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Axiomatizations of Peano Arithmetic:

A truth-theoretic view

Ali Enayat∗Mateusz 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 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 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 CT−JδKfor δ∈∆∗22

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 I∆0+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)|n∈ω},

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 p→qis

translated to p≤q.

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 τ∈SchPA 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,≤CT−iis 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 ≤CT−is deﬁned by:

τ1≤CT−τ2iﬀ CT−[EA]`T[τ1]→T[τ2].

In Section 4.2, we prove similar results about the partial ordering h∆,≤CT−i, 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∆,≤CT−i,

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 I∆0+Exp of PA, where I∆0is 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

a∆0-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 B⊇EA. 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 x∈ClTermLPA is the arithmetical formula that expresses "xis (the code of)

a closed term of LPA"; x∈SentLPA is the arithmetical formula that expresses "xis (the code of)

a sentence of LPA", x∈Var expresses "xis (the code of) a variable", and x∈Form≤1

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.

CT1∀s, t ∈ClTermLPA T(s=t)↔s◦=t◦.

CT2∀ϕ, ψ ∈SentLPA T(ϕ∨ψ)↔T(ϕ)∨T(ψ).

CT3∀ϕ∈SentLPA T(¬ϕ)↔ ¬T(ϕ).

CT4∀ϕ(v)∈Form≤1

LPA T(∃vϕ)↔ ∃x T (ϕ[x/v]).

CT5∀ϕ(¯v)∈FormLPA ∀¯s, ¯

t∈ClTermSeqLPA (¯

s◦=¯

t◦→T(ϕ[¯s/¯v]) ↔T(ϕ[¯

t/¯v])) .

In CT5above sand tdenote ﬁnite tuples of terms; and ¯

s◦,¯

t◦refer 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 CT−provided 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|n∈ω}. 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:

∀ϕ∈SentLTPrT(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<ω(PA).

In the above REF0(T) := T, REFn+1(T) := REF(REFn(T)), REF<ω(T) := Sn∈ωREFn(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. ∀c”ccodes a set of sentences”∧T(Wϕ∈cϕ)→ ∃ϕ∈c 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

CT−for 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)∈Form≤1

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 n∈ωthere 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 n∈ωthe 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)|n∈ω}, 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

n∈ωlet 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∈ω}.

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

n∈ω,PA proves ConZF(n). On the other hand, ZF "knows" that PA holds in the standard model

of arithmetic, so for all n∈ω,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 ↔τ∈Sch−

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,· · ·, xn−1there is an object

ysuch that for all objects t,t∈yiﬀ (t=x0or ... or t=xn−1)” holds for each n∈ω(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(δ≤pt 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 δ∼pt 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

δ∼p.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:= {n∈ω|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

CT−JϕK:= CT−[EA] + T[ϕ].

•We use ∆to denote the subset of ∆∗consisting of formulae δ∈∆∗such 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∈∆∗|δ≤p.t. PA}.

On ∆−and ∆we shall consider the relation ≤CT−given by

δ≤CT−δ0⇐⇒ CT−[EA]`T[δ]→T[δ0].

Convention 18. Simplifying things a little bit, when talking about the structures h∆,≤CT−iand

h∆−,≤CT−i, we shall assume that ∆is replaced by the quotient set ∆/∼, where ∼is the least

equivalence relation that makes ≤CT−antisymmetric, to wit:

8

δ∼δ0iﬀ δ≤CT−δ0and δ0≤CT−δ.

•Let us stress an important diﬀerence between CT−[PA]and CT−JδPAK: 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∆,≤CT−iand h∆−,≤CT−iare 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 δ⊕δ0and δ⊗δ0of ∆such that over CT−[PA]we have:

T[δ]∧T[δ0]↔T[δ⊕δ0].(1)

T[δ]∨T[δ0]↔T[δ⊗δ0].(2)

It can be readily seen that if we deﬁne:

δ⊕δ0(x) := δ(x)∨δ0(x),

then δ⊕δ0∈∆and (1) is satisﬁed. For (2) it is suﬃcient to deﬁne:

δ⊗δ0(x) := ∃y, z < xδ(y)∧δ0(z)∧x=y∨z,

where x=y∨zexpresses that xis a disjunction of yand z. To see that (2) holds and δPA ≤p.t. δ⊗δ0

one simply applies reasoning by cases; the proof of δ⊗δ0≤p.t. δPA is trivial.

Remark 20. If δ∈∆corresponds 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 CT−JδK

is a conservative extension of PA by Theorem 26. In contrast, even for very natural δ∈∆,CT−JδK

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

CT−JδK` ∀ϕPrEA(ϕ)→T(ϕ).

However, by a theorem of Cieśliński [3], over CT−[EA], the above consequence of CT−JδKimplies

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 CT−Jδ0K

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 := {τ∈Sch−

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 ≤CT−on Sch−

PA as follows:

τ≤CT−τ0⇐⇒ CT−`T[τ]→T[τ0].

When talking about the structural properties of hSchPA,≤CT−iwe 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. hSch−

PA,≤CT−iand hSch,≤CT−iare 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 0∈SchPA:

T[τ]∧T[τ0]↔T[τ⊕τ0](3)

T[τ]∨T[τ0]↔T[τ⊗τ0](4)

The case of ⊕is trivial. We put:

τ⊕τ0:= τ∧τ0.

The case of ⊗is (a little bit) harder. We put:

τ⊗τ0:= τ∨(τ0[Q/P ]),

where Qis a fresh unary predicate. As remarked earlier (compare footnote 4) thanks to the coding

apparatus, τ⊗τ0can be expressed as a scheme with a single unary predicate P. Then we obtain

CT−[EA]`T[τ⊗τ0]≡ ∀φ∀ψ 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

τ∈Sch−

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:

A≤mBiﬀ there is a total recursive function fsuch that: ∀n∈ω(n∈A⇔f(n)∈B).

The proof will be complete once we demonstrate the following four assertions:

(i)TrueN

Π2≤mSch−

PA.8

(ii)Sch−

PA ≤mSchPA .

(iii)Sch−

PA ≤mSchT

PA.

(iv)TrueN

Π2≤mCons. To prove (i), suppose π=∀x∃y δ(x, y)is a Π2-statement, where δ(x, y)is

∆0. We ﬁrst observe that by Σ1-completeness of PA:

(∗)π∈TrueN

Π2iﬀ ∀n∈ωPA ` ∃y δ(n, y).

On the other hand, R={∃y δ(n, y ) : n∈ω}is a recursive set of sentences, so by Theorem 12 there

is τsuch that τ∈Sch−

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:

π∈TrueN

Π2iﬀ f(g(π)) ∈Sch−

PA.

The proof of (ii)is based on the observation that τ∈Sch−

PA iﬀ h(τ)∈SchPA , where h(τ) := τ∧τPA,

and τPA is deﬁned as follows:

τPA := Q∧[P(0) ∧ ∀x(P(x)→P(S(x))) → ∀xP(x)].

To verify (iii), we claim that τ∈Sch−

PA iﬀ (τ∧τPA )∈SchT

PA. The implication τ∈Sch−

PA ⇒

(τ∧τPA)∈SchT

PA follows directly from Theorem 3 (since PA proves Sτ∧τPA if τ∈Sch−

PA).On the

other hand, if (τ∧τPA)∈SchT

PA, then by the deﬁnition of SchT

PA,PA proves Sτ, so τ∈Sch−

PA.

Finally, to establish (iv)suppose π=∀x∃y 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:

π∈TrueN

Π2⇐⇒ f(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 τ∈Sch−

PA

iﬀ ϕτ∈Cons (the direction ⇒follows from Theorem 8; and the direction ⇐follows from the

relevant deﬁnitions). Therefore:

π∈TrueN

Π2⇐⇒ h(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(ϕ∨P(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),T∗θ(x)will abbreviate the formula T(θ[x/v]). Hence T∗θ(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 CT−JτKis 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 τ∈SchPA,ψis interpretable in CT−JτK, 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=∃xϕ0(x),

where ϕ0(x)∈∆0such that the following equivalence is provable in CT−[PA]:

ϕ↔CT−JV(∀z≤y¬ϕ0(z),δPA)K ψ.

Similarly to the Pakhomov-Visser argument, we argue that ϕis false. Suppose not and take the

least n∈ωsuch that ϕ0(n)holds. Then, in Q,∀z≤x¬ϕ0(z)is equivalent to x < n, hence the

following is provable in CT−[PA]:

∀θ(x)TV(∀z≤y¬ϕ0(z),δPA)[θ]↔TV(y<n,δPA)[θ].

We claim that:

CT−[PA]` ∀θ(x)TV(y<n,δPA)[θ].(∗)

Indeed, working in CT−[PA]ﬁx θ∈Form≤1

LPA. By compositional conditions TV(y<n,δPA)[θ]is

equivalent to:

^

i<nT∗True(i, θ)→ ∀x(δPA(x)∧pdepth(x)≤i)→T∗θ(x).

12

However, once again by compositional conditions imposed on T,T∗True(i, θ)is equivalent to:

True(i, T ∗θ(x)), hence to the assertion that T∗θ(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∗θ(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(∀z≤y¬ϕ0(z),δPA)[P]is a scheme template, such that the scheme associated with

it axiomatizes PA. Moreover, CT−[PA] + TV(∀z≤y¬ϕ0(z),δPA)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 τ∈SchPA

such that CT−JτKdoes 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 τ∈SchPA such that CT−JτKdoes not interpret Q+ConPA.

Corollary 33. For every scheme template τ∈SchPA there is a scheme template τ0∈SchPA such that

CT−JτKinterprets CT−Jτ0K, but not vice versa.

Proof. Fix τand apply Corollary 31 to ψ:= CT−JτK. 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 CT−Jτ00Kdoes not relatively interpret CT−JτK. Now it is suﬃcient to take

τ0:= τ⊗τ00, 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 δ⊆δ0as 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 δ⊆δ0from 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 CT−JτK, being a ﬁnite theory, would be interpretable in a ﬁnite fragment of PA, call it T.

But then, since CT−JτKextends PA and PA is reﬂexive, CT−JτK`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 T∗θ0(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 T∗θ(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:

∀x∀θ∈Form≤1

LPA∀ϕ∈SentLPA True(x, T ∗θ)∧depth(ϕ)≤x→T∗θ(ϕ)↔T(ϕ).(∗)

Fix an arbitrary (M, T )|=CT0. For an arbitrary c∈M, 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 θ∈Form≤1

LPA. Assume that the depth of θis b

and let c= max{a, b}. Assume True(a, T ∗θ), i.e. the formula T∗θis 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∗θ)holds. Moreover, it is suﬃcient to show that

∀xTc∗θ(x)↔Tc(x).

In other words, it is suﬃcient to prove that

(M, Tc)|=∀xT∗θ(x)↔T(x)

The above can be demonstrated by a routine induction on the build-up of formulae. More pre-

cisely, let

Ξ(y) := ∀ϕ∈depth(y)T∗θ(ϕ)↔T(ϕ).

Then Ξ(0) and ∀x < aΞ(x)→Ξ(x+ 1)hold (in (M, Tc)), because both Tc∗θand 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)|=∀y≤aΞ(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)→T∗θ(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,

T∗θ(ϕ)is equivalent to T(ϕ). Hence T∗θ(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,

T∗θ(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 ,≤CT−i, where

≤CT−is deﬁned by: τ1≤CT−τ2iﬀ CT−[EA]`T[τ1]→T[τ2]. We show the crucial application:

Theorem 38. hSchPA,≤CT−iis 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,≤CT−i.

Proof. Suppose that satisﬁes the assumptions. Firstly, we build a family of consistent theories

{σn}n∈ωsuch that the following hold for all m, n ∈ω:

1. If mn, then PA `σn⊆σm.

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<ω(PA),

i.e. for every Π1-formula θ(x), the following theory is consistent:

REF<ω(PA) + ∀xπ(x)↔θ(x).

For each n∈ωlet 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`Conσm→Conσn. Let θ(x) := mx. By ﬂexibility there exists model Msuch that

M |=REF<ω(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 |=∀xmx→π(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}n∈ωas above and

for each m∈ωchoose δm∈∆∗to be the natural elementary deﬁnition of the following set of

sentences:

PA +{Conσm(n)|n∈ω},

where Conσm(n)asserts that there is no proof of contradiction of σmwith Gödel code ≤n. Since

for every m∈ω,σmis consistent, δmis really an axiomatization of PA, hence Vδm∈SchPA . We

check that the map

m7→ Vδm

is an embedding of hω, i into hSchPA,≤CT−i.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 i∈ω. Hence CT0`Conσm→Conσn, 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 CT−JδK

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−JδKfor δ∈∆are never too strong.

Proposition 40. For every δ∈∆,CT0`CT−JδK.

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 δ∈∆such that Tand

CT−JδKhave the same arithmetical theorems.

Proposition 40 and Theorem 41 when put together, yield the following charactertization of

arithmetical theories provable in REF<ω(PA).

Corollary 42. For every arithmetical recursively enumerable theory Textending PA the following are

equivalent:

1. REF<ω(PA)` T

2. There exists a δ∈∆such that Tand CT−JδKcoincide on arithmetical theorems.

To prove Theorem 41 we need to arrange δsuch that:

16

•δ∈∆.

•CT−JδKdoes 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)∧ ¬θ→ϕ.

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 T⊆Msuch 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 CT−JδPAK.

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 CT−JϕK` T . Indeed, it is suﬃcient to show that for every

sentence ψwe have:

CT−JϕK`ϕ0(ψ)→T(ψ).

17

However, over EA,ϕ0(ψ)implies ϕ(Wψ), which in turn, over CT−JϕKimplies 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 )|=CT−JϕKbecause 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 CT−JδKis conservative over T. To see this, ﬁx an arbitrary model M |=T

and let (N, T )|=CT−JδPAKbe a model from Theorem 44. Then (N, T )|=CT−JδKsince, reasoning

by cases as in the proof of Proposition 46, for every ϕ∈Nsuch that N |=(ϕ)we have:

(N, T )|=T_ϕ.

Now, we argue that CT−JδK` T . Let ϕbe an arbitrary –axiom of T. We claim:

CT−JδK`ϕ.

To see why the last claim holds, reason in CT−JδK. We have:

∀θ T (σWϕ[θ]).

By the axioms of CT−the 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:

τ∈SchPA 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:= ∀xϕ(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,CT−JδπK`θ(n). Indeed, work in CT−JδπKand 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:

π∈TrueN

Π2⇐⇒ CT−JδπKis conservative over PA.

Assume ﬁrst that π∈TrueN

Π2and π=∀x ϕ(x), for some ϕ(x)∈Σ1. In particular ϕ(n)is a true Σ1

sentence for every n∈ω, hence:

PA `ϕ(n)for every n∈ω.

As usual, ﬁx any model M |=PA and take its elementary extension (N, T )|=CT−JδPAKin which

every disjunction of nonstandard length is true. As previously, it follows that (N, T )|=CT−JδπK.

Conversely, assume that CT−JδπKis conservative over PA. Then for every n∈ω,PA `θ(n).

In particular, for every n∈ω,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∆,≤CT−i.

Proof. To each [ϕ]PA we assign δϕ∈∆as 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 CT−JδPA K,CT−JδϕKis equivalent to ϕ. Indeed, working in CT−JδPAKas-

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 CT−JδPAK, assume CT−JδϕK. 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 CT−JδϕKand CT−JδψKprove

CT−JδPAK, the following are equivalent for arbitrary arithmetical formulae ϕ, ψ that are provable

in CT0:

•PA `ϕ→ψ.

•CT−[PA]`T[δϕ]→T[δψ].

Moreover, let us observe that for ϕ,ψas above, CT−Jδϕ∨ψKis equivalent to CT−JδϕK∨CT−JδψK

and the same with ∧.Consequently the map

ϕ7→ CT−JδϕK

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∆−,≤CT−ithat 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 a∈CT0/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)∧ ¬θ→ϕ|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 ψ∨θ

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 [ψ∨θ]PA ∈CT0/PA

is not the greatest element. Consequently, T[ι(ψ∨θ)] = T[δψ∨θ]is below the Tarski Boundary.

However, since ψdoes not prove ϕ(over PA), a fortiori ψ∨θdoes not prove ϕ. Hence CT−Jδψ∨θK

does not prove CT−[PA] + α. Additionally, CT−[PA] + α`CT−Jδψ∨θK, since ψ∨θfollows from

α.

Proposition 54. There are recursive inﬁnite antichains in h∆,≤CT−i.

Proof. We shall make use of a Π1-formula that is PA-independent, i.e., for every binary sequence

sof length n∈ωthe following sentence is unprovable in PA:

π(0)s(0) ∧π(1)s(1) ∧. . . ∧π(n−1)s(n−1) ,

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)}k∈ωis an inﬁnite antichain in CT0/PA. By Proposition 52 this

implies that {δπ(k)}k∈ωis an inﬁnite antichain in ∆. These considerations show that it suﬃces to

verify:

CT0`π(k)for each k∈ω. (∗)

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∆,≤CT−i.

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, δ0∈∆such that CT−JδKand CT−Jδ0Kare non-conservative extensions of

PA, but CT−JδK∨CT−Jδ0Kis 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 `ϕ∨ψ.

Hence the desired δ, δ0∈∆can be chosen as δ:= δϕand δ0:= δψ(deﬁned as in the proof of

Proposition 52).

21

5 Coda: The arithmetical reach of CT−JδKfor 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 CT−JδK, where δ∈∆∗.

Theorem 57. sup PA is deductively equivalent to TrueN

Π2+REF<ω(PA).

Proof. First note that REF<ω(PA)⊆sup PA is an immediate corollary to Theorem 41. Also, the

proof of TrueN

Π2⊆sup PA is morally contained in the proof of Theorem 26: for every true Π2-

sentence π:= ∀x∃yϕ(x, y), the theory

PA ∪ {∃yϕ(n, y)|n∈ω}

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∈∆,CT−JδK`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<ω(PA) + π.13 Hence

EA +REF<ω(PA) + π`ϕ. Since EA +πis a true Π2-sentence the proof is complete.

6 Open problems

(I) Are the lattices hSchPA,≤CT−iand h∆,≤CT−idense? Does h∆,≤CT−ihave maximal or min-

imal elements? Does hSchPA,≤CT−ihave minimal elements (by the proof of Theorem 30 no

≤CT−-maximal element exists)?

(II) Are the lattices hSchPA,≤CT−iand h∆,≤CT−iuniversal for countable distributive lattices?14

(III) How do hSchPA,≤CT−iand h∆,≤CT−iﬁt in the Lindenbaum algebra of CT−[EA]?

(IV) Is the Lindenbaum algebra of Cons dense?

(V) Do hSchPA,≤CT−iand h∆,≤CT−ihave decidable copies? If not, how undeciable are they?

(VI) How close can we get to the Tarski Boundary from below using theories CT−JδK, where δ∈∆?

In other words, if CT−[PA] + αis nonconservative over PA, is there some δ∈∆such that

CT−JδKis nonconservative over PA, and CT−[PA] + α`T[δ]?

(VII) How close can we get to the Tarski Boundary from above using theories CT−JδK, where δ∈∆?

In other words, if CT−[PA] + αis conservative over PA, is there some δ∈∆such that CT−JδK

is conservative over PA, and CT−[PA] + T[δ]`α?

(VIII) Do the answers to Questions (VI) and (VII) change if CT−JδKis 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<ω (PA)there is a model Nwhich is

elementarily equivalent to Mand T⊆Nsuch 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 k∈ω,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(n−1)s(n−1) .

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)→ ∃q≤p (x, 0, π, q ).

By metainduction on n∈ω, we show that for every n∈ω,CT0`len(s) = n+ 1 → ¬PrPA(¬πs).

Observe that this implies that for every n∈ω,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 ∀q≤p¬(n, 1, π, q )hold. Since is a ∆0-formula, we have:

PrPA(n, 0, π, p)∧ ∀q≤p¬(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(πsn→π(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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