Axiomatizations of Peano Arithmetic: A truth-theoretic view

Abstract

We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic (PA){\textsf {(PA)}} . More specifically, let Elementary Arithmetic (EA){\textsf {(EA)}} be the fragment IΔ0+Exp\mathsf {I}\Delta _0 + \mathsf {Exp} of PA{\textsf {PA}} , and let CT−[EA]{\textsf {CT}}^-[{\textsf {EA}}] be the extension of EA{\textsf {EA}} by the commonly studied axioms of compositional truth CT−{\textsf {CT}}^- . We investigate both local and global properties of the family of first order theories of the form CT−[EA]+α{\textsf {CT}}^-[{\textsf {EA}}] +\alpha , where α\alpha is a particular way of expressing “ PA{\textsf {PA}} is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to PA{\textsf {PA}} and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of PA{\textsf {PA}} .

Full text

The Journal of Symbolic Logic,Page1of30
AXIOMATIZATIONS OF PEANO ARITHMETIC:
A TRUTH-THEORETIC VIEW
ALI ENAYAT AND MATEUSZ ŁEŁYK
Abstract. We employ the lens provided by formal truth theory to study axiomatizations of Peano
Arithmetic (PA). More specifically, let Elementary Arithmetic (EA) be the fragment IΔ0+Exp of PA,and
let CT–[EA] be the extension of EA by the commonly studied axioms of compositional truth CT–.We
investigate both local and global properties of the family of first order theories of the form CT–[EA]+α,
where αis a particular way of expressing “PA is true” (using the truth predicate). Our focus is dominantly
on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to
PA and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of PA.
§1. Introduction. Logicians have long known that different sets of axioms can
have the same deductive closure and yet their arithmetizations might exhibit
marked differences, e.g., by Craig’s trick every recursively enumerable set of
axioms is deductively equivalent to a primitive recursive set of axioms. Feferman’s
pivotal paper [8] on the arithmetization of metamathematics revealed many other
dramatic instances of this phenomenon relating to Peano Arithmetic. Let PA be the
usual axiomatization of Peano Arithmetic obtained by augmenting Q(Robinson
Arithmetic) with the induction scheme, and consider the theory that has come to be
known as Feferman Arithmetic, which we will denote by FA. The axioms of FA are
obtained by an infinite recursive process of “weeding out” applied to PA as follows:
enumerate the proofs of PA until a proof of 0 = 1 is arrived, and then discard the
largest axiom used in deriving 0 = 1; we then proceed to enumerate proofs using
only axioms of PA smaller than the one discarded. If we arrive at another proof of
0 = 1 from the reduced axiom system, we proceed in the same manner. By definition,
FA consists of the axioms of PA that remain upon the completion of this recursive
infinite process. Thus FA =PA in a sufficiently strong metatheory that can prove the
consistency of PA.1However, the consistency of FA is built into its definition and
PA can readily verify this fact; thus the equality of FA and PA is not provable in PA
even though this equality is provable in a sufficiently strong metatheory.
In this paper we employ the lens provided by formal truth theory to study axioma-
tizations of PA. Our focus is on two types of axiomatizations, namely: (1) schematic
Received August 17, 2021.
2020 Mathematics Subject Classification. Primary 03F30, 03F25, Secondary 03F25, 03C62.
Key words and Phrases. Peano arithmetic, axiomatic theories of truth, axiomatization, schemes,
conservativity.
1Recall that the consistency of PA is provable within Zermelo–Fraenkel set theory ZF; indeed the
consistency proof can be carried out in the small fragment of second order arithmetic obtained by
augmenting ACA0with the induction scheme for Σ1
1-formulae.
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic. This is an Open
Access article, distributed under the terms of the Creative Commons Attributionlicence (https://creativecommons.org/licenses/by/4.0/),
which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
0022-4812/00/0000-0000
DOI:10.1017/jsl.2022.83
1
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

2ALI ENAYAT AND MATEUSZ ŁEŁYK
axiomatizations that are deductively equivalent to PA, and (2) axiomatizations that
are proof-theoretically equivalent to the canonical axiomatization of PA . More
specifically, let Elementary Arithmetic (EA) be the fragment IΔ0+Exp of PA,and
CT–[EA] be the extension of EA by the commonly studied axioms of compositional
truth CT–(as in Definition 3). We investigate the family of first order theories of the
form CT–[EA]+α,where αeither uses a schematic description of PA to express “PA
is true,” or αuses a proof-theoretically equivalent formulation of PA to express “PA
is true” (in the sense of Definition 16).
Several problems can be posed about the aforementioned finitely axiomatized
theories of the form CT–[EA]+α, the most prominent of which is the determination
of their position with respect to the Tarski Boundary, i.e., the boundary that
demarcates the territory of truth theories that are conservative over PA.2For
example, the pioneering work of [14]showsthatCT–[EA]+α1is on the conservative
side of the Tarski Boundary, where α1is the sentence that expresses “each instance
of the induction scheme is true” (see Definition 7). On the other hand, let
PA+:= PA +{Con(n)|n∈},
where Con(n) is the arithmetical sentence that expresses “there is no proof of
inconsistency of PA whose code is below n”andis the set of natural numbers. It is
easy to see that PA+is deductively equivalent to PA (provably in EA). However, if we
consider a natural arithmetical definition of PA+,callit(x), and then we choose
α2to be the sentence
T[]:=∀x((x)→T(x)),(where Tis the truth predicate),
then CT–[EA]+α2is on the nonconservative side of the Tarski Boundary since
CT–[EA]+α2can prove the consistency of PA.
We now briefly discuss the highlights of the paper. In Theorem 26 we show that
the set Cons consisting of the (codes of) sentences αsuch that CT–[EA]+αis
conservative over PA is Π2-complete; which shows, a fortiori, that the collection
of sentences αsuch that CT–[EA]+αis conservative over PA is not recursively
enumerable. Another main result of the paper pertains to the strengthening CT0
of CT–[EA] obtained by augmenting CT–[EA] with the scheme of Δ0-induction
(in the extended language containing the truth predicate). It is known that the
arithmetical strength of CT0far surpasses that of PA, e.g., CT0can prove ConPA ,
ConPA+ConPA ,etc.(seeTheorem6). In Theorem 42 we show that given any r.e.
extension Uof PA such that CT0U, there is an axiomatization of PA which is
proof-theoretically equivalent to the usual axiomatization of PA andwhichhasthe
property that the arithmetical consequences of the (finitely axiomatized) theory3
CT–[EA]+T[] coincides with the deductive closure of U. (Note that Theorem 6
provides us with an ample supply of theories Uthat Theorem 42 is applicable to.)
2We will refer to the conservative (respectively nonconservative) side of the Tarski Boundary as
the region that is above (respectively below) the Tarski Boundary; this is in step with the traditional
Lindenbaum algebra view, where p→qis translated to p≤q.
3Throughout the whole text we systematically employ the word “theory” to refer to an arbitrary set
of sentences. In particular theories in our sense need not be closed under logical consequence.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 3
Our other main results are structural. In Section 3.2, we focus on the collection
SchPA consisting of the scheme templates such that PA is deductively equivalent to
the scheme generated by (see Definitions 7and 22). For example, in Theorem 30
we show that from the point of view of relative interpretability, theories of the
form CT–[EA]+T[], where ∈SchPA and T[] is the sentence asserting that every
instance of is true, have no maximal element.4In the same section we also prove that
the partially ordered set SchPA,≤CT–is universal for countable partial orderings
(in particular, it contains infinite antichains, and also contains a copy of the linearly
ordered set Qof the rationals), where the partial ordering ≤CT–is defined by
1≤CT–2iff CT–[EA]T[1]→T[2].
In Section 4.2, we prove similar results about the partial ordering Δ,≤CT–, where Δ
is the collection of elementary presentations of PA that are proof-theoretically equiv-
alent to (the canonical axiomatization of) PA. In particular we show that there is an
embedding CT0/PA →Δ,≤CT–,whereCT0/PA is the end segment of the Linden-
baum algebra of PA generated by the collection of arithmetical consequences of CT0.
Finally, in Theorem 58 of the last section of the paper we give a precise description
of the set sup PA consisting of arithmetical sentences that are provable in some theory
of the form CT–[EA]+T[], where (x) is an elementary formula (in the sense of
Definition 2) that defines an axiomatization of PA in the standard model Nof
arithmetic.
Our results are motivated by (1) seeking a better understanding of the contours of
the Tarski Boundary; (2) exploring the extent to which the statement “PA is true” is
determinate in the context of the basic compositional truth theory CT–[EA], and (3)
further investigating structural aspects of finite axiomatizations of infinite theories,
a topic initiated in the work of Pakhomov and Visser [22].
§2. Preliminaries.
2.1. CT–,CT0, and the Tarski Boundary.
Definition 1. Peano Arithmetic (PA) is the theory formulated in the language
{0,S,+,×} whose axioms consist of the axioms of Robinson’s Arithmetic Qtogether
with the induction scheme. We will denote the standard model of arithmetic by N
and its universe of discourse by .
Definition 2. Elementary Arithmetic (EA) is the fragment IΔ0+Exp of PA ,
where IΔ0is the induction scheme for Δ0-formulae (i.e., formulae with only bounded
quantifiers), and Exp asserts the totality of the function exp(x)=2
x(it is well-known
that the graph of exp can be described by a Δ0-formula). An elementary formula is
an arithmetical formula whose quantifiers are bounded by terms built from the
function symbols S,+,×,andexp. The family of (Kalm´
ar) elementary functions
is a distinguished subfamily of the primitive recursive functions.5It is well-known
that the provably recursive functions of EA are precisely the elementary functions;
4Once again, we treat the interpreted theory as greater in this ordering.
5Elementary functions occupy the third layer (E3) of the Grzegorczyk hierarchy of primitive recursive
functions {En|n∈}. It is often claimed that almost all number theoretical functions that arise in
mathematical practice are elementary.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

4ALI ENAYAT AND MATEUSZ ŁEŁYK
and that a function fis elementary iff fis computable by a Turing machine with a
multiexponential time bound.
Definition 3. We say that Bis a base theory if Bis formulated in LPA with
B⊇EA.WeuseLTto refer to the language obtained by adding a unary predicate T
to LPA .CT–[B] is the theory extending Bwith the LT-sentences CT1 through CT5
below.
We follow the notational conventions from [5]. In particular x∈ClTermLPA is
the arithmetical formula that expresses “xis (the code of) a closed term of LPA ”;
x∈ClTermSeqLPA is the arithmetical formula that expresses “xis (the code of)
a sequence of closed terms of LPA ”; x∈FormLPA is the arithmetical formula that
expresses “xis (the code of) a formula of LPA ”; x∈SentLPA is the arithmetical
formula that expresses “xis (the code of) a sentence of LPA ”; x∈Var expresses “x
is (the code of) a variable”; x∈VarSeq expresses “xis (the code of) a sequence
of variables”; x∈Form≤n
LPA expresses “xis a (the code of) formula of LPA with at
most ndistinct free variables”(nis not a variable in Form≤n
LPA ); xis (the code of) the
numeral representing x;ϕ[x/v] is (the code of) the formula obtained by substituting
the variable vwith the numeral representing xand ϕ[¯
s/¯v] has an analogous meaning
for simultaneous substitution of terms from the sequence ¯
sfor variables from the
sequence ¯v;s◦denotes the value of the term s,and ¯
s◦denotes the sequence of
numbers that correspond to values of terms from the sequence of terms ¯
s.
Finally, for better readability we will sometimes skip formulae denoting syntactic
operations and write the effect of the operations instead. Thus, for example, we
will write T(¬ϕ) to denote “There exists which is the negation of the sentence ϕ
and T().”. For similar reasons, we shall often identify formulae with their G¨
odel
codes. Where it is helpful to distinguish between the two, ϕwill denote the G¨
odel
number of ϕor the numeral naming this number (depending on the context).
CT1∀s, t ∈ClTermLPA T(s=t)↔s◦=t◦.
CT2∀ϕ,  ∈SentLPA T(ϕ∨)↔T(ϕ)∨T().
CT3∀ϕ∈SentLPA T(¬ϕ)↔¬T(ϕ).
CT4∀v∈Var∀ϕ∈Form≤1
LPA T(∃vϕ)↔∃xT(ϕ[x/v]).
CT5∀ϕ(¯v)∈FormLPA ∀¯
s, ¯
t∈ClTermSeqLPA ¯
s◦=¯
t◦→T(ϕ[¯
s/¯v])↔T
(ϕ[¯
t/¯v]).
The axiom CT5 is sometimes called generalized regularity,orgeneralized term-
extensionality, and is not included in the accounts of CT–provided in the
monographs of [3,10]. The conservativity of this particular version of CT–[PA]
can be verified by a refinement of the model-theoretic method introduced in [7], as
presented both in [5,12]. Moreover, the following strengthening of the conservativity
result in [5].
Theorem 4. There is a polynomial-time computable function fsuch that for every
CT–[PA]-proof of an arithmetical sentence ϕ,f()is a PA-proof of ϕ.Moreover
the correctness of fis verifiable in PA.
The above result shows that CT–[PA]isfeasibly reducible to PA.Inparticular,
the basic truth theory CT–[PA] admits at most a polynomial speed-up over PA.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 5
Moreover, as shown in [5], PA proves the consistency of every finitely axiomatizable
subtheory of CT–[PA], which together with the arithmetized completeness theorem
and Orey’s compactness theorem shows that CT–[PA] is interpretable in PA.
Theorem 4witnesses the “flatness” of CT–[PA] over its base theory PA. The so-
called Tarski Boundary project, seeks to map out the extent of this phenomenon.
More concretely, given a metamathematical property of theories Pwhich is exhibited
by CT–[PA] we are interested in determining which extensions of CT–[PA] also exhibit
P. In particular P(x) can stand for any of the properties below:
•xis conservative over PA.
•xis relatively interpretable in PA.
•xadmits at most a polynomial speed-up over PA.
There is an obvious way of obtaining a natural strengthening of CT–[PA] which
fails to have any of the above properties. To describe this strengthening, given a
theory Tlet PrT(ϕ) be the arithmetical formula that expresses “ϕis provable from
T,” where the axioms of Tare given by some arithmetical formula. The Global
Reflection for Tis the following truth principle:
∀ϕ∈SentLTPrT(ϕ)→T(ϕ).(GRP(T))
We stress that GRP(T) depends not only on Tbut also on the particularly chosen
formula, which represents the axiom set of T.BelowPA denotes the canonical
formula which naturally represents the set of axioms of PA (as in Definition 1). Note
that CT–[EA]+GRP(PA) is non-conservative over PA since ConPA is provable in
CT–[EA]+GRP(PA). However, CT–[EA]+GRP(PA ) is much stronger, as indicated
by the following result.
Theorem 5 (Kotlarski [13]–Smory´
nski [29], Łełyk [16]). The arithmetical
consequences of CT–[EA]+GRP(PA)coincides with REF<(PA).
In the above REF0(T):=T,REFn+1 (T):=REF(REFn(T)), REF< (T):=
n∈REFn(T), where REF(T) denotes the extension of Twith all instances of
the Uniform Reflection Scheme for T, i.e., REF(T) consists of all sentences of the
following form, where ϕranges over LT-formulae with at most one free variable:
∀xPrT(ϕ(x)) →ϕ(x).
Interestingly enough, over CT–[EA], GRP(PA) lends itself to many different
characterisations, some of which express very basic properties of the truth predicate.
Theorem 6. Over CT–[EA]the following are all equivalent to GRP(PA ):
1. Δ0-induction scheme for LT(see [16,17]).
2. GRP(∅), i.e., ∀ϕPr∅(ϕ)→T(ϕ)(see [2])
3. ∀c“ccodes a set of sentences”∧T(ϕ∈cϕ)→∃ϕ∈cT(ϕ)(see [4]).
Theorem 6reveals the surprising robustness of the theory CT–[EA]+GRP(PA ).
Out of the three above principles, the third one looks especially modest, being
only one direction of a straightforward generalisation (often dubbed disjunctive
correctness) of the compositional axiom CT2ofCT–for disjunctions.6
6The last part of Theorem 6refines the main result of [6], which shows that CT0can be axiomatized
by simply adding the disjunctive correctness axiom to CT–[EA].
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

6ALI ENAYAT AND MATEUSZ ŁEŁYK
This shows that conceptually CT–[PA] is closer to the Tarski Boundary than
previously conceived. One of the achievements of the current research is the discovery
of the remarkable fact that this “conceptually small” area is populated by very
different natural theories of truth, each of which “merely” expresses that PA is
true.
•Note that by part (1) of Theorem 6,CT–[EA]+GRP(PA) is also axiomatizable
by the theory CT0[EA], which is obtained by augmenting CT–[EA]withΔ
0-
induction scheme for LT. Since this theory plays a very important role in our
paper, for the sake of convenience we omit the reference to the base theory in
CT0[EA] and refer to it as CT0. This is additionally justified by the fact that
CT0[EA]=CT0[B] for any base theory B(i.e., any subtheory of PA that extends
EA).
As mentioned already in Section 1, our main focus in the current paper is on
finite extensions of CT–[EA] that expresses “PA is true.” As shown in Theorem 58,
if we admit all elementary presentations of PA,theneachtrueΠ
2-statement can
be proved in a theory of this form. Hence, it is natural to look for some intuitive
restrictions on “admissible” presentations of PA. We investigate two such admissible
families of axiomatizations: schematic axiomatizations (introduced in Section 2.2)
and prudent axiomatizations (introduced in Section 2.3). The former family is well-
known; the latter family is defined in this paper as consisting of axiomatizations
whose deductive equivalence to PA is verifiable in the weak, finitistically justified
metatheory Primitive Recursive Arithemtic (PRA).
2.2. Schematic axiomatizations.
Definition 7. Atemplate (for a scheme) is given by a sentence [P] formulated in
the language obtained by augmenting LPA with a predicate P,wherePis unary.7An
LPA -sentence is said to be an instance of if is of the form ∀v[ϕ(x, v)/P], where
[ϕ(x, v)/P] is the result of substituting all subformulae of the form P(t), where
tis a term, with ϕ(t, v) (and re-naming bound variables of ϕto avoid unintended
clashes). We use Sto denote the collection of all instances of , and we refer to S
as the scheme generated by .
•We will use T[] to refer to the LT-sentence that says that each instance of S
is true; more formally:
T[]:=∀v, w ∈Var ∀ϕ(v, w )∈Form≤2
LPA ∀zT([ϕ(v, z/w )/P]).
In the above, the quantification ∀ϕ(v, w)∈Form≤2
LPA expresses “for all formulae
with at most two free variables v, w.” (∀ϕ(v)∈Form≤1
LPA below has an
analogous meaning.) We note that, over CT–[EA], T[] is equivalent to the
assertion
∀v∈Var ∀ϕ(v)∈Form≤1
LPA T([ϕ(v)/P]).
7Thanks to the coding apparatus available in arithmetic, we can limit ourselves to a single unary
predicate P. In other words, the notion of a schematic axiomatization presented here is not affected in
our context if the template is allowed to use finitely many predicate symbols P1,...,Pnof various finite
articles.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 7
We sometimes write “Tis -correct” instead of T[].
As mentioned in Section 1, the special case of the following theorem was
established for B=PA by Kotlarski, Krajewski, and Lachlan [14], and in full
generality by Enayat and Visser [7], and Leigh [15].
Theorem 8. CT–[B]+T[]is conservative over Bfor every base theory Band every
scheme template such that BS.
We will need the following definition and classical result about partial truth
definitions in the proof of Theorem 12 below.
Definition 9. The depth of a formula ϕis defined recursively by setting the depth
of an arbitrary atomic formula to be zero, putting the depth of ¬ϕand ∀xϕ to be
one plus the depth of ϕ, the depth of ϕ∨to be one plus the maximum of depths
of ϕand . The depth of a term is defined similarly: the depth of a variable or a
constant is zero and the depth of t+sand t·sis one plus the maximum of depths
of t, s.Thepure depth of the formula ϕis the defined analogously to depth of ϕ,
except for the condition for atomic formulae: the pure depth of a formula s=tis
one plus the maximum of the depths of s, t. The depth of a formula ϕwill be denoted
with depth(ϕ), whereas its pure depth by pdepth(ϕ). Observe that the depth of ϕis
always bounded above by its pure depth. We will write
True(y, P),
where Pis a unary predicate and yis a variable, for the formula obtained from the
conjunction of CT1 through CT4 of Definition 3in which (1) the predicate Tis
replaced by P, and (2) the universal quantifiers on ϕand are limited to formulae
of depth at most y. Intuitively speaking, True(y, P) says that Psatisfies the Tarskian
compositional clauses for formulae of depth at most y.
Example 10. The depth of an atomic formula is 0, whereas its pure depth can
be arbitrarily large. The depth of ∃xx=S(S(0)) ∨¬x=xis 3, whereas its pure
depth is 6.
The following theorem is classical; see [9] for a proof.
Theorem 11 (Partial Truth Definitions). For each n∈there is a unary LPA -
formula Truen(x)such that the formula obtained by replacing ywith nand Pwith
Truen(x)in the formula True(y, P)is provable in EA.
Theorem 12 (Vaught [31], Visser [32]). Let Tbe an r.e. theory with enough coding8,
and let LTbe the language of T. There is a primitive recursive function f(indeed fis
elementary) such that given any unary Σ1-formula that defines a set of LT-sentences
Φin N,f()is a scheme template such that the deductive closures of T+Sf()and
T+Φcoincide.
8Visser [32] showed that supporting a pairing function is “enough coding” in this context; this
improved the main result of Vaught’s paper [31], in which “enough coding” meant being able to interpret
an ∈-relation for which the statement: For all objects x0,...,x
n–1 there is an object ysuch that for all
objects t,t∈yiff (t=x0or ...or t=xn–1)” holds for each n∈(sequential theories support such an
∈-relation).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

8ALI ENAYAT AND MATEUSZ ŁEŁYK
Proof outline for T=EA.Suppose (x)isaΣ
1-formula that defines a set Φ
of sentences of LPA in the standard model of arithmetic. (By Craig’s trick, can
be chosen to be an elementary formula, this does not play a role in this proof, but
it will come handy in the proof of Proposition 21, which is based on this one.) Let
True(y, P) be as in Definition 9. The desired scheme template is
∀yTrue(y, P)→∀x(x)∧pdepth(x)≤y)→P(x).
We note that:
(1) EA +SΦ, because for each n∈the truth predicate for formulae of depth
at most nis definable by Theorem 11;and
(2) EA +ΦS. To see this, suppose to the contrary that some instance of S
is not provable in EA + Φ. Then by the completeness theorem of first order logic
there is a model Mof EA +Φ+¬. Since by the definition of Sthere is a formula
ϕ(x, v) such that is a sentence of the form ∀v[ϕ(x, v )/P], we have
M|=EA +Φ+¬(∀v[ϕ(x, v)/P]).
Thus Mis a model of EA + Φ in which the sentence ∃v¬[ϕ(x, v)/P]) holds, i.e.,
M|=∃v∃y (v, y ),where
(v, y):=Tr u e (y, ϕ (x, v)/P)∧∃x(x)∧pdepth(x)≤y)∧¬ϕ(x, v ).
Let aand bbe elements in Msuch that M|=(a, b). The key observation at
this point is that bcannot be a standard element since M|=Φ.(It is precisely at
this step that the argument would have broken down if we had used depth instead
instead of pure depth in our formulation of the scheme template .) Together with
the fact that M|=True(b, ϕ(x, a)/P), this implies that the formula ϕ(x, a) defines
a subset of Mthat satisfies Tarski’s compositional clauses for all standard formulae,
thus contradicting Tarski’s undefinability of truth theorem. 
Remark 13. The proof of the above theorem would not go through, if in
the definition of ,pdepth was changed to depth. Indeed, assume is modified
accordingly. It is enough to take Φ := {ConEA (n)|n∈},whereConEA(x)
expresses “there is no proof of inconsistency of EA whose code is below x.” Let
be the natural elementary definition of Φ, i.e.,
(x):=∃y<x
x=ConEA(y).
Observe that each sentence in Φ has the same, standard depth, call it k. Assume that
is a truth predicate for formulae of depth k. Then the sentence
∀yTrue(y, )→∀z(z)∧depth(z)≤y)→(z)
clearly implies ConEA, hence Sis, over EA, properly stronger than Φ.
The above is the main reason for introducing both depth and pure depth of a
formula into the picture. On the one hand, the natural definition of partial truth
predicates involves the notion of depth. On the other, we need pure depth to make
the proof of Theorem 12 work. The crucial difference between the two notions of
depth is that in an arbitrary model M|=EA and for an arbitrary standard number
n,ifM|=SentLPA (ϕ)∧pdepth(ϕ)≤n, then ϕis “almost” a standard sentence:
there is a standard sentence of the same pure depth as ϕ(differs with ϕonly
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 9
w.r.t. the indices of bounded variables) such that for every formula P(x) such that
M|=Tr u e (n, P (x)) we have M|=P(ϕ)↔P().
Remark 14. Note that by coupling Theorem 12 with the KKL Theorem we
can readily obtain the so called Kleene–Vaught Theorem for extensions of EA that
asserts that every r.e. extension of EA can be finitely axiomatized in an extended
language. For another line of reasoning, see the proof of Proposition 47.
Remark 15. Let ConZF be the arithmetical statement asserting the consistency
of ZF, and for each n∈let ConZF(n) be the restricted consistency statement for
ZF (that expresses “there is no proof of inconsistency of ZF whose code is below
n”). Consider the following extension PA+of PA:
PA+:= PA +{ConZF(n)|n∈}.
Then provably in ZF:
“PA+is conservative over PA”iffConZF .
To see that the above holds, we reason in ZF. Suppose PA+is conservative over PA.
Then for all n∈,PA proves ConZF (n). On the other hand, ZF “knows” that PA
holds in the standard model of arithmetic, so for all n∈,nis really not a proof
of inconsistency of ZF, i.e., ConZF holds. On the other hand, if ConZF holds, then by
Σ1-completeness of PA,PA+is conservative over PA.
Moreover, by invoking Theorem 12, there is a scheme whose instances are
provable inPA (assuming ConZF), but ZF cannot verify this. Moreover, coupled
with Theorem 8, and using part(c) of Definition 22, this also shows that there is a
scheme template such that
ZF ConZF ↔∈SchT
PA .
2.3. Prudent axiomatizations. In Section 4we will investigate another intuitive
restriction on “admissible” axiomatizations of PA, namely axiomatizations that are
prudent in the sense that their correctness can be verified in a finitistic metatheory.
To formalize this intuition we use the well-entrenched notion of proof-theoretic
reducibility.
Definition 16. Let ,range over elementary formulae with one free variable.
We say that is proof-theoretically reducible to (≤pt )if
IΣ1∀ϕPr(ϕ)→Pr(ϕ).
In the above Pr(x) is the canonical provability predicate that expresses “There
is a proof of xin First-Order Logic using the sentences from the set of axioms
described by as additional assumptions.” We write PA for the elementary formula
representing the usual axiomatization of PA (as in Definition 1), i.e., PA(x)
expresses: xis either (the code of) an axiom of Qor (the code of) an instance of the
induction scheme. We say that is proof-theoretically equivalent to PA (written as
∼pt PA )if
IΣ1∀ϕPr(ϕ)↔PrPA (ϕ).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

10 ALI ENAYAT AND MATEUSZ ŁEŁYK
It is a classical fact due to Parsons [24,25] that IΣ1and the system of
Primitive Recursive Arithmetic, known as PRA, have the same Π2-consequences.
In particular it follows that whenever ∼p.t. , then in fact and are deductively
equivalent provably in PRA. As a consequence there are primitive recursive proof
transformations mapping proofs in to proofs with the same conclusions in and
vice-versa.
•For the purposes of the results obtained in this paper, we do not need the full
power of the proof-theoretic equivalence of and to be verifiable in IΣ1since
a theory as weak as Buss’s S1
2would be sufficient. (Thus we can require that
there are polynomial-time computable proof transformations mapping proofs in
to proofs with the same conclusions in and vice-versa.) However, we decided
to stick to the more well-known notion of proof-theoretic reducibility rather
than feasible reducibility, especially since the former notion is philosophically
well-motivated by Hilbert’s finitism, as argued forcefully by Tait [30].
•We focus primarily on elementary presentations, rather than on, possibly more
natural, r.e. axiomatizations for two reasons. First of all, for most of our main
results, the simpler the axiomatizations, the better. (The results concerning
them, mostly, have the form “For every xthere is a prudent axiomatization 
such that ....”) Secondly, from the philosophical perspective, the elementary
formulae, being decidable in multi-exponential time and hence absolute
between models of EA, guarantee (or at least come closer to guaranteeing)
that the notion of an axiom of the given theory is determinate. From these
perspectives, feasible axiomatizations, i.e., P-Time decidable, axiomatizations
would be even better, but we leave the investigation of such axiomatizations for
further research.
Definition 17. We use Δ∗to denote the collection of unary elementary formulae
(x) such that N:= {n∈|N|=(n)}codes an LPA-theory that is deductively
equivalent to PA. We sometimes refer to the members of Δ∗as elementary
presentations of PA.
•Given any arithmetical formula ϕwith exactly one free variable,
T[ϕ(x)] := ∀xϕ(x)→T(x),
where xis the unique free variable of ϕ.SoT[ϕ]istheLT-sentence expressing
that the theory described by ϕis true. Moreover, we put
CT–ϕ:= CT–[EA]+T[ϕ].
•We use Δ to denote the subset of Δ∗consisting of formulae ∈Δ∗such that
is proof-theoretically equivalent to PA .Thus Δis the collection of (defining
formulae of) prudent axiomatizations of PA . Occasionally we also need the
extension of Δ, denoted Δ–, defined
Δ–:= ∈Δ∗|≤pt PA.
On Δ–and Δ we shall consider the relation ≤CT–given by
≤CT–⇐⇒ CT–[EA]T[]→T[].
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 11
Convention 18. Simplifying things a little bit, when talking about the structures
Δ,≤CT–and Δ–,≤CT–, we shall assume that Δis replaced by the quotient set Δ/∼,
where ∼is the least equivalence relation that makes ≤CT–antisymmetric, to wit:
∼iff ≤CT–and ≤CT–.
•Let us stress an important difference between CT–[PA]andCT–PA : the latter
but not the former includes the sentence “All induction axioms are true.” In
particular, the latter is finitely axiomatizable, while the former is known to be
reflexive and therefore not finitely axiomatizable.
•Note that the meaning of T[··· ] depends on whether the object within the
brackets is a scheme template, in which case T[··· ] is interpreted as in Definition
7, or an arithmetical formula, in which case T[··· ] has the meaning given in
Definition 17. We shall try to reserve the use of variables ,, etc. in this context
for schematic templates and ϕ,,, etc. for formulae.
Proposition 19. Both Δ,≤CT–and Δ–,≤CT–are distributive lattices.
Proof. We only present the proof for the case of Δ as it is (1 + ε)-times harder.
For showing that both structures are distributive lattices, it is enough to show that
given , ∈Δ, one can find elements ⊕and ⊗of Δ such that over CT–[PA]
we have
T[]∧T[]↔T[⊕],(1)
T[]∨T[]↔T[⊗].(2)
Indeed, this is because the Lindenbaum algebra of CT–is a distributive lattice. It
can be readily seen that if we define
⊕(x):=(x)∨(x),
then ⊕∈Δ and (1) is satisfied. For (2) it is sufficient to define
⊗(x):=∃y, z < x(y)∧(z)∧x=y∨z,
where x=y∨zexpresses that xis a disjunction of yand z. To see that (2) holds and
PA ≤pt ⊗one simply applies reasoning by cases; the proof of ⊗≤pt PA is
trivial. 
Remark 20. If ∈Δ corresponds to a schematic axiomatization of PA (i.e., for
some template [P], (x) says that xis the result of substituting Pwith some unary
arithmetical formula), then CT–is a conservative extension of PA by Theorem 26.
In contrast, even for very natural ∈Δ, CT–may be a highly non-conservative
extension of PA. For example, consider
REFEA =∀xPrEA(ϕ(x)) →ϕ(x)|ϕ(x)∈L
PA .
By a classical theorem of Kreisel, the union of EA and REFEA is deductively
equivalent to PA (see, e.g., [1, p. 39]). Let (x) be a natural elementary definition of
EA ∪REFEA. Then, in fact ∈Δ. An easy argument shows that
CT–∀ϕPrEA(ϕ)→T(ϕ).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

12 ALI ENAYAT AND MATEUSZ ŁEŁYK
However, by a theorem of [2], over CT–[EA], the above consequence of CT–
implies the Global Reflection Principle for PA .
Proposition 21. Every LPA-theory Textending EA whose axioms are described by
an elementary formula (in the standard model of arithmetic) has a proof-theoretically
equivalent presentation such that CT–is a conservative extension of T.
Proof. Fix Tand as in the assumptions. Let be the natural elementary
definition of the set S,whereis the template defined as in the proof of Theorem 12.
This works, since in the proof of Theorem 12, the verification of the fact that
the deductive closure of EA +Sand EA + Φ coincide formalizes smoothly in
the subsystem WKL∗
0of second order arithmetic, which is well-known to be a
conservative extension of EA, as first shown in [28]. More explicitly, the verification
of EA +SΦ requires only the existence of well-behaved partial truth predicates
(that can be developed within EA, as demonstrated, e.g., in [1, Proposition 2.6]).
On the other hand, the verification of EA +ΦSrequires the completeness
theorem of first order logic (which is readily available in WKL∗
0) together with
Tarski’s undefinability of truth theorem. Although Tarski’s theorem presupposes
the consistency of Φ, this can be assumed, because if Φ is inconsistent, so is Sby
the proof of the first implication, and in such a scenario the two theories clearly
coincide. Hence is indeed proof-theoretically reducible to . However, in this case
CT–is trivially equivalent to CT–[PA]+T[], hence is a conservative extension
of T, due to Theorem 8.
§3. Schematically correct axiomatizations.
3.1. Complexity.
Definition 22. In the following definitions ranges over scheme templates and
Sis the corresponding scheme (in the sense of Definition 7) generated by .
(a) Sch–
PA := {:PA S},i.e., Sch–
PA is the collection of templates whose
corresponding scheme is PA-provable.
(b) SchPA := {∈Sch–
PA :SPA}, i.e., SchPA is the collection of templates
whose corresponding scheme is an axiomatization of PA .
(c) SchT
PA is the collection of templates such that the arithmetical consequences
of CT–[EA]+T[] coincides with PA (recall that T[] says that Tis -correct,
as in Definition 7).
(d) Cons := {ϕ∈L
T:CT–[PA]+ϕis conservative over PA}.
Recall that in Section 1we defined ≤CT–on Sch–
PA as follows:
≤CT–⇐⇒ CT–T[]→T[].
Note that at this point ≤CT–denotes both the ordering on scheme templates and
the ordering on prudent axiomatizations. As the notation “≤CT–” will never be
used in isolation this shouldn’t lead to serious confusion. When talking about the
structural properties of SchPA,≤CT–we shall tacitly assume that SchPA is factored
out by an appropriate equivalence relation, so as to make ≤CT–a partial order (as
in Convention 18).
Proposition 23. Sch–
PA ,≤CT–and SchPA ,≤CT–are distributive lattices.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 13
Proof. As previously we do the case of a smaller structure, with SchPA as the
universe. Arguing as previously in Proposition 19, it is enough to define ⊕and ⊗
such that CT–[PA] proves the following for all , ∈SchPA :
T[]∧T[]↔T[⊕],(3)
T[]∨T[]↔T[⊗].(4)
Thecaseof⊕is trivial. We put
⊕:= ∧.
Thecaseof⊗is (a little bit) harder. We put
⊗:= ∨([Q/P]),
where Qis a fresh unary predicate. As remarked earlier (compare footnote 4) thanks
to the coding apparatus, ⊗can be expressed as a scheme with a single unary
predicate P. Then we obtain
CT–[EA]T[⊗]≡∀ϕ∀T
[ϕ/P]∨[/Q].
It is very easy now to check that (4) is satisfied. 
We note that the above proof adapts to the case of SchT
PA ,≤CT–. Quite in the
opposite direction, it can be shown that Cons,≤CT–is not even a lattice as there
are two sentences ϕ,  ∈Cons such that CT[PA]+ϕ+is a non-conservative
extension of PA. First examples of such sentences were discovered by Bartosz Wcisło
(unpublished). We plan to present a family of such examples in the forthcoming
sequel [18] to the current paper.
Theorem 24 (KKL-Theorem, first formulation). CT–[PA]+T[]is conservative
over PA for each ∈Sch–
PA .
Let Θ be the union of sentences of the form T[] (expressing that Tis -correct)
as ranges in Sch–
PA . Since the union of two schemes is axiomatizable by a single
scheme, the KKL-theorem can be reformulated as:
Theorem 25 (KKL-Theorem, second formulation). CT–[PA]+Θis conservative
over PA .
The above formulation naturally suggests the question: How complicated is Θ
(viewed as a subset of )? Is it recursively enumerable? The result below shows that Θ
is Π2-complete, since Θ is readily seen to be recursively isomorphic to Sch–
PA (indeed
the isomorphism is witnessed by an elementary function). Therefore, Θ is pretty far
from being recursively enumerable.
Theorem 26. The sets Sch–
PA ,SchPA ,SchT
PA , and Cons are all Π2-complete.
Proof. Each of the four sets is readily seen to be definable by a Π2-formula, so
it suffices to show that each is Π2-hard, i.e., the complete Π2-set TrueN
Π2consisting
of (G¨
odel numbers of) Π2-sentences that are true in the standard model Nof PA is
many-one reducible (denoted ≤m) to each of them. Recall that ≤mis defined among
subsets of via:
A≤mBiff there is a total recursive function fsuch that: ∀n∈(n∈A⇔f(n)∈B).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

14 ALI ENAYAT AND MATEUSZ ŁEŁYK
The proof will be complete once we demonstrate the following four assertions:
(i)TrueN
Π2≤mSch–
PA .9
(ii)Sch–
PA ≤mSchPA .
(iii)Sch–
PA ≤mSchT
PA .
(iv)TrueN
Π2≤mCons.
To prov e (i), suppose =∀x∃y(x, y )isaΠ
2-statement, where (x, y)isΔ
0.We
first observe that by Σ1-completeness of PA:
(∗)∈TrueN
Π2iff ∀n∈PA ∃y(n,y).
On the other hand, R={∃y(n,y):n∈}is a recursive set of sentences, so by
Theorem 12 there is such that ∈Sch–
PA iff PA R. To finish the proof, it remains
to observe that the transition from to the Σ1-formula that defines Rin Nis given
by a recursive function g, therefore if fis the total recursive function as in Theorem
12 then we have
∈TrueN
Π2iff f(g()) ∈Sch–
PA .
The proof of (ii) is based on the observation that ∈Sch–
PA iff h()∈SchPA ,where
h():=∧PA ,andPA is defined as follows:
PA := Q∧[P(0) ∧∀x(P(x)→P(S(x))) →∀xP(x)].
To veri f y (iii), we claim that ∈Sch–
PA iff (∧PA )∈SchT
PA . The implication
∈Sch–
PA ⇒(∧PA )∈SchT
PA follows directly from Theorem 3 (since PA proves
S∧PA if ∈Sch–
PA ).On the other hand, if (∧PA )∈SchT
PA , then by the definition
of SchT
PA ,PA proves S,so∈Sch–
PA .
Finally, to establish (iv) suppose =∀x∃y(x, y)isaΠ
2-statement, where
(x, y)isΔ
0. In the proof of part (i) we showed that there are recursive functions f
and gsuch that
∈TrueN
Π2⇐⇒ f(g()) ∈Sch–
PA .
Let hbe the function that takes a template as input, and outputs the sentence
T[]∈L
Texpressing “Tis -correct.” Clearly his a recursive function. Also, it is
evident that ∈Sch–
PA iff T[]∈Cons (the direction ⇒follows from Theorem 8;
and the direction ⇐follows from the relevant definitions). Therefore,
∈TrueN
Π2⇐⇒ hf(g()))∈Cons.
Proposition 27. Let ϕsbe the single LT-sentence that expresses “every PA-
provable scheme is true.” Then CT0can be axiomatized by CT–[EA]+ϕs.
Proof. By Theorem 6,CT0can be axiomatized by CT–[EA]+GRP(PA). This
makes it clear that ϕsis provable in CT0.For the other direction, working in
CT–[EA]+ϕs, suppose is PA-provable. Then the scheme given by ∀x(∨P(x))
is PA-provable, so the instance of this scheme in which Pis replaced with x=x
9The proof of (i)showsthatSch–
Tis Π2-complete for any extension Tof Robinson’s Qthat is
Σ1-sound, and which also supports a pairing function.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 15
is true, but since T(∀x(x=x)), we have T(). Thus, since was arbitrary,
CT–[EA]+ϕsGRP(PA).
3.2. Structure of schematically correct extensions. In this subsection we take a
closer look at the structure of SchPA. In particular, we look at interpretability
properties of its elements, where by “interpretability” we always mean relative
interpretability, as described in [9]. The most basic tool we shall use is a modification
of the Vaught operation from the proof of Theorem 12. Let us introduce the relevant
definition:
Definition 28. For arithmetical formulae ϕ(x),(x) with at most one free
variable let the ϕ-restricted Vaught schematization of be the scheme template.
V(ϕ,)[P]:=∀yϕ(y)∧True(y, P)→∀x((x)∧pdepth(x)≤y)→P(x).
For a single formula ,V[P]abbreviatesV(x=x,)[P] and we often omit the reference
to P. Similarly Vϕ,[(x)] abbreviates Vϕ,[(x)/P(x)].
Convention 29. Working in CT–[EA]and having fixed an ( possibly nonstandard )
arithmetical formula with one free variable (v),T∗(x)will abbreviate the formula
T([x/v]). Hence T∗(x)says that xsatisfies . This notation was first introduced
in [19] and is very successful in decreasing the number of brackets and improving
readability.
Below, we shall borrow a notation used in the context of prudent axiomatizations:
CT–is the theory CT–[EA]+T[], i.e., CT–[EA] together with the assertion that
Tis -correct. In such contexts the variables such as ,,orV–– below will always
denote scheme templates.
Theorem 30. If ∈L
Tis such that for every ∈SchPA ,is interpretable in
CT–,thenis interpretable in CT–[PA ].
Proof. We prove the contrapositive. Fix which is not interpretable in CT–[PA].
We modify the Pakhomov–Visser diagonalization from [22, Theorem 4.1]. Observe
that for two finite theories α,, the condition “αinterprets ”isΣ
1.Letα
denote the formalization of this relation. Consider a Σ1-sentence ϕ=∃xϕ (x),
where ϕ(x)∈Δ0such that the following equivalence is provable in CT–[PA]:
ϕ↔CT–V(∀z≤y¬ϕ(z),PA )
.
Similarly to the Pakhomov–Visser argument, we argue that ϕis false. Suppose
not and take the least n∈such that ϕ(n) holds. Then, in Q,∀z≤x¬ϕ(z)is
equivalent to x<n
, hence the following is provable in CT–[PA]:
∀(x)TV(∀z≤y¬ϕ(z),PA )[]↔TV(y<n,PA )[].
We claim that
CT–[PA]∀(x)TV(y<n,PA )[].(∗)
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

16 ALI ENAYAT AND MATEUSZ ŁEŁYK
Indeed, working in CT–[PA]fix∈Form≤1
LPA . By compositional conditions
TV(y<n,PA )[]is equivalent to

i<n T∗True(i,)→∀x(PA (x)∧pdepth(x)≤i)→T∗(x).
However, once again by compositional conditions imposed on T,T∗True(i,)is
equivalent to True(i,T∗(x)), hence to the assertion that T∗(x) is a compositional
truth predicate for formulae of depth at most i. Assuming that this is the case, since
iis standard, every induction axiom of pure depth at most iis true in the sense of
T∗(x). This concludes our proof of (∗).
Now, since ϕis true, it follows that
CT–[PA]+∀(x)TV(y<n,PA )[]interprets .
However, by the above argument it would mean that CT–[PA] interprets , contrary
to the assumption.
Since ϕis false, V(∀z≤y¬ϕ(z),PA )[P] is a scheme template, such that the scheme
associated with it axiomatizes PA.Moreover,CT–[PA ]+TV(∀z≤y¬ϕ(z),PA )does
not interpret .
Since CT–[PA] is interpretable in PA (see [7,15]), we obtain the following corollary.
Corollary 31. For every ∈L
Tsuch that PA does not interpret there is a
scheme template ∈SchPA such that CT–does not relatively interpret .
Since PA Q+ConPA (see [26]) we obtain the following corollary. It is of interest
because it gives an example of a natural theory that is not interpretable in PA (because
it is finite) but this is not due to the reason that the theory interprets the consistency
of PA (like most known finite extensions of PA).
Corollary 32. There is a scheme template ∈SchPA such that CT–does not
interpret Q+ConPA .
Corollary 33. For every scheme template ∈SchPA there is a scheme template
∈SchPA such that CT–interprets CT–, but not vice versa.
Proof. Fix and apply Corollary 31 to := CT–. This is legal, since the latter
theory is a finitely axiomatizable extension of PA, hence it is not interpretable in
PA.10 So there is a scheme  ∈SchPA such that CT–does not relatively interpret
CT–. Now it is sufficient to take := ⊗, as in the proof of Proposition23.
Next we will consider more structural properties of SchPA . These properties will
be shown to be transferable to the Lindenbaum Algebra of CT0.
•For the rest of this section and are arbitrary elementary formulae that,
provably in EA, specify arithmetical theories, i.e., possibly infinite sets of
arithmetical sentences. We will write ⊆as an abbreviation of ∀x((x)→
10This is because otherwise CT–, being a finite theory, would be interpretable in a finite fragment
of PA,callitT. But then, since CT–extends PA and PA is reflexive, CT–ConT. Hence Twould
interpret Q+ConT, which is impossible by the interpretability version of the Second Incompleteness
Theorem, see [9] (we owe this argument to Albert Visser).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 17
(x)). Recall (from Definition 17)thatT[] is the following sentence expressing
“Tis correct”:
∀x(x)→T(x).
Note the difference between T[V]andT[].
The first result is immediate:
Proposition 34. For every and ,CT–[PA ]∀x(x)→(x)→T[V]→
T[V].
For many applications, the condition ⊆from the antecedent is too restrictive.
One would like to relax it to , however, this one is too weak to guarantee
(over CT–[PA]) that the implication T[V]→T[V] holds. This is because the
truth predicate axiomatized by pure CT–[PA ] is far from being closed under logic
(compare with Theorem 6). The next proposition is a fair compromise between the
two solutions.
•Given a unary arithmetical formula ϕ(x), in the proposition below we use the
convention of using ϕ(x) to refer to the formula that defines the set of (codes
of) sentences of the form ϕ(n) (in the standard model Nof arithmetic).
Proposition 35. For arbitrary arithmetical formulae ϕ(x)and (x),
CT–[PA]∀xϕ(x)→(x)→T[Vϕ]→T[V].
Proof. Fix arbitrary arithmetical formulae and ϕwith exactly one free
variable. Let and ϕbe defined in the bullet point above the current proposition.
Without loss of generality, assume that the variable xoccurs in . Working in
CT–[PA] assume that ∀xϕ(x)→(x)and T[Vϕ] hold. We argue that T[V]
holds as well. Fix arbitrary a,,bsuch that True(a, T ∗)andpdepth((b)) ≤a.
It follows that for some standard n,pdepth(ϕ(b)) ≤a+n, hence there exists a
formula (x) such that
True(a+n, T ∗).
By T[Vϕ] we conclude T∗(ϕ(b)).However, since ϕ(x) is of standard depth, it
follows that ϕ(b) holds. Hence (b) holds as well. Since (b) is also of standard
depth, we conclude that T∗((b)), which ends the proof. 
The proposition below is an important tool for discovering various patterns in
SchPA . It enables us to switch from somewhat less readable Vaught schematizations
of elementary presentations of theories to more workable presentations themselves.
It says that over CT0,-correctness is equivalent to V-correctness.
Proposition 36. For every ,CT0T[]↔T[V].
Proof. We start by showing that provably in CT0all arithmetical partial truth
predicates are coextensive, i.e., the following is provable in CT0:
∀x∀∈Form≤1
LPA ∀ϕ∈SentLPA (Tru e (x, T ∗)∧depth(ϕ)≤x)→T∗(ϕ)↔T(ϕ).
(∗)
Fix an arbitrary (M,T)|=CT0. For an arbitrary c∈M,letTcdenote the
((M,T)-definable) restriction of Tto all sentences of depth at most c. Then
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

18 ALI ENAYAT AND MATEUSZ ŁEŁYK
(M,T
c)|=True(c, T ). However, as proved in [20, Fact 32], (M,T
c) satisfies full
induction scheme for LT. Hence Tcis a fully inductive truth predicate for formulae
of depth at most c. Using this we argue that (∗) holds in (M,T). Working in the
model, fix an arbitrary aandanarbitrary∈Form≤1
LPA . Assume that the depth of 
is band let c=max{a, b}. Assume Tr u e (a, T ∗), i.e. the formula T∗is a partial
truth predicate for formulae of depth ≤a. Since for every formula ϕof depth at most
c,Tc(ϕ) is equivalent to T(ϕ), we conclude that True(a, Tc∗) holds. Moreover, it
is sufficient to show that
∀xTc∗(x)↔Tc(x).
In other words, it is sufficient to prove that
(M,T
c)|=∀xT∗(x)↔T(x).
The above can be demonstrated by a routine induction on the build-up of formulae.
More precisely, let
Ξ(y):=∀ϕ∈depth(y)T∗(ϕ)↔T(ϕ).
Then Ξ(0) and ∀x<a
Ξ(x)→Ξ(x+1)
hold (in (M,T
c)), because both Tc∗
and Tcare partial truth predicates for formulae of depth at most a. Since Ξ(y)isa
formula of LT,in(M,T
c) we have an induction axiom for it, and we can conclude
(M,T
c)|=∀y≤aΞ(y).
This completes the proof of (∗).
We show that over CT–[PA], T[] implies T[V]. We fix an arbitrary and
working in CT0assume that ∀x(x)→T(x). We show that Tis V-correct,
i.e., for every arithmetical formula (possibly nonstandard) T(V[]) holds. By the
compositional conditions, T(V[]) is equivalent to
∀xTrue(x, T ∗)→∀y(y)∧pdepth(y)≤x)→T∗(y).
Fix x, assume True(x, T ∗) and fix an arbitrary ysuch that pdepth(y)≤xand
(y). By -correctness T(y) holds, hence yis a formula and since pdepth(y)≤x,
yis a formula of depth at most x. Then, by the previous claim (∗) we know that
for every ϕwhose depth is at most x,T∗(ϕ) is equivalent to T(ϕ). Hence T∗(y)
holds as well.
For the converse direction, we assume Tis V-correct. Fix an arbitrary xand
assume that (x) holds. In particular xis a formula. Let ybe the depth of xand
let be any arithmetical truth predicate such that PrPA (Tru e (y,)) holds. By the
Global Reflection in CT0,T(True(y,)) holds as well, and this in turn implies, by
compositional conditions, True(y, T ∗). Consequently, by V-correctness, T∗(x)
holds. Finally, it follows that T(x) holds by our claim (∗). This concludes the proof
of -correctness and the whole proof. 
Corollary 37. For every ,  ,ifCT–[EA]T[V]→T[V],thenCT0T[]→
T[].
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 19
The above corollary yields a versatile tool for studying the structure of
SchPA ,≤CT–,where≤CT–is defined by: 1≤CT–2iff CT–[EA]T[1]→T[2].
We show the crucial application:
Theorem 38. SchPA ,≤CT–is a countably universal partial order.
We recall that a partial order P, ≤Pis said to be countably universal if every
countable partial order Q, ≤Qcan be embedded into P, ≤P, i.e., there is an
injection f:Q→Psuch that for every a, b ∈P,f(a)≤Pf(b)⇐⇒ a≤Qb.
The above theorem reduces immediately to the one below.11 This is thanks to
the work of Hubiˇ
cka and Neˇ
setˇ
ril [11, Corollary 2.6], where a particular countably
universal partial order is defined. It is clear from the presentation that the order
W,≤Wis decidable and provably a partial order in PA.
Theorem 39. Suppose that is a decidable partial order on such that PA proves
that is a partial order. Then there is an embedding ,  →SchPA ,≤CT–.
Proof. Suppose that satisfies the assumptions. Firstly, we build a family of
consistent theories {n}n∈such that the following hold for all m, n ∈:
1. If mn, then PA n⊆m.
2. CT0Conm.
3. If mn, then CT0+ConmConn.
Asshownin[21, Section 2.3, Theorem 11], there is a Π1-formula (x) that is flexible
over REF< (PA ), i.e., for every Π1-formula (x), the following theory is consistent:
REF<(PA)+∀x(x)↔(x).
For ea c h n∈let nbe the natural Σ1-definition of the following set of
sentences12:
PA +{(k)|nk}.
Now, condition (1) easily follows from the (PA-provable) transitivity of .
Condition (2) easily reduces to Condition (3), so let us now show the latter. Aiming
at a contradiction assume mnand CT0Conm→Conn.Let(x):=mx.
By flexibility there exists model Msuch that
M|=REF< (PA )+∀x(x)↔(x).
By the choice of Mit follows that M|=¬(n). As a consequence, by provable Σ1-
completeness of PA,M|=PrPA (¬(n)),and M|=¬Conn. However, since M|=
REF(PA), as viewed in M,PA is consistent with Π1-truth (of M). Consequently,
since M|=∀xmx→(x), it follows that M|=Conm. Hence M|=Connas
well, which contradicts our previous conclusions.
We are ready to construct the promised embedding. Fix the family {n}n∈as
above and for each m∈choose m∈Δ∗to be the natural elementary definition
of the following set of sentences:
PA +{Conm(n)|n∈},
11We are grateful to Fedor Pakhomov for pointing our this more general result.
12Observe that since need not be elementary; also nneed not be elementary either. However, is
not our final axiomatization.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

20 ALI ENAYAT AND MATEUSZ ŁEŁYK
where Conm(n) asserts that there is no proof of contradiction of mwith G¨
odel
code ≤n. Since for every m∈,mis consistent, mis really an axiomatization of
PA, hence Vm∈SchPA .Wecheckthatthemap
m→ Vm
is an embedding of ,  into SchPA ,≤CT–.Fix m, n ∈and assume mn. Then
clearly PA ∀xConm(x)→Conn(x). Consequently, applying Proposition 35 to
ϕ(x):=Conm(x)and(x):=Conn(x), we obtain
CT–[PA]T[Vm]→T[Vn].
Suppose now mnand aiming at a contradiction, assume that CT–[PA ]
T[Vm]→T[Vn]. Then, by Corollary 37,CT0T[m]→T[n]. However, since
CT0T[PA ], CT0T[i]↔Conifor every i∈. Hence CT0Conm→Conn,
which is impossible by our previous considerations, since mn.
Corollary 40. The following partial orders are countably universal (we take the
ordering ≤CT–on Cons to be inherited from the Lindenbaum Algebra of CT–):
•Sch–
PA ,≤CT–.
•SchT
PA ,≤CT–.
•Cons,≤CT–.
Proof. This follows since SchPA,≤CT–can be easily embedded into each of the
above partial orders. 
§4. Prudently correct axiomatizations. Recall (from Definition 17) that Δ is the
collection of prudent axiomatizations of PA. In the first subsection we classify the
extensions of PA that can be axiomatized by theories of the form CT–and measure
the complexity of the Tarski Boundary problem for such theories.
4.1. Universality and complexity. As indicated by the proposition below, theories
of the form CT–for ∈Δ are never too strong.
Proposition 41. For every ∈Δ,CT0CT–.
Proof. This follows immediately from Theorem 6that CT0∀ϕPrPA (ϕ)→
T(ϕ).
Therefore, the theory CT0provides an upper-bound for the strength of theories
in question. The following theorem is this section’s main result.
Theorem 42. For any r.e. LPA -theory Textending PA such that CT0T there
exists a ∈Δsuch that Tand CT–have the same arithmetical theorems.
Proposition 41 and Theorem 42 when put together, yield the following
characterization of arithmetical theories provable in REF<(PA ).
Corollary 43. For every arithmetical recursively enumerable theory Textending
PA the following are equivalent:
1. REF<(PA)T.
2. There exists a ∈Δsuch that Tand CT–coincide on arithmetical theorems.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 21
To prove Theorem 42 we need to arrange such that
•∈Δ.
•CT–does not overgenerate, i.e., its arithmetical consequences do not
transcend those of T.
To satisfy the first condition we recall that by (Cie´
sli´
nski’s) Theorem 6, uniform
reflection over logic is an example of a principle which is provable in PA and whose
“globalized” version is equivalent to CT0. We shall often use the notation described
in the following definition. We recall that, for heuristic reasons, we sometimes write
ϕto denote either the G¨
odel number of ϕor the numeral naming this number
(depending on the context).
Definition 44. For two sentences and ϕ,ϕ[] abbreviates the sentence
(Pr∅()∧¬)→ϕ.
The map ϕ,  → ϕ[] is clearly elementary and we shall identify it with its
elementary definition.
To satisfy the second condition we could use Vaught’s theorem on axiomatizability
by a scheme, as we did earlier (see Remark 14). However, we prefer to introduce
an original method of finding “deductively weak” axiomatizations of arithmetical
theories. The very essence of our method was noted already in the original
KKL-paper [14]: there are models of CT–[PA] in which nonstandard pleonastic
disjunctions of obviously false statements are deemed true by the truth predicate.
For example, if Mis a countable recursively saturated model of PA and ais
any nonstandard element, then there is a truth class T⊆Msuch that (M,T)|=
CT–[PA] and the sentence
0=0∨(0 =0∨(... ∨0=0)...)
 
amany disjuncts
is deemed true by T. This phenomenon was quite recently pushed to the extreme by
the following result of Bartosz Wcisło that appears in [4].
Theorem 45. If M|=EA, then there is an elementary extension Nof Mthat
has an expansion (N,T)|=CT–[EA], which has the property that every disjunction of
nonstandard length in Nis deemed true by T.Moreover,ifM|=PA,then(N,T)can
be taken to be a model of CT–PA .
The above theorem provides us with a new method of finding finite conservative
axiomatizations of arithmetical theories extending EA.
Definition 46. Given an arithmetical sentence ϕ,thepleonastic disjunction of ϕ
is the sentence
ϕ∨(ϕ∨(... ∨ϕ)...)
 
ϕtimes
.
The pleonastic disjunction of ϕwill be denoted with ϕ.
Note that the above definition formalizes smoothly in EA (in which case ϕis
identified with ϕand treated both as a number and as a formula) and that in a
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

22 ALI ENAYAT AND MATEUSZ ŁEŁYK
nonstandard model of this theory ϕhas standard length if and only if ϕis (coded
by) a standard number.
Proposition 47. Every r.e. T⊇EA can be finitely axiomatized by a theory of the
form CT–[EA]+T[ϕ], for some elementary formula ϕ(x).
We recall that, by our conventions, CT–[EA]+T[ϕ]andCT–ϕmean the same
thing.
Proof. Let ϕ(x) formalize an elementary axiomatization of T(which exists by
Craig’s trick). Define
ϕ(x):=∃<x
ϕ()∧x=.
That is to say that xsatisfies ϕif it is a pleonastic disjunction of a formula from
an elementary axiomatization of T. Observe first that CT–ϕT. Indeed, it is
sufficient to show that for every sentence we have
CT–ϕϕ()→T().
Observe that, over EA,ϕ() implies ϕ(), which in turn, over CT–ϕ
implies T(). However, over pure CT–[EA] the last sentence implies T()by
compositional conditions, since is a disjunction of length and hence is
standard.
We show conservativity: pick any model M|=T. By Theorem 45 there is
a(N,T)|=CT–[EA] such that Nis an elementary extension of Mand every
disjunction of nonstandard length is made true by T. It follows that (N,T)|=
CT–ϕ. Indeed, firstly observe that if N|=ϕ(a) then there exists such that
N|=ϕ()∧a=. Now the argument splits into two cases:
1. is a standard sentence. In this case is standard and N|=,by
elementarity. Consequently (N,T)|=T() by compositional clauses; or
2. is not a standard sentence. In this case is a disjunction of nonstandard
length, hence is made true in (N,T). 
We shall recycle the above conservativity argument in the proof of Theorem 42,
which we now turn to.
Proof of Theorem 42. Fix Tsuch that
CT0T.
Let be an arbitrary elementary axiomatization of T.Letϕ[] denote the map
from Definition 44. We now observe that the proof of the reflexive property of
PA is formalizable in IΣ1. This follows from two well-known facts: firstly, the cut-
elimination theorem formalizes in IΣ1(thus provably in IΣ1, every provable sentence
has a proof that has the subformula property) and secondly, IΣ1is enough for
the formalization of the proof of existence of partial truth predicates in PA.Asa
consequence, we obtain
IΣ1∀PrPA (¬(Pr∅()∧¬)).
Hence (x)∈Δ, where is defined as follows:
(x):=PA (x)∨∃, ϕ < x (ϕ)∧x=ϕ[].
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 23
Observe that naturally defines the following set of sentences:
Q∪{Ind(ϕ)|ϕ∈L
PA }∪(Pr∅()∧¬)→ϕ|∈L
PA ,ϕ ∈T.
We argue first that CT–is conservative over T. To see this, fix an arbitrary
model M|=Tand let (N,T)|=CT–PA be a model from Theorem 45. Then
(N,T)|=CT–since, reasoning by cases as in the proof of Proposition 47,for
every ϕ∈Nsuch that N|=(ϕ)wehave
(N,T)|=Tϕ.
Now, we argue that CT–T.Letϕbe an arbitrary –axiom of T. We claim
CT–ϕ.
To see why the last claim holds, reason in CT–.Wehave
∀T(ϕ[]).
By the axioms of CT–the above is equivalent to
∃Pr∅()∧¬T() →Tϕ.(∗)
Now we reason by cases: either ∀Pr∅()→T()or not. If the latter holds,
we have T(ϕ) by Modus Ponens applied to (∗). Hence ϕholds by compositional
conditions, because ϕis a disjunction of standard length and ϕis a standard
sentence. If the former holds, we have CT0by Theorem 6and ϕholds, because we
assumed that CT0T.
We conclude this subsection with complexity results that complement
Theorem 26.
Proposition 48. The set Δ∗is Π2-complete.
Proof. Clearly Δ∗is Π2-definable. Consider the map fthat takes a scheme
template as input and outputs the formula (x) that expresses “xis an instance
of .” fis clearly recursive (indeed elementary) and satisfies
∈SchPA iff ∈Δ∗.
Therefore SchPA is many-one reducible to Δ∗, which in light of the Π2-completeness
of SchPA (established in Theorem 26), completes the verification of Π2-completeness
of Δ∗.
Proposition 49. The sets Δand Δ–are both Σ1-complete.
Proof. Both sets are clearly r.e., because both definitions require just the
provability of a particular sentence in IΣ1. To verify completeness we sketch the
reduction fof the set of true Σ1-sentences to Δ (an analogous reduction works for
Δ–). Given a Σ1-sentence ϕwe define a formula f(ϕ):=PA (x)∨x=ϕ. It follows
that
N|=ϕ⇐⇒ f(ϕ)∼pt PA .
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

24 ALI ENAYAT AND MATEUSZ ŁEŁYK
The left-to-right direction follows by IΣ1-provable Σ1-completeness of PA.Theright-
to-left direction follows by the soundness of IΣ1and PA.
Theorem 50. The set {∈Δ|T[]∈Cons}is Π2-complete.
In what follows, Π2-REF(PA) denotes the extension of EA with all sentences of
the form
∀xPrPA (ϕ(x))→ϕ(x)
for ϕ(x)∈Π2. It is a folklore result [1] that this theory is finitely axiomatizable. We
need the following folklore lemma, proved, e.g., in [23]:
Lemma 51. PA +¬Π2-REF(PA)is Π2-sound.
Proof of Theorem 50.Fix a Π2-sentence := ∀xϕ(x), where ϕ(x)isΣ
1.Let
be the formula in Δ that describes the union of (the canonical axiomatization of)
PA with the following set of sentences:
Pr∅()∧¬→ϕ(n)|∈L
PA ,n ∈.
The function → is clearly recursive, and ∈Δ. Let (x):=Π
2-REF(PA)∨
ϕ(x) and observe that for every n,CT–(n). Indeed, work in CT–and
assume ¬Π2-REF(PA). Then clearly ¬CT0and consequently, as in the proof of
Theorem 42 we get Tϕ(n)). Finally, the latter implies ϕ(n), since it is a standard
sentence.
Let TrueN
Π2be the set of Π2-statements that are true in N. We will prove
∈TrueN
Π2⇐⇒ CT–is conservative over PA.
Assume first that ∈Tru e N
Π2and =∀xϕ(x), for some ϕ(x)∈Σ1.Inparticular
ϕ(n) is a true Σ1sentence for every n∈, hence,
PA ϕ(n) for every n∈.
As usual, fix any model M|=PA and take its elementary extension (N,T)|=
CT–PA in which every disjunction of nonstandard length is true. As previously, it
follows that (N,T)|=CT–.
Conversely, assume that CT–is conservative over PA. Then for every n∈
,PA (n). In particular, for every n∈,PA +¬Π2-REF(PA)ϕ(n). By the
soundness of this theory we conclude that is true. 
4.2. Structure of prudent axiomatizations. Theorem 42 allows us to transfer
results about the fragment of the Lindenbaum algebra of PA consisting of sentences
provable in CT0to results about the structure of Tarski Boundary. Let us isolate the
former structure: put
CT0/PA := {[ϕ]PA |ϕ∈L
PA ∧CT0ϕ},
where [ϕ]PA denotes ϕ-equivalence class modulo PA-provable equivalence, i.e., the
element of the Lindenbaum algebra of PA that contains ϕ. Then, it is fairly easy to
see that the following holds:
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 25
Observation 52. CT0/PA with the operations inherited from the Lindenbaum
algebra of PA is a lattice with a greatest but not a least element (obviously assuming
the consistency of CT0). The lack of the least element follows from the fact that the
arithmetical consequences of CT0is a theory in LPA which extends PA . In particular
this theory is not finitely axiomatizable by essential reflexivity of PA. Moreover the
greatest element has no immediate predecessors. This follows by the classical fact that
the Lindenbaum Algebra of PA is dense (see [27]) for a proof.
The following is an easy corollary to Theorem 42.
Proposition 53. There exists a lattice embedding CT0/PA →Δ,≤CT–.
Proof. To each [ϕ]PA we assign ϕ∈Δ as in the proof of Theorem 42.(x)
is now simply x=ϕ. Hence by compositional axioms, and the fact that ϕis
standard we have
CT–[EA]∀T(ϕ[]) ↔T(ϕ[]).
Consequently, ϕcan be taken to axiomatize the (natural definition of the) following
set of sentences:
PA ∪{(Pr∅()∧¬)→ϕ|∈L
PA }.
We claim that for an arbitrary ϕ∈L
PA ,overCT–PA ,CT–ϕis equivalent
to ϕ. Working in CT–PA assume first that ϕholds. Then for every we have
T(ϕ[]),
since T(ϕ[]) is equivalent to an implication with a true conclusion. Hence every
sentence satisfying ϕis true. For the converse implication, working over CT–PA ,
assume CT–ϕ.Wearguebycases:
•If CT0holds, then ϕholds, by assumption.
•If CT0fails, then, as in the proof of Theorem 42,ϕholds.
We show that the mapping ϕ→ ϕis a lattice embedding. Firstly, we show that
the mapping preserves the partial ordering. To this end, we prove that the following
are equivalent for arbitrary arithmetical formulae ϕ,  that are provable in CT0:
•PA ϕ→.
•CT–[PA]T[ϕ]→T[].
Indeed the top-to-bottom direction follows easily, since for an arbitrary ϕ∈L
PA ,
CT–ϕis equivalent to ϕand CT–ϕproves CT–PA . The bottom-to-up
direction uses the same observations plus additionally the conservativity of CT–[PA]
over PA . Finally, we show that the mapping preserves infima and suprema. It
is enough to observe that for ϕand as above, CT–ϕ∨is equivalent to
CT–ϕ∨CT–and the same with ∧.This concludes the proof. 
The next proposition slightly lies on the margins of our considerations as it does
not concern axiomatizations of PA, but rather concerns the set of theorems of PA.
However, we include it, since it reveals an interesting feature of the Tarski Boundary.
Proposition 54. There is an embedding :CT0/PA →Δ–,≤CT–that is cofinal
in the region below (i.e., the nonconservative side of ) the Tarski Boundary. More
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

26 ALI ENAYAT AND MATEUSZ ŁEŁYK
precisely, for every α∈L
Tsuch that CT–[PA]+αis non-conservative over PA,there
is an a∈CT0/PA such that T[(a)] is strictly above α(i.e., is logically weaker ) and
CT–[PA]+T[(a)] is non-conservative over PA .
Proof. The embedding is defined as in the proof of the previous proposition
with the only exception that we do not add PA to . More concretely, if [ϕ]PA ∈
CT0/PA, then we put ([ϕ]PA ) to be the natural elementary definition of the following
set of sentences:
{(Pr∅()∧¬)→ϕ|∈L
PA }.
Denote the canonical elementary definition of this set with ϕ. As in the proof
of the previous proposition, we obtain that for every [ϕ]PA ∈CT0/PA,provablyin
CT–[PA], ϕis equivalent to T[ϕ]. Consequently, is a lattice embedding. Now
we claim that is cofinal with the Tarski Boundary in the sense explained. Pick
any α∈L
Tsuch that CT–[PA]+αis non-conservative over PA (but consistent).
By definition, CT–[PA]+αϕfor some PA - unprovable sentence ϕ∈L
PA . Then,
since the Lindenbaum algebra of PA is atomless there is a sentence ∈L
PA , which is
logically strictly weaker than ϕ. Then there is a sentence such that []PA ∈CT0/PA
and ∨is unprovable in PA. This holds, since it is known that over PA,REF(PA)
(which is a consequence of CT0) does not follow from any finite, consistent, set
of sentences. Hence [∨]PA ∈CT0/PA is not the greatest element. Consequently,
T[(∨)] = T[∨] is below the Tarski Boundary. However, since does not
prove ϕ(over PA), a fortiori ∨does not prove ϕ. Hence CT–∨does not
prove CT–[PA]+α. Additionally, CT–[PA ]+αCT–∨, since ∨follows
from α.
Proposition 55. There are recursive infinite antichains in Δ,≤CT–.
Proof. We shall make use of a Π1-formula that is PA-independent, i.e., for every
binary sequence sof length n∈the following sentence is unprovable in PA:
(0)s(0) ∧(1)s(1) ∧... ∧(n–1)s(n–1) ,
where for any formula ϕ,ϕ0:= ϕ, and ϕ1:= ¬ϕ. We will use the construction
of such a Π1-formula described in [21, Theorem 9, Chapter 2]. Let (x) be such
a formula. Assuming that each (k)isprovableinCT0,{(k)}k∈is an infinite
antichain in CT0/PA. By Proposition 53 this implies that {(k)}k∈is an infinite
antichain in Δ. These considerations show that it suffices to verify
CT0(k),for each k∈. (∗)
The verification of (∗) is a straightforward formalization of the reasoning in [21,
Theorem 9, Chapter 2], so it is delegated to the Appendix. 
Proposition 56. There is an embedding (Q,<)→Δ,≤CT–.
Proof. This is an immediate consequence of the existence of an embedding (Q,
<)→CT0/PA, which in turn follows from the well-known fact that the Lindenbaum
Algebra of PA is dense (see [27] for a proof). 
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 27
Proposition 57. There are , ∈Δsuch that CT–and CT–are non-
conservative extensions of PA, but CT–∨CT–is a conservative extension of
PA.
Proof. Consider ϕ:= ConPA+¬ConPA and := ConPA →ConPA+ConPA . Both ϕ
and generate different non-zero elements in CT0/PA butitiseasytoseethat
PA ϕ∨.
Hence the desired , ∈Δ can be chosen as := ϕand := (defined as in the
proof of Proposition 53). 
§5. Coda: The arithmetical reach of CT–for ∈Δ∗.Recall from Definition 17
that Δ∗is the collection of elementary presentations of PA, i.e., elementary formulae
that define (in N) a theory that is deductively equivalent to PA.Wearenowina
position to fulfill our promise given in the introduction and characterize the set
denoted sup PA of arithmetical sentences that are provable in some theory of the
form CT–,where∈Δ∗.
Theorem 58. sup PA is deductively equivalent to TrueN
Π2+REF<(PA).
Proof. First note that REF<(PA )⊆sup PA is an immediate corollary to
Theorem 42. Also, the proof of Tr u eN
Π2⊆sup PA is morally contained in the proof
of Theorem 26:foreverytrueΠ
2-sentence := ∀x∃yϕ(x, y ), the theory
PA ∪{∃yϕ(n,y)|n∈}
is deductively equivalent to PA, hence the natural arithmetical definition of the
above set witnesses that sup PA . To prove the converse inclusion13, assume that
for some ∈Δ, CT–ϕ.Letbe the true Π2-sentence
∀xPr(x)→PrPA (x),
expressing that every theorem of is provable already in PA. Then it is easy to
observe that
CT–[PA]++GRP(PA)ϕ.
However, by any of the proofs of Theorem 5,thetheoryCT–[PA]++GRP(PA )is
arithmetically conservative over EA +REF<(PA)+.14 Hence EA +REF< (PA)+
ϕ. Since EA +is a true Π2-sentence the proof is complete. 
§6. Open problems.
(I) Are the lattices SchPA ,≤CT–and Δ,≤CT–dense? Does Δ,≤CT–have
maximal or minimal elements? Does SchPA ,≤CT–have minimal elements
(by the proof of Theorem 30 no ≤CT–-maximal element exists)?
13This proof is due to Fedor Pakhomov and appears here with his kind permission.
14The crucial lemma in all the known proofs states that for every model M|=REF<(PA )thereisa
model Nwhich is elementarily equivalent to Mand T⊆Nsuch that (N,T)|=CT–[PA]+GRP(PA ).
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

28 ALI ENAYAT AND MATEUSZ ŁEŁYK
(II) Are the lattices SchPA ,≤CT–and Δ,≤CT–universal for countable
distributive lattices?15
(III) How do SchPA ,≤CT–and Δ,≤CT–fit in the Lindenbaum algebra of
CT–[EA]?
(IV) Is the Lindenbaum algebra of Cons dense?
(V) Do SchPA ,≤CT–and Δ,≤CT–have decidable copies? If not, how
undecidable are they?
(VI) How close can we get to the Tarski Boundary from below using theories
CT–,where∈Δ? In other words, if CT–[PA]+αis nonconservative
over PA , is there some ∈Δ such that CT–is nonconservative over PA,
and CT–[PA]+αT[]?
(VII) How close can we get to the Tarski Boundary from above using theories
CT–,where∈Δ? In other words, if CT–[PA]+αis conservative over
PA, is there some ∈Δ such that CT–is conservative over PA, and
CT–[PA]+T[]α?
(VIII) Do the answers to Questions (VI) and (VII) change if CT–is required
to be a subtheory of CT0?
§7. Appendix.
Proof Verification of (∗) of the proof of Proposition 55.To lighten the
notation, we will identify numerals with their denotations, and formulae with
their codes. We wish to show that if (x) is the Π1-formula (x) constructed in
[21, Theorem 9, Chapter 2], then for every k∈,CT0(k). Let us revisit the
construction of (x). Given a finite binary sequence sof length n, and a unary
arithmetical formula ϕ(x), let ϕsabbreviate the following sentence:
ϕ(0)s(0) ∧ϕ(1)s(1) ∧···∧ϕ(n–1)
s(n–1).
For a unary formula ϕ,let(x, i, ϕ, p)express:
there is a binary sequence sof length x+1suchthats(x)=ian pis a proof in PA of ¬ϕs.
Finally, let (x) be a formula assured to exist by the diagonal lemma such that the
following is provable in PA:
(x)↔∀p(x, 1,,p)→∃q≤p(x, 0,,q).
By metainduction on n∈, we show that for every n∈,CT0len(s)=n+1→
¬PrPA (¬s).Observe that this implies that for every n∈,(n)isprovablein
CT0. We first show that (0) is provable in CT0. Working in CT0, assume that ¬(0)
holds. It follows that for some p,(0,1,,p) holds, hence in particular, PrPA ((0))
holds. However, in CT0the theorems of PA are true, so (0) holds, contrary to
the assumption. Hence CT0¬PrPA (¬(0)). Moreover, since (0) holds, for every
PA-proof of (0) there exists a smaller PA-proof of ¬(0). Consequently, since CT0
proves the consistency of PA,forn=0,CT0∀slen(s)=n+1→¬PrPA (¬s).
Now, assume n=k+1,CT0∀slen(s)=n→¬PrPA (¬s). Working in CT0
assume for some sof length n+1,PrPA (¬s).Fix ssuch that the proof of sin PA is
15This question was communicated to us by Fedor Pakhomov.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

AXIOMATIZATIONS OF PEANO ARITHMETIC:A TRUTH-THEORETIC VIEW 29
the least possible (among s’s of length n+ 1). Denote (the code of) this proof with
p. We distinguish two cases:
1. s(n) = 0. Then, by the definition of s,wehavePrPA (sn→¬(n)).
Moreover, both (n, 0,,p)and∀q≤p¬(n, 1,,q) hold. Since is a Δ0-
formula, we have
PrPA (n, 0,,p)∧∀q≤p¬(n, 1,,q).
In particular, PrPA((n)). Hence PrPA (¬sn), which is impossible by the
induction step, since snhas length n.
2. s(n) = 1. Then, as before, PrPA (sn→(n)).Moreover, by minimality of p,
we have (n, 1,,p)and∀q<p ¬(n, 0,,q). Hence, as before we obtain
PrPA (¬(n)), which contradicts the induction assumption.
This concludes the proof of the induction step and the whole proof. 
§8. Acknowledgements. We have both directly and indirectly benefitted from
conversations with several colleagues concerning the topics explored in this paper,
including (in reverse alphabetical order of last names) Bartosz Wcisło, Albert
Visser, Fedor Pakhomov, Carlo Nicolai, Roman Kossak, Cezary Cie´
sli´
nski, Lev
Beklemishev, and Athar Abdul-Quader. The research presented in this paper
was supported by the National Science Centre, Poland (NCN; Grant Number
2019/34/A/HS1/00399).
REFERENCES
[1] L. Beklemishev,Reflection principles and provability algebras in formal arithmetic.Russian
Mathematical Surveys, vol. 60 (2005), no. 2, pp. 197–268.
[2] C. Cie´
sli ´
nski,Deflationary truth and pathologies.Journal of Philosophical Logic, vol. 39 (2010),
no. 3, pp. 325–337.
[3] ———, The Epistemic Lightness of Truth: Deflationism and its Logic, Cambridge University Press,
Cambridge, 2018.
[4] C. Cie´
sli ´
nski,M. ŁeŁyk,andB. Wc isŁo,The two halves of disjunctive correctness.Journal of
Mathematical Logic (2022), https://doi.org/10.1142/S021906132250026X.
[5] A. Enayat,M. ŁeŁyk,andB. WcisŁo,Truth and feasible reducibility,thisJournal, vol. 85 (2020),
pp. 367–421.
[6] A. Enayat and F. Pakhomov,Truth, disjunction, and induction.Archive for Mathematical Logic,
vol. 58 (2019), nos. 5–6, pp. 753–766.
[7] A. Enayat and A. Visser,New constructions of satisfaction classes,Unifying the Philosophy of
Truth (T. Achourioti, H. Galinon, J. M. Fern ´
andez, and K. Fujimoto, editors), Springer, Dordecht, 2015,
pp. 321–325.
[8] S. Feferman,Reflecting on incompleteness,thisJournal, vol. 56 (1991), no. 1, pp. 1–49.
[9] P. H ´
ajek and P. Pudl ´
ak,Metamathematics of First-Order Arithmetic, Springer, Berlin, 1993.
[10] V. Halbach,Axiomatic Theories of Truth, Cambridge University Press, Cambridge, 2011.
[11] J.Hubi ˇ
cka and J. Neˇ
setˇ
ril,Some examples of universaland generic partial orders,Model Theoretic
Methods in Finite Combinatorics, Contemporary Mathematics, vol.558, American Mathematical Society,
Providence, 2011, pp. 293–317.
[12] R. Kossak and B. Wci sŁo,Disjunctions with stopping conditions.Bulletin of Symbolic Logic,
vol. 27 (2021), no. 3, pp. 231–253.
[13] H. Kotlarski,Bounded induction and satisfaction classes.Zeitschrift f ¨
ur matematische Logik und
Grundlagen der Mathematik, vol. 32 (1986), pp. 531–544.
[14] H. Kotlarski,S. Krajewski,andA. Lachlan,Construction of satisfaction classes for
nonstandard models.Canadian Mathematical Bulletin, vol. 24 (1981), pp. 283–293.
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press

30 ALI ENAYAT AND MATEUSZ ŁEŁYK
[15] G. Leigh,Conservativity for theories of compositional truth via cut elimination,thisJournal,
vol. 80 (2015), no. 3, pp. 845– 865.
[16] M. ŁeŁyk,Axiomatic theories of truth, bounded induction and reflection principles, Ph.D. thesis,
University of Warsaw, 2017.
[17] ———, Model theory and proof theory of the global reflection principle.The Journal of Symbolic
Logic, First View, (2022), pp. 1–42, https://doi.org/10.1017/jsl.2022.39.
[18] ———, Axiomatizations of Peano arithmetic: A truth-theoretic perspective, pt. 2, in preparation,
forthcoming.
[19] M. ŁeŁyk and B. Wci sŁo,Models of positive truth.Review of Symbolic Logic, vol. 12 (2018),
pp. 144–172.
[20] ———, Local collection and end-extensions of models of compositional truth.Annals of Pure and
Applied Logic, vol. 172 (2021), no. 6, Article no. 102941.
[21] P. Lindstr ¨
om,Aspects of Incompleteness, Lecture Notes in Logic, Cambridge University Press,
Cambridge, 2017.
[22] F. Pakhomov and A. Visser,On a question of Krajewski’s,thisJournal, vol. 84 (2019), no. 1,
pp. 343–358.
[23] F. Pakhomov and J. Walsh ,Reflection ranks and ordinal analysis,thisJournal, vol. 86 (2020),
no. 4, pp. 1350–1384.
[24] C. Parsons,On a number theoretic choice schema and its relation to induction,Intuitionism and
Proof Theory: Proceedings of the Summer Conference at Buffalo N.Y. 1968 (A. Kino, J. Myhill, and R.E.
Vesley, editors), Studies in Logic and the Foundations of Mathematics, vol. 60, Elsevier, Amsterdam,
1970, pp. 459–473.
[25] ———, On n-quantifier induction,thisJournal, vol. 37 (1972), no. 3, pp. 466–482.
[26] P. Pudl ´
ak,Cuts, consistency statements and interpretations, this Journal,
vol. 50 (1985), pp. 423–441.
[27] V. Y. Shavrukov and A. Visser,Uniform density in Lindenbaum algebras.Notre Dame Journal of
Formal Logic, vol. 55 (2014), no. 4, pp. 569–582.
[28] S. G. Simpson and R. L. Smith,Factorization of polynomials and 0
1induction.Annals of Pure
and Applied Logic, vol. 31 (1986), nos. 2–3, pp. 289–306.
[29] C. Smory´
nski,-consistency and reflection,Colloque International de Logique (Colloq. Int.
CNRS), CNRS Inst. B. Pascal, Paris, 1977, pp. 167–181.
[30] W. W. T a i t ,Finitism.Journal of Philosophy, vol. 78 (1981), no. 9, pp. 524–546.
[31] R. Vaught,Axiomatizability by a schema,thisJournal, vol. 32 (1967), pp. 473–479.
[32] A. Visser,Vaught’s theorem on axiomatizability by a scheme.Bulletin of Symbolic Logic,vol.18
(2012), no. 3, pp. 382–402.
DEPARTMENT OF PHILOSOPHY, LINGUISTICS
AND THEORY OF SCIENCE
UNIVERSITY OF GOTHENBURG
GOTHENBURG, SWEDEN
E-mail:ali.enayat@gu.se
DEPARTMENT OF PHILOSOPHY
UNIVERSITY OF WARSAW
WARSAW, POLAND
E-mail:mlelyk@uw.edu.pl
https://doi.org/10.1017/jsl.2022.83 Published online by Cambridge University Press