Math. Log. Quart. 46 (2000) 1, 17 – 23

Mathematical LogicQuarterly

c© WILEY-VCH Verlag Berlin GmbH 2000

An Effective Conservation Result for Nonstandard Arithmetic

Erik Palmgren1)

Department of Mathematics, Uppsala University,P. O. Box 480, S – 75106 Uppsala, Sweden2)

Abstract. We prove that a nonstandard extension of arithmetic is effectively conservativeover Peano arithmetic by using an internal version of a definable ultrapower. By the samemethod we show that a certain extension of the nonstandard theory with a saturation prin-ciple has the same proof-theoretic strength as second order arithmetic, where comprehensionis restricted to arithmetical formulas.

Mathematics Subject Classification: 03F30, 03H15.

Keywords: Nonstandard arithmetic, Proof-theoretic strength, Bounded ultrapowers.

1 Introduction

While it is trivially true that nonstandard methods used inside ZFC will not in-crease the set of theorems that ZFC proves, the addition of axioms correspondingto transfer and saturation principles to weaker theories may, or may not, increasethe strength of the theory. A conservation result for a higher order theory weakerthan ZFC was obtained by Kreisel [7]. The proof-theoretic strength of various ax-ioms for nonstandard arithmetic, especially saturation axioms, has been investigatedby Henson, Kaufmann and Keisler [4]. In this paper we solve a problem leftopen in [4], namely to determine the strength of ∗PA+∗Π∞-ind+∗Π1

0-comp (see Sec-tion 4). Our main result is however an effective conservation result for ∗PA+∗Π∞-indover PA. For both results we use Skolem’s method of definable ultrapowers (cf. [5]),but internalise the construction in the respective theories. Although this construc-tion is relatively straightforward, arithmetised (bounded) ultrapowers have to ourknowledge not been used in the literature. An application of potential interest is todevelop nonstandard analysis built solely on hyperfinite notions (Harthong [3]) in anonstandard arithmetic. Effective content can then be obtained by the conservationresult, and a negative translation into intuitionistic arithmetic. Effective conservationresults have been achieved by Dragalin [2] for weaker nonstandard theories usingconstructive model theory. For the case of intuitionistic arithmetic, effective conser-vation results have been obtained by I. Moerdijk and the author [9] (using ideasfrom Moerdijk [8]). Coquand and Smith [1] gives an elegant conservation resultfor a slightly weaker theory.

1)This research was supported by grants from the Swedish Research Councils for EngineeringSciences (TFR) and Natural Sciences (NFR). I am grateful to Prof. Dr. P. Zahn for suggestingimprovements of the presentation.

2)e-mail: palmgren@math.uu.se

18 Erik Palmgren

2 Nonstandard extensions of arithmetic

We use the variant of Peano arithmetic (PA) presented in [4] which has function sym-bols for all primitive recursive functions. The language of the theory is denoted L(PA).Our basic nonstandard theory ∗PA is formulated in an extension of this language by apredicate symbol St( · ), with the intuitive interpretation as a delimitation of the stan-dard numbers. We denote the relativisation of a formula ϕ of PA to St by ϕ st. Forquantifiers, we write (∀ stx)ψ for (∀x) (St(x) → ψ) and (∃ stx)ψ for (∃x) (St(x) ∧ ψ).The axioms of ∗PA are:

(I) All the universal closures of axioms of PA relativised to St. Moreover axiomsstating that St(x) is closed under the formation of terms, i. e., for each n-ary functionsymbol f of PA, (∀x1 . . .xn) (St(x1) ∧ · · · ∧ St(xn)→ St(f(x1, . . . , xn))),(II) (∃x)¬St(x),(III) T r a n s f e r P r i n c i p l e . For any formula ϕ(x) in L(PA),

(∀ stx) (ϕ st(x) ↔ ϕ(x)),(IV) E n d E x t e n s i o n A x i o m . (∀xy) (St(y) ∧ x < y → St(x)).

This defines ∗PA.We shall also consider theP r i n c i p l e o f E x t e r n a l I n d u c t i o n (or ∗Π∞-ind in the terminology of [4]):

For any formula ϕ(y, x) in L(∗PA),

(∀x) (ϕ(x, 0) ∧ (∀ sty) (ϕ(x, y) → ϕ(x, s(y))) → (∀ sty)ϕ(x, y)).

The End Extension Axiom can be proven using External Induction and (I) – (III).Let e(x, y) denote the exponent of the yth prime in the prime number factorisation

of x. We write y ∈ x for e(x, y) = 1. An axiom considered by Henson et al. [4] isthe following comprehension scheme ∗Π1

0-comp: For any PA-formula ϕ(n, x),

(∀x)(∃a)(∀ stn) (n ∈ a ↔ ϕ st(n, x)).

We first give a typical (non-constructive) model-theoretic proof of the followingP r o p o s i t i o n 2.1. ∗PA is a conservative extension of PA.P r o o f . Let ϕ be a PA-formula provable in ∗PA. Take any model M � PA. By

the MacDowell-Specker Theorem (cf. Kaye [6]) there exists a proper, elementary endextensionM′ ofM. Thus (M′,M) � ∗PA withM , the universe ofM, interpreting St.Hence M′ � ϕ. But M ≺ M′ so M � ϕ. By the completeness theorem it followsthat PA ` ϕ. 2

To verify also the External Induction Principle, would require thatM is standardrelative toM′. But it seems that the MacDowell-Specker Theorem does not guaranteethis. It would be interesting to find a simple, external model-theoretic argumentthat proves ∗PA + ∗Π∞-ind to be conservative over PA. However there is an internalargument for this (Section 3) which yields effective conservation.

R e m a r k . The O v e r s p i l l P r i n c i p l e is the following proposition

(OS) (∀x) ((∀ stn)ϕ(n, x) → (∃ infn)ϕ(n, x)),

where ϕ(x, n) is a formula of L(PA) and (∃ infm)ψ abbreviates (∃m) (¬St(m) ∧ ψ).This principle is in fact provable in ∗PA.

An Effective Conservation Result for Nonstandard Arithmetic 19

3 Effective conservation

The conservation result is proved by an internal model construction, where a boundedultrapower is built for each quantifier complexity. We recall some terminology. Asusual, a PA-formula ϕ(y) is called a Σn-formula if it is of the form

(∃x1)(∀x2)(∃x3) . . . (Pxn)ψ(x1, . . . , xn, y),

where ψ is quantifier free and P is ∀, if n is even, and ∃ if n is odd. The formula is aΠn-formula if it has the same form but with ∀ and ∃ interchanged. The class of PA-formulas with at most q quantifiers is denoted Γq. A PA-formula ϕ(x) provably equalto a Σn-formula (Πn-formula) is called a Σn(PA)-formula (Πn(PA)-formula). A PA-formula which is both a Σn(PA)-formula and a Πn(PA)-formula is called a ∆n(PA)-formula. We refer to Kaye [6] for standard results about the closure properties ofthese classes of formulas. For each complexity class Γq, a ∆0

q+1-ultrapower will be builtin PA. The ultrapower is constructed from ∆q+1(PA)-sequences and a nonprincipal∆q+1(PA)-ultrafilter given by three PA-formulas H(n), F (n, x, y) and U(p, i), to bedefined in Section 3.2. Below we state the requirements (F1 – F3, U1 – U5) onthese formulas, a primitive recursive function d and a constant i0. In Section 3.1 theultrapower is assembled and the conservation result proved.

We use the abbreviations n = n1, . . . , nk, x = x1, . . . , xk, d(n) for d(n1), . . . , d(nk)and F (n, i, x) for F (n1, i, x1) ∧ · · · ∧ F (nk, i, xk). We write i ∈ Up for U(p, i), andn ∈ H for H(n). The quantifier combination (∃p)(∀i ∈ Up) is abbreviated by Qi. Thefollowing properties (F1 – F3, U1 – U5) are required to hold in PA.

The formulas H and F code functions

(F1) (∀n ∈ H)(∀x)∃!y F (n, x, y),

(F2) (∀nx) (d(n) ∈ H ∧ F (d(n), x, n)),(F3) i0 ∈ H ∧ (∀n)F (i0, n, n),

where i0 is a code for the identity function and d(n) is a code for the constant nfunction. The formula U codes a nonprincipal ultrafilter for Γq-formulas.

(U1) (∀p)(∀k)(∃i ≥ k) i ∈ Up,

(U2) (∀i)(∃p) i /∈ Up,(U3) (∀kp)(∃q)(∀i) (i ∈ Uq ↔ i ∈ Uk ∧ i ∈ Up),

(U4) (∀n ∈ H)((Qi)ψ(n, i) ∨ (Qi)¬ψ(n, i)), where ψ(n, i) ≡ (∃y) (F (n, i, y) ∧ ϕ(y))and ϕ is a Γq-formula,

(U5) (∀n ∈ H) ((Qi)(∃y)ψ(n, i, y) ↔ (∃m ∈ H)(Qi)(∃y) (F (m, i, y) ∧ ψ(n, i, y))),where ψ(n, i, z) ≡ (∃y) (F (n, i, y) ∧ ϕ(y, z)) and ϕ is a Γq-formula.

As an easy consequence of (F1 – F3) and (U1), (U3) and (U4) we haveL e m m a 3.1. Let ψk(n, i) ≡ ∃y (F (n, i, y) ∧ ϕk(y)), where ϕk ∈ Γq, k = 1, 2.

Then the following are provable in PA:

(i) (∀n ∈ H) (¬(Qi)ψ1(n, i)↔ (Qi)¬ψ1(n, i)),(ii) (∀n ∈ H) ((Qi) (ψ1(n, i) ∧ ψ2(n, i))↔ (Qi)ψ1(n, i) ∧ (Qi)ψ2(n, i)). 2

20 Erik Palmgren

3.1. The bounded ultrapower MDefine, inductively, for each ∗PA-formula ϕ(x1, . . . , xk) a PA-formula

M � ϕ(x1, . . . , xk)[n1, . . . , nk],

with free variables among n1, . . . , nk.M � ⊥(x)[n] ≡⊥,

M � St(t(x))[n] ≡ (∃m)(Qi)(∃x) (F (n, i, x) ∧ t(x) = m),M � r(x) = t(x)[n] ≡ (Qi)(∃x) (F (n, i, x) ∧ r(x) = t(x)),M � (ϕ ◦ ψ)(x)[n] ≡ (M � ϕ(x)[n]) ◦ (M � ψ(x)[n]), where ◦ ∈ {∧,∨,→},M � (Py ϕ)(x)[n] ≡ (Pm ∈ H)M � ϕ(x, y)[n, m], where P ∈ {∀, ∃}.

Note that, for all ϕ(x) ∈ L(∗PA), the following is provable in PA:

(∀m, n ∈ H)((Qi)(∃x) (F (m, i, x) ∧ F (n, i, x)) ∧M � ϕ(x)[m]→M � ϕ(x)[n]).(1)

It is clear that M � (∀x1 . . .xn) (St(x1) ∧ · · · ∧ St(xn) → St(f(x1 , . . . , xn))) holdsin PA for each function symbol f of PA. Using (1) it follows easily that for anyϕ(x, y) ∈ L(∗PA), and any P ∈ {∀, ∃}, the following holds in PA:

(∀n ∈ H)(M � P sty ϕ(x, y)[n]↔ (Pm)M � ϕ(x, y)[n, d(m)]

).(2)

Then by induction on PA-formulas ϕ(x), one proves using (2) that in PA:

(∀n)(M � ϕ st(x)[d(n)]↔ ϕ(n)

).(3)

More interestingly, we have a formalised, bounded version of Los’ Theorem.L e m m a 3.2. For each PA-formula ϕ(x) in Γq, the following is provable in PA:

For all n ∈ H,M � ϕ(x)[n]↔ (Qi)(∃x) (F (n, i, x) ∧ ϕ(x)).P r o o f . By induction on the complexity of ϕ. For atomic ϕ the equivalence is

clear since the filter is non-trivial (U1). The inductive cases are straightforwardlyproven using the properties (U1), (U3) – (U5). 2

T h e o r e m 3.3.

(i) If ϕ is a (universal closure of a) PA-axiom, then M � ϕ st is provable in PA.(ii) If ϕ(x) ∈ Γq is a PA-formula, then M � (∀ stx) (ϕ st(x) ↔ ϕ(x)) is provable

in PA.(iii) If ϕ(x, y) is a ∗PA-formula, then

M � (∀x) (ϕ(x, 0) ∧ (∀ sty) (ϕ(x, y)→ ϕ(x, s(y))) → (∀ sty)ϕ(x, y))is provable in PA.(iv) PA proves M � (∃x)¬St(x).

P r o o f . Part (i) follows directly from (2) and (3). Part (ii) follows from Lem-ma 3.2 and (3). Part (iii) is proved by ordinary induction using (2). Part (iv) is aconsequence of (F3) and (U2). 2

C o r o l l a r y 3.4. The theory ∗PA + ∗Π∞-ind is a conservative extension of PA.P r o o f . Let ϕ be a closed PA-formula provable from T = ∗PA + ∗Π∞-ind. Thus

for some finite ∆ ⊆ T , ∆ ` ϕ. Let q be the total number of quantifiers in the set∆ ∪ {ϕ}. Consider the ultrapowerM constructed from F , H , U of complexity q. ByTheorem 3.3 and the discussion above,M � ∆ is provable in PA. Hence by soundness,alsoM � ϕ. From Lemma 3.2 follows PA ` ϕ. 2

An Effective Conservation Result for Nonstandard Arithmetic 21

3.2 Construction of the formulas F , H and U

It is well-known that partial truth-definitions are available in PA (see e. g. [6]). Foreach n ≥ 1, there is a Σn-formula SatΣ,n(x, y) such that for Σn-formulasϕ(v1, . . . , vm),and a fixed Godel coding p · q, we have

PA ` SatΣ,n(pϕ(v1, . . . , vm)q, [y1, . . . , ym]) ↔ ϕ(y1 , . . . , ym),

and similarly for Πn-formulas. Moreover there are primitive recursive predicatesFormΣ,n(x, k) and FormΠ,n(x, k) defining the syntactic classes of Σn- and Πn-for-mulas with k parameters. Let 〈 · , · 〉 be a surjective pairing function with projectionsπ1, π2. We say that y = 〈x, x′〉 is a ∆n-index if FormΣ,n(x, 1), FormΠ,n(x′, 1) and(∀u) (SatΣ,n(x, [u]) ↔ SatΠ,n(x′, [u])). The formula expressing this will be denotedby Dn(y). Write Sn(y, i) for SatΣ,n(π1(y), [i]).

Define inductively Vm:

V0 = ∅,

Vm+1 ={Vm ∪ {m} if Dn(m) and (∀k)(∃i ≥ k)(Sn(m, i) ∧ (∀j ∈ Vm)Sn(j, i)),Vm otherwise.

Formally, we can obtain this Vm in PA as follows: Prove by induction on p that thereexists a unique finite 0-1 sequence s of length p such that

(∀m < p)([s]m = 1 ↔ Dn(m) ∧ (∀k)(∃i ≥ k) (Sn(m, i)

∧ (∀j < m)([s]j = 1→ Sn(j, i)))).

The whole statement is denoted (∀p)(∃s)A(p, s) and we clearly have (∀p)(∃!s)A(p, s).Let j ∈ Vm ≡ j < m ∧ (∃s) (A(m, s) ∧ [s]j = 1). The formula defining the ∆n(PA)-ultrafilter is now given by U(p, i) ≡ i ≥ p ∧ (∀j ∈ Vp)Sn(j, i). Thus, informallyspeaking, for a fixed p, Up is the intersection of finitely many ∆n(PA)-classes.

Now we define:F (z, x, y)≡ SatΣ,n(z, [x, y]),

H(z) ≡ FormΣ,n(z) ∧ (∀x)(∃!y)F (z, x, y),d(k) = p(∃x1)(∀x2) . . . (Pxn) v2 = sk(0)q,i0 = p(∃x1)(∀x2) . . . (Pxn) v1 = v2q,

where P = ∃ if n is odd, and P = ∀ otherwise.The following lemma is immediate.L e m m a 3.5. F , H, d, i0 satisfy properties (F1) – (F3). 2

L e m m a 3.6. U and F satisfy properties (U1) – (U3).P r o o f . The properties (U2) and (U3) are obviously true. (U1) is proved by a

simple induction on p. 2

L e m m a 3.7. U and F satisfy properties (U4) and (U5) with q = n− 1.P r o o f . We work within PA. First we show that for z with Dn(z) it holds

(Qi)Sn(z, i) ∨ (Qi)¬Sn(z, i).(4)

Suppose ¬(Qi)Sn(z, i), i. e. (∀p)(∃i ∈ Up)¬Sn(z, i). Using (U2) and (U3) one provesthat

(∀p)(∀k)(∃i ≥ k) (i ∈ Up ∧ ¬Sn(z, i)).(5)

22 Erik Palmgren

By arithmetisation of the syntax there are u, v with FormΣ,n(u, 1), FormΠ,n(v, 1) and(∀y)(¬SatΠ,n(π2(z), [y]) ↔ SatΣ,n(u, [y])), (∀y)(¬SatΣ,n(π1(z), [y]) ↔ SatΠ,n(v, [y])).We have Dn(〈u, v〉), since Dn(z). By (5) and the above equivalences it follows that(∀p)(∀k)(∃i ≥ k) (i ∈ Up ∧ Sn(〈u, v〉, i)). Put p = 〈u, v〉. Hence

(∀k)(∃i ≥ k) (Sn(〈u, v〉, i) ∧ (∀j ∈ V〈u,v〉)Sn(j, i)),

i. e. 〈u, v〉 ∈ V〈u,v〉+1 . So for i ∈ U〈u,v〉+1 we have Sn(〈u, v〉, i), i. e. ¬Sn(z, i). Thus theright disjunct of (4) must hold. We now prove the property (U4). Let ϕ(y) ∈ Γn−1.This is a ∆n(PA)-formula. Consider the formula

ψ(n, i) ≡ (∃y) (F (n, i, y) ∧ ϕ(y)).

For fixed n ∈ H , there is a ∆n-index z with ψ(n, i)↔ Sn(z, i), since each F (ni, · , · )is functional. So (4) gives (Qi)ψ(n, i) ∨ (Qi)¬ψ(n, i).

As for the property (U5), the direction “(←)” is trivial. To prove “(→)” supposethat for n ∈ H , there is a p such that

(∀i ∈ Up)(∃y)ψ(n, i, y),

where ψ(n, i, z) ≡ (∃y) (F (n, i, y) ∧ ϕ(y, z)) and ϕ ∈ Γn−1. For fixed n ∈ H , we haveψ(n, i, z)↔ Sn(y, 〈i, z〉) for some ∆n-index y. Moreover, there is a (number coding)0-1 sequence s such that (∀j < p)([s]j = 1 ↔ j ∈ Vp). Using this fact, it followsby induction on p that there is a ∆n-index y′ so that i ∈ Up ↔ Sn(y′, i). Define aformula by

γ(p, n, i, z) ≡ (i ∈ Up ∧ ψ(n, i, z) ∧ (∀u < z)¬ψ(n, i, u)) ∨ (i /∈ Up ∧ z = 0).

For fixed n and p, there is then a Σn-index u so that γ(p, n, i, z)↔ Sn(u, [i, z]), andsince γ(p, n, · , · ) is functional, there exists m ∈ H with F (m, i, z) ↔ γ(p, n, i, z).Hence (∀i ∈ Up)(∃z) (F (m, i, z) ∧ ψ(n, i, z)). 2

4 Arithmetical comprehension

The theory of arithmetical comprehension PA2 + Π1∞-IND + Π10-CA is obtained by

adding second order logic to PA, extending the induction schema to all second orderformulas and introducing the comprehension schema

(∀x)(∀X)(∃Y )(∀y) (y ∈ Y ↔ ϕ(x, X, y)),

where ϕ ∈ L(PA2) contains neither second order quantifiers, nor the set variable Y .P r o p o s i t i o n 4.1 (Henson et al. [4]). PA2 + Π1∞-IND + Π1

0-CA is interpretablein ∗PA + ∗Π∞-ind + ∗Π1

0-comp.P r o o f (Sketch). The idea is that first order quantifiers are interpreted by stan-

dard quantifiers ∀ st, ∃ st, whereas second order quantifiers are interpreted by unre-stricted quantifiers. Moreover (x)n = 1 is the interpretation of n ∈ x. It is nowobvious that Π1

0-CA can be interpreted using ∗Π10-comp. 2

We prove the converse using formalised arithmetical ultrapowers.T h e o r e m 4.2. ∗PA + ∗Π∞-ind + ∗Π1

0-comp is interpretable in PA2 + Π1∞-IND +Π1

0-CA.P r o o f (Sketch). Using the comprehension schema and induction we may con-

struct a truth definition A(x, y) for arithmetical formulas. Using this it possible

An Effective Conservation Result for Nonstandard Arithmetic 23

to construct arithmetical ultrafilters in complete analogy with Section 3, and usearithmetical functions as objects of the ultrapower. We have predicates F , H andU satisfying properties (F1) – (F3) and (U1) – (U5), but for arbitrary formulas ϕof arithmetic. Analogously to Theorem 3.3, we prove that ∗PA + ∗Π∞-ind can beinterpreted. It remains to check the comprehension principle. Let n ∈ H . Then

ψ(i) ≡M � ϕ st(y, x)[d(i), n]

is indeed arithmetical, and has index p, say. We define an arithmetical function

γ(x, y) ≡ (∀i) (e(y, i) = 1↔ ψ(i) ∧ i < x) ∧ (∀i) (e(y, i) = 0 ∨ e(y, i) = 1)

coding finite parts of the characteristic function for ψ(i). We can compute its index kfrom p. Since the ultrafilter is nonprincipal, we readily see that for

ψ(v, x) ≡ (∀ stu) (u ∈ v ↔ ϕ st(u, x))

we have M � ψ(v, x)[k, n]. 2

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(Received: September 8, 1998; Revised: January 7, 1999)