPA is inconsistent let B
be IΣk,wherekis the first nsuch that IΣn+1 is inconsistent, otherwise let B=PA.
Thus Con(B) holds in M. By Mostowski’s Reflection Theorem, Bextends IΣnfor each
n∈, and therefore Bextends PA (from an external point of view). Thus Theorem
4.1 applies.
Corollary 4.3. PA Con(CT–[B]) for every finitely axiomatized subtheory Bof PA
that extends IΔ0+Exp.
Proof. Let Bbe a finitely axiomatizable subtheory of PA. The formal consistency of
B0is verifiable in PA by Mostowski’s Reflection Theorem. Therefore by Theorem 4.1
PA can produce an internal model of CT–[B] for which it has a satisfaction predicate;
which in turn immediately implies that Con(CT–[B]) is provable in PA.
Corollary 4.4. CT–[PA ]is not finitely axiomatizable.
Proof. Put Corollary 4.3 together with G¨
odel’s second incompleteness
theorem.
We are now ready to present the main result of this section. Recall that the notions
of I-disjunctive correctness and I-deductive correctness were defined in Definition
2.11(d,f).
Theorem 4.5 (Joint with Albert Visser). Let Mbe a countable recursively saturated
model of PA.There are arbitrarily large cuts Iof Mfor which I≺Mand I∼
=M, and
for each such cut Ithere is a full truth class Ton Mthat is I-disjunctively and I-deductively
correct.
Proof. Let Mbe a countable recursively saturated model of PA.LetNbe as in the
proof of Stage 1 of the proof of Theorem 3.4, thus Nis an elementary extension of
Mthat is strongly interpreted in Mand SSy(M) = SSy(N).Thanks to Corollary
4.3 there is an elementary extension N∗of Nthat carries a truth class TN∗such that
(N∗,T
N∗)|=CT–[PA ],and (N∗,T
N∗) is strongly interpretable in M, which in turn
will allow us to conclude that (N∗,T
N∗) satisfies the following four properties:
5This fact is an immediate consequence of the cut-elimination theorem, which is provable
in PA; indeed it is well-known that cut-elimination is already provable in the fragment
IΔ0+Supexp of Primitive Recursive Arithmetic, where Supexp expresses the totality of
superexponentiation function (also known as tetration). See [2] for more detail.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

16 ALI ENAYAT
(1) TN∗is M-disjunctively and M-deductively correct. This is handled the by usual
argument by induction that shows that the classical Tarskian truth predicate
respects arbitrary disjunctions and is closed under deductions, as noted in
Remark 2.12.
(2) There is an M-definable embedding of Mas an initial segment of N∗.
(3) SSy(M) = SSy(N∗).
(4) N∗is recursively saturated.
Note that Th(M)=Th(N∗) since N∗elementarily extends N, and Th(M)=Th(N).
Together with (3), (4) and Theorem 2.2, this allows us to conclude that M∼
=N∗,
which implies that we ‘copy over’ TN∗on M, thus Mcarries a full truth Tclass that is
I-disjunctively and I-deductively correct for some cut Iof M. By using Theorem 2.5
and Remark 2.6 we can conclude that such cuts Ican be arranged to be arbitrarily large
in M. More specifically, by Remark 2.6(c) there are arbitrarily large strong cuts Iin M
such that I≺Mand I∼
=M, and by Remark 2.6(b) such cuts are not -coded from
above in M. Therefore by Theorem 2.5 forsuchagivencutIthere is an isomorphism
f:N∗→Mwith f(i)=ifor all i∈I. This makes it clear that T:= f(TN∗)isthe
desired full truth class on M.
Remark 4.6. Suppose M|=PA +¬Con(PA)and Mis countable and recursively
saturated. Then Mcarries an inductive satisfaction class S[18, corollary 15.12]. For
such an S, there is a topped nonstandard initial segment Iof Msuch that Ssatisfies the
Tarski conditions for all formulas in I.IfM, and ∈Mcodes up a proof of inconsistency
of PA in M,then/∈I. As shown by Abdul-Quader and Łełyk [1, proposition 43], the
methodology employed in [7] can be used to show that we cannot expect to build a full
satisfaction class Sthat is disjunctively correct on a cut Ithat contains if Ssatisfies
both the regularity axiom (also known as generalized term extensionality, as in Remark
3.5) and also internal induction. This is in contrast to Theorem 4.5 that shows that a full
satisfaction class that is I-disjunctively correct can always be arranged for arbitrarily high
cuts Iin M. Thus, even though the full satisfaction classes constructed in Theorem 4.5
can be arranged to satisfy the regularity axiom, in general they do not satisfy the internal
induction axiom.
Remark 4.7. Suppose T(x, y)is a binary predicate; we will write it as Tx(y)to indicate
that for a fixed x,Txis a truth predicate. Let CT∗be the sentence in the language of
PA augmented with Tx(y)that says that for all xthe axioms of CT–hold for Tx.Then
by putting Theorem 4.5 together with the completeness theorem of first-order logic, and
the fact that every countable model of PA has a countable elementary extension that is
recursively saturated we can conclude that the theory PA +CT∗+“∀xTxis disjunctively
correct for disjunctions whose number of disjuncts is at most x” is conservative over PA.
Remark 4.8. As pointed by the referee, one can give a “soft” proof of Theorem 4.5 for
the special case when Mis a model of True Arithmetic (the theory of the standard model of
PA). The argument goes as follows. It is known that if Iis a cut of a countable recursively
saturated model of PA that properly contains the Skolem closure of 0 in M, then for
every a∈Iand every c∈M, there is a b∈Msuch that c<band the type of ain M
coincides with the type of bin M. This observation allows one to build an automorphism
fof Msuch that f(a)=b, which in turn shows that (M,I)∼
=(M,J)for J=f(I).
It follows that if Mhas a full satisfaction class that is I-disjunctively correct, then M
also has a satisfaction class that is J-disjunctively correct (and the same for deductive
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS 17
correctness). Since the theory T(I, S)that says that Iis a cut and Sis an I-disjunctively
correct full satisfaction class is consistent with Th(M), by chronic resplendence Mhas a
cut Iand a full satisfaction class Ssuch that (M,I,S)is a resplendent model of T(I, S).
In particular Iis above the Skolem closure of 0and thus the aforementioned observation
applies.
Acknowledgments. I have benefitted from priceless feedback concerning the work
reported here from Lawrence Wong, Bartosz Wcisło, Albert Visser, Jim Schmerl,
Mateusz Łełyk, Roman Kossak, Cezary Cie´
sli´
nski, and Athar Abdul-Quader (in
reverse alphabetical order). Also many thanks to the anonymous referee whose detailed
comments were instrumental in reshaping the preliminary draft of this paper into its
current format.
Funding. The research presented in this paper was supported by the National
Science Centre, Poland (NCN), grant number 2019/34/A/HS1/00399.
BIBLIOGRAPHY
[1] Abdul-Quader, A., & Łełyk, M. (2024). Pathologies in satisfaction classes.
Annals of Pure and Applied Logic,175(2), Article no. 103387, 32 pp.
[2] Buss, S. (1998). An introduction to proof theory.InHandbook of Proof Theory.
S. R. Buss (ed), Amsterdam: North-Holland, pp. 1–78.
[3] Cie´
sli´
nski, C. (2010). Deflationary truth and pathologies. The Journal of
Philosophical Logic,39, 325–337.
[4] ———. (2017). The Epistemic Lightness of Truth: Deflationism and its Logic.
Cambridge: Cambridge University Press.
[5] ———. (2021). Interpreting the compositional truth predicate in models of
arithmetic. Archive for Mathematical Logic,60, 749–770.
[6] ———. (2021). On some problems with truth and satisfaction. In Trepczy ´
nski,
M., editor. Philosophical Approaches to the Foundations of Logic and Mathematics.
Leiden: Brill, pp. 175–192.
[7] Cie´
sli´
nski, C., Łełyk, M., & Wcisło, B. (2023). The two halves of disjunctive
correctness. Journal of Mathematical Logic,23(2), Article no. 2250026, 28 pp.
[8] Enayat, A. (2023). Curious satisfaction classes.Preprint,arXiv:2308.07463
[math.LO].
[9] Enayat, A., Łełyk, M., & Wcisło, B. (2020). Truth and feasible reducibility.
Journal of Symbolic Logic,85, 367–421.
[10] Enayat, A., & Pakhomov, F. (2019). Truth, disjunction, and induction. Archive
for Mathematical Logic,58, 753–766.
[11] Enayat, A., & Visser, A. (2012). Full satisfaction classes in a general setting,
privately circulated manuscript.
[12] ———. (2015). New constructions of full satisfaction classes. In Achourioti,
T., Galinon, H., Fujimoto, K., & Mart´
ınez-Fern´
andez, J., editors. Unifying the
Philosophy of Truth. New York: Springer, pp. 321–325.
[13] Engstr¨
om, F. (2002). Satisfaction classes in nonstandard models of first-order
arithmetic. Preprint, arXiv:math/0209408 [math.LO].
[14] Fujimoto, K. (2022). The function of truth and the conservativeness argument.
Mind,131, 129–157.
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press

18 ALI ENAYAT
[15] H´
ajek, P., & Pudl´
ak, P. (1993). Metamathematics of First-Order Arithmetic.
Berlin: Springer.
[16] Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge
University Press.
[17] Jockusch, C. G., & Soare, R. I. (1972). Π0
1classes and degrees of theories.
Transactions of the American Mathematical Society,173, 33–56.
[18] Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides. Oxford:
Oxford University Press.
[19] Kossak, R., & Kotlarski, H. (1988). Results on automorphisms of recursively
saturated models of PA. Fundamenta Mathematicae,129, 9–15.
[20] Kossak, R., & Schmerl, J. (2006). The Structure of Models of Arithmetic.
Oxford Logic Guides. Oxford: Oxford University Press.
[21] Kossak, R., & Wcisło, B. (2021). Disjunctions with stopping conditions.
Bulletin of Symbolic Logic,27, 231–253.
[22] Kotlarski, H. (1986). Bounded induction and satisfaction classes. Zeitschrift
f¨
ur matematische Logik und Grundlagen der Mathematik,32, 531–544.
[23] Kotlarski, H., Krajewski, S., & Lachlan, A. H. (1981). Construction of
satisfaction classes for nonstandard models. Canadian Mathematical Bulletin,24,
283–293.
[24] Krajewski, S. (1976). Nonstandard satisfaction classes. In Marek, W., et al.,
editors. Set Theory and Hierarchy Theory: A Memorial Tribute to Andrzej Mostowski,
537. Berlin: Springer, pp. 121–144.
[25] Leigh, G. (2015). Conservativity for theories of compositional truth via cut
elimination. Journal of Symbolic Logic,80, 845–865.
[26] Łełyk, M. (2023). Model theory and proof theory of the global reflection
principle. Journal of Symbolic Logic,88, 738–779.
[27] Łełyk, M., & Wcisło, B. (2021). Local collection and end-extensions of models
of compositional truth. Annals of Pure and Applied Logic,172, Article no. 102941,
22 pp.
[28] Robinson, A. (1963). On languages which are based on non-standard
arithmetic. Nagoya Mathematical Journal,22, 83–117.
[29] Smith, S. (1987). Nonstandard characterizations of recursive saturation and
resplendency. Journal of Symbolic Logic,52, 842–863.
[30] ———. (1989). Nonstandard definability. Annals of Pure and Applied Logic,
42, 21–43.
[31] Wcisło, B. (2024). Truth and collection. Preprint, arXiv:2403.19367 [math.LO].
[32] Wcisło, B., & Łełyk, M. (2017). Notes on bounded induction for the
compositional truth predicate. The Review of Symbolic Logic,10, 455–480.
DEPARTMENT OF PHILOSOPHY, LINGUISTICS, AND THEORY OF SCIENCE
UNIVERSITY OF GOTHENBURG
GOTHENBURG, SWEDEN
E-mail: ali.enayat@gu.se
https://doi.org/10.1017/S1755020324000182 Published online by Cambridge University Press