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Abstract

We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I∆0 + Exp of PA, and 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 α is 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.
Axiomatizations of Peano Arithmetic:
A truth-theoretic view
Ali EnayatMateusz Łełyk
August 18, 2021
Abstract
We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano
Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I0+Exp of
PA, and 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.
Contents
1 Introduction 2
2 Preliminaries 3
2.1 CT,CT0,andtheTarskiBoundary ........................... 3
2.2 Schematicaxiomatizations................................. 6
2.3 Prudentaxiomatizations.................................. 8
3 Schematically correct axiomatizations 10
3.1 Complexity.......................................... 10
3.2 Structure of schematically correct extensions . . . . . . . . . . . . . . . . . . . . . . 12
4 Prudently correct axiomatizations 16
4.1 Universality and complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Structure of prudent axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Coda: The arithmetical reach of CTJδKfor δ22
6 Open problems 22
7 Appendix 23
Department of Philosophy, Linguistics, and Theory of Science, University of Gothenburg, Sweden; ali.enayat@gu.se
Department of Philosophy, University of Warsaw, Poland; mlelyk@uw.edu.pl
1
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. Fefer-
man’s pivotal paper on the arithmetization of metamathematics [8] 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=1from 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 axiomatizations
of PA. Our focus is 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. More specifically, let EA (Elementary Arithmetic) be the
fragment I0+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 Kotlarski, Krajewski, and Lachlan
[14] shows that CT[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)|nω},
where Con(n)is the arithmetical sentence that expresses "there is no proof of inconsistency of PA
whose code is below n" and ωis 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+, call it δ(x), and then we choose α2to be the sentence
T[δ] := x(δ(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
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.
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 pqis
translated to pq.
2
conservative over PA is not recursively enumerable. Another main result of the paper pertains
to the strengthening CT0of CT[EA]obtained by augmenting CT[EA]with the scheme of 0-
induction (in the extended language containing the truth predicate). It is known that the arith-
metical strength of CT0far surpasses that of PA, e.g., CT0can prove ConPA,ConPA+ConPA , etc. (see
Theorem 6). In Theorem 41 we show that given any r.e. extension Uof PA such that CT0`U, there
is an axiomatization δof PA which is proof-theoretically equivalent to the usual axiomatization of
PA and which has the property that the arithmetical consequences of the (finitely axiomatized)
theory CT[PA] + T[δ]coincides with the deductive closure of U(note that Theorem 6 provides
us with an ample supply of theories Uthat Theorem 41 is applicable to).
Our other main results are structural. In Section 3.2, we focus on the collection SchPA consisting
of the scheme templates τsuch that PA is deductively equivalent to the scheme generated by τ(see
Definitions 7 and 22). For example, in Theorem 30 we show that from the point of view of relative
interpretability, theories of the form CT[EA] + T[τ], where τSchPA and T[τ]is the sentence
asserting that every instance of τis true, have no maximal element.3In the same section we also
prove that the partially ordered set hSchPA,CTiis universal for countable partial orderings (in
paricular, it contains infinite antichains, and also contains a copy of the linearly ordered set Qof
the rationals), where the partial ordering CTis defined by:
τ1CTτ2iff CT[EA]`T[τ1]T[τ2].
In Section 4.2, we prove similar results about the partial ordering h,CTi, where is the col-
lection of elementary presentations of PA that are proof-theoretically equivalent to (the canonical
axiomatization of) PA. In particular we show that there is an embedding CT0/PA → h,CTi,
where CT0/PA is the end segment of the Lindenbaum algebra of PA generated by the collection
of arithmetical consequences of CT0.
Finally, in Theorem 57 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[δ], where δ(x)is an elementary formula (in the sense of Definition 2) that defines an axiomati-
zation 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 con-
text of the basic compositional truth theory CT[EA], and (3) further investigating structural as-
pects of finite axiomatizations of infinite theories, a topic initiated in the work of Pakhomov and
Visser [20].
Acknowledgements. We have both directly and indirectly benefitted from conversations with
several colleagues concerning the topics explored in this paper, including (in reverse alphabet-
ical order of last names) Bartosz Wcisło, Albert Visser, Fedor Pakhomov, Carlo Nicolai, Roman
Kossak, Cezary Cieśliński, 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.
2 Preliminaries
2.1 CT,CT0, and the Tarski Boundary
Definition 1. PA (Peano Arithmetic) 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 Nand its universe of discourse by ω.
3Once again, we treat the interpreted theory as greater in this ordering.
3
Definition 2. EA (Elementary Arithmetic) is the fragment I0+Exp of PA, where I0is the
induction scheme for 0-formulae (i.e., formulae with only bounded quantifiers), and Exp asserts
the totality of the function exp(x)=2x(it is well-known that the graph of exp can be described by
a0-formula). An elementary formula is an arithmetical formula whose quantifiers are bounded
by terms built from the function symbols S,+,×, and exp. The family of (Kalmár) elementary
functions is a distinguished subfamily of the primitive recursive functions.4It is well-known that
the provably recursive functions of EA are precisely the elementary functions; 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 BEA. We use LT
to refer to the language obtained by adding a unary prediate Tto LPA.CT[B]is the theory
extending Bwith the LT-sentences CT1through CT5below.
In what follows xClTermLPA is the arithmetical formula that expresses "xis (the code of)
a closed term of LPA"; xSentLPA is the arithmetical formula that expresses "xis (the code of)
a sentence of LPA", xVar expresses "xis (the code of) a variable", and xForm1
LPA is the
arithmetical formula that expresses "xis a (the code of) formula of LPA with at most one free
variable”, xis (the code of) the numeral representing x, and ϕ[x/v]is (the code of) the formula
obtained by substituting the variable vwith the numeral representing x.
CT1s, t ClTermLPA T(s=t)s=t.
CT2ϕ, ψ SentLPA T(ϕψ)T(ϕ)T(ψ).
CT3ϕSentLPA T(¬ϕ)↔ ¬T(ϕ).
CT4ϕ(v)Form1
LPA T()↔ ∃x T (ϕ[x/v]).
CT5ϕv)FormLPA ¯s, ¯
tClTermSeqLPA (¯
s=¯
tT(ϕs/¯v]) T(ϕ[¯
t/¯v])) .
In CT5above sand tdenote finite tuples of terms; and ¯
s,¯
trefer to the corresponding valuations of
sand t. The axiom CT5is sometimes called generalized regularity, or generalized term-extensionality,
and is not included in the accounts of CTprovided in the monographs of Halbach [10] and
Cieśliński [4]. The conservativity of this particular version of CT[PA]can be verified by a re-
finement of the model-theoretic method introduced in [6], as presented both in [7] and [12].
Moreover, [7] establishes the following strengthening of the conservativity-result.
Theorem 4. There is a polynomial-time computable function fsuch that for every CT[PA]-proof πof an
arithmetical sentence ϕ,f(π)is a PA-proof of ϕ. Moreover the correctness of fis verifiable in PA.
The above result shows that CT[PA]is feasibly reducible to PA. In particular, the basic truth
theory CT[PA]admits at most a polynomial speed-up over PA. Moreover, as shown in [7], 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 4 witnesses 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.
4Elementary functions occupy the third layer (E3) of the Grzegorczyk hierarchy of primitive recursive functions
{En|nω}. It is often claimed that almost all number theoretical functions that arise in mathematical practice are
elementary.
4
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:
ϕSentLTPrT(ϕ)T(ϕ).(GRP(T))
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]–Smoryński [25], 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) := SnωREFn(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:
xPrT(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. 0-induction scheme for LT(see [16] and [17]).
2. GRP(), i.e. ϕPr(ϕ)T(ϕ)(see [3])
3. cccodes a set of sentencesT(Wϕcϕ)→ ∃ϕc T (ϕ)(see [2]).
Theorem 6 reveals 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 straight-
forward generalisation (often dubbed disjunctive correctness) of the compositional axiom CT2of
CTfor disjunctions.5
This shows that conceptually CT[PA]is closer to the Tarski Boundary than previously con-
ceived. 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 the-
ory CT0[EA], which is obtained by augmenting CT[EA]with 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 the introduction, our main focus in the current paper is on finite
extensions of CT[EA]that expresses "PA is true". As shown in Theorem 57, if we admit all ele-
mentary presentations of PA, then each true Π2-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 (in-
troduced in Section 2.2) and prudent axiomatizations (introduced in 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
PRA (Primitive Recursive Arithemtic).
5The last part of Theorem 6 refines the main result of Enayat and Pakhomov’s paper [5], which shows that CT0can be
axiomatized by simply adding the disjunctive correctness axiom to CT[EA].
5
2.2 Schematic axiomatizations
Definition 7. Atemplate (for a scheme) is given by a sentence τ[P]formulated in the language
obtained by augmenting LPA with a predicate P, where Pis unary.6An LPA-sentence ψis said to
be an instance of τif ψis of the form y τ[ϕ(x, y)/P ], where τ[ϕ(x, y)/P ]is the result of substituting
all subformulae of the form P(t), where tis a term, with ϕ(t, y)(and re-naming bound variables
of ϕto avoid unintended clashes). We use Sτto denote the collection of all instances of τ, and we
refer to Sτas the scheme generated by τ.
We will use T[τ]to refer to the LT-sentence that says that each instance of Sτis true; more
formally:
T[τ] := ϕ(x, y)z T (τ[ϕ(x, z)/P ]).
We note that, over CT[EA],T[τ]is equivalent to the assertion
ϕ(x)Form1
LT(τ[ϕ(x)/P ]).
We sometimes write "Tis τ-correct" instead of T[τ].
As mentioned in the introduction, the special case of the following theorem was first estab-
lished (for B=PA ) by Krajewski, Kotlarski, and Krajewski [14] and in full generality by Enayat
and Visser [6], and Leigh [15].
Theorem 8. CT[B] + T[τ]is conservative over Bfor every base theory Band every scheme template τ
such that B`Sτ.
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 understood as the maximal number of connectives and
quantifiers on a path in the syntactic tree of ϕ. The pure depth of the formula ϕis the maximal
length of a path in the syntactic tree of ϕ, assuming that each vertex of the syntactic tree is la-
belled with exactly one symbol from the alphabet or a variable. 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 CT1through CT4of Definition 3 in which (1) the predicate Tis replaced by P, and (2) the
universal quantifiers on ϕand ψare 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 xx=S(S(0))¬x=xis 3, whereas its pure depth is 6(the vertices in the longest
path are labelled by x,,=,S,S,0.)
The following theorem is classical; see [9] for a proof.
Theorem 11 (Partial Truth Definitions).For each nωthere 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.
6Thanks 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 τis allowed
to use finitely many predicate symbols P1,...,Pnof various finite arities.
6
Theorem 12 (Vaught [27], Visser [28]).Let Tbe an r.e. theory with enough coding7, and let LTbe
the language of T. There is a primitive recursive function f(indeed fis elementary) such that given any
unary Σ1formula σthat defines a set of LT-sentences Φin N,f(σ)is a scheme template such that Sf(σ)
axiomatizes Φ.
Proof outline for T=EA.Suppose σ(x)is a Σ1-formula that defines a set Φof sentences of LPA in
the standard model of arithmetic. (By Craig’s trick, σcan be chosen to be an elementary formula.)
Let True(y, P )be as in Definition 9. The desired scheme template τis:
y[True(y, P )[z(σ(z)pdepth(z)y)P(z)]] .
We note that:
(1) EA +Sτ`Φ, because for each nωthe truth predicate for formulae of depth at most nis
definable by Theorem 11; and
(2) EA + Φ `Sτ, thanks to Tarski’s undefinablity of truth theorem.
Remark 13. The proof of the above theorem would not go through, if in the definition of τ,
pdepth was changed to depth. Indeed, assume τis modified accordingly. It is enough to take
Φ := {ConEA(n)|nω}, where ConEA (x)expresses "there is no proof of inconsistency of EA
whose code is below x". Let σbe the natural elementary definition of Φ, i.e.
σ(x) := y < xx=pConEA (y)q.
Observe that each sentence in Φhas the same, standard depth, call it k. Assume that θis a truth
predicate for formulae of depth k. Then the sentence
y[True(y, θ)[z(σ(z)depth(z)y)θ(z)]] .
clearly implies ConEA, hence Sτis, 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 Vaught’s argument work.
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 46.
Remark 15. Let ConZF be the arithmetical statement asserting the consistency of ZF, and for each
nωlet 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)|nω}.
Then provably in ZF :
PA+is conservative over PA” iff ConZF.
To see that the above holds, we reason in ZF. Suppose PA+is conservative over PA. Then for all
nω,PA proves ConZF(n). On the other hand, ZF "knows" that PA holds in the standard model
of arithmetic, so for all nω,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 in PA (as-
suming ConZF), but ZF cannot verify this. Coupled with Theorem 8 this also shows that there is
a scheme template τsuch that
ZF `ConZF τSch
PA.
7Visser [28] showed that supporting a pairing function is "enough coding" in this context. For Vaught [27] "enough
coding" meant being able to interpret an -relation for which the statement: For all objects x0,· · ·, xn1there is an object
ysuch that for all objects t,tyiff (t=x0or ... or t=xn1)” holds for each nω(sequential theories support such
an -relation).
7
2.3 Prudent axiomatizations
In Section 4 we 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 δ,δ0range over elementary formulae with one free variable. We say that δis
proof-theoretically reducible to δ0(δpt δ0) if
IΣ1` ∀ϕPrδ(ϕ)Prδ0(ϕ).
We write
δPA
for the elementary formula representing the usual axiomatization of PA (as in Definition 1), i.e.,
δPA(x)expresses: xis either (the code of) an axiom of Qor (the code of) an instance of the induc-
tion scheme. We say that δis proof-theoretically equivalent to δPA (written as δpt δPA ) if
IΣ1` ∀ϕPrδ(ϕ)PrδPA(ϕ).
It is a classical fact due to Parsons ([22], [23]) that IΣ1and the system of Primitive Recursive
Arithmetic, known as PRA, have the same Π2-consequences. In particular it follows that whenever
δp.t. δ0, then in fact δand δ0are deductively equivalent provably in PRA. As a consequence
there are primitive recursive proof transformations mapping proofs in δto proofs with the same
conclusions in δ0and 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 δand δ0to 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 δto proofs with the same conclusions in δ0and
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 philosoph-
ically well-motivated by Hilbert’s finitism, as argued forcefully by Tait [26].
Definition 17. We use to denote the collection of unary elementary formulae δ(x)such that
δN:= {nω|N|=δ(n)}codes an LPA-theory that is deductively equivalent to PA. We sometimes
refer to the members of as elementary presentations of PA.
Given any arithmetical formula ϕ(x),
T[ϕ(x)] := xϕ(x)T(x).
So T[ϕ]is the LT-sentence expressing that the theory described by ϕis true. Moreover, we
put
CTJϕK:= CT[EA] + T[ϕ].
We use to denote the subset of consisting of formulae δsuch that δis proof-
theoretically equivalent to δPA .Thus is the collection of (defining formulae of) prudent axiom-
atizations of PA.Occasionally we also need the extension of , denoted , defined
:= {δ|δp.t. PA}.
On and we shall consider the relation CTgiven by
δCTδ0CT[EA]`T[δ]T[δ0].
Convention 18. Simplifying things a little bit, when talking about the structures h,CTiand
h,CTi, we shall assume that is replaced by the quotient set /, where is the least
equivalence relation that makes CTantisymmetric, to wit:
8
δδ0iff δCTδ0and δ0CTδ.
Let us stress an important difference between CT[PA]and CTJδPAK: 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[x]depends on whether xis a scheme template,
in which case T[x]is interpreted as in Definition 7, or an arithmetical formula, in which case
T[x]has the meaning given in Definition 17.
Proposition 19. Both h,CTiand h,CTiare distributive lattices.
Proof. We only present the proof for the case of as it is (1+ε)-times harder. It is enough to show
that given δ, δ0, one can find elements δδ0and δδ0of such that over CT[PA]we have:
T[δ]T[δ0]T[δδ0].(1)
T[δ]T[δ0]T[δδ0].(2)
It can be readily seen that if we define:
δδ0(x) := δ(x)δ0(x),
then δδ0and (1) is satisfied. For (2) it is sufficient to define:
δδ0(x) := y, z < xδ(y)δ0(z)x=yz,
where x=yzexpresses that xis a disjunction of yand z. To see that (2) holds and δPA p.t. δδ0
one simply applies reasoning by cases; the proof of δδ0p.t. δPA is trivial.
Remark 20. If δcorresponds to a schematic axiomatization of PA (i.e., for some template τ[P],
δ(x)says that xis the result of substituting Pwith some unary arithmetical formula), then CTJδK
is a conservative extension of PA by Theorem 26. In contrast, even for very natural δ,CTJδK
may be a highly non-conservative extension of PA. For example, consider:
REFEA =xPrEA(ϕ(x)) ϕ(x)|ϕ(x)∈ LPA.
By a classical theorem of Kreisel, the union of EA and REFEA is deductively equivalent to PA (see,
e.g., [1, p. 39]). Let δ(x)be a natural elementary definition of EA REFEA. Then, in fact δ.
An easy argument shows that
CTJδK` ∀ϕPrEA(ϕ)T(ϕ).
However, by a theorem of Cieśliński [3], over CT[EA], the above consequence of CTJδKimplies
the Global Reflection Principle for PA.
Proposition 21. Every theory Textending EA whose axioms are described by an elementary formula δ
(in the standard model of arithmetic) has a proof-theoretically equivalent presentation δ0such that CTJδ0K
is conservative over T.
Proof. The proof is based on the observation that in the proof of Theorem 12, the verification that
Sτand Φcoincide, formalizes smoothly in EA. More explicitly, the implication Sτ`Φrequires
only the existence of well-behaved partial truth predicates (that can be developed within EA, as
demonstrated e.g. in [1, Proposition 2.6]). The implication Φ`Sτrequires Tarski’s undefinability
of truth theorem. Although the latter presupposes the consistency of Φ, this can be assumed,
because if Φis inconsistent, so is Sτby the proof of the first implication, and in such a scenario
the two theories clearly coincide.
9
3 Schematically correct axiomatizations
3.1 Complexity
Definition 22. In the following definitions τranges over scheme templates and Sτis the corre-
sponding scheme (in the sense of Definition 7) generated by τ.
(a) Sch
PA := {τ:PA `Sτ},i.e. Sch
PA is the collection of templates whose corresponding scheme
is PA-provable.
(b) SchPA := {τSch
PA :Sτ`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 τsuch that the arithmetical consequences of CT[EA]+T[τ]
coincides with PA (recall that T[τ]says that Tis τ-correct, as in Definition 7).
(d) Cons := {ϕ∈ LT:CT[PA] + ϕis conservative over PA}.
Recall that in the Introduction we defined CTon Sch
PA as follows:
τCTτ0CT`T[τ]T[τ0].
When talking about the structural properties of hSchPA,CTiwe shall tacitly assume that SchPA
is factored out by an appropriate equivalence relation, so as to make CTa partial order (as in
Convention 18.)
Proposition 23. hSch
PA,CTiand hSch,CTiare distributive lattices.
Proof. As previously we do the case of a smaller structure, with Sch as the universe. It is enough
to define and such that CT[PA]proves the following for all τ, τ 0SchPA:
T[τ]T[τ0]T[ττ0](3)
T[τ]T[τ0]T[ττ0](4)
The case of is trivial. We put:
ττ0:= ττ0.
The case of is (a little bit) harder. We put:
ττ0:= τ(τ0[Q/P ]),
where Qis a fresh unary predicate. As remarked earlier (compare footnote 4) thanks to the coding
apparatus, ττ0can be expressed as a scheme with a single unary predicate P. Then we obtain
CT[EA]`T[ττ0]≡ ∀φψ T τ[φ/P ]τ0[ψ/Q].
It is very easy now to check that (4) is satisfied.
Theorem 24 (KKL-Theorem, first formulation).CT[PA] + T[τ]is conservative over PA for each
τSch
PA.
Let Θbe the union of sentences of the form T[τ](expressing that Tis τ-correct) as τranges
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.
10
The above formulation naturally suggests the question: How complicated is Θ(viewed as a subset
of ω)? Is it recursively enumerable? The result below shows that Θis Π2-complete, since Θis
readily seen to be recursively isomorphic to Sch
PA (indeed the isomorphism is witnessed by an
elementary function). Therefore, Θis 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 (Gödel 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 ωvia:
AmBiff there is a total recursive function fsuch that: nω(nAf(n)B).
The proof will be complete once we demonstrate the following four assertions:
(i)TrueN
Π2mSch
PA.8
(ii)Sch
PA mSchPA .
(iii)Sch
PA mSchT
PA.
(iv)TrueN
Π2mCons. To prove (i), suppose π=xy δ(x, y)is a Π2-statement, where δ(x, y)is
0. We first observe that by Σ1-completeness of PA:
()πTrueN
Π2iff nωPA ` ∃y δ(n, y).
On the other hand, R={∃y δ(n, y ) : nω}is a recursive set of sentences, so by Theorem 12 there
is τsuch that τSch
PA iff PA `R. To finish the proof, it remains to observe that the transition
from πto the Σ1-formula σthat defines Rin Nis given by a recursive function g, therefore if fis
the total recursive function as in Theorem 2:
πTrueN
Π2iff f(g(π)) Sch
PA.
The proof of (ii)is based on the observation that τSch
PA iff h(τ)SchPA , where h(τ) := ττPA,
and τPA is defined as follows:
τPA := Q[P(0) ∧ ∀x(P(x)P(S(x))) → ∀xP(x)].
To verify (iii), we claim that τSch
PA iff (ττPA )SchT
PA. The implication τSch
PA
(ττPA)SchT
PA follows directly from Theorem 3 (since PA proves SττPA if τSch
PA).On the
other hand, if (ττPA)SchT
PA, then by the definition of SchT
PA,PA proves Sτ, so τSch
PA.
Finally, to establish (iv)suppose π=xy δ(x, y)is a Π2-statement, where δ(x, y)is 0. In
the proof of part (i) we showed that there are recursive functions fand gsuch that:
πTrueN
Π2f(g(π)) Sch
PA.
Let hbe the function that takes a template τas input, and outputs the sentence T[τ]∈ LTex-
pressing "Tis τ-correct" . Clearly his a recursive function. Also, it is evident that τSch
PA
iff ϕτCons (the direction follows from Theorem 8; and the direction follows from the
relevant definitions). Therefore:
πTrueN
Π2h(f(g(π))) Cons.
8The proof of (i)shows that Sch
Tis Π2-complete for any extension Tof Robinson’s Qthat is Σ1-sound, and which
also supports a pairing function.
11
Proposition 27. Let σbe the single LT-sentence that expresses "every PA-provable scheme is true". Then
CT0can be axiomatized by CT[EA] + σ.
Proof. By Theorem 6, CT0can be axiomatized by CT[EA] + GRP. This makes it clear that σis
provable in CT0.For the other direction, suppose ϕis PA-provable, then the scheme given by
x(ϕP(x)) is PA-provable, so the instance of this scheme in which Pis replaced with x6=xis
true, but since T(x(x=x)), we have T(ϕ).
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 inter-
pretability properties of its elements, where by "interpretability" we always mean relative inter-
pretability, 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), δ(x)with at most one free variable let the ϕ-restricted
Vaught schematization of δbe the scheme template
V(ϕ,δ)[P] := yϕ(y)True(y, P )→ ∀x(δ(x)pdepth(x)y)P(x).
For a single formula δ,Vδ[P]abbreviates V(x=x,δ)[P]and we often omit the reference to P. Simi-
larly Vφ,δ[θ(x)] abbreviates Vφ,δ [θ(x)/P (x)].
Convention 29. Working in CT[EA]and having fixed an (possibly nonstandard) arithmetical
formula with one free variable θ(v),Tθ(x)will abbreviate the formula T(θ[x/v]). Hence Tθ(x)
says that xsatisfies θ. This notation was first introduced in [18] and is very successful in decreas-
ing the number of brackets and improving readability.
Recall from Definition 17 that CTJτKis the theory CT[EA] + T[τ], i.e. CT[EA]together with
the assertion that Tis τ-correct.
Theorem 30. If ψ∈ LTis such that for every τSchPA,ψis interpretable in CTJτK, then ψis
interpretable in CT[PA].
Proof. Fix ψas in the assumption of the theorem. We modify the Pakhomov-Viser diagonalization
from [20, Theorem 4.1]. Observe that for two finite theories α,β, the condition "αinterprets β"
is Σ1. Let α  β denote the formalization of this relation. Consider a Σ1-sentence ϕ=0(x),
where ϕ0(x)0such that the following equivalence is provable in CT[PA]:
ϕCTJV(zy¬ϕ0(z)PA)K ψ.
Similarly to the Pakhomov-Visser argument, we argue that ϕis false. Suppose not and take the
least nωsuch that ϕ0(n)holds. Then, in Q,zx¬ϕ0(z)is equivalent to x < n, hence the
following is provable in CT[PA]:
θ(x)TV(zy¬ϕ0(z)PA)[θ]TV(y<n,δPA)[θ].
We claim that:
CT[PA]` ∀θ(x)TV(y<n,δPA)[θ].()
Indeed, working in CT[PA]fix θForm1
LPA. By compositional conditions TV(y<n,δPA)[θ]is
equivalent to:
^
i<nTTrue(i, θ)→ ∀x(δPA(x)pdepth(x)i)Tθ(x).
12
However, once again by compositional conditions imposed on T,TTrue(i, θ)is equivalent to:
True(i, T θ(x)), hence to the assertion that Tθ(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 Tθ(x). This concludes our proof of ().
Now, since ϕis true, it follows that:
CT[PA] + θ(x)TV(y<n,δPA)[θ]interprets ψ.
However, by the above argument it would mean that CT[PA]interprets ψ, contrary to the as-
sumption.
Since ϕis false, V(zy¬ϕ0(z)PA)[P]is a scheme template, such that the scheme associated with
it axiomatizes PA. Moreover, CT[PA] + TV(zy¬ϕ0(z)PA)does not interpret ψ.
Since CT[PA]is interpretable in PA (see [6] and [15]), we obtain the following corollary.
Corollary 31. For every ψ∈ LTsuch that PA does not interpret ψthere is a scheme template τSchPA
such that CTJτKdoes not relatively interpret ψ.
Since PA 7Q+ConPA ([24]) 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 not due to
the consistency of PA being interpretable.
Corollary 32. There is a scheme template τSchPA such that CTJτKdoes not interpret Q+ConPA.
Corollary 33. For every scheme template τSchPA there is a scheme template τ0SchPA such that
CTJτKinterprets CTJτ0K, but not vice versa.
Proof. Fix τand apply Corollary 31 to ψ:= CTJτK. This is legal, since the latter theory is a
finitely axiomatizable extension of PA, hence it is not interpretable in PA.9So there is a scheme
τ00 SchPA such that CTJτ00Kdoes not relatively interpret CTJτK. Now it is sufficient to take
τ0:= ττ00, as in the proof of Proposition 23.
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 δand δ0are arbitrary elementary formulae that, provably in EA,
specify arithmetical theories, i.e. possibly infinite sets of arithmetical sentences. We will
write δδ0as an abbreviation of x(δ(x)δ0(x)). Recall (from Definition 17) that T[δ]is
the following sentence expressing "Tis δcorrect":
xδ(x)T(x).
Note the difference between T[Vδ]and T[δ].
The first result is immediate:
Proposition 34. For every δand δ0,CT[PA]` ∀xδ(x)δ0(x)T[Vδ0]T[Vδ].
For many applications, the condition δδ0from the antecedent is too restrictive. One would
like to relax it to δ`δ0, however this one is too weak to guarantee (over CT[PA]) that the im-
plication T[Vδ0]T[Vδ]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.
9This is because otherwise CTJτK, being a finite theory, would be interpretable in a finite fragment of PA, call it T.
But then, since CTJτKextends PA and PA is reflexive, CTJτK`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).
13
Given a unary arithmetical formula ϕ(x), in the proposition below we use the convention
of using δϕ(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 ψ(x)
CT[PA]` ∀xϕ(x)ψ(x)T[Vδϕ]T[Vδψ].
Proof. Fix ψand ϕ,δψ, δϕ,nas described in the assumptions. Without loss of generality, as-
sume that the variable xoccurs in ψ. Working in CT[PA]assume that xϕ(x)ψ(x)and
T[Vδϕ]hold. We argue that T[Vδψ]holds as well. Fix arbitrary a,θ,bsuch that True(a, T θ)and
pdepth(ψ(b)) a. It follows that for some standard n,pdepth(ϕ(b)) a+n, hence there exists a
formula θ0(x)such that
True(a+n, T θ0).
By T[Vδϕ]we conclude Tθ0(ϕ(b)).However, since ϕ(x)is of standard depth, it follows that ϕ(b)
holds. Hence ψ(b)holds as well. Since ψ(b)is also of standard depth, we conclude that Tθ(ψ(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,δ-correctness is
equivalent to Vδ-correctness.
Proposition 36. For every δ,CT0`T[δ]T[Vδ].
Proof. We start by showing that provably in CT0all arithmetical partial truth predicates are coex-
tensive, i.e. the following is provable in CT0:
xθForm1
LPAϕSentLPA True(x, T θ)depth(ϕ)xTθ(ϕ)T(ϕ).()
Fix an arbitrary (M, T )|=CT0. For an arbitrary cM, let Tcdenote the ((M, T )-definable)
restriction of Tto all sentences of depth at most c. Then (M, Tc)|=True(c, T ). However, as proved
in [29, Fact 32], (M, Tc)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 aand an arbitrary θForm1
LPA. Assume that the depth of θis b
and let c= max{a, b}. Assume True(a, T θ), i.e. the formula Tθis 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, Tcθ)holds. Moreover, it is sufficient to show that
xTcθ(x)Tc(x).
In other words, it is sufficient to prove that
(M, Tc)|=xTθ(x)T(x)
The above can be demonstrated by a routine induction on the build-up of formulae. More pre-
cisely, let
Ξ(y) := ϕdepth(y)Tθ(ϕ)T(ϕ).
Then Ξ(0) and x < aΞ(x)Ξ(x+ 1)hold (in (M, Tc)), because both Tcθand Tcare partial
truth predicates for formulae of depth at most a. Since Ξ(y)is a formula of LT, in (M, Tc)we have
an induction axiom for it, and we can conclude
(M, Tc)|=yaΞ(y).
This completes our claim.
14
Now we fix an arbitrary δand working in CT0assume that xδ(x)T(x). We show that T
is Vδ-correct, i.e. for every arithmetical formula θ(possibly nonstandard) T(Vδ[θ]) holds. By the
compositional conditions the last sentence is equivalent to
xTrue(x, T θ)yδ(y)pdepth(y)x)Tθ(y).
Fix x, assume True(x, T θ)and fix an arbitrary ysuch that pdepth(y)xand δ(y). By δ-
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,
Tθ(ϕ)is equivalent to T(ϕ). Hence Tθ(y)holds as well.
Now assume Tis Vδ-correct. Fix an arbitrary xand assume that δ(x)holds. In particu-
lar xis a formula. Let ybe the depth of xand let θbe any arithmetical truth predicate such
that PrPA(True(y, θ)) holds. By the Global Reflection in CT0,T(True(y , θ)) holds as well, and
this in turn implies, by compositional conditions, True(y , T θ). Consequently, by Vδ-correctness,
Tθ(x)holds. Finally, it follows that T(x)holds by our claim (). This concludes the proof of
δ-correctness and the whole proof.
Corollary 37. For every δ, δ0, if CT[EA]`T[Vδ]T[Vδ0], then CT0`T[δ]T[δ0].
The above corollary yields a versatile tool for studying the structure of hSchPA ,CTi, where
CTis defined by: τ1CTτ2iff CT[EA]`T[τ1]T[τ2]. We show the crucial application:
Theorem 38. hSchPA,CTiis a countably universal partial order.
The above theorem reduces immediately to the one below.10 This is thanks to the results of
[11, Corollay 2.1], where a particular countably universal partial order is defined. It is clear from
the presentation that the order hW,Wiis decidable and provably a partial order in PA.
Theorem 39. Suppose that is a decidable partial order on ωsuch that PA proves that is a partial order.
Then there is an embedding hω, i → hSchPA,CTi.
Proof. Suppose that satisfies the assumptions. Firstly, we build a family of consistent theories
{σn}nωsuch that the following hold for all m, n ω:
1. If mn, then PA `σnσm.
2. CT00Conσm.
3. If mn, then CT0+Conσm0Conσn.
As shown in [19, Section 2.3, Theorem 11], there is a Π1-formula π(x)that is flexible over REF(PA),
i.e. for every Π1-formula θ(x), the following theory is consistent:
REF(PA) + xπ(x)θ(x).
For each nωlet σnbe the natural Σ1-definition of the following set of sentences11:
PA +{π(k)|nk}.
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 mn
and CT0`ConσmConσn. Let θ(x) := mx. By flexibility there exists model Msuch that
M |=REF(PA) + xπ(x)θ(x).
10We are grateful to Fedor Pakhomov for pointing our this more general result.
11Observe that since need not be elementary; also σnneed not be elementary either. However, σis not our final
axiomatization.
15
By the choice of Mit follows that M |=¬π(n). In particular, M |=PrPA(¬π(n)) and M |=
¬Conσn. However, since M |=REF(PA), as viewed in M,PA is consistent with Π1-truth (of M).
Consequently, since M |=xmxπ(x), it follows that M |=Conσm. Hence M |=Conσnas
well, which contradicts our previous conclusions.
We are ready to construct the promised embedding. Fix the family {σn}nωas above and
for each mωchoose δmto be the natural elementary definition of the following set of
sentences:
PA +{Conσm(n)|nω},
where Conσm(n)asserts that there is no proof of contradiction of σmwith Gödel code n. Since
for every mω,σmis consistent, δmis really an axiomatization of PA, hence VδmSchPA . We
check that the map
m7→ Vδm
is an embedding of hω, i into hSchPA,CTi.Fix m, n ωand assume mn. Then clearly
PA ` ∀xConσm(x)Conσn(x). Consequently, applying Proposition 35 to ϕ(x) := Conσm(x)
and ψ(x) := Conσn(x), we obtain:
CT[PA]`T[Vδm]T[Vδn].
Suppose now mnand aiming at a contradiction, assume that CT[PA]`T[Vδm]T[Vδn].
Then, by Corollary 37, CT0`T[δm]T[δn]. However, since CT0`T[δPA],CT0`T[δi]
Conσifor every iω. Hence CT0`ConσmConσn, which is impossible by our previous
considerations, since mn.
4 Prudently correct axiomatizations
Recall (from Definition 17) that 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 CTJδK
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 CTJδKfor δare never too strong.
Proposition 40. For every δ,CT0`CTJδK.
Proof. This follows immediately from Theorem 6 that CT0` ∀ϕPrPA(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 41. For any r.e. theory T ⊆ LPA such that CT0` T there exists a δsuch that Tand
CTJδKhave the same arithmetical theorems.
Proposition 40 and Theorem 41 when put together, yield the following charactertization of
arithmetical theories provable in REF(PA).
Corollary 42. For every arithmetical recursively enumerable theory Textending PA the following are
equivalent:
1. REF(PA)` T
2. There exists a δsuch that Tand CTJδKcoincide on arithmetical theorems.
To prove Theorem 41 we need to arrange δsuch that:
16
δ.
CTJδKdoes not overgenerate, i.e. its arithmetical consequences do not transcend those of
T.
To satisfy the first condition we recall that by (Cieśliński’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.
Definition 43. For two sentences θ,ϕ,σϕ[θ]abbreviates the sentence
Pr(pθq)∧ ¬θϕ.
The map hϕ, θi 7→ σϕ[θ]is clearly elementary and we shall identify it with its elementary defini-
tion.
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 TMsuch that (M, T )|=CT[PA]and the sentence
06= 0 (0 6= 0 (. . . 06= 0) . . .)
|{z }
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 [2].
Theorem 44. 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, if M |=PA, then (N, T )can be taken to be a model of CTJδPAK.
The above theorem provides us with a new method of finding finite conservative axiomatiza-
tions of arithmetical theories extending EA.
Definition 45. Given an arithmetical sentence ϕ, the pleonastic disjunction of ϕis the sentence
ϕ(ϕ(. . . ϕ). . .)
| {z }
pϕqtimes
.
The pleonastic disjunction of ϕwill be denoted with Wϕ.
Note that the above definition formalizes smoothly in EA (in which case ϕis identified with
pϕqand treated both as a number and as a formula) and that in a nonstandard model of this
theory Wϕhas standard length if and only if ϕis (coded by) a standard number.
Proposition 46. 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).
Proof. Let ϕ0(x)formalize an elementary axiomatization of T(which exists by Craig’s trick). De-
fine
ϕ(x) := ψ < xϕ0(ψ)x=_ψ.
That is to say that xsatisfies ϕif it is a pleonastic disjunction of a formula from an elementary
axiomatization of T. Observe first that CTJϕK` T . Indeed, it is sufficient to show that for every
sentence ψwe have:
CTJϕK`ϕ0(ψ)T(ψ).
17
However, over EA,ϕ0(ψ)implies ϕ(Wψ), which in turn, over CTJϕKimplies T(Wψ). However,
over pure CT[EA]the last sentence implies T(ψ)by compositional conditions, since Wψis a
disjunction of length pψqand hence is standard.
We show conservativity: pick any model M |=T. By Theorem 44 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 )|=CTJϕKbecause if N |=ϕ(a)then there exists ψsuch that
N |=ϕ0(ψ)a=Wψ. We now distinguish two cases:
1. ψis a standard sentence. In this case Wψis standard and N |=Wψ, by elementarity. Con-
sequently (N, T )|=T(Wψ)by compositional clauses; or
2. ψis not a standard sentence. In this case Wψis a disjunction of nonstandard length, hence
is made true in (N, T ).
We shall recycle the above conservativity argument in the proof of Theorem 41, which we now
turn to.
Proof of Theorem 41. Fix Tsuch that
CT0` T .
Let be an arbitrary elementary axiomatization of T. Let σϕ[θ]denote the map from Definition
43. Observe that by cut-elimination being formalizable in PA
IΣ1` ∀θPrPA(¬(Pr(pθq)∧ ¬θ)).
Hence δ(x), where δis defined as follows:
δ(x) := δPA(x)∨ ∃θ, ϕ < x(ϕ)x=σWϕ[θ].
Observe that δnaturally defines the following set of sentences:
Q∪ {Ind(ϕ)|ϕ∈ LPA} ∪ nPr(pθq)∧ ¬θ_ϕ|θ∈ LPA, ϕ ∈ T o.
We argue first that CTJδKis conservative over T. To see this, fix an arbitrary model M |=T
and let (N, T )|=CTJδPAKbe a model from Theorem 44. Then (N, T )|=CTJδKsince, reasoning
by cases as in the proof of Proposition 46, for every ϕNsuch that N |=(ϕ)we have:
(N, T )|=T_ϕ.
Now, we argue that CTJδK` T . Let ϕbe an arbitrary –axiom of T. We claim:
CTJδK`ϕ.
To see why the last claim holds, reason in CTJδK. We have:
θ T (σWϕ[θ]).
By the axioms of CTthe above is equivalent to:
θPr(pθq)∧ ¬T(θ)T_ϕ.()
Now we reason by cases: either θPr(pθq)T(θ)or not. If the latter holds, we have T(Wϕ)
by Modus Ponens applied to (). Hence ϕholds by compositional conditions, because Wϕis a
disjunction of standard length and ϕis a standard sentence. If the former holds, we have CT0by
Theorem 6 and ϕholds, because we assumed that CT0` T .
18
We conclude this subsection with complexity results that complement Theorem 26.
Proposition 47. The set is Π2-complete.
Proof. Clearly is Π2-definable. Consider the map fthat takes a scheme template τas input
and outputs the formula δτ(x)that expresses "xis an instance of τ". fis clearly recursive (indeed
elementary) and satisfies:
τSchPA iff δτ.
Therefore SchPA is many-one reducible to , which in light of the Π2-completeness of SchPA
(established in Theorem 26), completes the verification of Π2-completeness of .
Proposition 48. The sets and are both Σ1-complete.
Proof. Straightforward and left to the reader.
Theorem 49. The set {δ|T[δ]Cons}is Π2-complete.
In what follows, Π2-REF(PA)denotes the extension of EA with all sentences of the form
xPrPA(pϕ(x)q)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 [21]:
Lemma 50. PA +¬Π2-REF(PA)is Π2-sound.
Proof of Theorem 49. Fix a Π2-sentence π:= (x), where ϕ(x)is Σ1. Let δπbe the formula in
that describes the union of (the canonical axiomatization of) PA with the following set of sen-
tences: nPr(pχq)∧ ¬χ_ϕ(n)|χ∈ LPA, n ωo.
The function π7→ δπis clearly recursive, and δπ. Let θ(x) := Π2-REF(PA)ϕ(x)and observe
that for every n,CTJδπK`θ(n). Indeed, work in CTJδπKand assume ¬Π2-REF(PA). Then
clearly ¬CT0and consequently, as in the proof of Theorem 41 we get T(Wϕ(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:
πTrueN
Π2CTJδπKis conservative over PA.
Assume first that πTrueN
Π2and π=x ϕ(x), for some ϕ(x)Σ1. In particular ϕ(n)is a true Σ1
sentence for every nω, hence:
PA `ϕ(n)for every nω.
As usual, fix any model M |=PA and take its elementary extension (N, T )|=CTJδPAKin which
every disjunction of nonstandard length is true. As previously, it follows that (N, T )|=CTJδπK.
Conversely, assume that CTJδπKis conservative over PA. Then for every nω,PA `θ(n).
In particular, for every nω,PA +¬Π2-REF(PA)`ϕ(n). By the soundness of this theory we
conclude that πis true.
19
4.2 Structure of prudent axiomatizations
Theorem 41 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∈ LPA 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:
Observation 51. CT0/PA with the operations inherited from the Lindenbaum algebra of PA is a
lattice with a greatest but not a least element. Moreover the greatest element has no immediate
predecessors.
The following is an easy corollary to Theorem 41.
Proposition 52. There exists a lattice embedding CT0/PA → h,CTi.
Proof. To each [ϕ]PA we assign δϕas in the proof of Theorem 41 (note that we shall locally
change our previous convention used in Proposition 35. (x)is now simply x=pϕqand, by
compositional axioms, we have
CT[EA]` ∀θT(σWϕ[θ]) T(σϕ[θ]).
Consequently, δϕcan be taken to axiomatize the (natural definition of the) following set of sen-
tences
PA ∪ {(Pr(pθq)∧ ¬θ)ϕ|θ∈ LPA}.
Observe first that, over CTJδPA K,CTJδϕKis equivalent to ϕ. Indeed, working in CTJδPAKas-
sume first that ϕholds. Then for every θwe have:
T(σϕ[θ]),
since T(σϕ[θ]) is equivalent to an implication with a true conclusion. Hence every sentence satis-
fying δϕis true. For the converse implication, working over CTJδPAK, assume CTJδϕK. Consider
cases:
CT0holds. Then ϕholds, by assumption.
CT0fails. Then, as in the proof of Theorem 41, ϕholds.
Consequently, since CT[PA]is conservative over PA, and both CTJδϕKand CTJδψKprove
CTJδPAK, the following are equivalent for arbitrary arithmetical formulae ϕ, ψ that are provable
in CT0:
PA `ϕψ.
CT[PA]`T[δϕ]T[δψ].
Moreover, let us observe that for ϕ,ψas above, CTJδϕψKis equivalent to CTJδϕKCTJδψK
and the same with .Consequently the map
ϕ7→ CTJδϕK
is a lattice embedding.
The next proposition is slightly 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.
20
Proposition 53. There is an embedding ι:CT0/PA → h,CTithat is cofinal in the region be-
low (i.e., the nonconservative side of) the Tarski Boundary. More precisely, for every α∈ LTsuch that
CT[PA] + αis non-conservative over PA, there is an aCT0/PA such that T[ι(a)] is strictly above α
(i.e. is logically weaker) and CT[PA] + T[ι(a)] is non-conservative over PA.
Proof. The embedding ιis defined as in the proof of the previous proposition with the only ex-
ception that we do not add PA to δψ. More concretely, if [ϕ]PA CT0/PA, then we put ι([ϕ]PA)to
be the natural elementary definition of the following set of sentences
{Pr(pθq)∧ ¬θϕ|θ∈ LPA}.
Denote the canonical elementary definition of this set with δϕ. As in the proof of the previous
proposition, we obtain that for every [ϕ]PA CT0/PA, provably in CT[PA],ϕis equivalent to
T[δϕ]. Consequently, ιis a lattice embedding. Now we claim that ιis cofinal with the Tarski
Boundary in the sense explained. Pick any α∈ LTsuch that CT[PA] + αis non-conservative
over PA (but consistent). By definition, CT[PA] + α`ϕfor some PA - unprovable sentence
ϕ∈ LPA. Then, since the Lindenbaum algebra of PA is atomless there is a sentence ψ∈ LPA, which
is logically strictly weaker than ϕ. Then there is a sentence θsuch that [θ]PA CT0/PA and ψθ
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 [ψθ]PA CT0/PA
is not the greatest element. Consequently, T[ι(ψθ)] = T[δψθ]is below the Tarski Boundary.
However, since ψdoes not prove ϕ(over PA), a fortiori ψθdoes not prove ϕ. Hence CTJδψθK
does not prove CT[PA] + α. Additionally, CT[PA] + α`CTJδψθK, since ψθfollows from
α.
Proposition 54. There are recursive infinite antichains in h,CTi.
Proof. We shall make use of a Π1-formula that is PA-independent, i.e., for every binary sequence
sof length nωthe following sentence is unprovable in PA:
π(0)s(0) π(1)s(1) . . . π(n1)s(n1) ,
where for any formula ϕ,ϕ0:= ϕand ϕ1:= ¬ϕ. We will use the construction of such a Π1-
formula described in [19, Theorem 9, Chapter 2]. Let π(x)be such a formula. Assuming that
each π(k)is provable in CT0,{π(k)}kωis an infinite antichain in CT0/PA. By Proposition 52 this
implies that {δπ(k)}kωis an infinite antichain in . These considerations show that it suffices to
verify:
CT0`π(k)for each kω. ()
The verification of ()is a straightforward formalization of the reasoning in [19, Theorem 9, Chap-
ter 2], so it is delegated to the Appendix.
Proposition 55. There is an embedding (Q, <)→ h,CTi.
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 an atomless
boolean algebra.
Proposition 56. There are δ, δ0such that CTJδKand CTJδ0Kare non-conservative extensions of
PA, but CTJδKCTJδ0Kis a conservative extension of PA.
Proof. Consider ϕ:= ConPA+¬ConPA and ψ:= ConPA ConPA+ConPA. Both ϕand ψgenerate
different non-zero elements in CT0/PA but it is easy to see that
PA `ϕψ.
Hence the desired δ, δ0can be chosen as δ:= δϕand δ0:= δψ(defined as in the proof of
Proposition 52).
21
5 Coda: The arithmetical reach of CTJδKfor δ
Recall from Definition 17 that is the collection of elementary presentations of PA, i.e., elemen-
tary formulae that define (in N) a theory that is deductively equivalent to PA. We are now in a
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 CTJδK, where δ.
Theorem 57. sup PA is deductively equivalent to TrueN
Π2+REF(PA).
Proof. First note that REF(PA)sup PA is an immediate corollary to Theorem 41. Also, the
proof of TrueN
Π2sup PA is morally contained in the proof of Theorem 26: for every true Π2-
sentence π:= x(x, y), the theory
PA ∪ {∃(n, y)|nω}
is deductively equivalent to PA, hence the natural arithmetical definition of the above set witnesses
that sup PA `π. To prove the converse inclusion12 , assume that for some δ,CTJδK`ϕ. Let
πbe the true Π2-sentence
xPrδ(x)PrPA(x),
expressing that every theorem of δis provable already in PA. Then it is easy to observe that
CT[PA] + π+GRP(PA)`ϕ,
where GRP(PA)is the global reflection for PA. However, by any of the proofs of Theorem 5, the
theory CT[PA] + π+GRP(PA)is arithmetically conservative over EA +REF(PA) + π.13 Hence
EA +REF(PA) + π`ϕ. Since EA +πis a true Π2-sentence the proof is complete.
6 Open problems
(I) Are the lattices hSchPA,CTiand h,CTidense? Does h,CTihave maximal or min-
imal elements? Does hSchPA,CTihave minimal elements (by the proof of Theorem 30 no
CT-maximal element exists)?
(II) Are the lattices hSchPA,CTiand h,CTiuniversal for countable distributive lattices?14
(III) How do hSchPA,CTiand h,CTifit in the Lindenbaum algebra of CT[EA]?
(IV) Is the Lindenbaum algebra of Cons dense?
(V) Do hSchPA,CTiand h,CTihave decidable copies? If not, how undeciable are they?
(VI) How close can we get to the Tarski Boundary from below using theories CTJδK, where δ?
In other words, if CT[PA] + αis nonconservative over PA, is there some δsuch that
CTJδKis nonconservative over PA, and CT[PA] + α`T[δ]?
(VII) How close can we get to the Tarski Boundary from above using theories CTJδK, where δ?
In other words, if CT[PA] + αis conservative over PA, is there some δsuch that CTJδK
is conservative over PA, and CT[PA] + T[δ]`α?
(VIII) Do the answers to Questions (VI) and (VII) change if CTJδKis required to be a subtheory
of CT0?
12This proof is due to Fedor Pakhomov and appears here with his kind permission.
13The crucial lemma in all the known proofs states that for every model M |=REF(PA)there is a model Nwhich is
elementarily equivalent to Mand TNsuch that (N, T )|=CT[PA] + GRP(PA).
14This question was communicated to us by Fedor Pakhomov.
22
7 Appendix
Verification of ()of the proof of Proposition 54. To lighten the notation, we will identify numerals
with their denotations, and formulae with their codes. We wish to show that if π(x)is the Π1-
formula π(x)constructed in [19, Theorem 9, Chapter 2], then for every kω,CT0`π(k). Let
us revisit the construction of π(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(n1)s(n1) .
For a unary formula ϕ, let (x, i, ϕ, p)express:
there is a binary sequence sof length x+ 1 such that s(x) = ian pis a proof in PA of ¬ϕs.
Finally, let π(x)be a formula assured to exist by the diagonal lemma such that the following is
provable in PA:
π(x)↔ ∀p(x, 1, π, p)→ ∃qp (x, 0, π, q ).
By metainduction on nω, we show that for every nω,CT0`len(s) = n+ 1 → ¬PrPA(¬πs).
Observe that this implies that for every nω,π(n)is provable in CT0. We first show that π(0) is
provable in CT0. Working in CT0, assume that ¬π(0) holds. It follows that for some p,(0,1, π, p)
holds, hence in particular, PrPA (π(0)) holds. However, in CT0the theorems of PA are true, so π(0)
holds, contrary to the assumption. Hence CT0` ¬PrPA(¬π(0)). Moreover, since π(0) holds, for
every PA-proof of π(0) there exists a smaller PA-proof of ¬π(0). Consequently, since CT0proves
the consistency of PA, for n= 0,CT0` ∀slen(s) = n+ 1 → ¬PrPA(¬πs).
Now, assume n=k+ 1,CT0` ∀slen(s) = n→ ¬PrPA(¬πs). Working in CT0assume for
some sof length n+1,PrPA(¬πs).Fix ssuch that the proof of πsin PA is 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 πs, we have PrPA (πsn→ ¬π(n)). Moreover, both
(n, 0, π, p)and qp¬(n, 1, π, q )hold. Since is a 0-formula, we have:
PrPA(n, 0, π, p)∧ ∀qp¬(n, 1, π, q ).
In particular, PrPA (π(n)). Hence PrPA(¬πsn), which is impossible by the induction step,
since snhas length n.
2. s(n)=1. Then, as before, PrPA(πsnπ(n)).Moreover, by minimality of p, we have
(n, 1, π, p)and q < p ¬(n, 0, π, q). Hence, as before we obtain PrPA(¬π(n)), which con-
tradicts the induction assumption.
This concludes the proof of the induction step and the whole proof.
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This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth.