New Constructions of Satisfaction Classes

Abstract

We use model-theoretic ideas to present a perspicuous and versatile method of constructing full satisfaction classes on models of Peano arithmetic. We also comment on the ramifications of our work on issues related to conservativity and interpretability.

Full text

NEW CONSTRUCTIONS OF SATISFACTION CLASSES
ALI ENAYAT AND ALBERT VISSER
Abstract. We use model-theoretic ideas to present a perspicuous and
versatile method of constructing full satisfaction classes on models of
Peano arithmetic. We also comment on the ramifications of our work
on issues related to conservativity and interpretability.
1. Introduction
In our forthcoming paper [2] we explore satisfaction classes over a wide
variety of ‘base theories’ ranging from weak fragments of arithmetic to sys-
tems of ZF set theory and beyond. This note provides a synopsis of some
of this work in the context of the most popular base theory adopted in in-
vestigations of axiomatic theories of truth, namely PA (Peano Arithmetic).
The notion of a satisfaction class was first introduced and investigated by
Krajewski in his 1976 paper [11]. Two noteworthy accomplishments of [11]
are the following results:
(1) If a countable model of a ‘base theory’ (such as PA) carries at least
one full satisfaction class, then it carries continuum-many full satisfaction
classes.
(2) Every model of ZF has an elementary extension that carries a full satis-
faction class.
The question whether the analogue of (2) holds for PA remained open until
the appearance of the joint work [10] of Kotlarski, Krajewski, and Lachlan in
1981, in which the rather exotic proof-theoretic technology of ‘M-logic’ (an
infinitary logical system based on a nonstandard model M), was invented
to construct ‘truth classes’ over countable recursively saturated models of
PA.1This model-theoretic result can be used to show that the analogue of
(2) does indeed hold for PA,which in turn can be used to show that PAFT
is conservative over PA,where PAFT =PA + “Tis a full truth class”. The
This research was partially supported by a grant from the Descartes Center of Utrecht
University, which supported the first author’s visit to Utrecht to work closely with the
second author.
1As explained in Section 4, a truth class is essentially a well-behaved kind of satisfaction
class. The M-logic methodology was further elaborated to establish refined constructions
of full truth classes by Smith [15,16,17], Kaye [9], and Engstr¨om [3].
1

2 ALI ENAYAT AND ALBERT VISSER
conservativity of PAFT over PA has attracted considerable philosophical at-
tention, especially in relation to the grand debate concerning deflationism.2
In this paper we present a perspicuous method for the construction of full
satisfaction classes that is dominantly based on model-theoretic techniques
(e.g., expanding the language, compactness, and elementary chains). As
we shall see, our construction method is quite versatile and can be used
to construct many (if not all) of the results that have hitherto been only
possible to establish with the use of M-logic machinery. Furthermore, the
method can also be employed to build new types of full satisfaction classes
(see Section 6).
We present the necessary preliminaries in Section 2, and then in Section 3
we concentrate on the basic form of our new construction of full satisfaction
classes, where it is used to show that every model of PA has an elemen-
tary extension that carries a full satisfaction class. The versatility of the
methodology of Section 3 is illustrated in Section 4, in which an appropriate
modification of the method is used to construct truth classes for models of
PA. As explained in Section 5, certain arithmetizations of our construc-
tion can also be employed to establish that (1) PAFT is interpretable in PA;
and (2) the conservativity of PAFT over PA can be verified in PRA (Primi-
tive Recursive Arithmetic). Finally, in Section 6 we briefly describe further
applications of the methods introduced in this paper.
Acknowledgments. We are grateful to the editors of this volume for their
interest in our work. Thanks also to Volker Halbach, Fredrik Engstr¨om, and
James Schmerl for helpful feedback on preliminary drafts of this paper. We
are particularly indebted to Schmerl for catching an inaccuracy in an earlier
formulation of Lemma 3.1, and for his suggestion to distill the results of our
paper [2] in this form for wider dissemination.
2. Preliminaries
2.1. Definition. Throughout this paper PA refers to Peano arithmetic for-
mulated in a relational language LPA using the logical constants {¬,∨,∃,=}.
Note that in this formulation PA has no constant symbols; the arithmetical
operations of addition and multiplication are construed as ternary relations;
and conjunction, universal quantification, and other logical constants are
taken as defined notions in the usual way.
It is well known that PA has more than sufficient expressive machinery to
handle syntactic notions. The following list of LPA-formulae will be useful
here.3
2A recent noteworthy paper in this connection is McGee’s [13].
3All of the formulae in the list can be arranged to be Σ1-formulae in the sense of
Definition 2.4.

SATISFACTION CLASSES 3
•Form(x) is the formula expressing “xis the code of an LPA -formula
using variables {vi:i∈N},and the non-logical symbols available in
LPA”.
•Asn(x) is the formula expressing “xis the code of an assignment”,
where an assignment here simply refers to a function whose domain
consists of a (finite) set of variables. We use αand its variants (α0,
α0, etc.) to range over assignments.
•y∈FV(x) is the formula expressing “Form(x) and yis a free variable
of x”.
•y∈Dom(α) is the formula expressing “the domain of αincludes y”.
•Asn(α, x) is the following formula expressing “αis an assignment for
x”:
(Form(x)∧Asn(α)∧ ∀y(y∈Dom(α)↔y∈FV(x)) .
•xCyis the formula expressing “xis the code of an immediate
subformula of the LPA -formula coded by y”, i.e., xCyabbreviates
the conjunction of Form(y) and the following disjunction:
(y=¬x)∨ ∃z((y=x∨z)∨(y=z∨x)) ∨ ∃i(y=∃vix).4
The theory PAFS (read as “PA with full satisfaction”) is formulated in an
expansion of the language LPA by adding a new binary predicate S(x, y).
The binary/unary distinction is of course not an essential one since PA has
access to a definable pairing function. However, the binary/unary distinction
at the conceptual level marks the key difference between satisfaction classes
and truth classes (the latter are discussed in Section 4).PAFS is defined
below with the help of a collection of sentences Tarski(S,F).
When reading the definition below it is helpful to bear in mind that
Tarski(S,F) expresses:
Fis a subset of Form that is closed under immediate sub-
formulae; each member of Sis an ordered pair of the form
(x, α), where x∈Fand αis an assignment for x; and S
satisfies Tarski’s compositional clauses.
2.2. Definition. PAFS := PA ∪Tarski (S,Form), where Tarski(S,F) is the
conjunction of the universal generalizations of the formulae tarski0(S,F)
through tarski4(S,F) described below, all of which are formulated in LPA ∪
{F(·),S(·,·)}, where Sand Fdo not appear in LPA.
4Technically speaking, this formula should be written so as to distinguish the logical
operations of the meta-langauge with those of the object-language. For example, using
Feferman’s commonly used ‘dot-convention’, one would write:
y=¬
•x∨ ∃zy=x∨
•z∨y=z∨
•x ∨ ∃iy=∃
•vix.
However, since the difference between the two kinds of operation will be always clear from
the context, we have opted for the lighter notation.

4 ALI ENAYAT AND ALBERT VISSER
In the following formulae Rranges over the relations in LPA;
t, t0, t1,· · · are metavariables, e.g, we write R(t0,···, tn−1)
instead of Rvi0,···, vin−1; and α0⊇αabbreviates
(Dom(α0)⊇Dom(α)) ∧ ∀t∈Dom(α)α(t) = α0(t).
•tarski0(S,F) := (F(x)→Form(x)) ∧(S(x, α)→(F(x)∧Asn(α, x))) ∧
(yCx∧F(x)→F(y)) .
•tarski1,R(S,F) :=
F(x)∧(x=pR(t0,···, tn−1)q)∧Asn(α, x)∧V
i<n
α(ti) = ai→
(S(x, α)↔R(a0,···, an−1)) .
•tarski2(S,F) := (F(x)∧(x=¬y)∧Asn(α, x)) →
(S(x, α)↔ ¬S(y, α)) .
•tarski3(S,F) := (F(x)∧(x=y1∨y2)∧Asn(α, x)) →
(S(x, α)↔(S(y1, α FV(y1)) ∨S(y2, α FV(y2)))) .
•tarski4(S,F) := (F(x)∧(x=∃t y)∧Asn(α, x)) →
(S(x, α)↔ ∃α0⊇αS(y, α0)) .
2.3. Definition. Suppose M |=LPA,F⊆M, and Sis a binary relation
on M.5
(a) Sis an F-satisfaction class if (M, S, F )|=Tarski(S,F).6
(b) Let ωMbe the well-founded initial segment of Mthat is isomorphic to
the ordinal ω. We say that Fis the set of standard LPA-formulae of Mif
F=FormM∩ωM.
In this case there is a unique F-satisfaction class on M, known as the
Tarskian satisfaction class on M.
(c) Sis a full satisfaction class on Mif Sis an F-satisfaction class for
F:= FormM.This is equivalent to (M, S)|=PAFS .
2.4. Definition. Σ0= Π0= the collection of LPA-formulae all of whose
quantifiers are of the form ∃x < y ϕ or ∀x < y ϕ; Σn+1 consists of formulae
of the form ∃x0· · · ∃xk−1ϕ, where ϕ∈Πn; and Πn+1 consists of formulae
of the form ∀x0· · · ∀xk−1ϕ, where ϕ∈Σn.Here kranges over ω, with
the understanding that k= 0 corresponds to an empty block of quantifiers;
this convention leads to the pleasant consequence that Σn⊆Σn+1 and
Πn⊆Πn+1 for all n.
5Throughout the paper we use the convention of using M,M0, N , etc. to denote the
universes of discourse of structures M,M0,N,etc.
6Note that the closure of Funder direct subformulae does not guarantee that Fshould
also contain ‘infinitely deep’ subformulae of a nonstandard formula in F.

SATISFACTION CLASSES 5
2.5. Theorem. (Mostowski [9], [6]) For each nonzero n < ω there is a
binary Σn-formula Satn(x, y)such that
PA `Tarski(Satn,Σn),
where Σnis (the arithmetization of )the set of codes of formulae in Σn.
3. The Basic Construction
In this section we explain the basic methodology of building satisfaction
classes using tools from model theory. The following lemma lies at the heart
of the main result of this section.
3.1. Lemma. Let N0|=PA,F1:= FormN0,F0⊆F1,and suppose S0is an
F0-satisfaction class. Then there is an elementary extension N1of N0that
carries an F1-satisfaction class S1⊇S0and (c, α)∈S0whenever c∈F0,
α∈N0,and (c, α)∈S1.
Proof: Let L+
PA(N0) be the language obtained by enriching LPA with con-
stant symbols for each member of N0, and new unary predicates Ucfor each
c∈FormN0. It helps to have in mind that the intended interpretation of Uc
is {α∈Ac:S1(c, α)}, where Ac:= {α:N1|=Asn(α, c)}.
We first wish to describe a new set of axioms
Θ := {θc:c∈F1}
formulated in L+
PA(N0), where θcstipulates ‘local Tarskian behavior’ for Uc.
If R∈ LPA and N0|=c=pR(t0,···, tn−1)q, then
θc:= ∀α(Uc(α)↔Asn(α, c)∧R(α(t0),···, α(tn−1))).
If N0|=c=¬d, then
θc:= ∀α(Uc(α)↔Asn(α, c)∧ ¬Ud(α)) .
If N0|=c=d1∨d2, then
θc:= ∀α(Uc(α)↔Asn(α, c)∧(Ud1(αFV(d1)) ∨Ud2(αFV(d2)))) .
If N0|=c=∃vab, then
θc:= ∀α(Uc(α)↔Asn(α, c)∧ ∃α0⊇αUb(α0)∧Asn(α0, b)) .

6 ALI ENAYAT AND ALBERT VISSER
Let
Γ := {Uc(α) : c∈F0and (c, α)∈S0} ∪ {¬Uc(α) : c∈F0and (c, α)/∈S0},
and let
Th+(N0) := Th(N0, a)a∈N0∪Θ∪Γ.
We now proceed to show that Th+(N0) is consistent by demonstrating that
each finite subset of Th+(N0) is interpretable in (N0, S0).To this end, sup-
pose T0is a finite subset of Th+(N0) and let Cconsist of the collection
of c∈F1such that Ucappears in T0.If C=∅,T0is readily seen to be
consistent, so we shall assume that C6=∅for the rest of the argument.
Our goal is to construct subsets {Uc:c∈C}of N0such that the following
two conditions hold when Ucis interpreted by Uc:
(1) (N0, Uc)c∈C|={θc:c∈C},and
(2) For c∈C∩F0, Uc={α∈N0: (c, α)∈S0}.
We shall construct {Uc:c∈C}in stages, beginning with the simplest
formulae in C, and working our way up using Tarski rules for more complex
ones. Recall that cCdexpresses “cis a direct subformula of d”. Define C∗
on Cby:
cC∗diff (cCd)N0and θd∈T0∩Θ.
Note that whenever cC∗d, then for all c0Cdwe have c0∈Cand c0C∗d.
The finiteness of Cimplies that (C, C∗) is well-founded, which in turn helps
us define a useful measure of complexity for c∈Cusing the following
recursive definition:
rankC(c) := sup{rankC(d) + 1 : d∈Cand dC∗c}.
Note that for c∈C,rankC(c) = 0 precisely when θc/∈T0∩Θ. Next, let
Ci:= {c∈C:rankC(c)≤i}.
Observe that C06=∅(since Cis finite and nonempty), and that if c∈Ci+1,
then the codes of all immediate subformulae of the formula coded by care
in Ci.This observation ensures that the following recursive clauses yield a
well-defined Ucfor each c∈C.
•If c∈C0then Uc:= {α: (c, α)∈S0},if c∈F0;
Uc:= ∅,if c /∈F0..
•If c∈Ci+1\Ciand c=¬d, then
Uc:= {α∈Ac:α /∈Ud}.
•If c∈Ci+1\Ciand c=a∨b, then
Uc:= {α∈Ac:αFV(a)∈Uaor αFV(b)∈Ub}.

SATISFACTION CLASSES 7
•If c∈Ci+1\Ciand c=∃vab, then
Uc:= α∈Ac:∃α0∈N(α⊆α0and α0∈Ub).
Note that in the first item above, the choice of Uc:= ∅when c∈C0and
c /∈F0is completely arbitrary.7Also, in the third item above where c=a∨b,
both aand bwill be in Ci, thanks to the properties of C∗.
It is routine to verify, using induction on rankC(c),that (1) and (2) hold
for (N0, Uc)c∈C. More specifically, if rankC(c) = 0, then θc/∈T0∩Θ, so
(1) is vacuously satisfied, and (2) is satisfied by design. On the other hand,
when rankC(c)>0 then (1) is satisfied since Ucis defined so as to comply
with Tarski conditions; and (2) is satisfied since S0is an F0-satisfaction
class. This concludes the proof of the consistency of arbitrary finite subsets
T0of Th+(N0), which in turn shows that Th+(N0) has a model, i.e., some
elementary extension N1of N0has an expansion N+
1of the form
N+
1:= (N1, Uc)c∈F1
with the property that N+
1|=Th+(N0).Let S1be the binary relation on
N1defined via
S1(c, α)⇔α∈Uc.
It is evident that S1is an F1-satisfaction class, S1⊇S0and (c, α)∈S0
whenever c∈F0,α∈N0, and (c, α)∈S1.
3.2. Theorem. Let M0be a model of PA of any cardinality.
(a) If S0is an F0-satisfaction class on M0,then there is an elementary
extension Mof M0that carries a full satisfaction class that extends S0.
(b) There is an elementary extension Mof M0that carries a full satisfac-
tion class.
Proof: Note that (b) is an immediate consequence of (a) since we may
choose F0to be the set of atomic M0-formulae and S0to be the obvious
satisfaction predicate for F0.To establish (a), we note that by Lemma 3.1
there is an elementary extension M1of M0that carries an F1-satisfaction
class, where F1:= FormM0. Lemma 3.1 allows this argument to be carried
out ω-times to yield two sequences hMi:i∈ωiand hSi:i∈ωithat satisfy
the following properties for each i∈ω:
(1) Mi≺ Mi+1;
(2) Si+1 is an Fi+1-satisfaction class on Mi+1 with Fi+1 := FormMi; and
(3) Si=Si+1 ∩ {(c, α) : c∈Fi,Mi|=Asn(α, c)}.
Let M:= S
i∈ω
Mi,and S:= S
i∈ω
Si. Tarski’s elementary chain theorem and
(1) together imply that Melementarily extends M0. It is easy to see, using
(2) and (3), that Sis a full satisfaction class on M.
7As shown in [2] this feature can be exploited to construct ‘pathological’ satisfaction
classes, such as the one mentioned at the end of Section 6 of this paper.

8 ALI ENAYAT AND ALBERT VISSER
Theorem 3.2, when coupled with the completeness theorem of first order
logic, immediately yields the following conservativity result.
3.3. Corollary. PAFS is a conservative extension of PA.
Proof: Suppose not. Then for some arithmetical sentence ϕwe have:
(1) PAFS `ϕ, and
(2) PA 0ϕ.
Since (2) implies that PA ∪ {¬ϕ}is consistent, by the completeness theorem
for first order logic, there is a model M0|=PA ∪ {¬ϕ}.On the other hand,
by part (b) of Theorem 3.2 there is an elementary extension M1of M0
that carries a full satisfaction class, and therefore by (1) M1|=ϕ. This
contradicts the fact that M1elementarily extends M0.
3.4. Corollary. Every resplendent model of PA carries a full satisfac-
tion class. In particular, every countable recursively saturated model of PA
carries a full satisfaction class.
Proof: The first claim directly follows from the definition of a resplendent
model. The second claim follows from the first claim, when coupled with
the key result that countable recursively saturated models are resplendent
(see [9, Section 15.2] for more detail). 
4. Truth Classes
With the exception of Krajewski’s original paper [11], what we refer to as a
‘truth class’ here has been dubbed ‘satisfaction class’ in the model-theoretic
literature. More specifically, Krajewski [11] employed the framework of sat-
isfaction classes over base theories formulated in relational languages as in
this paper, however, the later series of papers [10], [16], and [17] all used the
framework of truth classes over Peano arithmetic formulated in a relational
language, augmented with ‘domain constants’. Later, Kaye [9] developed
the theory of satisfaction classes over models of PA in languages incorpo-
rating function symbols; his work was extended by Engstr¨om [3] to truth
classes over models of PA in functional languages.
As explained in this section, there is a simple canonical correspondence
between truth classes over models of PA (in a relational language) and cer-
tain types of satisfaction classes, here referred to as ‘extensional’. The main
aim of this section is to demonstrate that the method of building satisfac-
tion classes in the previous section can be conveniently modified so as to
yield full extensional satisfaction classes (and thereby: full truth classes)
over appropriate models of PA.
Within PA one can easily define an injective function cthat yields the code
for a constant symbol xfor each member xof the domain. This enables PA to
internally represent the language L+
PA =LPA+ ‘domain constants’. We can
then add a unary predicate T(x) denoting a truth class (instead of a binary

SATISFACTION CLASSES 9
predicate S(x, y) for a satisfaction class) to LPA, whose intended interpre-
tation is “xis the code of a true sentence σ”, where σis an arithmetical
sentence formulated in a language L+
PA. We will make this more precise in
the following definition.
4.1. Definition.PAFT := PA∪Tarski(T), where Tarski(T) is the conjunction
of the universal generalizations of tarski0(T) through tarski4(T),all formu-
lated in the language LPA ∪ {T(·)}, as described below.8In what follows
Sent(x) is the LPA-formula that expresses “xis a formula of L+
PA with no
free variables”, and Rranges over relations symbols in LPA.
•tarski0(T) := (T(x)→Sent(x)) .
•tarski1,R(T) := pR(t0,···, tn−1)q=x→(R(t0,···, tn−1)↔T(x)) .
•tarski2(T) := (x=¬y)→(T(x)↔ ¬T(y1)) .
•tarski3(T) := (x=y1∨y2)→(T(x)↔(T(y1)∨T(y2))) .
•tarski4(T) := (x=∃viϕ)→(T(x)↔ ∃zT(ϕ(z)))).
Tis a full truth class on Mif (M, T )|=PAFT .
4.2. Definition. A substitution for a formula ψof LPA is a function
σ:FV(ψ)→Var
such that σrespects substitutability in the ‘usual way’, i.e., if xis a free
variable of ψ, then xis not in the scope of any quantifier that binds σ(x).
Given ψand σas above, let ψ∗σbe the formula obtained from ψby
applying the substitution σ, and Abe the set of pairs (ϕ, α) such that αis
an assignment for the formula ϕ. This allows us to define a key equivalence
relation ∼on Aby decreeing that (ϕ0, α0)∼(ϕ1, α1) iff there is some
(ψ, β)∈A, and there are substitutions σ0and σ1for ψ, with
ϕi=ψ∗σiand β=αi◦σi, for i= 0,1.
In the above, αi◦σiis the composition of αiand σi.It is important to bear
in mind that, intuitively speaking, (ϕ0, α0)∼(ϕ1, α1) means that ϕ0and ϕ1
are the same except for their free variables, and for all variables xand y, if x
occurs freely in the same position in ϕ0as ydoes in ϕ1, then α0(x) = α1(y).
•An F-satisfaction class Sis extensional if for all ϕ0and ϕ1in F,
M |= (ϕ0, α0)∼(ϕ1, α1) implies (ϕ0, α0)∈Siff (ϕ1, α1)∈S.9
The following proposition describes a canonical correspondence between
extensional satisfaction classes and truth classes. The routine but laborious
proof is left to the reader.
8PAFT is the relational analogue of the theory of CTin Halbach’s monograph [8]. The
base theory of CTis PA formulated in a functional language. The conservativity of CT
over the functional language version of PA can also be established using the techniques of
this paper (see Section 6).
9Note that an extensional satisfaction predicate need not be closed under re-naming of
bound variables.

10 ALI ENAYAT AND ALBERT VISSER
•In what follows cis the M-definable injection m7→cmthat desig-
nates a constant symbol mfor each m∈M, and ϕ(c◦α) is the sen-
tence in the language L+
PA obtained by replacing each occurrence of
a free variable xof ϕwith the constant symbol m, where α(x) = m.
4.3. Proposition. Suppose M |=PA,Tis a full truth class on M, and S
is an extensional full satisfaction class on M.
(a) S(T)is an extensional satisfaction class on M, where S(T)is defined
as the collection of ordered pairs (ϕ, α)such that ϕ(c◦α)∈T .
(b) T(S)is a truth class on M, where T(S)is defined as the collection
of ϕ∈ L+
PA such that for some ψ∈ L+
Band some assignment αfor ψ,
ϕ=ψ(c◦α)and (ψ, α)∈S.
(c) S(T(S)) = S,and T(S(T)) = T.
Before describing the construction of extensional satisfaction classes we
need the preliminaries presented in Definition 4.4 and Lemma 4.5.
4.4. Definition.
(a) Given formulae ϕ0and ϕ1of LPA, we write ϕ0≈ϕ1if there is a formula
ψ, and substitutions σ0and σ1for ψsuch that ϕi≈ψ∗σifor i= 0,1.
(b) Given c∈FormM, let TCM(c) be the externally defined transitive
closure of cwith respect to the direct subformula relation, i.e.,
TCM(c) := S
n<ω
TCM(c, n),
where TCM(c, 0) := {c}and
TCM(c, n + 1) := {x∈M:xCMdfor some d∈TCM(c, n)}.
The following lemma presents salient features of the two equivalence re-
lations ∼and ≈.
4.5. Lemma.Let ∼be as in Definition 4.2; and TCM(c)and ≈be as in
Definition 4.4.
(i)If d∈TCM(c)and d6=c, then ¬(c≈d).
(ii)≈preserves the principal connectives, i.e., it relates negations to nega-
tions, disjunctions to disjunctions, and existential formulae to existential
formulae with the same bound variable. Moreover, if ¬c≈¬d, then c≈d;
if c∨d≈c0∨d0, then c≈c0and d≈d0;and if ∃t c ≈∃t0c0, then t=t0
and c≈c0.
(iii)If (ϕ0, α0)∼(ϕ1, α1), then ϕ0≈ϕ1.
(iv)If (¬ϕ0, α0)∼(¬ϕ1, α0),then (ϕ0, α0)∼(ϕ1, α1).
(v)If (ϕ0∨ϕ1, α)∼(ϕ0
0∨ϕ0
1, α0), then (ϕ0, α FV(ϕ0)) ∼(ϕ0
0, α0FV(ϕ0
0))
and (ϕ1, α FV(ϕ1)) ∼(ϕ0
1, α0FV(ϕ0
1)) .

SATISFACTION CLASSES 11
(vi)If ϕ=∃t ψ, and ϕ0=∃t0ψ0,and (ϕ, α)∼(ϕ0, α0),then t=t0and for
some e
(ϕ, α[t:e]) ∼(ϕ0, α0[t0:e]).10
The next Lemma presents a variant of Lemma 3.1 that is our main tool
for constructing extensional satisfaction classes.
4.6. Lemma. Let N0|=PA,F1:= FormN0,F0⊆F1,and suppose S0is
an extensional F0-satisfaction class. Then there is an elementary extension
N1of N0that carries an extensional F1-satisfaction class S1⊇S0and
(c, α)∈S0whenever c∈F0,α∈N0,and (c, α)∈S1.
Proof. Let Θ and Γ be as in the proof of Lemma 3.1, and let
Th+(N0) := Th(N0, a)a∈N0∪Θ∪Γ∪∆,
where ∆ := {δcc0:c, c0∈F1},and
δcc0:= ∀α∀α0((c, α)∼(c0, α0)→(Uc(α)↔Uc0(α0))) .
The proof of the lemma would be complete once we verify that Th+(N0)
has a model. To this end, we shall demonstrate that every finite subset T0
of Th+(N0) is interpretable in N. Let Cbe the collection of c∈F1such
that cappears in T0.Also, let C∗and rankC(c) be precisely as in the proof
of Lemma 3.1.
We can extend Cto another finite set Cso that it satisfies a certain
closure property, namely: whenever we have c≈c0and dC∗c, where c,c0
and dare all in C, then there is some d0∈Csuch that d0C∗c0with d≈d0.
This can be done simply by adding any missing direct subformulae d0by an
appropriate recursion.11 By replacing Cby Cwe may therefore additionally
assume:
(#) If cand c0are both in Cwith c≈c0, then rankC(c) = rankC(c0).
As in the proof of Lemma 3.1, we then recursively construct {Uc:c∈C}
such that:
(1) (N0, Uc)c∈C|={θc:c∈C}and
(2) For c∈C∩F0, Uc={α∈N0: (c, α)∈S0}.
10Here α[t:e] is the assignment obtained by redefining the value of αat the variable
tto be eif t∈Dom(α); note that α[t:e] = αif t /∈Dom(α).
11More specifically, first define C◦on Cby d0C◦ciff d0C∗c0≈c, for some c0∈C. Since
C◦is cycle-free, Cis well-founded, and therefore lends itself to a ranking function rank◦
C(c).
Let n= max {rank◦
C(c) : c∈C},and for 0 ≤i≤ndefine Di:= {c∈C:rank◦
C(c) = i}.
Next use a ‘backward’ recursion to define En, En−1,· · ·, E0via:
•En:= Dn;
•En−(i+1) := Dn−(i+1) ∪ {d:dCN0cfor some c∈En−i}.
Finally, let C:= En∪···∪E0.It is easy to see that Cis finite, extends C, and has the
desired closure property.

12 ALI ENAYAT AND ALBERT VISSER
It remains to show:
(3) (N0, Uc)c∈C|={δcc0:c, c0∈C}.
We establish (3) by using induction on rankC(c) to show that ∀c∈C P (c),
where P(c) abbreviates:
∀c0∈C(N0, Uc)c∈C|=∀α∀α0((c, α)∼(c0, α0)→(Uc(α)↔Uc0(α0))) .
If rankC(c) = 0 and (c, α)∼(c0, α0), then by part (iii) of Lemma 4.5 we
have c≈c0,which in turn by (#) assures us that rankC(c0) = 0. This makes
it clear that P(c) holds when rankC(c) = 0 since S0is assumed to be an
extensional F0-satisfaction class.
To verify the inductive step, suppose:
(4) P(x) holds for all x∈Cwith rankC(x) = i.
Let c∈Cwith rankC(c) = i+1,and suppose (c, α)∼(c0, α0), where c=∃t d.
Then c0=∃t0d0, and d≈d0by part (ii) of Lemma 4.5. Observe that thanks
to (#) we have:
(5) rankC(c0) = i+ 1 and rankC(d) = rankC(d0) = i.
Now if α∈Uc,then α[t:e]∈Udfor some eby (1), and therefore by part
(vi) of Lemma 4.5, we obtain:
(6) (d, α[t:e]) ∼(d0, α0[t0:e]).
Using (4), (5), and (6) we may now conclude that α0[t:e]∈Ud0, which by
(1) yields α0∈Ud, thus completing the verification of the quantificational
case (by symmetry). A similar reasoning can be carried out for propositional
cases. This concludes the proof of consistency of T0.
The rest is precisely as before: the consistency of Th+(N0) implies that
there is an elementary extension N1of N0that has an expansion N+
1:=
(N1, S)|=Th+(N0),and the binary relation S1on N1defined via
S1(c, α)⇔α∈Uc
has the property that S1is an extensional F1-satisfaction, S1⊇S0, and
(c, α)∈S0whenever c∈F0,α∈N0, and (c, α)∈S1.
4.7. Theorem. Let M0|=PA.There is an elementary extension Mof
M0that carries a full extensional satisfaction class.
Proof: Since the satisfaction class S0on the collection F0of atomic formulae
of M0is extensional, we may use Lemma 4.6 instead of Lemma 3.1 in order
to carry out the elementary chain argument of Theorem 3.2. 
By coupling Theorem 4.7 with part (b) of Proposition 4.3 we obtain:
4.8. Corollary. Every model of PA has an elementary extension that
carries a full truth class.
Finally, the line of reasoning employed in the proof of Corollary 3.3 shows,
using Corollary 4.8, that:

SATISFACTION CLASSES 13
4.9. Corollary. PAFT is a conservative extension of PA.
5. Arithmetization, Interpretability, and Conservativity
Here we briefly discuss the arithmetization of the constructions of the pre-
vious two sections, with an eye towards issues connected with interpretabil-
ity and conservativity. As explained in [2, Section 4] the compactness and
elementary chain argument employed in the proofs of Theorems 3.2 and 4.7
can be implemented in the fragment IΣ2of PA with the help of the ‘Low
Basis Theorem’ of Recursion Theory. Coupled with Orey’s Compactness
Theorem, this can used to establish the following:
5.1. Theorem. [2] PAFT is interpretable in PA.12
On the other hand, the technology of LL1-sets13 of [6, Theorem 4.2.7.1,
p. 104] can be used to show that the proofs of both theorems 3.2 and 4.7
can even be implemented in the fragment IΣ1of PA. In light of the fact that
the statement “PAFT is conservative over PA” is a Π2-statement, and IΣ1is
well known14 to be Π2-conservative over PRA, we obtain the following:
5.2. Theorem. [2] The conservativity of PAFT over PA can be verified in
PRA.15
5.3. Remark. The verification of the conservativity of PAFT over PA within
PRA was first claimed by Halbach in [7], using cut-elimination.16 Later, Fis-
cher [4] gave a proof, based on the cut-elimination argument in [7], to show
that PAFT is interpretable in PA.Unfortunately, a gap was discovered re-
cently (by Fujimoto) in the cut-elimination argument in [7], which in turn
impaired Fischer’s interpretability claim. Happily, Leigh [12] has succeeded
in developing a proof-theoretic demonstration of the conservativity of PAFT
over PA that is implementable in PRA.Moreover, [12, Theorem 1] can be
12Indeed BFS turns out to be interpretable in Bfor all base theories Bthat have ac-
cess to the full scheme of induction over their ambient ‘numbers’. In particular, ACAFS is
interpretable in ACA.On the other hand, as shown in [2, Section 8], ACAFS
0is not inter-
pretable in ACA0(more generally, BFS is shown to be not interpretable in B, if Bis finitely
axiomatizable).
13LL1-sets are a special type of ‘low’ sets.
14This classical result was independently established by Mints, Parsons, and Takeuti,
using proof-theoretic methods. The work of Paris and Kirby (described in [14, IX.3]),
and more recently Avigad [1] has also provided model-theoretic demonstrations of this
conservativity result.
15Indeed, by using the technique of Friedman [5], this conservativity result is already
verifiable in the fragment SEFA (Superexponential Arithmetic) of PRA.
16Halbach’s base theory in his work is the usual version of PA that is formulated in a
functional language.

14 ALI ENAYAT AND ALBERT VISSER
used to verify the interpretability of PAFT over PA, by using Fischer’s strat-
egy in [4].17 Therefore, Theorems 5.1 and 5.2 can be arrived at via two
completely different routes.
6. Further Results
In Section 4 we saw that the core methodology of Section 3 can be fine-
tuned to build full extensional satisfaction classes. Indeed, as shown in [2]
one can strengthen Theorem 4.7 by imposing further desirable conditions
on the satisfaction class S. For example, every model M0of PA has an
elementary extension Mthat carries a full extensional satisfaction class S
that satisfies all of the following additional properties:
(1) SatM
n⊆Sfor all n∈ω(see Theorem 2.5 for Satn).
(2) If c∈FormMand M |= “cis an axiom of PA”, then Sdeems cto be
‘true’.18
(3) If cand c0are M-formulae such that M |= “c0is an alphabetic variant19
of c”, then (c, α)∈Siff (c0, α)∈S.
Furthermore, the third condition above can be strengthened by accomodat-
ing a combination of extensional equivalence and alphabetic equivalence,
thereby yielding truth classes that are closed under alphabetic equivalence.
We have also shown that a small dose of condition (1) can be used to build
full truth classes over models of arithmetical theories formulated in func-
tional languages (this result will appear in the projected sequel to [2]).
One can also use the method of Section 3 to build bizarre satisfaction
classes. For example, as shown in [2, Section 5], every model M0of PA has
an elementary extension Mthat has a full satisfaction class Sthat exhibits
the following pathology:
{a∈M: (σa, αNull)∈S}=ωM,
where ωMis the well-founded initial segment of Mthat is isomorphic to ω,
and σais defined for all a∈Mby a recursion within Mvia the following
clauses:
•σ0:= ∃v0(v0=v0) (or σ0= any other logically valid sentence);
•σn+1 := (σn∨σn).
17We are grateful to Graham Leigh for his kind permission to quote his unpublished
work here.
18As remarked in the last sentence of [10], this condition can also be arranged using
the machinery of M-logic. Note that ‘axioms of PA’ in the sense used here do not include
the logical axioms.
19c0is an alphabetic variant of cif c0is obtainable from cby the usual rules of re-naming
the bound variables of c.

SATISFACTION CLASSES 15
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Ali Enayat
Department of Philosophy, Linguistics, and Theory of Science,
University of Gothenburg,Box 200, SE 405 30,Gothenburg,
Sweden
ali.enayat@gu.se
Albert Visser
Department of Philosophy, Bestuursgebouw, Heidelberglaan 6,
584 CS Utrecht, The Netherlands.
albert.visser@phil.uu.nl