Indiscernibles and satisfaction classes in arithmetic

We investigate the theory PAI (Peano Arithmetic with Indiscernibles). Models of PAI are of the form (M, I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M, I) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B below. Theorem A. Let M be a nonstandard model of PA of any cardinality. M has an expansion to a model of PAI iff M has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of PA: Corollary. A countable model M of PA is recursively saturated iff M has an expansion to a model of PAI. Theorem B. There is a sentence s in the language obtained by adding a unary predicate I(x) to the language of arithmetic such that given any nonstandard model M of PA of any cardinality, M has an expansion to a model of PAI + s iff M has a inductive full satisfaction class.


INTRODUCTION
We investigate an extension of PA (Peano Arithmetic), denoted PAI, which is equipped with a designated unbounded class of indiscernibles (see Section 3 for the precise definition).The motivation to study PAI arose from the study [E-2] of the set-theoretic counterpart ZFI < of PAI, where it is shown that there is an intimate relationship between ZFI < and large cardinals, thus indicating that the set-theoretical consequences of ZFI < go well beyond ZFC.
In light of the results obtained in [E-2] it is natural to investigate PAI since it is well-known [KW] that PA is bi-interpretable with the theory ZF −∞ + TC, where ZF −∞ is the system of set theory obtained from ZF by replacing the axiom of infinity by its negation, and TC is the sentence asserting that every set is contained in a transitive set (which in the presence of the other axioms implies that the transitive closure of every set exists).The aforementioned proof of the bi-interpretability of PA and ZF −∞ + TC can be readily extended to show the bi-interpretability of PAI and ZFI −∞ + TC.
Our main results are Theorems A and B of the abstract that relate PAI to the well-studied notions of (a) inductive partial satisfaction classes and (b) inductive full satisfaction classes, which are intimately connected (respectively) with the axiomatic theories of truth known as UTB and CT (see Section 2.2).After presenting preliminaries in Section 2, (refinements of) Theorems A and B are established in Section 3. In Section 4 we examine PAI through the lens of interpretability, and in Section 5 we probe the model-theoretic behavior of fragments of PAI. 1

PRELIMINARIES
In this section we present the relevant notations, conventions, definitions, and results that are needed in the subsequent sections.
2.1.Theories and models 2.1.1.Definition.The language of arithmetic, L A , is {+, •, S, <, 0}.We use the convention of writing M , M 0 , N , etc. to (respectively) denote the universes of discourse of structures M, M 0 , N , etc.In (a) through (g) below, L ⊇ L A and M and N are L-structures.
(a) Σ 0 (L) = Π 0 = ∆ 0 (L) = the collection of L-formulae all of whose quantifiers are bounded by L-terms (i.e., they are of the form ∃x ≤ t, or of the form ∀x ≤ t, where t is an L-term not involving x.More generally, Σ n+1 (L) consists of formulae of the form ∃x 0 • • • ∃x k−1 ϕ, where ϕ ∈ Π n (L); and Π n+1 (L) consists of formulae of the form ∀x 0 • • • ∀x k−1 ϕ, where ϕ ∈ Σ n (with the convention that k = 0 corresponds to an empty block of quantifiers).We shall omit the reference to L if L = L A , e.g., Σ n := Σ n (L A ). Also, we shall write Σ n (X) instead of Σ n (L A ∪{X}), where X is a new predicate symbol.We often conflate formal symbols with their denotations (if there is no risk of confusion).
(b) PA (Peano Arithmetic) is the result of adding the scheme of induction for all L A -formulae to the finitely axiomatizable theory known as (Robinson's) Q. PA(L) is the theory obtained by augmenting PA with the scheme of induction for all L-formulae.IΣ n (L) is the fragment of PA(L) with the induction scheme limited to Σ n (L)-formulae.Given a new predicate X, we write PA(X) and IΣ n (X) (respectively) instead of PA(L A ∪{X}) and IΣ n (L A ∪{X}).
(c) If ϕ(x) is an L-formula, ϕ M := {m ∈ M : M |= ϕ(x)}.For X ⊆ M , then we say that X is M-definable if X is first order definable (parameters allowed) in M.
and m ∈ Ack c is shorthand for the formula expressing "the m-th bit of the binary expansion of c is 1".
(e) A subset X of M is said to be piecewise-coded (in M) if {x ∈ M : x < m and x ∈ X} is M-finite for each m ∈ M .
(f ) M is rather classless if any piecewise-coded subset of M is already M-definable (By a theorem of Kaufmann, every extension of PA has a recursively saturated rather classless model [KS,Theorem 10.1.5]).
(g) We identify the longest well-founded initial submodel of models of PA with the ordinal ω.
The following result was established by Proposition 3.2] for models M of PA; the generalization to models of I∆ 0 + Exp appears in [EP,Lemma 4.2] (note that Exp is the axiom stating the totality of the exponential function).
2.1.2.Theorem.Let M |= I∆ 0 + Exp, and X ⊆ M. The following are equivalent : (b) S is said to be a partial satisfaction class if S satisfies Tarski's recursive conditions for a satisfaction predicate for all standard formulae.Thus a typical member of S is of the form ϕ, a , where ϕ ∈ Form M m = the set of L A -formulae in M with m free variables, where m ∈ M (note that ϕ need not be standard) and a ∈ M is an m-tuple in the sense of M.
(c) S is said to be a full satisfaction class if S satisfies Tarski's recursive conditions for a satisfaction predicate for all formulae in M.
• For better readability we will often write ϕ, a ∈ S or ϕ(a) ∈ S instead of the more official S( ϕ, a ).Also, if ϕ is a sentence (i.e., has no free variables), we will write ϕ ∈ S instead of ϕ, ∅ ∈ S (where ∅ is the empty tuple).
The theories UTB (Uniform Tarski Biconditionals) and CT (Compositional Truth) described below are well studied in the literature of axiomatic theories of truth (see, e.g., the monographs by  and Halbach [H]).
• Note that for the purposes of this paper, satisfaction and truth are interchangeable, but in general there are subtle differences between the two, see [C-2].

Definition.
In what follows T (x) is a new unary predicate, c is a new constant symbol, Form 1 is the set of (Gödel-numbers of) L A -formulae with exactly one free variable, and

•
x is the arithmetically definable function that outputs the numeral for x given the input x.
(c) CT = PA(T ) + θ, where θ is a single sentence that stipulates that T satisfies Tarski's inductive clauses for a truth predicate for arithmetical sentences.
The following proposition is well-known and easy to prove; the nontrivial direction of part (a) is the right-to-left part, which employs a routine overspill argument; part (b) follows easily from part (a) and the definitions involved.The proofs of (c) and (d) are routine but somewhat laborious.

Proposition.
The following holds for every model M of PA of any cardinality.The concepts of recursive saturation and satisfaction classes are intimately tied, as witnessed by the following classical result of Barwise and Schlipf whose proof invokes the resplendence property of countable recursively saturated models (for a proof, see Corollary 15.12 of [Ka]).Note that the right-to-left implication in the Barwise-Schlipf theorem holds for uncountable models M as well (and is proved by a simple overspill argument).However, by Tarski's undefinability of truth theorem the left-to-right direction fails for 'Kaufmann models' (i.e., recursively saturated rather classless models).
2.2.4.Theorem.(Barwise-Schlipf) A countable model M of PA is recursively saturated iff M has an inductive partial satisfaction class.
2.3.2.Definition.Given a structure M, some linear order (I, <) where I ⊆ M , we say that (I, <) is a set of order indiscernibles in M if for any L(M)-formula ϕ(x 1 , • • •, x n ), and any two n-tuples i and j from [I] n , we have: 2.3.3.Definition.Suppose M has parameter-free definable Skolem functions, and (I, < I ) is a set of order indiscernibles in M, and I 0 ⊆ I.We use the notation M I0 to denote the elementary submodel of M generated by I 0 (via the parameter-free definable functions of M).
• Note that the universe M I0 of M I0 consists of the elements of M that are pointwise definable in (M, i) i∈I0 .
2.4.Interpretability 2.4.1.Definition.Suppose U and V are first order theories, and for the sake of notational simplicity, let us assume that U and V are theories that support a definable pairing function.We use L U and L V to respectively designate the languages of U and V .
(a) An interpretation I of U in V , written: is given by a translation τ of each L U -formula ϕ into an L V -formula ϕ τ with the requirement that V ⊢ ϕ τ for each ϕ ∈ U , where τ is determined by an L V -formula δ(x) (referred to as a domain formula), and a mapping P → τ A P that translates each n-ary L U -predicate P into some n-ary L V -formula A P .The translation is then lifted to the full first order language in the obvious way by making it commute with propositional connectives, and subject to the following clauses: (∀xϕ) τ = ∀x(δ(x) → ϕ τ ) and (∃xϕ) τ = ∃x(δ(x) ∧ ϕ τ ).
Note that each interpretation U I V gives rise to an inner model construction that uniformly builds a model I(M) |= U for any M |= V .
(b) U is interpretable in V (equivalently: V interprets U ), written U V , iff U I V for some interpretation I.
(c) Given arithmetical theories U and V , U is ω-interpretable in V if the interpretation of 'numbers' and the arithmetical operations of the interpreted theory U are the same as those of the interpreting theory V .
(e) U and V are mutually interpretable when U V and V U.
(f ) U is a retract of V iff there are interpretations I and J with U I V and V J U , and a binary U -formula F such that F is, U -verifiably, an isomorphism between id U (the identity interpretation on U ) and J • I.In model-theoretic terms, this translates to the requirement that the following holds for every M |= U : (g) U and V are bi-interpretable iff there are interpretations I and J as above that witness that U is a retract of V , and additionally, there is a V -formula G, such that G is, V -verifiably, an isomorphism between the ambient model of V and the model of V given by I • J .In particular, if U and V are bi-interpretable, then given M |= U and N |= V , we have (h) The above notions can also be localized at a pair of models; in particular suppose N is an L U -structure and M is an L V -structure.For example, we say that N is interpretable in M, written N M (equivalently: M N ) iff the universe of discourse of N , as well as all the N -interpretations of L U -predicates are M-definable.Similarly, we say that M and N are bi-interpretable if there are parametric interpretations I and J , together with an M-definable F and an N -definable map G such that: • Recall that a theory U (with sufficient coding apparatus) is reflexive if the formal consistency of each finite fragment of U is provable in U .

THE BASICS OF PAI
3.1.Definition.PAI is the theory formulated in L A (I) whose axioms are (1) through (3) below.Note that we write x ∈ I instead of I(x) for better readability.
(3) The scheme Indis LA (I) = {Indis ϕ (I) : ϕ is an L A -formula} stipulating that I forms a class of order indiscernibles for the ambient model of arithmetic.More explicitly, for each n-ary formula is the following sentence: PAI • is the weakening of PAI in which the scheme Indis LA (I) is weakened to the scheme Indis is the following sentence:

Proposition.
Let N be the standard model of PA.Proof.(a) is an immediate consequence of the fact that the standard model of PA is pointwise definable, and therefore it does not even have a distinct pair of indiscernibles.
To see that (b) holds, fix some enumeration ϕ n : n ∈ ω of all arithmetical formulae, and use Ramsey's theorem to construct a sequence H n : n ∈ ω of subsets of ω such that for each n ∈ ω the following three conditions hold: (1) H n is infinite.
Then recursively define i n : n ∈ ω by: i 0 = max{min{H 0 }, ϕ 0 }, and i n+1 is the least i ∈ H n+1 that is greater than both i n and ϕ n .It is easy to see that (M, I ′ ) |= PAI • , where Since (c) readily follows from the definitions involved, the proof is complete.
• In light of part (c) of Theorem 3.2, most results in this paper about PAI has a minor variant in which PAI is replaced by PAI • .
3.3.Theorem.Each finite subtheory of PAI has an ω-interpretation in PA.Consequently: (a) PAI is a conservative extension of PA.
(b) PAI is interpretable in PA, hence PA and PAI are mutually interpretable.
(c) PAI is interpretable in ACA 0 (but not vice-versa).3 Proof.The ω-interpretability of any finite subtheory of PAI in PA is an immediate consequence of the well-known schematic provability of Ramsey's theorem ω → (ω) n 2 in PA for all metatheoretic n ≥ 2 [HP,Theorem 1.5,Chapter II].This makes it evident that (a) holds, and together with Orey's Compactness Theorem 2.4.3, yields (b).Finally, (c) follows from (b) since PA is trivially interpretable in ACA 0 .The parenthetical clause of (c) is an immediate consequence of (b) and the classical fact that PA is not interpretable in ACA 0 (the ingredients whose proof are Mostowski's reflection theorem for PA, finite axiomatizability of ACA 0 , and Gödel's second incompleteness theorem).
3.4.Remark.Standard techniques can be used to show that the proof of Theorem 3.3(b) yields a feasible reduction of PAI in PA.In other words, there is a polynomial-time function f such that, given the (binary code) of a proof π of an arithmetical sentence ϕ in PAI, f (π) is the (the binary code of) a proof f (π) of ϕ in PA.In particular, PAI has at most polynomial speed-up over PA.
• In what follows Form k is the set of L A -formulae with precisely k free variables.
3.5.Theorem.The following schemes are provable in PAI: (a) The apartness scheme: where Apart ϕ is the following formula: (b) The diagonal indiscernibility4 scheme: where Indis + ϕ (I) is the following formula: Proof.Let (M, I) |= PAI.To verify that the apartness scheme holds in (M, I), fix some i 0 ∈ I and some ϕ(x, y) ∈ Form n+1 .Then, since the I is unbounded and the collection scheme holds in (M, I), and I is unbounded in M, there is some j 0 ∈ I with i 0 < j 0 such that: The above, together with the indiscernibility of I in M, makes it evident that (M, I) |= Apart ϕ .
To verify that Indis + ϕ (I) holds in (M, I), we will first establish a weaker form of diagonal indiscernibility of I in which all j n < k 1 (thus all the elements of j are less than all the elements of k).Fix some ϕ ∈ Form n+1+r and i 0 ∈ I. Within M consider the function f : Since (M, I) satisfies the collection scheme and I is unbounded in M, this shows there are y 1 < • • • < y 2r in I such that: By the indiscernibility of I in M, the above implies the following weaker form of Indis + ϕ (I): We will now show that the above weaker form of Indis + ϕ (I) already implies Indis + ϕ (I).Given i ∈ I, a ∈ [I] r and b ∈ [I] r , with i < a 1 and i < b 1 , choose y ∈ [I] r with y 1 > max {a n , b n } .Then by the above we have: and which together imply: • Note that the diagonal indiscernibility scheme for L A -formulae ensures that if (M, I) |= PAI and i ∈ I, then I ≥i is a set of indiscernibles over the expanded structure (M, m) m<i , where

MAIN RESULTS
In this section we prove refinements of Theorems A and B of the abstract (as in Theorems 4.6 and 4.12).

4.1.
Theorem.There is a formula σ(x) in the language L A (I) such that S = σ M is an inductive partial satisfaction class on M for all models (M, I) |= PAI.
Proof.We first define a recursive function that transforms each formula ϕ(x) ∈ Form n into a ∆ 0formula ϕ * (x, z 1 , • • •, z k ), where {z i : 1 ≤ i ∈ ω} is a fresh supply of variables added to the syntax of first order logic (the definition of ϕ * below will make it clear that k is the ∃-depth of ϕ).In what follows x and y range over the set of variables before the addition of the fresh stock of z i s.We assume that the only logical constants used in ϕ are {¬, ∨, ∃} and none of the fresh variables z i occurs in ϕ.
(2) (¬ϕ) * = ¬ϕ * . ( that there is some j ∈ I with j < i 1 and a s < j for each 1 ≤ s ≤ n.Then M satisfies: Proof.We use induction of the complexity of ϕ.The only case that needs an explanation is the existential case, the others go through trivially.Thus, it suffices to verify that if and there is some j ∈ I with j < i 1 and a s < j for each 1 ≤ s ≤ k, then: where (ϕ(x, y)) * = ϕ * (x, y, z 1 , • • •, z k ).To establish the left-to-right direction of (∇), suppose M |= ∃y ϕ(a, y).By the veracity of the apartness scheme and the assumption that a s < j for each 1 ≤ s ≤ n, there is b < i 1 such that M |= ϕ(a, b).Thus since b, as well as a 1 , • • •, a n are all below i 1 , i 1 can serve as the element "j" of the inductive assumption, hence allowing us to conclude that ), as desired.The right-to-left direction of (∇) is trivial.This concludes the proof of the claim (∇).
We are now ready to show that there is an (M, I)-definable S ⊆ M such that S is an inductive satisfaction class over M. The following procedure takes place in (M, I), in particular, the variables n and k in (P) range in M and need not be standard: (P) Given any ϕ(x) ∈ Form n and any n-tuple a, calculate (ϕ(x)) , and let j ∈ I be the first element of I such that ϕ(x) < j and a s < j for each 1 ≤ s ≤ n, and then let and i 1 , • • •, i k to be the first k elements of I that are above j.Then define S by: where Sat ∆0 is the canonical Σ 1 -definable satisfaction predicate for ∆ 0 formulae of arithmetic.
Thus the desired formula σ is given by 4.2.Remark.Three remarks are in order concerning the proof of Theorem 4.1.
(a) If ϕ(x) is a standard formula, (M, I) |= PAI, and j ∈ I, then the condition ϕ(x) < j in the procedure (P) is automatically satisfied since every element of I is nonstandard.The role of the condition ϕ(x) < j will become clear in the proof of Lemma 4.10 and Theorem 4.11.
(b) If I ′ is a cofinal subset of I such that (M, I ′ ) |= PAI and S ′ is the partial satisfaction class on M as defined by σ in (M, I ′ ), then thanks to the diagonal indiscernibility property of I, S = S ′ .This fact comes handy in the proof of Theorem 5.5.
(c) The transformation ϕ → ϕ * given in the proof of Theorem 4.1 can be reformulated in the following more intuitive way: Given ϕ(x) ∈ Form n , find an equivalent formula ϕ pnf (x) in the prenex normal form: and then define (ϕ(x)) * to be: A similar transformation is found in the proof of the Paris-Harrington Theorem [PH].Consider the function where h is defined in (M, I) by h(m) := the (Gödel number of) the least L A -formula ϕ(x, y) such that, as deemed by S, m is defined by ϕ(x, i) for some tuple i of parameters from I, i.e., S contains the sentences ϕ(m, i) and ∃!x ϕ(x, i).Note that the set of standard elements of M is definable in (M, I) as the set of i such that i < j for some j in the range of h.Thus (M, I) is a nonstandard model of PAI, in which the standard cut ω is definable, which is impossible.[KS]) can be 5 An alternative, more direct proof of (a) invokes diagonal indiscernibility.Suppose to the contrary that (M, I) |= PAI and I is definable in M by a formula ϕ(x, m) for some m ∈ M. Let i 1 < i 2 < i 3 be the first three elements of I above M.Note that i 2 and i 3 is each pointwise definable in (M, m, I).Hence i 2 and i 3 are discernible in (M, m, I), and therefore they are also discernible in (M, m) (since I is definable in M with parameter m).On the other hand, by the diagonal indiscernibility property of I, for any arithmetical formula θ(x, y), M satisfies θ(m, i 1 ) ↔ θ(m, i 2 ).We have arrived at a contradiction.
readily adapted to show that every countable recursively saturated model M of PA has a pointwise definable expansion (M, I) |= PAI.4.6.Theorem.The following are equivalent for a model M of PA of any cardinality: (i) M has an expansion to a model of UTB(c).
(ii) M has an expansion to a model of PAI.
Consequently, M has an expansion to a model of UTB iff M has an expansion to a model of PAI • .
Proof.Since (ii) ⇒ (i) is justified by Theorem 4.1, it suffices to show that (i) ⇒ (ii).6By Proposition 2.2.3(b) there is an inductive partial satisfaction class S on M. Consider the L A (S)formula ψ(x) that expresses: "there is a definable (in the sense of S) unbounded set of indiscernible for L A -formulae of Gödel-number at most x".
More specifically, ψ(x) is the formula ∃θ ∈ Form 1 (U (θ) ∧ H(θ, x)), where U (θ) is the following L A (S)-sentence: and H(θ, x) is the following L A (S)-sentence: where Indis ϕ (θ) is the following L A -sentence: By the schematic provability of Ramsey's theorem in PA, (M, S) |= ψ(n) for each n ∈ ω, so by overspill, (M, S) |= ψ(c) holds for some nonstandard c ∈ M .Hence there is some θ 0 ∈ Form M 1 such that (M, S) |= H(θ, c), and thus (M, I) |= PAI, where: This concludes the proof of the equivalence of (i) and (ii).The 'consequently' clause readily follows from Proposition 3.2 and the equivalence of (i) and (ii).
• Going back to Theorem 4.1, one might wonder if it is possible for σ M to be a full satisfaction class on M.There are certainly many models (M, I) of PA for which σ M is not a full satisfaction class since the existence of a full inductive satisfaction class on a model M implies that Con(PA) holds in M (and much more, see the remarks following Theorem 4.9).The results of the rest of this section are informed by this question.
4.7.Definition.α is the L A (I)-sentence the sentence α expressing "σ defines a full satisfaction class", where σ(x) is the formula given in the proof of Theorem 4.1.
Proof.It is well-known [H,Section 8.6] that the arithmetical consequences of PA(S) + FS(S) coincide with the arithmetical consequences of ACA (the extension of ACA 0 by the full induction scheme).It has long been known that the arithmetical consequences of ACA can be axiomatized by PA + RFN ε0 (PA), a result which has been recently given a new proof in the work of Beklemishev and Pakhomov [BP,Sec. 8.3].7 The following lemma, which will come handy at the end of the proof of Theorem 4.11, shows that if (M, I) |= PAI + α, and ϕ ∈ Form M (note that ϕ is allowed to be nonstandard), then as viewed by S, a tail of I satisfies diagonal ϕ-indiscernibility.4.10.Lemma.Suppose (M, S, I) |= PAI + PA(S, I) + FS(S).If ϕ ∈ Form M n+r+1 (where n, r ∈ M ), then: where θ(i, ϕ) is the following L A (S, I)-formula: Proof.The strategy of establishing the diagonal indiscernibility of I in the proof of Theorem 3.5(b) can be readily carried out in this context, thanks to the fact that (M, S, I) |= PA(S, I).

Theorem.
There is a formula ι(x) in the language L A (T, c) such that for all models (M, T, c) of CT(c), (M, I) |= PAI + α for I = ι (M,T,c) .
Proof.We will describe the formula ι(x) by working in an arbitrary model (M, T, c) |= CT(c).Since PA(S) + FS(S) and CT are well-known to be bi-interpretable, we do most of our work with the model (M, S) |= PA(S) + FS(S), and at the end will take advantage of the nonstandard element c.The basic idea is that PA can verify that the formalized (infinite) Ramsey theorem is provable in PA, so using the inductive full satisfaction class S we can follow the strategy of the proof of part (b) of Theorem 3.2 to define a set I in (M, S) such that not only (M, I) |= PAI + α.More specifically, Ramsey's theorem's can be fine-tuned by asserting that if an arithmetically definable coloring f of m-tuples is of complexity Σ n , then there is an arithmetical infinite monochromatic subset for f of complexity Σ n+m+1 [HP].Therefore where Indisc ϕ (θ) is as in the proof of Theorem 4.6.On the other hand, it is well-known that PA(S) + FS(S) proves the global reflection principle8 ∀ϕ(Prov PA (ϕ) → S(ϕ)).

Hence
Reasoning in (M, S), fix some enumeration ϕ m : m ∈ M of all arithmetical formulae, and use ( * ) to construct a recursive sequence θ m : m ∈ M of elements of Form M 1 such that the following three conditions hold: ∈ S}, and within (M, S) recursively define i m : m ∈ M by: i 0 = max{min{H 0 }, ϕ 0 }, and i m+1 is the least i ∈ H m+1 that is greater than both i m and ϕ m .It is easy to see that (M, I) |= PAI • , where I = i m : m ∈ M , and therefore as noted in Proposition 3.2(c) (M, I >c ) |= PAI.The procedure described for constructing I makes it clear that I is definable by an L A (T, c)-formula ι(x).
It remains to show that (M, I) |= PAI + α.Note that (M, S, I) |= PA(S, I), this is precisely where Lemma 4.10 comes to the rescue, since together with the veracity of PAI + PA(S, I) + FS(S) in (M, S, I) it allows us verify the following nonstandard analogue (∇ * ) of (∇) from the proof of Theorem 4.1 (in what follows the map ϕ → ϕ * is defined as in the proof of Theorem 4.1 within M).
(∇ * ).Suppose ϕ = ϕ(x) ∈ Form M r for some r ∈ M (NB: r need not be standard), that there is some j ∈ I with j < i 1 and a s < j for each 1 ≤ s ≤ r.Then (M, S, I) satisfies the following: Proof.This follows from putting the completeness theorem of first order logic together with Theorems 4.9 and 4.12.4.14.Remark.In contrast to Theorem 4.6, in Theorem 4.12 CT(c) cannot be weakened to CT 0 (c), where CT 0 (c) is the fragment of CT(c) in which the extended induction scheme is limited to ∆ 0 (T )formulae.This is because the arithmetical consequences of CT 0 form a tiny fragment of the arithmetical consequences of CT.More explicitly, it has been long known by the cognoscenti that by combining results of Kotlarski [Ko] and Smoryński [Sm], the arithmetical consequences of CT 0 can be shown to be axiomatized by RFN ω (PA).The recent work of Le lyk [L-2] provides a modeltheoretic proof of this axiomatizability result, and culminates earlier results obtained in Kotlarski's aforementioned paper, Wcis lo and Le lyk's [WL] (ii) M is recursively saturated and satisfies RFN ε0 (PA).
Proof.By Theorem 4.14, there is a (uniform) ω-interpretation I c of a model (M, I >c ) of PAI + α within any model (M, T, c) of CT(c).On the other hand, the definition of PAI + α makes it clear that there is a (uniform) ω-interpretation J of (M, T ) within (M, I >c ).A slight variant of this argument (without the use of the constant c) shows that CT is a retract of PAI • + α, but we need a variation of the interpretation I c in order to show that CT(c) is a retract of PAI + α because it not clear that the element c is definable in (M, I >c ).We can get around this problem by modifying the interpretation I as follows: Given any model (M, T, c) of CT(c), we first define I >c as in the interpretation I, and then we define the modified interpretation I c given by: Thanks to the diagonal indiscernibility property of I >c , (M, I c ) is a model of PAI.Moreover, (M, I c ) can be shown to be a model of PAI + α with the same argument used in the proof of Theorem 4.14 (relying on Lemma 4.11).We can now readily define an interpretation J ′ that inverts I c by letting J ′ (M, I c ) = (M, T, c), where T is the unique truth predicate corresponding to the partial satisfaction class given by the formula σ (of Theorem 4.1), and c is defined as "the first coordinate of the ordered pair canonically coded by any member of I c ". Thus CT(c) is a retract of PAI + α. 5.10.Theorem.PAI • + α is not a retract of CT.
Proof.We begin with observing that Proposition 5.3 and Theorem 5.5 show that the ω-interpretation I σ of CT in PAI • given by the formula σ of Theorem 4.1 is not 'invertible', in the sense that there is no interpretation of PAI in CT such that I σ and J witness that PAI is retract of CT.We next note that if there are interpretations I and J that witness that PAI + α is a retract of CT, then Theorem 5.4 assures us that verifiably in PAI • , the interpretation I is the same as I σ up to a definable permutation of the ambient universe.This shows that I is not invertible either, thus concluding the proof.
5.11.Question.Is PAI + α is a retract of CT(c)?5.12.Question.Is either of the pair of theories {PAI, UTB(c)} a retract of the other one?5.13.Question.Is either of the pair of theories {PAI + α, CT(c)} a retract of the other one?
• We have not succeeded in ruling out that PAI and UTB(c) are not bi-interpretable (ditto for PAI + α and CT(c)).We conjecture that the above questions all have negative answers.As partial evidence for our conjecture, let us observe that Theorem 5.2 and 5.5 show that the ωinterpretation I σ of UTB(c) in PAI, and CT(c) in PAI + α, given by the formula σ of Theorem 4.1 is not 'invertible', in the sense that there is no interpretation of PAI in UTB(c) such that I σ and J witness that PAI is retract of UTB(c).

FRAGMENTS OF PAI
In this section we briefly examine the model-theoretic behavior of subsystems PAI n (n ∈ ω) and PAI − of PAI.The section is concluded with two open questions.
6.1.Definition.For n ∈ ω, PAI n is the subsystem of PAI in which the extended induction scheme involving I is weakened to Σ n (I)-formulae, i.e., the axioms of PAI n consist of PA plus the fragment IΣ n (I) of PA(I), plus axioms (2) and (3) of Definition 3.1 asserting the unboundedness and indiscernibility of I. PAI − is the subsystem of PAI 0 with no extended induction scheme involving I, so the axioms of PAI − consist of PA plus axioms (2) and (3) of Definition 3.1.
• Given M |= PA, it is evident that (M, I) |= PAI − iff I is an unbounded set of indiscernibles in M; and by Theorem 2.1.2,(M, I) |= PAI 0 iff I is a piecewise-coded unbounded set of indiscernibles in M.
6.2.Theorem.Every model of PA has an elementary end extension that has an expansion to a model of PAI 0 , but not to a model of PAI.
Proof.Recall that a type p(x) is said to be an 'unbounded indiscernible type' if it is a nonprincipal type satisfying: (1) there is no constant Skolem term c such that x ≤ c is in p(x), and (2) for any model M of PA, if I ⊆ M is a set of elements each realizing p(x), then I is a set of indiscernibles in M. It is well-known that unbounded indiscernible types exist in abundance (continuum-many), and that a type p(x) is minimal (in the sense of Gaifman) iff p(x) is an unbounded indiscernible type (see theorems 3.1.4and 3.2.10 of [KS]).Fix a minimal type p(x) and any model M 0 of PA, and let M be an ω-canonical extension of M 0 using p(x) as in section 3.3 of [KS].Thus M is obtained by an ω-iteration of the process of adjoining an element satisfying p(x).Since p(x) is an unbounded indiscernible type, this makes it clear that M carries an unbounded indiscernible subset I, and additionally the order-type of I is ω.The latter feature makes it clear that I is piecewise-coded in M, and thus (M, I) |= PAI 0 in light of Theorem 2.1.2.
It remains to show that M does not have an expansion to PAI.Suppose not, and let (M, I) |= PAI.It is easy to see that from the point of view of (M, I), the order-type of (I, <) is the same as the order-type of (M, <) (where < is the ordering on M given by M).Recall that M can be written as the union of elementary initial submodels M n of M (as n ranges in ω), where M n is obtained by n-repetitions of the process of adjoining an element satisfying p(x).By minimality of p(x) this assures us that: ( * ) For each n ∈ ω, and each choice of c n ∈ M n+1 \M n , (M n , c 1 , • • •, c n ) is pointwise definable.
The existence of the above isomorphism f makes it clear that there is some k ≥ 1 such that I ∩ (M k \M k−1 ) is infinite.In particular we can pick distinct i 1 and i 2 in I ∩ (M k \M k−1 ), together with elements {c s : 1 ≤ s ≤ k} such that c s ∈ M s+1 \M s for each s, and moreover c k is below both i 1 and i 2 (since M k \M k−1 has no least element).By the diagonal indiscernibility property of I, this implies that i 1 and i 2 are indiscernible in the structure (M k , c 1 , • • •, c k , m) m∈M0 .But this indiscernibility contradicts ( * ), and thereby completes the proof.

Theorem.
If M is a model of countable cofinality of PA that is expandable to a model of PAI − , then M is expandable to a model of PAI 0 .However, every countable model of PA has an uncountable elementary end extension that is expandable to a model of PAI − , but not to PAI 0 .
Proof.Suppose (M, I) |= PAI − , where M has countable cofinality.The countable cofinality of M allows us to construct an unbounded subset I 0 of I of order type ω.Since every subset of M of order-type ω is piecewise-coded, by Theorem 2.1.2,(M, I 0 ) |= PAI 0 .To demonstrate the second assertion of the theorem, let M 0 be a countable model of PA and M be an ω 1 -canonical extension of M 0 using some minimal type p(x), i.e., M is obtained by an ω 1 -iteration of the process of adjoining an element satisfying p(x) (as in [KS,Section 3.3]).By Theorem 2.2.14 of [KS], M is rather classless, i.e., every piecewise-coded subset of M is definable in M. Thus if (M, I) |= PAI 0 , then (M, I) |= PAI, and I is M-definable, which contradicts Corollary 4.3(a).
We close the paper with the following open questions: 6.4.Question.Does Theorem 6.2 lend itself to a hierarchical generalization?In other words, is it true that for every n ∈ ω, every model of PA has an elementary end extension that has an expansion to a model of PAI n , but not to a model of PAI n+1 ?(It is not even clear how to build a model (M, I) of PAI n for n ∈ ω that is not a model of PAI n+1 .)6.5.Question.Is there a model M of PA such that M has an expansion to a model of PAI n for each n ∈ ω, but M has no expansion to a model of PAI?
(a) M has an inductive partial satisfaction class iff M has an expansion to a model of UTB.(b) M is nonstandard and has an inductive partial satisfaction class iff M has an expansion to a model of UTB(c).
(c) M has an inductive full satisfaction class iff M has an expansion to a model of CT.(d) M is nonstandard and has an inductive full satisfaction class iff M has an expansion to a model of CT(c).
(a) N does not have an expansion to a model of PAI (equivalently: Every model of PAI is nonstandard ).(b) N has an expansion to PAI • .(c) If (M, I) is a nonstandard model of PAI • , and c is any nonstandard element of M, then (M, I >c ) |= PAI, where I >c = {i ∈ I : i > c}.
4.3.Corollary.The following hold for every model M of PA of any cardinality.(a) There is no M-definable subset I of M such that (M, I) |= PAI (therefore no rather classless recursively saturated model of PA has an expansion to a model of PAI).(b) If M has an expansion to a model of PAI, then M is recursively saturated; and the converse holds if M is countable.(c) If M has an expansion (M, I) |= PAI, then M = M I , where M I consists of elements of M that are definable in (M, i) i∈I .Proof.(a) follows by putting Theorem 4.1 together with Tarski's theorem on undefinability of truth. 5(b) follows directly by putting Theorem 4.1 with the Barwise-Schlipf Theorem 2.2.4.To verify (c) suppose M I = M for (M, I) |= PAI.Recall that M is nonstandard by Proposition 3.2(a).By Theorem 4.1 there is an inductive partial satisfaction class S on M that is definable in (M, I).

4. 4 .
Remark.As shown by Schmerl [Sc], every countable recursively saturated model M of PA carries a set of indiscernibles I such that M I = M. Thus, in light of part (c) of Corollary 4.3, such a set of indiscernibles I never has the property that (M, I) |= PAI.4.5.Remark.In contrast to part (c) of Corollary 4.3, the proof technique of the Kossak-Schmerl construction of prime inductive partial satisfaction classes (as in Theorem 10.5.2 of assures us that σ M coincides with S, and thus (M, I) |= PAI + α. 4.12.Theorem.The following hold for any model M of PA of any cardinality: (a) M has an expansion to CT(c) iff M has an expansion to PAI + α.(b) M has an expansion to CT iff M has an expansion to PAI • + α.Proof.We only verify (a) since the argument for (b) is similar.The left-to-right direction of (a) is justified by Theorem 4.11.The other direction is evident thanks to the axiom α. 4.13.Corollary.The arithmetical consequences of PAI • + α and PAI + α are axiomatized by PA + RFN ε0 (PA).
, and Le lyk's doctoral dissertation [L-1].4.15.Corollary.The following are equivalent for every countable model M of PA: (i) M has an expansion to a model of PAI + α.