Fragments of arithmetic

Annals of Pure and Applied Logic 28 (1985) 33-71

North-Holland

33

FRAGMENTS OF -TIC

Wilfried SIEG

Department of Philosophy, Columbia University, New York, NY 10027, USA

Communicated by D. van Dalen Received 22 August 1983

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Mint [14]; each has been shown to be of

considerable interest for both mathematical practice and me&mathematical investigations. The

foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems.

The results are generalized to relate a hierarchy of subsystems, all contained in the theory of

arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relation-

ships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman.

Introduction

The bulk of classical analysis can be developed in conservative extensions of elementary number theory (Z); that is the outcome of work by Takeuti, Fefer- man, and Friedman. Their investigations lie in a rather long tradition of persistent efforts to pursue mathematical analysis by ‘restricted’ means. The work of constructivists like Kronecker, Brouwer, and Bishop is clearly part of that tradition. Predicatively inclined mathematicians contributed also significantly; indeed, Weyl’s “Das Kontinuum” is an early landmark in this kind of research. Detailed investigations with sharp logical-mathematical focus were prodded by the foundational concerns of the Hilbert-school. Hilbert, in the early twenties, showed in lectures how to develop classical analysis straightforwardly in (a theory equivalent to) full second-order arithmetic.’ When the consistency problem even for elementary number theory turned out to be much more recalcitrant and difficult than had been expected, it was very natural to be concerned with subsystems of analysis in two ways: to prove their consistency and to establish their significance by developing substantial parts of mathematical analysis in them.’ That the latter can be done already in a conservative extension of (Z) is (prima facie) surprising and satisfying.

It is now an utterly trivial observation that such a development can be carried out partially in ad-hoc subsystems which are proof theoretically equivalent to proper fragments of (Z). The interesting question is whether it can be given for

0168~0072/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)

34 W. Sieg

coherent parts of mathematical practice in fixed subsystems conservative over informative fragments. Friedman [6] and Mint [14] introduced subsystems (WKLJ and (S’) respectively. These theories are of surprising ‘metamathematical strength’, as they allow for example the proof of Godel’s completeness theorem. Mint asserts that parts of recursive analysis can be developed in (S’). (WKI+), in contrast, is actually adequate for a good deal of ordinary mathematical practice in analysis and algebra. This claim is made by Simpson and is substantiated by detailed work due to him, Brown, Friedman, and Smith. For example, (WKI.+) proves that any continuous, real-valued function on the unit interval [0, l] is uniformly continuous, has a supremum and attains it; the Heine-Bore1 theorem and the Cauchy-Peano existence theorem for ordinary differential equations can be established in this theory. To mention one example from algebra, (W&) proves the existence of prime ideals in countable commutative rings3

Nevertheless, Friedman and Mint showed that their theories are conservative over primitive recursive arithemtic (PRA). Such a result, if established by elementary means, is of obvious foundational significance as it gives a direct finitist justification for parts of mathematical practice. It is also of mathematical, computational interest as it shows that the provably recursive functions of (WKLJ and (S’) are exactly the primitive recursive ones. (For refinements of such computa- tionally informative reductions see the Concluding Remarks.) Friedman’s and Mint’s results are obtained in Part 5 as corollaries of more general facts which are established by elementary, proof-theoretic means.

The main focus of this paper are conservative extensions of (PRA). The basic results, however, can be extended very easily to ‘hierarchy results’; for example, proper hierarchies of sub-systems are seen to be conservative (for classes of arithmetic sentences) over fragments of arithmetic (z:O,+l-IR), respectively (II:- IA). The former theory is obtained from the formulation of (Z) with the induction rule IR by restricting the application of IR to ~~+l-formulas; the latter theory is like (Z) except that only II:-instances of the induction axiom IA are available. The extensions follow directly from the basic results by the use of (Skolem-) operator theories. The technical set up is given in Parts 1 and 2. Part 1 presents general proof-theoretic facts concerning subsystems of analysis and suitable sequent calculi; Part 2 introduces the (Skolem-) operator theories and shows, how they can be interpreted in fragments of arithmetic.

The main work is done in the remaining three parts. Part 3 is concerned with the relative strength of syntactically restricted forms of IA and IR in number theory. The pivotal technical result, used in a generalized form also in later considerations, is the fact that (QF-IA) is closed under a I&reflection-rule for proper subtheories. This is Theorem 3.3; it implies immediately a theorem of Parsons [ 181, namely that (II$IR) is conservative over (PRA). Using the operator theories the relationships between (zO,+l-IR), (nt+,IR), and (J&:+,-IA) of [21] are easily obtained. In Part 4, I point out that these results are actually best possible, and give new proofs of conservation theorems due to Paris [16] and,

Fragments of arithmetic 35

independently, to Friedman. These results link theories with restricted forms of the number-theoretic collection principle CP (or, equivalently, the finite axiom of choice) to fragments. In particular we’ll see that (IT:-IR + Zy-CP) is conservative over (PRA). The proof-theoretic Lemma 4.9, central for the proof, allows the elimination of quantifier-free instances of the axiom of choice AC,

from certain kinds of normal derivations. To the two proof-theoretic tools (reflection-rule and QF-AC,-elimination) a third one is added in Part 5 which makes it possible to eliminate a weak form of Kbnig’s lemma WKL from certain kinds of normal derivations. The coordinated use of these tools yields the main conservation results mentioned above. The strengthening of Mint’s result, for example, is: (BT + Zg-ACb_ + L&IR- + WKL) is conservative over (PRA).

Though conservative extensions of (PRA) are the main focus of the paper, it is also a somewhat systematic proof-theoretic investigation of fragments of arithmetic; systematic with respect to results as well as with respect to methods.4 The methods, furthermore, seem to be particularly suited to eliminate function existence principles like AC, or WKL from proofs in purely universal theories. They exploit via Gentzen’s Hauptsatz and a form of Herbrand’s theorem the derivabil- ity of the antecedent of such a principle, i.e. of a Z$statement, to define explicitly functions (in the case of AC,, a function of the postulated kind). Thus, in a way, we use the constructive understanding of the quantifier combination (V’x)(Ely) in a classical context.

This work can serve as a basis for straightforward proofs of theorems charac- terizing the provably recursive functions, the provable ordinals, and the function- als needed for the no-counter-example-interpretation of fragments and related subsystems of analysis. That would complete the systematic investigation of fragments. To complete, in a different sense, the turn from foundationahly convincing) reductions to computationahly informative) ones, I intend to examine proper parts of (F) and (M), establish their conservativensss over proper parts of (PRA) and determine, among other things, their probably recursive functions. Some suggestive results are known; they are summarized at the end of the paper where I also discuss very briefly the interest of such further work.

1. General (proof-theoretfc) facts concerning subsystems of analysis

The (languages of) formal theories to be investigated below are described in Section 1.1, and elementary relations are formulated there. In Section 1.2 I discuss the basic properties of corresponding sequent calculi; they are crucial for the proof-theoretic work in Parts 3, 4, and 5.

36 W. Sieg

1.1. Languages and basic theories

The language LE(PRA) of primitive recursive arithmetic (PRA) is contained in the language of any theory considered here. It is extended by number-quantifiers and bound variables to the language 2’(Z) of elementary number theory (Z). Parameters ranging over unary number theoretic functions may be added; the resulting languages are denoted by 2?+(PRA) and 2?(Z). 2?‘, the language of analysis, is like 3’+(Z) but contains also quantifiers over the second sort. Finally, the languages may be expanded by additional operators (e.g. Skolem operators) and predicate symbols.

Formulas are built up from atomic and negated atomic formulas using the logical connectives &, v and, if appropriate, V, 3. The negation of complex formulas is defined; that can be done, as classical logic is treated throughout. The conditional + and biconditional * are also defined. I use 4, I,$ . . . as syntactic variables ranging over formulas. Prenex formulas are classified according to the structure of their quantifier prefix; they are called arithmetic, if no function quantifiers occur in them. If 4 is of the form (au,) + . * (au,,)+, where + is quantifier-free, Q either V or 3, and q either a number of function variable, the quantifiers (au,), . . . , (an,,) are called the leading quantifiers of 4. The 2”,- (II:-) formulas are those prenex arithmetic formulas which have at most k leading and alternating quantifiers and begin with an existential (universal) quantifier. The class of all prenex arithmetic formulas is denoted by II:, that of all quantifier-free ones by QF. Now let 4 be of the above form, but 4 is arithmetic and the indicated quantifiers are alternating function quantifiers. If the first quantifier is existential (universal), then 4 is said to be in _EA- (II:-) form.

Theories formulated in these languages will always include particular base theories. In the case of 3(PRA) it will be just (PRA) as formulated in [9, I]; i.e. the axiom 0’ # 0, the defining equations for all primitive recursive functions, and the axiom of induction for all (quantifier-free) formulas of (PRA) in the form

IA* 40 & (Vx < a)(& + 4~‘) + +a,

where the bounded quantifier has been introduced by primitive recursion as usual. Note that the induction principle can be formulated equivalently as a rule.

IR* ”

Clearly, if proofs of the hypotheses of IR” are available, the axiom IA* yields its conclusion 4a. To see that each instance of IA* can be obtained by IR*, apply the rule to the quantifier-free formula $a, i.e. 40 & (Vx < a)(& + 4~‘) + +a.

Proceeding to elementary number theory and its fragments, the induction principle is considered in the form

IA

Fragmentsofarithmetic 37

Here the base theory is (QF-IA), consisting of the axioms of (PRA); however, induction is taken in the form IA for all quantifier-free formulas. The theory obtained from (QF-IA) by replacing IA by IR* for QF-formulas is denoted by (QF-IR*). The relationship between these theories is straightforward-they are all equivalent. Furthermore, (QF-IA”) is conservative over (PRA); that is an immediate consequence of the cut-elimination theorem, and I shall point it out again in the next section. Full number theory (Z) is (I&IA). As in the case of (PRA), the induction principle can be formulated as a rule; and this leads to an equivalent theory (Z”). Two further equivalent formulations will play a role in Part 4; (QF-IA) is extended by number theoretic choice principles, namely the finite axiom of choice

FAC (Vx <a)PY) 4XY -+(3Y)wx<a)4x(YL

and the collection principle

CP (Vx< a)(3Y)~xY~(3z)(Vx<a)(3Y<z) &Y

for all arithmetic formulas 4. The theories are denoted by (II:-FAC), respectively

(IIOCP).

1.1. Proposition. (II:-FAC) and (II”CP) are equivalent to (Z).’

The strongest subsystems of analysis to be considered are actually conservative extensions of elementary number theory. The analytic principles assert the existence of functions:

AC W)(W) #M + (WW) &WC,

AC, (Vx)(3Y)~xY~(3f)(Vx)~xf(x),

CA (WW) (f(x) = I* 4x).

The base theory for subsystems is formulated in 2” and is called (BT); it includes the axioms of (QF-IA) (but possibly with second-order parameters in the defining equations for primitive recursive functions and the instances of IA) and the schema for explict definitions of functions

ED W)(Vx) f(x) = Lrxl,

where t is an arbitrary term of 2”; t,[x] indicates that a may occur in t and is replaced by x. Notice that (BT) proves each instance of QF-CA. Full analysis extends (BT) by the second order induction axiom

(W)(f(O) = 1 & (Vx)(f(x) = 1 ---, f(x’) = I) + Wx) f(x) = 1)

and one of the above function existence principles. It is well known that (CA) and (AC,,) are equivalent and that (AC) is a conservative extension of (CA) for

38 W. Sieg

#-sentences. Subystems are obtained by restricting the function existence principles to classes F of formulas of 3’; (F-AC) 1 for example denotes the part of (AC) in which only F-instances of the axiom of choice are available.

1.2. Theorem. (i) (IIOCA) 1 is equivalent to (@-CA) 1 . (ii) (IIZ-CA) 1 is equivalent to (II:-AC,) I . (iii) (II:-AC,) 1 is contained in (II:-AC) 1 . (iv) @i-AC) 1, (II:-AC) 1 and (@-AC) 1 are all equivalent.

These results are standard. Some more delicate facts are mentioned at the end of the next section; in particular, that even (z:-AC) r , the strongest of the above theories, is conservative over (Z).

1.2. Sequent calculi and conservation results

The logical calculi underlying the various theories are sequent calculi in which finite sets of formulas can be derived. r, A, . . . are syntactic variables for finite sets of formulas. If r is {&, . . . , c$,,}, then lr is {l&, . . . , -I+,,}. Sometimes I shall use r F A as an abbreviation for lr, A, i.e. +‘U A. All calculi contain the following axioms and rules:

LA c 4314 where 4 is atomic,

82 r, 4 c G

r,(4&$I) '

“i r3 4i r, (41v42)

for i = 1,2,

C(ut> r,4 r914 r *

When appropriate, rules for quantification are available. In the case of (Z), for example, we have these rules:

V r, 4a uw4x

where a does not occur in r;

3 r, 4t

cw4x'

The rules for function-quantification are formulated analogously. Rules for iden- tity are also added. As usual, derivations are built up in tree-form. 0, E, . . . range as syntactic variables over derivations.

All of this is completely standard, and I refer to [22] for notions like “length of a formula 4 (derivation D)” (abbreviated by 141, respectively 101) and basic


results like weakening, substitution, and inversion for &. The crucial facts used below are the cut-elimination theorem which is provable for all the calculi employed here and the subformula property of cut-free or normal derivations.

Two additional lemmata will be appealed to often; so I state them explicitly. Both are proved very easily by induction on (the length of) normal derivations. The first lemma concerns the invertibihty of V.

1.3. Lemma (V inversion). If D is a normal derivation of A, (V’x) +x, then there is a normal derivation E of A, &; c is a new parameter not used in D and (El s IDI.

The second lemma is a form of Herbrand’s theorem and concerns the invertibil- ity of 3 in restricted contexts. (I also refer to this lemma as the ‘Herbrand-type lemma’.) For its general formulation I use a crude classification of prenex formulas: 3, and V1 denote the class of purely existential and purely universal formulas respectively. a2 (V,) is used for the class of prenex formulas whose prefixes consist of existential (universal) quantifiers followed by universal (existential) ones. Z13 (V,) and 3, (V’,) in general are explained similarly.

1.4. Lemma (3 inversion). Let A contain only 3,-formulas and let +a be in QF; if D is a normal derivation of A, (~x)c#cc, then there is a finite sequence of terms

t l,***, t,, and a normal derivation E of A, &, . . . , @,,. Furthermore, lEl~lDl.

Up to now I considered only logical rules. But as I am working in the sequent calculus, it is necessary to give a sequent formulation of the induction rule, i.e.

IR A, 40 A,&-+&

A, &a

where a must not occur in A. (nz-IR) denotes the formulation of elementary number theory with this rule - when A U {C#KX} contains only arithmetic formulas.

1.5. Proposition. (LIZ-IA) is equivalent to (170IR).

Proof. (IT:-IR) is obviously contained in (nz-IA). To establish an arbitrary instance of IA using IR consider the following derivation (double lines indicate that a few logical steps have to be taken):

190, -&#JQ + &‘), (@ + da’)

l(VX)(N + W), -@Jo, 40 +Jo, 1(Vx)(& * W), (& + da’)

14o,-lwx)(~x+w),4a

Ma ~'x)(~~-+W)+-~

This completes the proof. El

40 W. Sieg

Clearly, (ZR) is also equivalent to (IIOIR). But this equivalence is preserved under restrictions of the induction rules to zO,-formulas only if the elements of A

are required to be in z”,.” IR thus restricted is called Xz-IR. II:-IR denotes IR when the formulas in A U{#a} are required to be in II”,.

1.6. Proposition. (i) (zz-IR) is contained in (soIA), and (CoIR) is contained in (LIZ-IA)

(ii) (II:-IA) and (X0,-IA) are both contained in (zi+i-IR). (iii) (20IR) is contained in (II:+,-IR).

These are the immediate relationships between fragments of arithmetic formulated with IA, respectively IR for subclasses of the arithmetic formulas. Comple- menting (and more delicate) conservation results are proved in Part 3. Here let me just mention some easy facts which follow almost immediately from the cut-elimination theorem.

1.7. Theorem. (i) (QF-IA) is equivalent to (QF-IR) and conservative over (PRA). (ii) (BT) is conservative over (QF-IA) and thus over (PRA). (iii) (HZ-CA) r is conservative over (Z).

Recall that (I&CA) - a theory like (II:-CA) ] except that IA is available for all formulas of LE2- is not conservative over (Z); indeed, the consistency of (Z) can be proved in (LIZ-CA).

Using the Herbrand-type lemma one can sharpen the conservation result for (QF-IA), equivalently (QF-IA*), and for extensions of (QF-IA) which prove at least the same Hz-sentences.

1.8. CoroIlary. If (QF-IA) proves a @-sentence (Vx)&) &y, then there is a primitive recursive function f, such that (PRA) proves &f(a).

The last conservation result I want to mention can also be established by purely proof-theoretic means. (See [5].)7

1.9. Theorem. (z$AC) 1 is conservative over (II:-CA) 1 for I&sentences.

However, in addition to cut-elimination and the Herbrand-type lemma, one uses Skolem functions and a special way of eliminating instances of the quantifier- free axiom of choice, i.e. QF-AC. This technique will be exploited below in Parts 4 and 5.

2. Skolem functions and operator theories

Some of the properties of sequent calculi hold only for derivations whose end-sequents contain syntactically restricted formulas; e.g. the Herbrand-type


lemma. To overcome such limitations operator theories are formulated in Section 2.1. These theories can be interpreted in suitable fragments of arithmetic; that will be taken up in Section 2.2.

Operator theories were introduced as auxiliary and, as it turned out, versatile tools for proof-theoretic investigations in [5]. They were subsequently used also in [3] and [2].’

2.1. Operator theories

In the remarks after Theorem 1.9 I hinted at the use of Skolem-functions for generalizing an elimination theorem for QF-instances of the axiom of choice to arbitrary arithmetic instances. Operator theories bring out this and similar generalizing steps most lucidly: they contain Skolem-functions and permit the elimination of quantifiers from (subclasses of) the arithmetic formulas. For example, an operator theory (OT,) will be formulated in which all II:-formulas can be shown to be equivalent to quantifier-free ones. This will be sufficient for the investigation of fragments of elementary number theory; for subsystems of analysis one needs also abstraction operators to be able to formulate the function existence principle ED as a purely universal axiom. All of this requires a detailed description of the syntax of operator theories. I will give that now for the first-order theories (OT) and (OT,) and subsequently indicate the necessary additions and changes for the formulations of appropriate second-order theories (OT*) and (OTZ).

The language 5?(OT) is LX’(Z) with an expanded inductive definition of terms:9 (i) All number constants and parameters are number terms.

(ii) If f is a function constant and tl, . . . , t,,, a sequence of number terms of appropriate arity, then f(tl, . . . , t,,,) is a number term.

(iii) If 4a is a quantifier-free formula, then VX.(C#JX) is a number term. To delimit parts of z(OT) a complexity measure for ‘u-expressions’ and

formulas is introduced. The measure for u-expressions corresponds directly to the rank of c-expressions as defined in [9, II]. v-expressions are finite sequences of symbols of L!Y(OT) which are either v-terms or are obtained from v-terms by replacing parameters by variables not occurring in them. ux.(x = a +b) is a v-term, whereas VX.(X = a + y) is a v-expression, but not a v-term. A v-expression t is subordinate to a v-expression VZ.($Z) iff t is a part of +z and z occurs in t. The v-depth of v-expressions t is defined recursively by

if t does not contain a v-expression subordinate

v-dp(t) = to t,

p + 1 if t contains v-expressions h, 1 s i G n, subordinate to t and p = max,,,_ v-dp(ti).

The v-depth of a formula 4, v-dp(+), is simply the maximum of the v-depths of u-expressions occurring in 4, or 0 in case no u-expression is part of 4.

42 W. Sieg

2.1. Remark. v-dp has the following obvious property: if t” is obtained from t by substituting for (all occurrences of) a term s another term s*, then v-dp(t*) = v-

dp(t).

9(OT,) is defined as that part of L!?(OT) which contains only formulas of v-depth less than or equal to n. Clearly, 3(OT,) is just L&‘(Z). Notice that because of Remark 2.1 the set of terms of 5?(OT,) is closed under substitution.

The theories (OT,,) serve as auxiliary theories; their crucial role is to allow to prove that any II:-formula of 3(Z) is equivalent to a QF-formula of 3?(OT,). (OT,) is just (QF-IA); for nf 0, the axioms of (OT,) are those of (OT,) together with the Skolem-axioms

SK, 4a --, 4vx44x)

where 4a is a QF-formula of %‘(OT,) with v-dp(4a) < n. The induction principle is available for quantifier-free formulas of 9(OT,). For emphasis, I also write (QF-OT,,) for (OT,,). If IA is included for all formulas in FE: L??(OT,), then the theory is denoted by (F-OT,). To establish that (OT,) does fulfill its ‘crucial role’ I define recursively a translation on which associates with each formula in 5?(Z) a formula in L?(OT,); in particular, it will associate with II:-formulas quantifier- free ones. cr, transforms a given 4 of L!?(Z) first into its special prenex normal form in which sequences of quantifiers of the same kind are contracted to a single one. (The obvious classes of formulas are denoted by *Ci and *II%.> These formulas are then treated as follows. a,, leaves atomic formulas unchanged and commutes with the sentential connectives; on quantified formulas it operates according to the rules:

(i) If I/J is (3x) 4x and u,(4a) is 4”a, then

o,(G) is 4”vx.(4”x) if 4a is in *II:, 0 < m < II,

(3x) 4”x otherwise.

(ii) If + is (Vx)4x and u,,(h) is 4”a, then

o,(4) is 4nvX.(14nx) if 4a is in *z”,, OG m < n,

wx)4nx otherwise.

Let us quickly consider one example. Take as J, the formula (Vx)(3y) 4xya with 4 quantifier-free. u,,((3y)4bya) is 4bvy.(4byu)u and a,,(+) is consequently the formula

4 vx.(14xvy.(4xyu)u) v2.(4vx.(14xvy.(4xyu)u)zu)u.

The v-depth of u,,($) is incidentally 2. The observations in the next proposition are immediate from the definition of

a,, and of (OT,,).


2.2. Proposition. For each formula Q, in a(Z):

(i) (OTJ proves (4 t-, o,(4)). (ii) If 4 is in II”, (s:“,), then cr,,(+) is in ITO,,, (.ZL_,), indeed in the more special

classes *II:,, (*JYE+).

So we actually have that all Hz-formulas of L?(Z) are in (OT,) probably equivalent to QF-formulas of L&‘(OT,).

Now I am going to formulate the second order version of the operator theories

(OT,). The language 9(OT’) is L?‘, where however the inductive definition of number

and function terms is expanded as follows: (i) All number [function] constants and parameters are number [function]

terms. (ii) If f is a function term and tl, . . . , t,,, a sequence of number terms of

appropriate arity, then f(tl, . . . , t,,,) is a number term. (iii) If & is in QF and does not contain f-expressions or A-expressions with

occurrences of the parameter a, then XL(+) is a number term. (iv) If t is a number term, then Ax.(t,[x]) is a function term. The additional restriction in (iii) ensures that VX.(+X) does not contain subordi-

nate f-or A -expressions; where an f-expression [A-expression] is an expression of the form f(t) with f a function parameter [hx.(s,[x]>(r>]. The v-depth of V- expressions [formulas] of 9(OT2) is defined as that for v-expressions [formulas] of 9(OT). L!?(OT$J is that part of .9?(OT2) which contains only formulas of v-depth less than or equal to n. Clearly, .3?(OTg) is just s2 expanded by A-terms.

Remark 2.1 applies also to 3’(OT~)); consequently, the set of terms of 3(OTz) is closed under substitution.

The theory (OT;) is (BT) except that the function existence principle ED is formulated with A-terms as a purely universal statement.

ED, WI AY.k[Yl)(x) = cx[xl.

I refer to this principle also as QF-AA. For n # 0, the axioms of (OT:) are those of (OT:) together with the Skolem-axioms

where 4a is of v-depth less than n and satisfies the additional restrictions formulated in (iii) above (so that VX.(~X) can be formed). Exactly as in the case of the first-order theories, the induction principle is available for quantifier-free formulas of Y(OT3. For emphasis, I also write (QF-OT”,) for (OTZ). If IA is included for all formulas in FE 3?(OT2, then the theory is denoted by (F-OT3.

Using an obvious modification of a,,, also denoted simply by a,,, the analogue of Proposition 2.2 is immediate.

44 W. Sieg

2.3. Proposition. For each formula C#J in Z(Z):

6) (OT? proves (++ c,,(4)). (ii) If 4 is in III”, (Z”,>, then CT,,(&) is in II:,, (X”,_,), indeed in the more special

classes *II”,;, (*CO,+).

We have consequently, as before, that all II:-formulas of 5’(Z) are in (OT”,) provably equivalent to QF-formulas of L?(OTi). Two further observations will also be important.

2.4. Remarks. (i) The theories (OT,,) and (OT2 are purely universal theories; i.e. more precisely, equivalent to purely universal theories by taking the induction principle in the form IA*.

(ii) The induction principle can be formulated using the induction rule; the resulting theories are denoted by (F-OTR,) and (F-OTRZ). In case F = QF, the theories (OT,) and (OTR,), respectively (OT3 and (OTR:) are equivalent.

2.2. Interpretation of operator theories

The operator theories serve as intermediaries: in the next parts it will be seen that fragments of arithmetic and weak subsystems of analysis can be embedded into (extensions of) (OT,), respectively (OT3; in this section I want to show that the operator theories are reducible to fragments of arithmetic. The straightforward considerations will be carried out for the (OT,) first. For this purpose I define a second translation T,, by induction on formulas of Pz’(OT,), interpreting the Skolem-operator as the least-number-operator. I give this definition, however, only for a certain class of atomic formulas, as r,, commutes with all logical connectives and as all formulas can be assumed to be normalized with respect to V. (I.e. they are equivalent to formulas in which v-terms occur only on the right- hand-side of equations). The basic rules are:

(i) If + is s = t and v occurs in neither s nor t, then

~~($11) is s = t.

(ii) If I,!I is s = t, v does not occur in s, t is of the form VX.(C#LX), and +“a is

r,, (&), then

T,(+) is (c$%& (Vy<s)i~“y)v((Vx)71$“x&s=O).

The syntactic complexity of r,-translations of QF-formulas in LE(OT,) is determined next.

2.5. Lemma. Let t+b be a QF-formula of s(OT,); if v-dp($) = m in, then T,,(G) is in At+,.

Proof (by induction on m). If m = 0, T,,(G) is I,IJ and the claim is trivially satisfied.

Fragments of arirhmetic 45

For the induction step (from m - 1 to m) assume first that + is normalized with respect to u. Thus I,!J is a propositional combination of equations of the forms described above. If v-dp($) = m, m # 0, at least one of these equations is of the form s = ux.(&), such that v does not occur in s and w.(c$x) contains a subordinate v-expression of v-depth m - 1. Thus v-dp(&) = m - 1 and 7,(&z) is in AO, by induction hypothesis. But then it is clear that ~,,(s = vx.(4x)> is in II:. The T,,-translations of the other equations are also in IT:; their propositional combination is consequently in _E”,+l and IIL+l, thus in AC,,. The general case is proved by a simple modification of this argument. Cl

The 7,-translations of quantifier-free formulas of v-depth less than n are in II:. The Skolem-axioms SK,, are consequently provable from the least number principle for II:-formulas, which in turn is implied by HZ-IA. (For a proof of this fact, see Section 4.1.) But notice that the universal closure of the T,,-translations of the SK,, are in 170,+1, and that the T,,-translations of QF-instances of IA are instances of ZTO,+l-IA.

2.6. Proposition. Let 4 be any formula in Z(OT,). If (QF-OT,) [(QF-OTR,,)] proves 4, then 7,,(4) is provable in (IT:+,-IA) [(ZO,+,-IR)].

The considerations leading to Proposition 2.6 can be modified to obtain an analogous, but restricted result for (QF-OTZ) and (QF-OTRZ). The obvious extension of the T,,-translation includes now as a first step the elimination of all h-terms; the latter occur only in contexts Ax.(t,[x])(s) and are replaced (under the usual precautions) by t,[s].

2.7. hwdion. Let 4 be any e+,-formula of Z(Z). If 4 is provable in (QF-OT’,) [(QF-OTR:)], then 4 is provable in (flz+l-IA) [(_Z,O+,-IR)].

Proof. “4 is provable in (QF-OT:)” means that a sequent lA, 4 has a (normal) derivation in the sequent calculus and that A consists only of QF-OT:-axioms. Because of Proposition 2.3(i) we can assume to have a derivation of -IA, a,,(4). By V-inversion (applied to u,,(4) to turn it into a _Zy-formula), g-inversion (applied to function quantifiers in -IA), and replacement of function parameters by a fixed primitive recursive function, we obtain a derivation of lA*, a,(4)*. But then it is easy to establish 1A *, 4. The T,,-translations of the elements of A * are (or can easily be derived from) axioms of (Hz+,-IA).

The argument for the bracketed part of the proposition is similar. 0

3. Rule and axiom of induction

As far as theorems are concerned, it does not matter whether full number theory is formulated with the induction axiom or the induction rule.” However,

46 W. Sieg

for fragments of number theory, obtained by considering only instances of induction with formulas in subclasses of n”,, there is a difference in terms of theorems. That phenomenon was investigated by Parsons [18, 221. The central result for establishing (non-trivial conservative extension) relations is the reduction of (Hz-IR) to (QF-IA). I give a new and, in my view, simpler proof of this fact; the proof can also be generalized to stronger (second-order) theories, and that will be important for Part 5. In Section 3.1, the main work is directed towards establishing a reflection rule, by means of which it is easy to obtain - in Section 3.2-P&sons’ central result and its generalizations. Finally, in Section 3.3, the most significant relations between fragments of arithmetic are proved.

3.1. Elementary relations and reflection

In Section 1.2 I defined (170IA) and (X0IR) as two types of fragments of arithmetic. Parsons [22] considered many more formulations of ‘n-quantifier induction’ and summarized their relationship as follows: “. . . all the natural formulations of ‘n-quantifier induction’ are reducible to one of two (for nf 0) non-equivalent forms: the axiom of induction restricted to n”, (or, equivalently, _Z”,) formulae and the rule of induction restricted to .X:-formulae” (p. 466). Proposition 1.6 stated the most immediate relationships between these standard formulations. Some further easy and helpful facts are formulated in the next proposition.

3.1. Proposition. (i) (II:-IA) is equivalent to (X:-IA) for all n EN.

(3 (%itl-IR) P ro2)es the consistency of (170IA), for all n EN, n > 0.

Proof. Ad(i). To show that (noIA) is contained in (Zt-IA) it suffices to prove each instance of II:-IA in (20IA). So let +b be in no,; l+(b z a) is then clearly in C”,. Consequently,

-$(b-0) & (Vx)(l~(b-x)-tl~(b-x’))~l~(b-a)

is an instance of ,X:-IA. Setting a = b one obtains

(*) i+a & (Vx)(-@(a Ax) --$ i+(a LX’)) + 740.

The second conjunct in the antecedent of (*) is in (QF-IA) equivalent to (VX>(C$X + 4x’), and the @-instance of the induction axiom has been established in (X:-IA).

The converse inclusion is established in a similar way. Ad(ii). Let Pr, be the canonical proof predicate for (II:-IA), Sn+l the predi-

cate expressing that a is (the code of) a Co,+,-formula and Tr,,, the partial truth-definition for Zz+l-formulas.ll The first two predicates are primitive recursive, the last one is in Z”,+l. Exploiting (the formalization of) the cut-elimination theorem one can prove in (TO,+,-IR).

WY)WX)@,+~X & kyx + Tr,++).


As the adequacy of Tr,,, is also provable in (Zz+l-IR), we actually established the X:“,+, -reflection principle for (170IA), i.e.

(3~) Pr,y r4’ + 4

for all Z”,,, -formulas C#J. l2 Indeed, (II:-IA) is provably closed under numerical substitution; thus the 170,+2 -reflection principle can be inferred. Under weak conditions on a formal theory T, certainly satisfied by (HE-IA), the ZIY-reflection principle is equivalent to the consistency statement for T. Thus, (Zz+i-IR) proves the consistency of (Lfz-IA). Cl

The (non-) containment relations expressed in Propositions 3.1 and 1.6 are indicated in the following diagram (where nf 0):

(Z”IR) - (ZI;-IA) _ @I:-IA) .

The precise relationship between (II”,,, -1R) and (Zz-IR) and that between (II:-IA) and (,Xz-IR) will be determined in Section 3.3. Here I want to establish the consistency of certain parts of (QF-IA) in (QF-IA). To begin with let me just point out that the restriction to n > 0 in Proposition 3.1 (ii) is necessary, as it will be shown below that (L$-IR) is conservative over (QF-IA).

The reason why the argument for 3.l(ii) fails in case n = 0 is this: one cannot introduce in (_Ey-IR), or for that matter in (ny-IA) or ($&IR), a valuation function for the terms of (QF-IA). But now consider the theories (QF,-IA), m > 2, which contain only function symbols and defining axioms for the functions of Grzegorczyk’s &,. (Thus all these theories contain at least the Kalmar- elementary functions.) %5m+1 contains a universal function for 8,,, and thus one has a valuation function for the terms of (QF,-IA) in (QF,+,-IA). This is the basis for an adequate quantifier-free partial truth-definition in (QF,+,-IA) for the quantifier-free formulas of (QF,-IA).

3.2. Theorem. Let r U (4) contain only formulas in the language of (QF,-IA); let the elements of r be in Ill;, let 4 be quantifier-free. If T:= (QF,-IA) +r and S: = (QF-IA) + r, then S proves (3~) Pr,y r4’ + 4.

As above one can use the provable closure under numerical substitution to infer the @-reflection principle. Notice that the theorem - together with the fact that (QF-IA) is conservative over (PRA) - implies the first, classical results obtained in the pursuit of Hilbert’s program; namely, the consistency theorems of Ackerl mann, von Neumann, and Herbrand for elementary number theory with quantifier-free induction and only a finite list of primitive recursive functions.

The theorem will be used now to show that S is closed under a IL&reflection rule.

48 W. Sieg

3.3. Theorem. Let S and T be as in Theorem 3.2, and let C&I be a Il$fotmula in

Z(QF,-IA). If S proves (Vx)@y) Pr,y r@‘, then S proves (Vx)@.

Proof. By the provable closure under numerical substituion it is really sufficient to establish the claim when 4a is a Zy-formula (3~) +ya. So I shall show how S establishes (Vx)(3y)$yx from the hypothesis that (3y)+ya is provable in T. (That my informal argument can be formalized in S will be quite obvious.) Assume now that 0, is a normal derivation of

+, (3~) ha;

A contains only axioms of T. T is a purely universal theory; consequently, the Herbrand-type lemma is applicable and yields a sequence of terms tl, . . . , t,, together with a normal derivation of

(*) A, %a, . . . , @,,a

Using definition by cases a term t, is obtained which satisfies

I 0 otherwise.

From this equation, (*) and perhaps some additional T-axioms one can get a derivation E, of

without loss of generality it can be assumed that (i) E, is normal, and (ii) r contains only T-axioms.

The above is a uniform and effective procedure of obtaining for arbitrary a from a T-derivatiion of 4a the term t, and the derivation E, satisfying (i) and (ii). Now observe that the terms occurring in E, must be contained in the class of terms which is obtained from the terms in the original derivation 0, by closing under substitution and definition by cases. Grzegorczyk’s 8, is closed under these operations; thus all the terms (in E,) are available in (QF,-IA), and $&a is provable in T- provably in S. By Theorem 3.2, $&a is provable in S; so are (Zly)rCrya and, finally, (Vx)(3y) I&X, i.e. (Vx) C&C. 0

3.4. Remarks. (i) The considerations establishing Theorems 3.2 and 3.3 can be carried out when (QF-IA) is replaced by (QF-OT,) and (QF,-IA) by (QF,-

OT,); the latter theory is defined in an obvious way. One just has to observe that


the arguments and notions used in them can be relativized to finitely many function-parameters and fixed (non-primitive recursive) functions. In particular, (the results concerning) the Grzegorczyk hierarchy and the partial truth-definition can be so relativized.

(ii) If T is further extended by a set A of _X$sentences, then the theorem (and its generalizations formulated in (i)) can be proved in the following form: if S proves (Vx)(3 y) Pr -r+ay r@ , then S +A proves (Vx)+x. For assume the II;- formula &r has been proved in S to be provable in T+ A ; i.e. I& &. . * & $,, +4a is proved in S to be provable in T, where $1, . . . , $,, are elements of A. This formula, however, is equivalent to a @!-formula and, by Theorem 3.2, provable in S. Consequently, (Vx) 4x is provable in S + A.

3.2. Consequences of reflection

The last theorem allows us to prove the first proof theoretically interesting fact; namely, that (II$IR)13 is conservative over (QF-IA) and thus over (PRA). This was proved by Parsons [17, 181 and later independently by h4inc [13]. Actually, I prove now a generalized form of Parsons’ theorem; more significant generalizations are described below. (See Section 3.3.)

3.5. Theorem. Let r be a set of X&sentences in 9?(Z). (II$IR)+r is conservative over (QF-IA) + IY In particular, (IT;-IR) is conservative over (PRA).

Proof. (by induction on derivations D in (III;-IR)+ r). All cases are perfectly routine except when the last rule applied in D is II:-IR; D must then be of the form

D, and 0: can be assumed by induction hypothesis to be derivations in S : = (QF-IA) + r, i.e. A contains only axioms of S. Indeed, only finitely many axioms are involved and A is contained in T:= (QF,-IA)+r’, where r” E r nL!?(QF,-IA). Consequently, S proves Pr, rDol r401, PrT rD:l r& + &i” ) and - employing the usual procedure - Pr, rD,l r~6l for arbitrary b. But then S proves also (gy)Pr= ryl ‘46’ and (Vx)(3y)PrTy r+Xl . The hypotheses of Theorem 3.3 are satisfied, and we can conclude that S proves &/x)4x and clearly also &. q

This theorem is used now to relate (@-IA) and (QF-IA); these two theories prove the same TI$-sentences.

50 W. Sieg

3.6. Theorem. Let r be a set of $$sentences in Y(Z). (II:-IA) + r is conservative over (QF-IA) +I? for @-sentences. In particular, (#-IA) is conservative over

mw.

The theorem is established by showing that fly-instances of the induction axiom can be analyzed away from proofs of @-sentences; they can be analyzed away in favor of L&instances of the induction rule and thus, in view of Theorem 3.5 and the fact that (QF-IA) and (QF-IR) are equivalent, in favor of QF-instances. For this purpose it is convenient to have instances of L&IA in the form

(VY) &JO & (Vx)(VzMy)(+yx + 44 --, (VY) ha.

Let us also agree that A[-Jly-IA] stands for A together with a finite number of negated instances of fly-IA (in the above form). The theorem follows immediately from Lemma 3.7.

3.7. Lemma. Let A contain only @-formulas; if A[l@-IA] has a normal derivation, then A has a II;-IR-derivation; i.e. A can be proved in the sequent calculus expanded by II:-IR.

Proof (by induction on the length of normal derivations). I concentrate on the central case when an instance of fly-IA is introduced by the last rule. The given normal derivations must then be of the form

D 1 1 A[lII:-IA], x NT@-IA], lWy)@a I E

A [-&I:-IA]

where x is the antecedent of the instance. of IA. From D one obtains by &-inversion normal derivations D1 and D2 of

A[+%IAl, WY> 4~0,

respectively of

A[++IA], (Vx)(VzMy)(+yx + ~JZX’).

D1 and D2 are not longer than D, and their endsequents satisfy the condition on the complexity of formulas. The induction hypothesis yields a Z’&IR-derivation of

(1) 4 W~)llryO

and of

(2) 4 WXWZ)(~Y)(~YX --, 4zx’).


From (1) and (2) we can infer by (a little logic and) II;-IR the sequent

(3) A, WY )~@a.

Now apply the induction hypothesis to E and obtain a II;-IR-derivation of

(4) A, ~(VY)XY~.

The cut-rule C allows us to infer A from (3) and (4); consequently, we have a @- IR-derivation of A. 0

3.3. Generalizations

In Remark 3.4(i) I pointed out that Theorems 3.2 and 3.3 can be suitably generalized for the operator theories. The consequences of these theorems, formulated in Section 3.2, can be generalized in a similar manner.14

3.8. Proposition. Let r be a set of lY$sentences in z(OT,). (U:-OTR,) +r is

conservative over (QF-OT,) + r.

Now we have a simple argument for the equivalence of @I:+,-IR) and

(z:O,+,-IR).

3.9. Theorem. Let r be a set of X:+,-sentences in 2?(Z). @I:+,-IR) + r is conser-

vative over (zO,+,-IR) + r. In particular (L!E+,-IR) is conservative over (,Yz+,-IR).

Proof. Let 4 be any formula in 5’(Z) and assume that 4 is provable in (ETo,+,-IR) + F. Thus a,, (4) is provable in (L$-OTR,) + a,, (I); indeed, by Proposi- tion 2.2 4 is provable in that theory and consequently in (QF-OT,) + r appealing to Proposition 3.8. By Remark 2.4(ii), (QF-OT,) and (QF-OTR,) are equivalent; so Proposition 2.6 yields the claim. 0

Proposition 3.8 may be used to establish the appropriate generalization of Theorem 3.6.

3.10. Proposition. Let r be a set of X$sentences in LL’(OT,,). (@-OT,)+r is

conservative over (QF-OT,) + r for I$sentences (in L??(OT,)).

The next theorem is proved in a similar way as Theorem 3.9: reduce (LIZ+,- IA) + I to @I:-OT,) + u,,(r); then use Propositions 3.10 and 2.6.

3.11. Theorem. Let r be a set of 2z+2-sentences in 2(Z). (ITS+,-IA)+r is

conservative over (S:“,+,-IR)+r for II:+,-sentences. In particular, (Ez+,-IA) and (_XO,+,-IR) prove the same Il0,+,-sentences.

52 W. Sieg

In Part 4 it will be shown that this conservation theorem is best possible. For the investigation of weak subsystems of analysis and their relation to fragments of arithmetic appropriate generalizations of propositions 3.8 and 3.10 to second- order operator theories will be crucial. Here we can note the following fact.

3.12. Proposition. Let r be a set of I$-sentences in 9(OTz). (i) (@-OTRZ) + r is conservative over (QF-OTZ) + r.

(ii) (17!$OT2,) + r is conservative over (QF-OT’,)+ r for @j-sentences (in

Jf(OT:N.

4. (Number-theoretic) choice and collection

Are the main conservation results of Part 3 best possible? That is an obvious question to ask; Parsons [ 181 introduced number-theoretic analogues of set- theoretic choice and collection principles to answer it positively. In Section 4.1 -which is almost purely expository following [18] and [15] - it is shown that (X”,+,-CP) is contained in (s”,,, -IA), but that there is an instance of ,X”,+,-CP not provable in (s”,,, -1R). The collection principle was later also used for (or motivated by) quite different purposes:

(i) to characterize elementary end extensions of countable models of weak fragments of (Z) in

(ii) to determine

WI; (iii) to formulate

r71. In any event, the

D51; the proof-theoertic complexity of combinatorial statements

weak theories, adequate for parts of mathematical practice

main technical result of these investigations was a conservation theorem: (z”,,, -CP) is conservative over (nt-IA) for II:+,-sentences; it was established by model-theoretic arguments. I establish this and closely related results for second-order theories by straightforward proof-theoretic means in Section 4.2.

4.1. Elementary relations

Elementary relations between restricted versions of (nz-IA), (nz-CP), and (I70FAC) will be explored here. The main facts are: (X:“,+,-CP) is equivalent to

(s:+l-FAC); G”,+, -IA) proves the consistency of (,X0,+,-CP); there is an instance of zz+,-CP which is not provable in (,Xz+1 -IR) and thus not in (zz-IA). The latter fact shows not only that the conservation results of Sections 3.3 and 4.2 are best possible, it will also be useful in Part 5. Let me first notice an obvious fact.

(*) If 4 is a _Xz-formula, then (Vx < a)4 is in (20FAC) and (20CP) provably equivalent to a zO,-formula. Secondly, I used the least element principle LEP and its relation to the induction principle already in a remark before Proposition 2.6.


This relationship will be proved now; the principle itself is given by

(3x)& + (3x)(+x & (VY <x)14y).

4.1. Proposition. (i) (Zz+,-CP) is contained in (XE+l-IA). (ii) (IT:-LEP) is equivalent to (20IA). (iii) (X”LEP) is equivalent to (170IA). (iv) (_Zz-IA) is contained in (Zz+,-CP).

Proof. (i) is established by induction on n. Assume, for n = 0, the antecedent of an instance of QF-CP (QF-CP is obviously equivalent to J$CP), i.e. (Vx <a) (3~) +xy. Let I@ be the formula b <Q --, @z)(Vx < b)(3y < z)&y, which is in (QF-IA) equivalent to a _Xy-formula. Clearly, $0 and t,hb ---, +b’ are provable in (QF-IA). X:-IA yields (Vx)+x; thus, the consequent (3z)(Vx < a)(3y < z)4xy, of QF-CP is provable in (J$-IA). The argument for n # 0 proceeds in the same way except that the induction hypothesis is used to see that $b is equivalent to a Co,+,-formula; here, one appeals to (*) above.

For (ii) I establish first that (CoIA) is contained in (II:-LEP). So assume 40 & (VX)(~X + 4x’) and +VX)~X, when 4 is in 25:. II:-LEP can be applied to the second assumption to yield a c such that -I&&C = 0 or +c- 1 & l&z, contradicting the first assumption. For the converse, consider (3x)+x, with 4 in II:. The formula +!KX defined as (Vz < a)-~& is (by (i) and (*) in case n Z= 1) in X”, and _ZE--IA yields

90 & (Vx)(* + W) + (Vx)*;

(Vx)+x is equivalent to (Vx) l&x and thus to 1(3x)4x. $0 is trivially provable; so one obtains first

and then -$VX)(+!JX + t/d). The latter formula is easily seen to be equivalent to the consequent of the LEP-instance with 4.

(iii) is established in a parallel fashion. (iv), finally, is proved by induction on n. If n = 0, just notice that (Zy-CP) is by

definition (QE-IA+Xy-CP). In the induction step (JZ~+,-IA) has to be shown to be contained in (Z”,+,-CP); or, equivalently, @I:+,-CP): by induction hypothesis we know that (2’0IA) and thus, by Proposition 3.1 and (iii), (Zz-LEP) is contained in (II:+,-CP). So assume

(3~)tiO & (Vx)((3y)W + @Y)+YX’) and -1(Vx)(3y)$yx,

where I,!J is in II:. If a is such that l(qy)&~, then

(Vx <a’)@y)(tix V(Y = 0 & ~(~Y)W))

and nz+,-CP gives

(3z)(Vx <a’)(3y < z)(lLyx v(y = 0 & l(VY)@X)).

54 W. Sieg

Clearly, we can infer

(3r)(Vx < a’)(@y)rlyx++ @Y < z)$Yx)

and thus

(32)-7(3y cz)+ya.

This formula is, using (*) again, equivalent to a Zz-formula. But _Xz-LEP is available and a can be chosen as the smallest number with T(~~)I+!Ju, contradicting the antecedent of the SO,+,-IA-instance considered. 0

The equivalence between the collection and the finite choice principle, restricted to ZO,-formulas, can be readily established now.

4.2. Proposition. (2”CP) is equivalent to (Xz-FAC).

Proof. Assume (Vx < a)(3y)&y with 4 in II”,_,; IIz_l-CP yields (3z)(Vx <a) x

(3y <z)&xy. By Lemma 4.1 (iv) and (ii), II:_,-LEP is available in (XOCP). So one can use the bounded p-operator to code up a sequence y, s.t. (Vx < a)+x(y),.

From the same assumption we can infer with Sz-FAC that (3y)(Vx < u)+x(y),. Let z be y and use the fact that (y), c y to obtain immediately (3z)(Vx < u)(3y < z)4xy. cl

The containment relations formulated in Lemma 4.1 (i) and (iv) are proper ones; I show this lirst for (i).

4.3. Theorem. There is an instance of _‘Zz+,-IA which is not provable in (IZ”,+,-

CP).

Proof. This follows immediately - via Godel’s second incompleteness theorem and Proposition 4.2-once the ZE+l-reflection principle for (Zz+,-FAC) has been established in (X”,,, -IA). Let me argue informally and assume that we are

given a normal derivation D of -rA[lIT”FAC], x-where x is a ZE+r-formula and where A contains only axioms of (QF-IA). From D one can obtain normal derivations of lA[A,], x and of -rA[lC,], x where Ai and Ci are the antecedents, respectively consequents of finitely many FAC-instances. Exploiting the inverti- bility of V we have normal derivations of -IA[A~], x and lA[lCI], x where now

all formulas are in _Z”,+, -form. This transformation can be carried out formally in

(QF-IA): using the partial truth-definition Tr,+l an easy XO,+r-induction shows

that the disjunctions of the formulas in the above sequents are true. By the

adequacy and a little logic we actually have that

is provable in (Zz+, -IA); consequently, x is provable in this theory. q


The main claim in this argument can obviously be proved when the theories are extended by _XE+,-sentences.

4.4. CoroUary. Let r be a set of Z11,2-sentences. (Z~+,-IA)+T proves the ZIz+,- reflection ptinciple for (Z”,+,-CP) + r.

Making use of Skolem-functions in a similar way as in the proof of the next theorem one can modify the argument for Theorem 4.3 and observe:

4.5. CoroIIary. Let r be a set of S’0,++entences. (Z”,+,-IA)+r proves the II”,+,- reflection principle for (QF-IA) + r.

Leivant [12] established, in a different way, a closely related observation, i.e. he showed that (II:+,-IA) proves the Zz+, -reflection principle for (QF-IA). Clearly, the provability of the II:+, -reflection principle follows immediately.

4.6. Remark. Leivant concluded that no consistent extension of (II:+,-IA) has a

II:+2 axiomatization; indeed, by our simple additional observation or by Corol- lary 4.5, such a theory cannot have a _Ez+,- axiomatization either. In terms of arithmetical complexity, induction is consequently optimal for axiomatizing fragments of arithmetic.

Now I show that the containment relation of Lemma 4.l(iv) is a proper one, too.

4.7. Theorem. There is an instance of 2’ ,+i-CP which is not provable in (Zz+,-IR) or, equivalently, (II”,+,-IR).

This is Theorem 1 in [18]; Parsons observed that there is an instance of II:-CP which cannot be proved in (QF-IA) from true II:+,-sentences. Clearly, the theorem follows immediately from the soundness of (nz+,-IR); this semantic condition can be replaced by uniform (n +2)-consistency, as will be apparent from the proof of Parsons’ main observation. I give a simplified proof of it now.

4.8. Proposition. There is an instance of ,ZE+,-CP which is not provable in (QF-IA) from true II:+,-sentences.

Proof. Let T;’ be Kleene’s T-predicate relative to (the characteristic function of) a complete ZO,-predicate A ; Tp is in X”,+,. Now consider the formula 4

(Vx < a)(3y) Teeaxy + (3w)(Vx < a)(3y < w) T$‘eaxy

and assume that JI is derivable in (QF-IA) from true I7:+,-sentences &, . . . , c#+.

CT,,+~($) is then provable from Us+,, 1 <i G r, in (QF-OT,,,). Le. we have a

56 W. Sieg

normal derivation of

where A consists of purely universal sentences which are either axioms of (QF-OT,,,) or among the u,,+~ -translations of the &. a,,+l($) is in (QF-OT,,,) obviously equivalent to 4” of the form

(3w)(3x)((x -= a & u,+~((VY) 1 T%xy)) v W’x < a)u,+l((3~ < w)%=Y)).

+!r* is in Eli-form and derivable in a purely universal theory; so the Herbrand-type lemma can be applied. It gives us first of all a sequence of terms sO, . . . , s,,-~,

to, . . * , t,,_1 which are built up from parameters including e and a and function- symbols of LZ?(OT,+,) needed for the a,+,-translation of the &‘s and I,!L” Secondly, the Herbrand-type lemma provides (the basis for obtaining) a normal derivation of

lA, W ((Si < a & (Vy) 1 T$eaSiy) v (VX < a)(3y < c) Teeaxy); 04<m

V-inversion yields a derivation of

(1) lA, W ((pi < a & lT$easici) v bi $ a v (3~ < ti)T$eabiy). O=G<m

Notice that the terms si and c do not contain the parameters ci and bi,

respectively cj, for 0 <j < m. So we can define a function (by primitive recursion) not depending on ci or bi by

b(0, a, e) = so if s,Ca,

0 otherwise,

c si+l if Si+l <a and

b(i + 1, a, e) = pks

si+ _1 # b(j, a, e) for OSj=Si,

i+ l.(k # b(O, a, e) & - . .&k# W, a, e>> 1 otherwise.

As the functions denoted by 6 and si are recursive in A, the recursion-theorem gives a Gijdel-number d of a function recursive in A, such that

{E}(a, i) = L

ti[a, i, a] if i = b(j, a, e’) c a,

0 otherwise.

Now one can falsify the disjunction in (1) for each ii > m as follows. For 0 s i < m, define lTi as ~,[ii, e’] and pi as b(i, ii, ~5); fin pi a~ &[c?, Gi, e’] and Ci as /.~y. T$G$y. Notice fkst of all that gi <ii; thus we have only to falsify (& < zi & iT~Ziis&) and (3y < 6) T$Z&y. The latter sentence is false by definition of e’ and the fact that gi < 6; the former sentence is false by definition of Zi. 0


The proof of Proposition 4.8 actually shows that there is a parameter-free instance x of sz+,-CP which is not derivable in (II:+,-IR) or, equivalently, (X0,+,-IR). Clearly, there is a corresponding parameter-free instance of 170,+r-IA which is not provable in (X”,+,- IR), namely that instance which is used to prove x in (II:+,-IA). So we have (in either case) a z0,+2-sentence which is provable in (Hz+,-IA) but not in (X”,+,- IR): by Theorem 3.11 this is best possible.

4.2. Elimination of choice and collection

The strategy for proving the conservation theorem formulated in the introduction to this part is quite straightforward. It is first shown that the quantifier-free axiom of choice, QF-A& can be eliminated from derivations of @-sentences in universal theories T. This covers in particular the case that T is (II:-OT$, II: understood w.r.t. 9(Z). Then one notices that (zz+,-FAC) is contained (via the a,,-translation) in (II:-OTz+ QF-A&) and that it proves the same IIz+2- sentences as (II:-OT3; it is in the last step that the elimination lemma is needed. Finally, the r,,-translation is invoked to observe that all the L!0,+,-theorems of (II:-OT:) are provable in (II:-IA). Thus, (Xz+,-FAC) is conservative over (II:-IA) for all II:+,-sentences. As (Xz+,-FAC) is equivalent to (z”,+,-CP), the Paris-Kirby-Friedman result is an immediate consequence. Indeed, as I shall point out below, it is obtainable in a more general form.

The elimination lemma is a direct adaptation of a general proof-theoretic fact established in [5]. The language underlying the sequent calculus here is 9(OT$.

4.9. Lemma (elimination of QF-AC,). Let A contain only 3,-formulas. If D is a normal derivation of A[-rQF-AC,], then there is a normal derivation E of A[lQF- AA].

Proof (by induction on the length of D). I focus on the crucial case when an instance of -IQF-AC,, has been introduced by the last rule in D. D has then the immediate subderivations D1 and D2 with endsequents

AC1QF-AGI, W)(~Y) NY and

41QF-A&I, +f>WxMxf(x)

respectively. By V-inversion one obtains from Dr and D2 normal derivations El and E2 of

(I) AC1QF--AGI, (~Y)&Y,

respectively

(2) 4Y~QF--AGI, +“x) &4x),

where c (u) is a new number (function) parameter. Clearly, I&] ~10~1, i = 1,2; (1) and (2) satisfy the condition on the complexity of formulas. So the induction

58 W. Sieg

hypothesis can be applied to yield normal derivations of

(1’) NlQF-AAl, (3yMcy,

and of

(2’) A[lQF-AA], -~(Vx)+xu(x).

By the Herbrand-type lemma one obtains from the derivation leading to (1’) a sequence of terms tl, . . . , t,, and a normal derivation of

(l+) A[lQF-AA], &t,v. - .v&t,,.

Using an additional instance of QF-AA that

( tl if &tl,

t2 if +ct* &

. .

Ax.(t[x, . . .])(c) = . r * . . L if &L &

we define a function hx.(t[x, . . .]), such

IO otherwise.

Abbreviating hx.(t[x, . . .1)(c) by u(c) and using (l’), we obtain a derivation of

A[lQF-AA], &U(C)

and thus of

(3) A[lQF-AA], (VX)C$XU(X).

Now replace the function parameter u throughout the derivation leading to (2’) by the A-term v and get a derivation of

(4) A[lQF-AA], ~(VX)+XV(X).

The desired normal derivation E of A [lQF-A A] is obtained by first applying the cut-rule C to (3) and (4) and subsequent normalizing. Cl

The lemma implies the following proposition.

4.10. Proposition. Let r be a set of Si- or V,-sentences of z(OTi). (QF-OTE+ QF-AC,) + r is conservative over (QF-OT:) + r for I&sentences of z(OT$. The claim holds also when (LIZ-OTZ)), II: w.r.t. 9(Z), replaces (QF-OT”,>.

Now we can, without any trouble, prove the main conservation result.

4.11. Theorem. Let r be a set of zt+,-sentences. (Si+l-FAC)+r is conservative over (LIZ-IA) + r for II:+,-sentences.


Proof. (170IA) is contained in (Zz,, -FAC) by Propositions 3.1(i), 4.l(iv), and 4.2. So consider a II:+,-sentence 4 which is a theorem of (II:-FAC)+ I’; the latter theory is obviously equivalent to (Xi,, -FAC) + r. a, (4) is a II&sentence in 9’(OTz). It is provable in (II$OTz+QF-AC,)+o,,(r) and thus, by Proposition 4.10, in (IT$OT;) + a,,(r). But then Ic, is provable in (II:-OTZ) + r and, indeed, in (II”IA) + l7 0

By Proposition 4.2, (_EE+l-FAC) is equivalent to (.X”,,,-CP); so we have:

4.12. Corollary. Let r be a set of 20,++entences. (Z”,+,-CP) + r is conservative over (II:-IA) + I for Il0,+,-sentences.

Disregarding the fact that (LIZ-IA) is contained in (Sz+,-CP), we can reformu- late the corollary as follows: (170IA+Zz+i -CP) is conservative over (II:-IA) for IITjl+z-sentences. The addition of ,Zz+,-CP to (Zz+,-IR) leads to a theory which is conservative for IIE+z-sentences over (Z”,,, -1R): indeed, that theory can be shown to be closed under the II:+, -induction rule. Techniques similar to those used in the proof of Theorem 4.8 show:

4.13. Corollary. Let r be a set of X:+,-sentences. @IO,+,-IR+Sz+,-CP) +r is conservative over (Zz+,-IR) + r for II:+,-sentences.

For n = 0, this is a strengthening of Theorem 3.5; this special case was also proved in [18] as theorem 2(a).

It has to be pointed out that the elimination lemma can be proved (directly) for QF-FAC or QF-CP; but I gave the formulation for QF-AC,, as that will be needed in the next part. We can already see here, how the elimination lemma and Proposition 4.10 can be exploited to show that some subsystems of analysis can be reduced to fragments of arithmetic. (Reduced here in the sense that a class of arithmetic statements provable in a subsystem can be proved in a fragment.) It has only to be ensured that their c,,-translation is contained in (QF-OTZ + QF-AC,) + r. This latter condition is satisfied e.g. by (BT+Z~+,-Aq)+r.

4.14. Corollary. Let r be a set of Z,0+2-sentences. (BT+,EE+,-AC;) +r is reducible to (LIZ-IA) + r for IT:+,-sentences.

In Part 5, much more interesting conservation relations between subsystems of analysis and fragments of arithmetic are established. But before turning to that topic, let me round off the discussion of relations between fragments- by drawing a diagram. The arrows indicate containment relations: the theory at the head is contained in that at the tail. In (ii) and (iv) we have conservation for

no,+, -sentences, and that is best possible; in case (iii) and (v), the stronger theory actually establishes the consistency of the other theory. Recall that (_ZE+,-IR+

60 W. Sieg

WO,+,-IN

(XO,+,-IR+ZO,+,-CP) (II”IA+~:+l-CP)

(ii)

1 It

(S:o,+rW E+r-cp>

A-l___/

(@-IA)

Zz+,-CP) is equivalent to (IT:+,-IR+ Xc,“,, -CP); thus this theory is closed under the n,0+,-induction rule, whereas (flz+,-IA) is not.16 So, to remark on (i), (17:+,-IA) is strictly stronger than (Z~+l-IR+Z~+l-CP) and it is also clear that Hz+,-sentences are conserved. But I do not know whether this is best possible.

5. Subsystems conservative over fragments

Theorems 1.7(iii) and 1.9 state that the subsystems of analysis (17oCA) 1 and (,X:-AC) ] are conservative over elementary number theory. These subsystems allow, as I mentioned in the introduction, the formal development of substantial parts of classical analysis. Inessential extensions of (@-CA) r and (Zy-AC,) ] are significant for refining portions of such a development and are, furthermore, conservative over (PRA). They contain most importantly K&rig’s lemma for binary trees; that principle is denoted by WKL. In Section 5.1 I shall show that WKL can be eliminated from normal derivations of 3,-sequents. This fact, together with the elimination of the quantifier-free axiom of choice and a generalized reflection rule, implies - in Section 5.2 - the most interesting metamathematical facts; namely, that

(F):=(BT+X’$AC,+Xy-IA+WKL) and

(M): = (BT + X,O-AC, + &IR- + WKL)

are conservative over (PRA). Two (weaker) theorems of Friedman and Mint follow immediately. Using the operator theories one can generalize these results to relate subsystems of (IT:-CA) ] systematically to fragments of arithmetic. This is done in Section 5.3.

5.1. Elimination of WKL

Let me tirst recall the formulation of Weak Konig’s Lemma (WKL), as given for example in [25]:

Tf & (Vx)(3y)(lh(y) = x &f(y) = 1) + (3d(Vx)f(E(x)) = 1,

Frugments of arithmetic 61

where Tf is (equivalent to) a purely universal statement expressing that the set of sequence numbers given by its characteristic function f forms a binary tree, i.e.

(Vx, YMX * Y) = 1+ f (xl = 1) CQ wx, Y)(f(X * (y>) = I+ y s 1).

WKL can be eliminated from normal derivations of 3,-sequents in a way similar to QF-AC,. Assume that the language underlying the sequent calculus is either 2” or Z(OTt). Without loss of generality for future applications, it can be assumed that A in Lemma 5.1 contains negations of axioms for (generalized) pairing and projection functions.

5.1. Lemma (elimination of WKL). Let A contain only 3,-formulas, if D is a normal derivation of A[lWKL], then there is a normal derivation E of A[lQF- hAI.

Proof. (by induction on the length of D). As always I concentrate on the central case when the last rule in D introduces an instance of 1WKL. In that case D has the immediate subderivations D, and D, with the endsequents

and A[lWKL], Tf & (Vx)@y)(lh(y) = x&f(y) = 1)

AClWKLl, +g)(Vx)f (g(x)) = 1.

Let (Vx) Rxf be the purely universal form of Tf; then one obtains by &- and V-inversion derivations E,, E2, and E3 of

A[lWKL], Raf,

A[lWKLl, (gy)(lh(y) = c&f(y)= 1,

and of

A[lWKL], l(Vx)f(ii(x)) = 1.

These endsequents satisfy the condition on the complexity of formulas. Thus the induction hypothesis allows us to eliminate WKL in favor of QF-hA. We have in particular a normal derivation E; of A[lQF-XA], Raf; indeed, using V we have a derivation of A[lQF - hAI, (Vz) Rzf and consequently an F1 of

(1) A[lQF-hA1, T(f).

We get also a normal derivation EJ of A[lQF-hAI, (3x) f (ii(x)) # 1, thus an F3 of

(2) A[lQF-AA], (3x)f(ii(x)) # 1.

Finally we see that the induction hypothesis applied to E2 yields a normal derivation F2 of

(3) A[lQF-AA], (3y)(lh(y) = c&f(y) = 1).

62 W. Sieg

From this derivation we obtain via the Herbrand-type-lemma a sequence of terms

h, - - *, t,,, and a normal derivation of

A[lQF-AAl, 1 y<, (lh(ti) = c&f (ti> = 1). si-_

Define as usual a function hx.t such that

A[lQF-hAI, lh(t[c]) = c &f(t[c]) = 1,

and thus

(*) A[-IQF-hAI, f(t[c]) = 1

are cut-free provable. (Note that hx.t yields sequences of arbitrary length in f.) Now apply the Herbrand-type-lemma to F3 and obtain terms sl, . . . , s,, and a normal derivation of

A[lQF-hAI, W f(c(si))# 1. l=SiSn

With QF- h A one can define a function hx.s and get a derivation of

(0) A[lQF-AAl, f(fi(s)) # 1.

(The right-most formula expresses that f is well-founded.) s is in general a term denoting a primitive recursive functional of u and f. By an easy modification of W. Howard’s construction to obtain a primitive recursive hereditarily majorizing functional for any primitive recursive functional of finite type we have a closed term 6, such that s[u, f] is majorized by p. (Both u and f can be assumed to be majorized by the function with constant value 1. For Howard’s construction see p. 458 in [26].) Now replace in the derivation leading to (*) c by 0 and obtain a derivation of

(**> NlQF-hA1, f(tCP1) = 1.

Assume that t[P] is (to, . . . , tp_1) and define a function u* by

u*(l) = {

tl if l< /3,

0 ifps1.

Obviously, ii*(p) = t[f3]. Substituting u * for u in the derivation leading to (0)

yields a derivation of

AClQF- AAl, fG*(sCu*, fB> # 1,

and thus, by the majorization and the fact that f is a tree, of

A~IQF-AA], f@*(P)) # 1.

ii*(p) = t[P], however, as noted above; thus we have a derivation of

(00) AClQF-AAl, f@CPl) # 1.


Use of the cut-rule C, applied to the derivations of (**) and (00), and subsequent normalizing yields a cut-free derivation of

A[lQF-XA]. 0

The lemma implies immediately the next proposition.

5.2. Proposition. Let r be a set of 25- or VI-sentences of ~(oT;). (Q&W",+ wa)+r is conservative over (QF-OT”,)+r for @-sentences of 2?(OT',>.

The proof of Lemma 5.1 can be carried out to establish directly a slightly more general form of Proposition 5.2 (without appealing to the transformation of finitely branching into binary trees).

5.3. Remark. Let BKL be Kiinig’s lemma for finitely branching trees, where a bound on the size of the immediate successors of a node x is given by a function g, depending only on x; i.e. (Vx, y) (f(x * (y)) = l--, y G 1) in the formulation of Tf is replaced by (Vx, y) (f(x * (y)) = 1 + y <g(x)). The considerations above can be

carried out to show that (QF-OTE + BKL) + r is conservative over (QF-OTa + r for @-sentences. Notice, however, that the analogue for ‘full’ KL is not correct. (For that see the Appendix of [ll] and [23].)

5.2. Conservative extensions of (PRA)

The possibilities of eliminating various principles from normal derivations with syntactically restricted endsequents are combined now to eliminate simultaneously instances of Sy-AC,, S’$IA and WKL. These principles are eliminated from derivations of Il$sentences in theories, whose purely universal ‘core’ may be extended by J?i- or VI-sentences. At first, I consider theories formulated in S*. The next proof theoretic fact is central (where I assume as above that A contains the negations of axioms for generalized pairing and projection functions).

5.4. Proposition. Let A contain only 3,-formulas. If D is a normal derivation of

A [lQF-AC,, +:-IA, lWKL], then there is a _Zy-IR-derivation E of A[lQF- AA].

Proof (by induction on the length of D). Focusing again on the crucial cases when the special principles are introduced by the last rule in D, one can argue as follows. In case 1WKL (or lQF-AC,,) is introduced, one proceeds as in the proof of Lemma 5.1, respectively 4.9., except that the induction hypothesis eliminates not only all instances of -IWKL (or lQF-AC&), but also of the other principles. Now consider the case of 55:-IA. Without loss of generality the instances of St-IA can be assumed to be of the form

(*) (3YMYO & w, Y)ez)MYx + @‘I + (3YMwY

64 W. Sieg

and the endpiece of D of the form

1 DI A[ I, (3z)(4ba + $4

A[ I, W, ~)@z)(dv -j $4 D3 A[ 1,x

A[ 1 (Note that A[ ] stands for A[lQF-AC,,, X$-IA, lWKL] and x for the antecedent of (*). In case D’s endpiece is not of this form apply appropriate inversion lemmas to obtain a derivation of the same or smaller length having an endpiece of this form.) The induction hypothesis applied to the Di yields derivations DT, 1 s i ~3, of A[lQF-AA], (3y)wO, of A[lQF-AA], (3z)(@a ---f $~a’), and of A[lQF-AA], 1(3y)+ya, respectively. From 0; obtain a derivation of

A[lQF-AA], (3yMya + (3yMya’.

Zy-IR applied to DT and this derivation yields an F with endsequent

A[lQF-AA], (3yhW.z.

C applied to F and DT gives the desired Zy-IR-derivation E. Cl

This proposition and Proposition 3.8 imply the next theorem.

5.5. Theorem. Let r be a set of S$sentences in 2’. (BT+ Xc-AC,+ X:-IA+ WKL)+ r is conservative over (BT)+r for I$sentences.

To mention this here once more: r may also contain VI-sentences; this remark applies to the further conservation results as well.

Now notice that X:-AC,, proves each instance of A;-CA, the recursive com- prehension principle. For assume that & is in Zy, +x in @, and (VX)(~X~, JIx); then one has by logic (Vx)(3y)[(y = 1 & +x)v (y = 0 & 1$x)] and by the equivalence between 4 and 1c,

WMYMY = 1& 4x) v (Y = 0 & Wx>l.

As the matrix of this statement is in Zy, Cy-AC, yields

elf>W)(f (x) = 1& 4x1 v (f (x) = 0 & 3x).

f is obviously the characteristic function of 4.

5.6. Corollary. Let r be a set of S$sentences in 2’. (BT+A’$CA+&IA+ WKL) + r is conseruative over (BT) + r for IT&sentences.

Friedman’s theory (W&) is contained in (BT + A:-CA + X:-IA + WKL) and includes (PRA). Thus, Corollary 5.6 and the conservativeness of (BT) over (PRA)


imply that (WKL,) is conservative over (PRA). This fact was established by model-theoretic techniques by Friedman (see [24]).

5.7. Corollary. (WKLJ is conservative over (PRA).

In (WKLJ, and thus in (F), one can develop a good deal of mathematical analysis and of metamathematics. One can, for example, prove Giidel’s completeness theorem. The crucial tool is WKL! Mint formulated in “What can be done in PRA?” a (somewhat artificial) theory which shares some of the features of (WI&J: it is strong enough to prove WKL for primitive recursive binary trees (and that suffices for many applications) and it is conservative over (PRA). Mint’s theory is (BT + @-CA- + II:-IR-); the conservation result just mentioned is a corollary of the following stronger result.

5.8. Theorem. Let r be a set of l$sentences in 2%‘. (BT+J$$AC;+IIz-IR-+ WKL) + r is conservative over (BT) + r for II$sentences.

As X!$AC; proves each instance of @-CA-, the theorem implies a strengthened version of the main conservation result of Mint [14].

5.9. Corollary. Let r be a set of Z$sentences in Z’*. (BT + fly-CA_ + II:-IR- + WKL) + r is conservative over (BT) + r for Il$sentences. Consequently, (BT + II:- CA- + II$IR- + WKL) is conservative over (PRA).

The proof of the theorem makes use of the elimination lemmas used above. However, a detour through a suitable operator theory is required here, and an analogue of Proposition 5.4 is an important proof-theoretic fact; in its formulation we make the same assumption concerning A as above. Now the language is 3(OT:).

5.10. Proposition. Let A contain only 3,-formulas. If D is a normal derivation of A[lQF-AC;, lWKL], then there is a normal derivation E of A[lQF-hAI.

Proof of 5.8. The auxiliary theory suitable for the present purpose is

(A) : = @I;--OTT + QF-AC0 + @!--IR + WKL),

where -and that is important - QF is understood in this proof w.r.t. T(OT,), Zig and II; w.r.t. 3(Z). Two main facts have to be established; namely,

(i) if (M) + r proves a ZI$sentence 4, then (A) + r proves 4 ;

(ii) if (A) + r proves 4, then (IZ&OT~) + r proves 4. It is helpful for obtaining (ii) and necessary for completing the argument for the

theorem to notice (iii) if (@-OT:)+ r proves 4, so does (BT)+ r.

66 W. Sieg

Z&AC, is trivially equivalent to ,Y$ACo and is implied by QF-AC0 in the presence of SK,.17 So we have (i). For (iii) it suffices to observe that the r,-translations of the axioms of (II:-OT:) are all provable in (BT). That is trivial except for the Skolem axioms SK,. By the remark after Lemma 2.5 their r,-translations are in II: and provable in (@-IA). Indeed, by Theorem 3.6 they are provable in (QF-IA) and thus in (BT).

Now I concentrate on (ii). If 4 is provable in (A), then there is a II;-IR- derivation F of a sequent -IA, c$, where A contains only axioms of (II:-OT: + QF- AC0 + WKL). The claim will be established by induction on the number n of n;- IR-applications in F. In case IZ = 0, use Proposition 5.10 to obtain a proof of 4 in (17$OT:). For the induction step from n to n + 1 consider a II:-IR-application such that the subderivation D determined by it does not contain another II:-IR-application. I.e. we have (normal) derivations D, and D2 such that D is of the form

+a is in II; w.r.t. 9(Z), say (Vx)+xa, A, contains only @-formulas, A2 axioms of (IIgOTz); it can be assumed that +a + +u’ is of the form (Vx)@y)(Vz)x and x is quantifier-free.

From Dr we obtain by repeated V-inversion a derivation of

(1) A I, lAz[ I, 4~0,

where A; contains only Xy-formulas. In the same way we obtain from D, a derivation of

AI, l&C I, (~Y)(~z)x,

and then, using additional SK,-axioms assumed to be in A2 already, of

(2) AI, -&[ I, adPy)(Vz)x).

Proposition 5.10 can be applied to the derivations leading to (1) and (2) and yields derivations of A{, lA,[lQF-AA], &O and of A;, -IA,[-IQF-AA], ar(@y)(Vz)x). Using a little logic derivations of

(3) Al, lA,ClQF--AA], $0

and

(4) Al, -rA,[lQF-AA], I@ * +a

are obtained. From (3) and (4) we have by observation (iii) that OTg+lA, proves & and Jla + +a’. X$extensions of OTZ are closed under II;-IR by (a version of) Proposition 3.12(i); consequently, $u is established in OTg+lA,. So we have a (normal) derivation G of Al, -IAN, +a, where A3 contains only OT$axioms and


includes A,[QF-AA]. Now replace D in F by G and weaken the resulting syntactic configuration appropriately with -rA3: that yields a II;-IR-derivation of a sequent -IA*, 4 with just n applications of II!&IR and A* containing only axioms of (A). The induction hypothesis ensures the main claim. 0

The conservation Theorems 5.5 and 5.8 and results from Section 4.2 give us information on the relative strength of (M) and (F).

5.11. Remark. (F) and (M) prove the same @-sentences; however, by Proposi- tion 4.8, there are instances of Zy-IA which are not provable in (QF-IA) and thus also not in (M). Clearly, these instances are provable in (F); II; is thus best possible. l8

5.3. Generalizations

The most direct way of isolating parts of (II!,-CA) 1 or equivalently (II:- AC,) ] by simply considering the hierarchy of systems (_ZO,+,-AC,) ] does not work: the hierarchy collapses into (Z$AC,) 1 . It is the presence of function parameters in the axiom of choice which is ‘responsible’ for this collapse; indeed, it is the only obstacle against obtaining a proper hierarchy. So I consider extensions in F-, respectively M-style of (170CA) 1 and (Zz+,-AC,) r ; the latter systems are defmed as (BT + II:-CA-), respectively (BT + Xz+2-ACJ.

5.12. Theorem. (BT+ IIE-CA- + Z’$ACO + Zy-IA + WKL) is conservative ouer

(SO,+,--IA) for fl:+z -sentences; this holds also when the theories are extended by S:0,,2-sentences.

This as well as the next theorem are easily established by using (QF-OT$ and carrying out arguments parallel to those in the first two sections of this part.

5.13. Theorem. (BT+ Zz+2-Aq + II:+,-IR-+ WKL) is conservative ouer (Z”,+,-

IN for n”,+, -sentences; this holds also when the theories are extended by Zzs3-

sentences.

So we have obviously generalizations of Corollaries 5.7 and 5.9.

5.14. Corollary. (i) (BT + IIZ-CA- + A Y-CA + J$-IA + WKL) is conservative over (S:I1+,-IA) for II0,+,-sentences.

(ii) (BT+ IIz+,-CA-+ IIz+,IR-+ WKL) is conseruative over (Zz+,-IR) for

Z+, -sentences.

68 W. Sieg

Concluding remarks

Harrington proved by a rather complicated model-theoretic argument that the extension of Friedman’s (RCA& i.e. roughly (BT+ A’$CA+ @-IA), by WKL is conservative for I7:-sentences. Can this result be established by (straightforward) proof-theoretic means? Harrington’s result is a significant strengthening of the logical relation between (RC&) and (WKL,): from Theorem 4.5 one can only infer that the theories establish the same flz-sentences.

It seems to me to be important to look for refinements of the main results of Part 5; namely can one weaken (M) or (F) in such a way that the weakened version still allows the main mathematical developments but is conservative over proper fragments of (PRA)? One candidate of such a fragment in (KEA), a theory like (PRA) but containing only (names and) axioms for the Kalmar-elementary functions.

Thus one is led quite naturally to the investigation of fragments of (Zy-IR) or equivalently (II:-IR). (27-IR) is conservative over (PRA), as we have seen, and its provably recursive functions are exactly the primitive recursive ones. Parsons and Mint showed that this is true, even if only the Kalmar-elementary functions are initially available. In other words; (J$-IR) is equivalent to the theory (Zy-IR(d)).” Indeed, M inc remarked that MatiyaseviE’s diophantine representa- tion of recursively enumerable sets allows a further restriction of the initial functions from 8 to (‘, +, e}. There are some other suggestive results. For example, Parsons proved in the unpublished manuscript mentioned in Note 0, that the provably recursive functions of (II:-IR(8)) are exactly the Kalmar-elementary ones; Parikh considered the theory (A,-IR((‘, +, m}))“” and showed that its provably recursive functions are bounded by polynomials. For the fascinating connec- tion with important problems in the theory of computational complexity, number theory, and with foundational issues I have to refer to the literature.21 In any event, it would be of special interest if one could answer the above question concerning refinements positively, using a theory whose provably recursive functions are contained in a small class of subelementary, ‘feasible’ functions.

I am convinced that refined proof-theoretic tools can be fruitfully applied here. Certainly, that requires to pay very close attention to points which are (by and large with good reason) still neglected in proof theory: the canonical presentation of elementary syntax and the complexity of operations on (restricted classes of) derivations. The step to the detailed proof-theoretic analysis of selected mathematical proofs is then minimal.22

Notes

’ Much of the research for this paper was done while I was visiting Stanford University in 1981/82. I had the benefit of reading an unpublished manuscript of C.D. Parsons, in which he completed to a


large extent his analyses of fragments of arithmetic; cp. in particular note 16 below and the concluding remarks.

After I had completed the almost-final version of this paper in May 1983, S. Feferman pointed out a mistake in my original argument for the WKL-elimination-lemma, and S. Simpson suggested a more detailed discussion of the mathematical development in (WKLJ.

1 (The formal system for) classical analysis is second-order number theory with the full comprehen- sion principle CA. The relationship to formulations with other set or function existence principles is discussed in [4]. Hilbert’s way of presenting the material is indicated with some detail in supplement Iv of [9, II].

*The (unfortunately) not published Stanford Report on the Foundations of Analysis is a marvelous source of reflections and information. As to published accounts, see [l], [lo], and Feferman’s contribution to the Handbook of Mathematical Logic.

Feferman has emphasized that theories of finite type (over the natural numbers) are a more natural framework for directly representing mathematical practice. After all, working in subsystems of (CA) requires the coding up of higher type objects.

It should not be difficult to isolate a part of Feferman’s finite type theory, extend it by WKL, and adapt the proof theoretic techniques used here to establish its conservativeness over (PRA).

‘For further development see Simpson’s Reverse Mathematics, his Which set existence axioms

are needed to prove the CauchylPeano theorem for ordinary differential equations? and Friedman, Simpson, Smith’s Countable algebra and set existence axioms. But notice that these investigations have a crucial additional aim; namely, to establish the necessity of function or set existence principles for the proof of particular mathematial theorems. This is done, in one way, by showing that the mathematical theorem in question is actually provably equivalent over a weak base theory to the abstract principle. All the theorems mentioned in the main text are equivalent to weak Kiinig’s lemma over (RCA&

41n contrast to the uniformity of my methods, observe that Parsons used the Dialectica- interpretation, Paris and Friedman non-standard models, Mint the no-counterexample-interpretation.

’ This claim follows immediately from the more refined results 4.1 and 4.2.

6By inspection of the proof of Proposition 1.5 one sees immediately: if A is allowed to contain

m:+, -formulas, HZ-IA is provable with that “X:-U?‘.

‘This result was established by model-theoretic arguments by Barwise and Schlipf.

a The material is presented nevertheless in great detail. First of all, this paper should be independent of the earlier one; but, secondly and more importantly, the notion of v-depth is incorrectly defined in the earlier paper.

9 Clearly, terms and formulas have to be defined simultaneously. I assume in the formulation of (iii) that x is a variable which does not occur in cpa. This is needed to

make the definition of ‘subordinate’ below sensible.

lo For model-theoretic considerations, in particular for independence proofs, the formulation of (Z) with the induction axiom is most appropriate. For proof-theoretic purposes, e.g. the Gentzen-type analysis of (Z), the rule formulation is more convenient.

l1 For details concerning these matters see Smorynski’s contribution to the Handbook of Mathemat-

ical Logic and the first chapter of Troelstra’s Metamathematical Investigation of Intuitionistic Arirhme- tic and Analysis.

I2 The proof requires the consideration of negated instances of II:-IA; they can be dealt with in a manner completely parallel to that of negated instances of Ilz-FAC in the proof of Theorem 4.3.

I3 And thus the one-quantifier-system built on PRA or the system without nested quantifiers; see [17] and [13].

I4 They were generalized by Parsons; see Theorem 2 of [21] which corresponds to my Theorem 3.9,

70 w. sieg

and Theorem 3 of the same paper which corresponds to Theorem 3.11 below. Theorem 3.9 is further strengthened in Corollary 4.13. That result was announced in [20].

isTo simplify matters notationally, we indicate only the parameters e and a.

i6 The idea for the proof of this fact is indicated in [18, p. 4721 for the case n = 0 and the proof is carried out in full generality in Parsons’ manuscript mentioned in Note 0.

Notice also that the proof of the fact that Ackermann’s function is recursive can be carried out in (@-IA) extended by II:-IR, but clearly neither in (@-IA) nor in (@-JR).

i’ One has to be a little careful, as the Skolem-axioms are available only for formulas without function-parameters. So assume the antecedent of @-AC,, i.e. (Vx)(3y)&xy; by SK, one obtains (Vx)(By)&*xy where 4* is quantifier-free. QF-AC, yields now (Bf)wx)&*xf(x), and by logic we have (Bf)(Vx)(vz)(f(x) = z + 4*xz). (This step can be taken as f(x) is not ‘bound’ in any v-expression.) Now it follows clearly, using SK, again, that (!lf)(Vx)(Vz)Cf(x) = z --f 4xz), and that (Bfl(Vx) +xf(x),

‘a Compare also with the remark after the proof of 4.8.

*’ (F-JR(S)) is (F-IR) where only names and defining axioms for the functions in 9 are available; 1B denotes the class of Kalmar-elementary functions.

z0 See his Existence and feasibility in arithmetic, J. Symbolic Logic 36 (1971) 494-508. The A,-formulas are built up using the sentential connectives and only bounded quantifiers.

*i See: Joseph and Young, A survey of some recent results on computational complexity in weak theories of arithmetic, in: Math. Found. Comp. Science, Lecture Notes in Computer Science 118 (Springer, Berlin, 1981) 46-60; Wilkie, Some results and problems on weak systems of arithmetic, Logic Colloquium 77 North-Holland, Amsterdam, 285-296; Paris and Wilkie, A, sets and induction (mimeographed); Gandy, Limitations to mathematical knowledge, Logic Colloquium 80 (to appear).

** and the obvious next one. Kreisel has emphasized the significance of such investigations for a long time, recently for example in his contribution to the Herbrand Symposium, Finiteness theorems in arithmetic: an application of Herbrand’s theorem for P,-formulas; Logic Colloquium 81 (North- Holland, Amsterdam) 39-55.

Note added in proof

Two of the papers referred to in Note 3 were published in the meantime. Simpson’s second paper in J. Symbolic Logic 49 (1984) 783-802. Friedman, Simpson, and Smith’s paper in Annals Pure Appl. Logic 25 (1983) 141-183.

References

[l] S. Feferman, Systems of predicative analysis, J. Symbolic Logic 29 (1964) l-30. [2] S. Feferman, Monotone inductive definitions in: A.S. Troelstra and D. van Dalen, eds., The

L.E.J. Brouwer Centenary Symposium (North-Holland, Amsterdam. 1982) 77-89. [3] S. Feferman and G. figer, Choice principles, the bar rule and autonomously iterated comprehen-

sion schemes in analysis, J. Symbolic Logic 48 (1983) 63-70. [4] S. Feferman and W. Sieg, Iterated inductive definitions and subsystems of analysis, Lecture Notes

in Math. 897 (Springer, Berlin, 1981) 16-77. [5] S. Feferman and W. Sieg, Proof-theoretic equivalences between classical and constructive

theories for analysis, Lecture Notes in Math. 897 (Springer, Berlin, 1981) 78-142. [6] H. Friedman, Some systems of second order arithmetic and their use, Proc. Int. Ccmg. Math.

(Vancouver, 1974), Vol. 1 (Canad. Math. Gong., 1975) 235-242. [7] H. Friedman, On fragments of Peano arithmetic, Mime0 (1979).


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1968/70). [lo] G. Kreisel, A survey of proof theory, J. Symbolic Logic 33 (1968) 321-388. [ll] G. Kreisel, G. Mint and S. Simpson, The use of abstract languages in elementary metamathema-

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(1983) 182-184. [13] G.E. Mint, Quantifier-free and one-quantifier systems, J. Soviet Math. I (1973) 71-84. [14] G.E. Mint, What can be done in PRA?, Zapiski Nauchuyh Seminarov, LOMI, Vol. 60 (1976)

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Ressayre, eds., Model Theory and Arithmetic, Lecture Notes in Math. 890 (Springer, Berlin, 1981) 251-262.

[17] C. Parsons, Reduction of inductions to quantifier-free induction, Notices AMS 13 (1966) 740. [18] C. Parsons, On a number-theoretic choice schema and its relation to induction, in: Kino, Myhill,

Vessley, eds., Intuitionism and Proof Theory, (North-Holland, Amsterdam, 1970) 459-473. [19] C. Parsons, Proof-theoretic analysis of restricted induction schemata, J. Symbolic Logic 36 (1971)

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Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977) 867-896. [23] W. Sieg, A note on Kiinig’s lemma, Abstract for the ASL-summer meeting in Aachen (1983). [24] S. Simpson, Reverse Mathematics, to appear in: Nerode and Shore, eds., Proc. AMS Summer

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Fragments of arithmetic