On -Quantifier Induction

On 퐧-Quantifier InductionAuthor(s): Charles ParsonsSource: The Journal of Symbolic Logic, Vol. 37, No. 3 (Sep., 1972), pp. 466-482Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272731 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 37, Number 3, Sept. 1972

ON n-QUANTIFIER iNDUCTION

CHARLES PARSONS

In this paper we discuss subsystems of number theory based on restrictions on induction in terms of quantifiers, and we show that all the natural formulations of '7n-quantifier induction' are reducible to one of two (for n # 0) nonequivalent normal forms: the axiom of induction restricted to IIn (or, equivalently, En) formulae and the rule of induction restricted to Tn formulae.

Let Zo be classical elementary number theory with a symbol and defining equations for each Kalmar elementary function, and the rule of induction

IR AO Aa = A(Sa)

Aa restricted to quantifier-free formulae. Given the schema

IA AO A Vx < a[Ax = A(Sx)] ' Aa, let IAn be the restriction of IA to formulae of Z0 with < n nested quantifiers, IA' to formulae with : n nested quantifiers, disregarding bounded quantifiers, IAn the restriction to YI1 formulae, IAU the restriction to S0 formulae. IRM, IRA, IRnn, IR; are analogous.

Then, we show that, for every n, IA=, IAM, IAn, and IA' are all equivalent modulo Zo. The corresponding statement does not hold for IR. We show that, if n ? 0, IR1'+1 is reducible to IRn; evidently IRn is reducible to IRn+ On the other hand, IR' is obviously equivalent to IA' [10, Lemma 2].

Thus we have only two distinct hierarchies, IR2 - IR- and IAn - IA - IA -

IA' - IR', which coincide at n = 0. It also follows that the hierarchy ASn - ASn- AS' discussed in [10] (see also below) coincides with the second and, that by the Corollary to Theorem 1 of [10], IAn is properly stronger than IRn for n # 0.

In ?3 we show that IA' is a conservative extension of IRI for n + 1 sentences. In ?4 we use the main theorems of ?2 to characterize the G6del functional interpretations of these subsystems, and in ?5 we prove the faithfulness of these interpretations.

?1. We consider classical first-order theories with = as sole predicate, the symbols 0 (zero), S (successor), and further function symbols. Let Z0 be as in [10]; it has the Peano successor axioms, for each definition of a Kalmar elementary function a symbol for the function with the defining equations as axioms, the usual equality axioms, and the rule of induction restricted to quantifier-free formulae. (For the work of this paper it would not matter if all primitive recursive functions were admitted, but then IR1 (and even IRK2) would reduce to IR0.)

Received November 6, 1970.

466



ON n-QUANTiFIER INDUCTION 467

We recall certain facts proved in [10]. Consider the schemata

AS 3cVx < a[cX = 0 A Ax . V . cx = I A -,Ax1,

FAC Vx < a3yAxy D 2cVx < a A(x, cs),

where c ranges over sequence numbers and c, is the xth term of the sequence. lh c is the length of c (so c = IL < h APs'+ 1)

(A) AS" is equivalent modulo ZO to IAn. (B) FACn, FACr+1, FACn are equivalent modulo ZO. (C) FACI' is derivable in IAM+1. We add the following: (D) IA" and IA' are equivalent modulo ZO. PROOF. If Ax is IlO, then A(b -- x) is equivalent in ZO to a 10 formula. In

IA7 we have

A(b -- 0) A Vx < a(-,A(b - x) = -,A(b -, Sx)) = -,A(b -a a). If we take b = a, this implies the induction axiom on A, since

Vx < a(-1A(a - x) = -,A(a -, Sx)) Vx < a(Ax = A(Sx))

is provable in ZO. The same argument shows that IAM is derivable from IA~' The results below with those of [10] show that the system IR:+1 + FACU is

properly stronger than IR1'+2 (etc.) and no stronger than IA'+1 (etc.). But for n # 0 we have no proof that it is weaker than IA' 11

?2. ToREm 1. ASu is derivable in IA=', and hence so is IA'. PROOF. The second statement follows from the first by (A). If n = 0 the state-

ments are trivial. If n # 0 let Axy be En- 1. Then AS with VyAxy as A is implied in ZO by

3cidVx < aVz{cx = 0 A Axz . V . c, = 1 A -,A(x, d,)}

which we write as AS*. We suppose AS* to be false and derive a contradiction by the 'substitution pro-

cedure' as formulated in [11, ?12.1], For each m, we define sequences Cm, dm with lh Cm = lh dm = a. If m =0, cm = dm =<0 *0>. Then cm, dm have the property that

(2.2) Vi < a[cm. i = 0 V . cmi = 1 A _A(i, di)].

Suppose we have defined Cm, dm satisfying (2.2). Then by --,AS*, there exist z, w such that

z < a A * Cm~z = 0 ) -,Azw .A acmnz = 1 D A(z, dmiz).

Then by (2.2), cm, z = 0 A -,Azw. If i < a we set c + , i = cm, i and dm + 1, i = dm, i if i # z, cm + 1, i = 1 and dm + 1, i = w if i = z. Then (2.2) continues to hold for m + 1.

1 Such a proof was announced at a meeting of the Association for Symbolic Logic in Beverly Hills, California, on March 26, 1971. See (16]. We also showed (what for n = 0 is obvious) that FACI does not include IR7+1. Thus these systems are incomparable and properly stronger than IA", but their union is weaker than IAI+ 1.



468 CHARLES PARSONS

Cm,+ has one more nonzero term than Cm. But if Cm = <1... 1>, then, by (2.2), cm, dm satisfy AS*. Hence for some m ? a, it is impossible to continue, and cm, dm satisfy AS*.

To formalize this argument in IA=, let B(m, a, c*, d*) be the formula

lh c* lh d* = Sm A Vy < m{Seq(c*) A Seq(d1*) A lh c* = lh d* = a

Vi < a[ct,, = 0 V . C*, = 1 A -nA(i, dyt.)] A.y < m = 3z3w[z < a A c*,, = 0 A -,Azw Vi < a(i # z C. c =C*, = A dsAm, = dy1 :A:

i = Z D:. CS*Y.t = I A dsvy.t = W)]}.

Since A is E . , the formula 3z3w[ * - * ] is equivalent in ZO to a Fn formula and hence so is the scope the quantifier Vy < m. But by (B), (C) and (D) we have FACm, and hence Vy < ... m } is equivalent in IA' to a Xn formula. Hence so is 3c*3d* B(m, a, c*, d*), and we can apply induction to it.

The informal argument yields

3c*3d*B(O, a, c*, d*),

-,AS* =. 3c*3d*B(m, a, c*, d*) = 3c*3d*B(Sm, a, c*, d*)

so that by induction we have

(2.3) --AS* =z 3c*3d*B(a, a, c*, d*).

Let ,b be an elementary function such that if Seq (c*), x < lh c*, Seq (c*), lh c* > a, akc*ax is the number of i < a such that c*, # 0. Then we have, in Zo,

B(m, a, c*, d*) A y < m = ic*a(Sy) = S[oc*ay]; and hence

B(a, a, c*, d*) O #c*aa = a DVi < a(Ca, i O )-

But this means that, for all i < a,

C* = 1 A -IA(i, da,)

which implies AS*. By (2.3), AS* follows. THEORM 2. Any formula provable in IR'+2 is provable in IR,+1. PROOF. This will follow by induction on the number of applications in the proof

of the given formula of TIIn+2 induction, from the following: LEMMA 1. Let Z* be the result of adding to ZO symbols and defining equations for

primitive recursive functions. Let A(a) be a lln +2 formula of ZO and suppose AO and Aa = A(Sa) have been derived in Z* from Il,,+ 2 sentences. Then Aa is derivable in IR1 + from these sentences.

PROOF. We can consolidate the premises of the derivations of AO and Aa D

A(Sa) into a single sentence. We suppose also n = 2m (otherwise the proof is similar) and that the sentence is

(2.4) Vxo3yo- * Vxm3ymB(xo ... Xm .o ... y.)

and that Aa is Vxo3yO ... Vxm3ymAo(axo *... Xm, yo* . ym)



ON n-QUANTIFIER INDUCTION 469

We add to Z* new function symbols go... gm (gj with i + 1 places) and the axiom

(2.5) B(xo *xm, goxo* gmxo ... xm) and further new function symbols fi- fm, h ..hm (fi with 2i - 1 places and hi with 2i places). Let Ao(a, xo* .Xm, Yo... *ym) be as above; let

A21+1(a, xo *x0-f, yo* Ym-fj-1)

be the result of substituting hm-j(a, x0... xm-j, yo * Ym-i-1) for Ym-I in

A~ata XO - * x- It Yo ... Ym-1);

let A2,+2(a, XO . * x i-j-,, yo ..Y.-i-i) be the result of substituting

fm _Xaj, xo-* *xm ->-, Yo-** Ym -j-j)

for X_-I in A21+1. Then we add the axioms

(2.6), A21(a, XO * Xm-ItYo ?Ym-i) ' A21+1(a, XO... Xm-itYo.. Ym-f-

(27) - mA21+1(a. XO .. eMJ x^ sYO ... Ym-1-1) =) -- A2 + 2(a, xO*** 'Xm -f - sYO * * ?Ym - f-1)P

forj = 0.. - 1. Thus h1+1(a, xo *.. x x+, YI * *yj) stands in effect for

eyi+lA2m- 2(i+l)(a, XO... Xi+1, Yo0* Yf1 and

f + (at xo -xjvyo... yj) for

,exj+j-A2M-2f-1(a9 X0 * Xf+19 YO- .. Ays

Finally we add a one-place function symbol f with no new axioms. From the axioms (2.6) and (2.7) we can derive by logic

A2M(a, XO YO) Y_ Vxj3yj *.. Vxm3ymAo(a x0 . Xm, Yoj *Ym)

and from (2.5) we can derive (2.4). Hence in the expanded ZO* we can derive

(2.8) Vxo3yoA2m(0, XO, Yo),

(2.9) Vxo3yoA24(a, XO, YO) = Vxo3yoA2m(Sa, XO, YO).

Since VxOA2m(a, xofxo) implies Vxo3y0A2m(a, xo, yo), by (2.9) we can derive

(2.10) VxOA24(a, x0,fxo) = VxoiyoA2M(Sa, X0, Yo).

(2.8) and (2.10) are derived in quantificational logic from quantifier-free formulae-(2.5), (2.6), (2.7), the axioms of ZO*, and the conclusions of inductions in Z*. By Herbrand's theorem there are terms to(xo), e(a, x) and t(a, x), composed from 0, S, parameters of A and the indicated variables, function symbols of Z*, go ... gm, f... *fm, h,... hm, and (in the case of 5 and t) f, such that

(2.11) A24(0, XO, to(XO)),

(2.12) A2m[a, s(a, x),f(sax)] O A(Sa, x0, tax)

are provable from the indicated quantifier-free formulae by substitution and propositional logic. Hence they are provable in the result of adding

f( go2) ( gm. fl (. fmi hr i a ri hme

(2.5), (2.6), and (2.7) to primitive recursive arithmetic.



470 CHARLES PARSONS

Now consider the recursion (parameters of A suppressed)

(2.13) (Px = tox, (p(Sa)x = t*(Ax9pax, a, x)

where t*(Axxpax, a, x) is the result of replacing in t(a, x) subterms ft of t by p a x.

This is a case of nested primitive recursion, which is reducible to ordinary primitive recursion (e.g. by [12]). This reduction can be carried out formally in PRA (see the proof of Lemma 4 of [10]) provided we allow ordinary primitive recursions

(2.14) 00 to, O/(Sa) =tl(a, Oba),

(perhaps with parameters) where the terms to and t, may contain go. ,- ? g f fin

hi ... hm. Hence if PRA is augmented by these symbols and recursions (2.14), then a q

can be explicitly defined so that (2.13) is provable. Then substituting Axrpax for f in (2.12) we have

(2.15) A2m[a, s*(Axipax, a, x), p(a, s*(Axcpax, a, x))] - A2m[Sa, x, t*(Axpax, a, x)].

We introduce Waxy by the recursion

Gbax0 = x, Oax(Sy) = s*(Ax'p[a * Sy, x], a -e Sy, Waxy),

which is of form (2.14), so that (2.15) yields

a < b A A2jmla, 0(b, x, b - a), (p(a, iJb(b, x, b * a))] =' A20[a,, 0(b, x, b *, Sa), (p(Sa, 0)(b, x, b -- Sa)]

so that by (2.11) and induction A2m[b, /(b, x, b * b), rp(b, /(b, x, b * b))] and therefore A24(b, x, grbx) follow. Thus the formula is provable in PRA augmented by got . .gm, fi . fm, hi... hm, the axioms (2.5), (2.6), (2.7), and the recursion (2.14).

We now translate this system into IR1+1 + (2.4). With each function symbol X we associate a En + 1 representing predicate Cxal ee aky such that 3!yCal ... aky is provable in IRn+1 + (2.4). Then we have in the obvious way a In+ representing predicate Cy, with free variables just those of t, for which 3 !yCty is provable.

(i) X is S. Cray is y = Sa.

(ii) X is introduced by primitive recursion (2.14). Write Ct0y as

3zCt(,(al .. *ak- 1, Y. z),

C..(. )IIn and Ct1y as 3zCt,(al . *ak- , x, w, y, z). Then the formula

(2.16) 3c3d{Cto(al... ak-1, C0, do) A Vi < akCt.(al

... ak-l, i, Ca, Cj+1, di+,) A y = Ca}

is equivalent to a In + formula 3zCal *. akyz; let that formula be Cx. By 3!yCt0y and 3!yCt1y we can prove, by IR;+1 on 3y3zC(al ... ak.1, a, y, z),

(2.17) 3yCxa, .. aky.

Evidently if c, d and c', d' satisfy (2.16), then by the uniqueness of y's satisfying Ctoy and Ctly, c, = c for x < ak. (This involves only applying the induction axiom to the predicate a < ak = Ca = ca.) Together with (2.17) this yields 3!yCal ... aky.

(iii) X is gi.



ON n-QuANTIFIER INDUCTION 471

First we observe that for any IIn formula By the least number principle

(2.18) By = 3y[By A Vw < y - Bw]

is provable in IA' and therefore in IRn+ 1. If n = 0 this is trivial. Otherwise write By as VzByz. Consider the induction axiom on 3cVw < y -,B(w, cm). This is equi. valent to a En formula. By this induction axiom we can prove

Vw < y-NVzB(w, z) _ 3cVw < y-,B(w, cW)

and hence induction on Vw < y-VzB(w, z), which yields (2.18). This also shows that the consequent of (2.18) is equivalent in IA! to

3y[By A 3cVw < y-,B(w, cm)]

which is equivalent to a I 1 formula. (Such an equivalence is trivial if n = 0.) By [10,Lemma 2], IA' is a subsystem of IRI+1. Clearly (2.18) implies

(2.19) By = 3!y[By A Vw < y-Bw].

Now let BI(xo xi, yo .* .y) be the formula

Vx,+ 13yj+ 1 e Vxm3ymB(xo ... xm, yo * *ym) A Vw < y-,Vx+3y,+l . .. *Vxm3ymB(rx0 *.y... 1 w.yj +1. ? Ym).

By formulae (2.19) we have, for each j,

(2.20) Vxj + 13yj+ 1 ... Vxm3ymB(xo ... Xm, Yo ... ym) = 3 !yjBj(xo *x yo x .. y.).

Then by (2.4) we have 3 !y0Bo(x0, yo)

and then assuming ..o.. 2.. !yj Ay oB,(xo ... xj, yo. .. y,), (2.20), for j = i + 1, yields 3!yo... 3!yi + 1 A` oBj(xo ... x,, yo* . yj). By induction we have, for each i m,

(2.21) 3!yo* 3 !yj A B,(xo . *. xi,, y0o* . yi) f=0

By the above observations, Bj is, for each j < m, equivalent in IR; + 1 to Z,, + 1 formula. Hence so is

2yo .* 23y -4A B(Xo *.. *xj, yo.* yj) A Bj(xo x.. y0.o .. yi-i)

Let C03(xl . xi, y) be a I + 1 equivalent of this formula. Then, by (2.21), 3 !y Cg,(xl . ?X y) is provable.

(iv) X is hi. Consider the formula

Vy Vx, + 13yj + 1 - * * VxmmymAo(a, Xo ... xm, Yo .. *ym) = Vxj + 13yj + 1 . VxmiymAo(a, x0 ... xm,Yo y * *t j _1, Y. Yi + 1 ** .Ym) A Vw < y--Vx+13Yt+l .. Vxm3ymAo(a,x * *.*.xm,Yo .. *Yj - 1, w, yj + 1 . y*m)

:A . Vw-Vxj + 13yj + 1 .. *Vxm3ymAo(a, xo .* *xm, yo *yj + 1, w, yj 1 . ym) = y =- 01.

Abbreviate this as Vy[Al '.A2 A Vw < yA3 A- .A4 > y = 01. Then since i ? 1, A1 is 1In-2, A2 is lln-2, A3 is Yn-2 and A4 is H.n-1. Then the whole is cer- tainly equivalent in ZO to a Iln formula. Let that formula be

Ch,(a, xo *.. xi, yo ..* *y - 1 y)-



472 CHARLES PARSONS

Using the same abbreviations, write A2 as A2(y) and A1 as A2(yi). A3 and A4 contain neither y nor yj free. A4 implies -,A2(yJ) for all yj and hence

Vyi[A2(Yi) =. A2(0) A Vw < OA3 A . A4 = 0 = 0] and hence

(2.22) 3 !yVyi[A2(yi) * A2(y) A Vw < yA3 :A .A4 = y = 0].

-,A4 implies A2(y) for some y and hence, by (2.18), for some y, A2(y) A Vw < yA3. Since this lacks free yj it implies (with ..A4)

VyJ[A2(yi) ' A2(y) A Vw < yA3 :A.A4 = y = 0].

By (2.18), y is unique, so (2.22) follows. Since both A, and -,A4 imply (2.22), it is provable and hence so is 3 !yCh(a, x... * xi, yj ..* y -1, y).

(v) X is Jj. Consider now the formula

-Vx3yj * * - Vxm3ymAo(xO ... xm3, Yo * Ym) ) A -- 3y1Vx + 13y1 + 1... Vxm3ymAo(a, xo ... xi - X1, y, xi xmyo Ym)

Vw < y3yjVx+1y + 1... Vxm3ymAo(a, xO... x-,,, wx+, ... xm, Yo, Ym) : A . vxj3y . Vxm3ymAO(a, xO... Xm, yo. * Ym) = Y = 0.

Abbreviating this as -,VxjA5(x1) =.A5(y) A Vw < yA6: A .VxjA5(x1) = y = 0, note that since i ? 1, Vx1A5(x1) is II,, A5(y) and A6 are zn-l1 so that the whole is equivalent in Z0 to a Zn+ 1 formula. Let that formula be

Cf4(a, xo .. xi-1, yo. y -1, Y)

The proof of 3 !yCfy is similar to case (iv). We thus have a natural translation of our free variable system into IR;+1 +

(2.4). We must show that it preserves theorems. For identity axioms, Peano successor axioms, and axioms of propositional logic, this is trivial. The equations (2.14) with parameters a, .* *ak.1 are translated into

3y[C*,(al. ak-l, Oy) A Ctoy],

3w3y[C*(al * ak-1, Sa, y) A Cu,(al ... ak-l, a, w) A Ctl(a, w, y)].

These are immediate from 3!yC*oy, 3!yCtlawy, (2.17), and the uniqueness of CO .Cak-l if c, d satisfy (2.16). Hence they are provable in IR;+1 + (2.4) if 3 !yCj0y and 3 !yCt1awy are, which we have shown.

(2.5) is translated into

3Yo. Ym{A Cg1(xO' ..x1, yj) A B(XO - Xm, Yo YM)

This is immediate from (2.20) and the definition of Cg4. Since AO is a formula of Zo it is translated into itself. Suppose A2, is translated

into A*. Then A21+l is translated into a formula equivalent to

(2.23) 3y{Ckm.,(a, Xo ..Xm-j, Yo . Ym-j-ij y) A A2*(a, Xo **Xm-j-f lYm-q-i, Y)}

and A2J+2 into a formula equivalent to

3w3y{Cfm I(a, x0** *xXo - Ym - j - 1 . w) A Chm k(as xo *.*.*xm-j- 1,wS yo **Ym.-J1,Y)

A A~,(ao. xo ... xm_,.1 ws yo ... ym.j1y Y)}.



ON nlQUANTFIER INDUCTION 473

By induction on j we prove in IRE+1 (for 0 < j < m)

(2.24) Vxm -j+ 13Yrm-j .... VXm3YmAo(XO * * YO X ... Ym) A* (a, XO .Xm.. eY

(2.25) 3Ym-jVxm-j+j3Ym-j+j ...

VXm3ymAo(x o .. xm y Yo * *Ym) ( @ ) -~~~~~~~~~A*f +1(ag Xo .. *xm-fg Yo .. *ym-1-1)-

(2.24) is an identity ifj = 0. Suppose (2.24) holds, and so does

VXm-j+13Ym-j+i ... VXmEymAo(xO ... Xm, Yo * Ym)- Then it holds for ym-y = the unique y such that

(2.26) Chm -,(a, X X*mjY0 x ..Y.--, Y) and hence by (2.24) so does

A*4(a, xO ... Yt A * Ym-f-1Y Thus (2.23) holds and hence A* +1. The converse is similar.

Suppose now (2.25) holds, and moreover -n3Ym-F ... Vxm3ymAo(xO ... xm, yo ... Ym).

Then it holds for xm -.j = the unique w such that

(2.27) Cfm (a, XO * *Xm-.-ji Yo * Ym-j-19 w) and hence by (2.25) so does

A*y +,l(a. xO .. **Xm -y j1 w9 Yo ... *Ym - j ). Since there is a unique y satisfying (2.26) for xm- j = w, we must have

-,A* (aO, XO ... xm _f -1 w9 yo .. eYm -j-19 y) and hence -nA* +2(a, xo- *eXmp_, Yo * Ym-fj_-). Hence A* +2 implies

VXm-j3Ym-i ... VxmgymAo(a, xo * Xm, yo ... ym). Conversely, VXm -j3Ym _g *... Vxm3ymAo(xo .X..m, yo ... ym) implies that

3Ym-j ... Vxm3ymAo(a, XO e xmg yo * Ym) holds for xm-j = the unique w satisfying (2.27). By (2.25),

A* 1(a, xo ... xm-,i w9 yo ... ym-,i) follows. Hence so does A* +2. This proves (2.24) forj + 1.

Then, by (2.25),

VXm -fj + 3Ym - +1* * Vxm3ymAo(a, Xo C *Xm, yo * Ym) A*f + l(a3 xo .. x m-j5s Yo .. ym -1-1)

which yields, by (2.24), A* (a. xO ... xm -fg YO .. *Ym-_j) ') A* +,l(a, xO .. * * X y -sYO**Ym_ --1)

the translation of (2.6),. The translation of (2.7), is proved similarly. This shows that all axioms are translated into theorems of IRn+1 + (2.4). As

for rules of inference, substitution is clearly preserved in view of 3 !yCty; modus ponens is trivial. For induction, we need only observe that every formula is translated into one equivalent to a Fn + 1 formula. If we translate s = t by 3y(Csy A Cty) then, since 3 !yCy and 3 !yCty, we have

I -n3y(Cy A Cty) _= y3z(Cy A Ctz A y # z) and this gives a Fn f + 1 equivalent for any propositional combination of equations.



474 CHARLES PARSONS

Thus we have a proof in IR + 1 + (2.4) of

(2.28) 3y[Cv(b, X, y) A A2m(b, X, y)].

By (2.24) with j = m we have

A*m(b, x, y) -Vx23yj - - *Vxm3ymAo(b, xx,x Xm, y, Yi y *Ym)

and hence (2.28) implies

Vxo3yo ... Vxm~ymA0(b, .. xm, yo. Ym)

which is therefore provable in IRI + + (2.4), proving Lemma 1.

?3. We now consider a formulation of classical number theory in the Gentzen sequential calculus G3 of [4, p. 481]. Induction is now expressed by the rule

GI r P->A, Ao rP->, Aa = A(Sa) r A, At

where a is not free in the formulae of r or A. GI is equivalent to IAn if A is restricted to have < n nested quantifiers, to IAn' if

A is restricted to be Hn, etc. If A and the formulae of r and A are restricted to have <n nested quantifiers, GI is equivalent to IRn. Let GIn be GI so restricted. If A is restricted to be Hn and the formula of r and A are restricted to be Hln or Tn then GI is equivalent to IR applied to formulae A z Bx where A and B are Hn and A does not contain free x. Call this rule IR1'+ and, the corresponding GI, GI". That IRnn+ is derivable by GI" is clear. Given premisses of GI", let r be B1 ... Bk, AC1 ... C1. Suppose that

Bl A *... ABk.=.ClV* V Cz v AO, Bi A ... A Bk. : Cl v ... v Clv V. Aa = A(Sa)

have been proved in IRn+. The premisses truth-functionally imply B1 A A Bp 'm C V ... V Cq' V AO,

B1 A ... A Bp = C' v ... v C.' v Aa:= B" A\ ... Bp .= C,' v ... V Cq V A(Sa)

where B- ? * Bp are those Bi which are Hn and those - Cj where Ct is En, and C,'* C,' are those C1 which are 1,n and those -,B1 where Bi is In. Since B' A ... A Bp and C, v ... v C' v Aa are equivalent to Hn formulae, by IRn i and truth-functional logic we have

B1 A *** A Bk C1 V *** V C, V At. Now we have LEMMA 2. If B is a rIn +1 sentence provable in IAn, B is provable in IRn. If B is

provable in IA1', it is provable in IRnn'. PROOF. B is derivable in ZO from induction axioms

AOy A Vx[Aixy D A1(Sx, y)] z: Ajay

which we can suppose to have each a single parameter y, for i = 1 * * *m. Call this formula IAi(a, y). Then in the Gentzen formulation of ZO we have a proof of

(3.1) VyVa IAj(a, y), * * *, VyVa IAm(a, y) -* B1



ON f-QUANTIFIER INDUCTION 475

where B1 is the result of dropping the outer universal quantifier from B. By Gent- zen's theorem, we have a proof in which the premisses of (quantifier-free) inductions have quantifier-free proofs, and otherwise there are cuts only on quantifier-free formulae.

Every sequent in this proof of (3.1) is of the form

(3.2) Cl ... Ck- Dl. * -DI

where C1... Ck are each of one of the forms (i) Vylda IAj(ag y),

(ii) Va IAj(a, t), (iii) IA#(9 t), (iv) As(62 t) (v) a subformula of some At(s, t),

(vi) a subformula of B1, (vii) quantifier-free,

and DI ... DI are all of the form (iv)-(vii) or

(viii) AjOt A Vx(Aixt = Aj(Sx, t)), (lx) Vx[Ajxt :D At(Sxg 0)], (x) Ajbt = Aj(Sbg t).

Let Cu, e * * Ci1 -> D1 * * * D1 result from (3.2) by dropping all Cj of forms (i)-(iii). We show that, for every sequent (3.2) in the proof of (3.1), Cile*? Cj, --> D - e e-D is provable in GIn, and in GI" if the Ai(a, y) are all HI,,

This is proved by induction according to the proof of (3.1). We need to check only inferences which introduce formulae of forms (i)-(iii), and in fact only (iii), since for (i) and (ii) the reduced sequent Cl... C -* * D 1 e * - DI will be the same for the conclusion as for the premiss.

Formulae of form (iii) must be introduced by Ho inferences, i.e.

C ...* Ck- D ... * Di, AgOt A Vx[Ajxt = Ai(Sx, t)],

A1ost, C2.? . Ck -# ? ?D Ait C . Ck> D ... Di

N#95 t)9 C2? ... Ck-- D. e - DI

By the hypothesis of induction we have proofs in GI, of

(3.3) Cil* Ce, -> D1. -. DI, A0t A Vx[Ajxt D A1(Sx, t)],

(3.4) ~~~~~Aist, Cil ... Cip ->DI ... Di.

Since B1 is M, g it is clear that C1 * * * Cjp are all n-quantifier or simpler formulae, and If, or Zn if the Atay are all [II,. From (3.3) we obtain

(3.5) Cil. C* > * ...* D1, Af0t,

(3.6) Cl... * -> D1 * - * DI, Ajbt = Aj(Sb, t)

(b not free in any other formula in (3.5)). For each D1 of the form (viii) or (ix), let Di' be A,(a,, t) and let D2 be A1(Sa1, t)

with the aj all distinct. (i and t are in general not the same as in (3.3)-(3.6).) For Dj of form (x), D' is Atat and D2 is Aj(Sa, t). If Dj is of form (viii) let D3 be AjOt. If



476 CHARLES PARSONS

Dj is not of one of these forms let Dy be empty and Dj1 = Df = Dj. Then from the proofs of (3.5) and (3.6) we have proofs of

.... ** Ci, ... DD1- * *D D,* A*bt D A9(Sb, t),

C ...* Ci,, D -* ... D , A b Z Ajbt v A(Sb, t).

Clearly Di ...* DI, r = 1, 2, 3, are all < n-quantifier formulae and HI or Z,, if all the Afay are. Hence by GI,, (GIn) we can infer

Cil ... Cj., Di .. D* *- D2 ... * D2, Aist, Cil .. *i, * C> D3- ... D33, Aefs.

Then we can obtain by inference -* z', --.V, --> A, Cil - C*, -> Di. . DI, Aist and then C.... C** -> D1 ... DI by cut with (3.4).

If now we take (3.2) as (3.1), we have that ->B1, and hence ->B, is provable in GI,, (GIg). Hence B is provable in IR,, (IR11). Q.E.D.

Now we have THEoREM 3. If B is a ll,,, sentence provable in IAn, then B is provable in IRE. PROOF. If n = 0 this is trivial. If n # 0, by Theorem 1, B is provable in IAnn and

by Lemma 2 it is provable in IRn' . Then, by Theorem 2, B is provable in IR;. Q.E.D.

(If n = 1, this strengthens Theorem 3 of [10].) Note that if Ax is H1n, the induction axiom on Ax is Zn+, and its closure is

?4. Go-del's interpretation. We consider the free variable theory T of primitive recursive functionals of finite type (the types are 0 and (au) for all types a, r) where primitive recursion is covered by having for each r a constant R, and the axioms

RfabO = a, Rrab(Sx) = bx(Rsabx)

(a of type r, b of type (0)(T)T). Let T,, be the subsystem in which R, is admitted only if the rank of r is < n. The rank of 0 is O; the rank of (a)T is max (rk(a) + 1, rk(r)). For the usual intensional version of T (as in [13]) we allow for arbitrary r in T,, the functional f with the defining equations

eabO = a, ftab(Sx) = b (a, b of type r).2

Then it is easy to verify that either the extensional [11] or intensional version of T, suffices for the Godel functional interpretation of IA+ 1. Let A - be the nega- tive translation of A and let A' be the translation of A into a formula

(4.1) 3fl .. * fkV91 e * *?ZA nfl * * fk, 91- ..

*f1)

2 $ is needed for the definitions by cases needed for the interpretation of the axioms of in-

tuitionistic logic. Let SI be the rule: for any terms s, t not containing free cl ... cm, if 5c, ... Cm = tc1 ... Cm has

been proved by definitional conversions, infer 5 = t. This rule holds vacuously in the Spector formulation of T and is sound under the reading of = as Tait's strong definitional equality [13, p. 205]. In a version of T which is closed under SI, t can be defined as

AabxcL * * * cntO(ac1 ... Cm)(bc, * * * cm)x,

if r is (a,) ... (am)O and cl ... cm are of types a, ... am.



ON f-QUANTIFIER INDUCTION 477

where' 1... fkg g,* * *g1 range over functionals of finite type and AO is quantifier-free. Then by induction on n it is clear that if A is I +I 1 then A-' is of the form (4.1) with f1 ... fkof rank ?nandg, ..*g, of rank <n.(Ifn = O,A-'is A.)

In the extensional version of T we can code ...efk g, .. *g, as single functionals, so we suppose k = I = 1.3 If Ax is Il + , to satisfy the induction axiom we must have r, G, X, Y, Z so that (suppressing]'0 and F as parameters) (4.2) AO(OfJ, GHga) A : XHga < a

::* A4[XHga, YHga, H(XHga)( YHga)(ZHga)] AO[S(XHga), F(XHga)(YHga), ZHga] . : Ao(a, ra, g).

We can set ro = Jo, r(sx) = Fx(rx),

G'HgaO = g, G'Hga(Sx) = H(a Sx)(r(a -e Sx))(G'Hgax), GHga= G'Hgaa,

XHga = fuw < a{AO[w, rw, G'Hga(a I- W)] A -,Ao[Sw, r(Sw), G'Hga(a -- Sw)]} YHga= r(XHga),

ZHga = G'Hga[a S(XHga)], and (4.2) follows easily.

In the intensional version, the recursions defining 1 and G' become simultaneous recursions. However, by slight modification of the technique of [1], they can be satisfied in Tn.

Then by Theorem I we have THEOREM 4. IA',1 has a Gidelfunctional interpretation in Tn. It is possible to obtain a stronger result for the rule of induction, as follows. Let

PR be the rule for introducing constants by primitive recursion: given a, t closed terms of types (.l) ...* ('rk)'r and (.l) .... (,rk)(O)('r)'r respectively, introduce a constant 9p of type (.) ....* (Trk)(O)r with the recursion equations

al ...* akO ea,... *ak,

.al ... *ak(Sx) = tal . akx(pal ... akx).

3 It is sufficient to have SI. Consider, g wherefis of type (a,) ... (am)O, g of type (a') ... (ap)O. Then let

<f, g> = Au,... * mUl1 * V? %(ful ... Um)(g~V *.. VP).

For h of type (al)** (a.)(a') *.. (a4)(O)O set

v'h = Au ... Umhul * * * UmO?1 .. *Opd,

v2h = AV1 * . ?0 .m.n,. * v*VI.

Then SI suffices to prove zl<f, g> = f and v2<f, g> = g. Note that the rank of <Kf g> is the maximum of the ranks of f and g.

W. A. Howard has described to me (in correspondence) a method of carrying out Gddel's intepretation so that A' has only one existential quantifier and one universal quantifier of higher type. This makes it possible to carry out the interpretation in a common subsystem of the extensional and intensional T, as formulated in [3].

ADDED IN PROOF. That Gddel's interpretation requires either coding of pairs or some other treatment of simultaneous recursion was pointed out by A. Grzegorczyk, Recursive objects of all finite types, Fundamenta Mathematicae, vol. 54 (1964), pp. 73-93. He gives a method of coding which requires SI (for A-conversion), without remarking on this fact. That strong A-conversion is sufficient and apparently also necessary for such coding has also been observed by W. W. Tait and by S. Stenland. See Proceeding of the Second Scandinavian Logic Symposium (J. E. Fen- stad, Editor), North-Holland, Amsterdam, 1971, pp. 173-174.



478 CHARLES PARSONS

Given RX, 'p can of course be explicitly defined as

Aal ...akR-#a1- -ak)(tal ...ak).

Let TS* be the subsystem of T,, which admits R1 only for types T of rank < n but for ' of rank n admits the rule PR provided that the types 71. - o of the parameters are of rank < n. (If n = 0, this is just number theoretic primitive recursion.) Then we have

LEMMA 3. IRY++1 has a Gidelfunctional interpretation in Tn*. PROOF. The rule IR1+1 is equivalent to

A r BO A :. Ba = B(Sa) A ZBa

with A, B, lIL + and A lacking free a. Assume first the extensional version of T*I. Let A' be 3FVfA0(FFf) and let B' be 3GVgBD(a, G, g). Then Fand G are of rank n.f and g or rank n - 1 (if n = 0, F, G are missing andfand g are of rank 0). If we have an interpretation in TI* of the premisses of this inference, then we have functionals bD, 01, TO, L, 'F2, and proofs in TI* of

(4.3) Ao(F, ToFf) = BO(0, 4oFsf),

(4.4) AO(F, 'FaFGf) A. BO(a, G. T2aFGf) : BO(Sa, SDaFGf)

(parameters of A and B, of type 0, suppressed). Then clearly we have in Tn*?, X such that

(4.5) 4DFO = 4oF, DF(Sx) = D1xF(41Fx),

(4.6) XFfaO = f, XFfa(Sy) T2(a -, Sy)F(DF(a 0 Sy))(XFfay).

(4.4) yields

b < a A AO[F, 'YbF((DFb)(XFfa(a Sb))] D=. BO[b, (Fb, XFfa(a * b)] O BO[Sb, 'DF(Sb), XFfa(a * Sb)]

and so by induction"

Vx < aA0[F, WxF(DFx)(XFfa(a In Sx))] A BO(O DoF, XFfaa) .: BO(a, JFaf). Hence, by (4.3),

Vx < aA0[F, 'T1xF(DFx)(XFfa(a Sx))] A Ao0[FJ TOF(XFfaa)] .z BO(a, 4Daf).

Hence we have

AO(F, T*Ffa) = BO(a, Fa, f) where we set

T*Ffa = TOF(XFfaa), if -,AO[F, TOF(XFfaa)], = '1xOF(0FxO)(XFfa(a -- Sxo)), otherwise,

where xO = ux < a-nAo[F, T1xF(?Fx)(XFfa(a -Sx))].

4 Since AO is a propositional combination of equations of type 0, it is equivalent in To* to a formula of the form t = 0. Then the bounded least number operator and bounded quantifiers can be defined in the usual way, and the definition by cases of T* can be reduced to an explicit definition by a functional Ce:



ON n-QUANTIFIER INDUCTION 479

In the intensional case, the recursions (4.5) and (4.6) are replaced by simultaneous recursions, which again can be handled by the technique of [1]. This proves Lemma 3.

THEORM 5. IR'+2 has a Gidelfunctional interpretation in T,*e. PROOF. By Theorem 2 and Lemma 3, it suffices to reduce IRn+1 to IR='+1.

Suppose AO and Aa = A(Sa) have been proved, with A n+ 1. Then we have

a < b o. -,A(b -, a) = -nA(b -- Sa)

and -,Ab : -,A(b 0). By IR++, with -,Ab as A and a < b : -7A(b -- a) as

Ba, we have --Ab =.b <b D--A(b -, b)

which, by AO, implies Ab. Q.E.D.5

?5. Faithfulness of Goidel's interpretation. An interpretation which associates with each formula A of a system S1, a sequence AO, A1, A2, * ? * of formulae of S2 such that from a proof of A in S1 a proof of some Ak in S2 can be obtained, is said to be faithful if A is provable in S1 whenever some Ak is provable in S2. (In Kreisel's earlier terminology 'complete', e.g. [5].)

In [7] (cf. [8, pp. 380-381]), Kreisel showed the faithfulness of the Godel interpretation of Z in Spector's formulation of T by interpreting the variables of T as ranging over effective operations. By working out this proof in detail, it is possible to show that the interpretations obtained in ?4 of subsystems of Z in subsystems of the extensional T are all faithful.6 Below, when we speak of T we mean the extensional formulation.

All that is necessary to prove this is to check carefully how much number theory

5 This theorem was announced to a meeting of the Association for Symbolic Logic in Chicago, Illinois, on May 4, 1967. See [15, Theorem 2]. As was done for n = 0 in [10], the proof was a direct construction of the interpretation of the conclusion of IR,+ 2 from assumed interpretations of the premisses. Then corresponding to the definition of (I is a definition by PR with r of rank n + 1 and r * * rk all 0. s and t are terms (by hypothesis of induction) composed from constants of rank n + 1. This can be regarded as a generalization to rank n of nested primitive recursion, and it is reducible to rank n recursion by a natural generalization of the method of [12] of reduc- ing nesting primitive recursion to ordinary primitive recursion. However, this reduction can be carried out formally (it seems) only in a version of T,* with a strong rule of extensionality which destroys the decidability of equality at higher types.

Although [15] does not make this clear, I envisaged at the time Spector's version of T. I did not see that the argument required strong extensionality.

However, this interpretation of IR~' 2 in T,* plus strong extensionality can be shown by the method of ?5 below to be faithful already for IR,,+1. This yields an alternative reduction of IRI+2 to IR + 1 but it seems not to yield the further reduction to IR +1 (if n # 0). A similar method yields an alternative (much more complex) proof of Theorem 1 above.

Theorem 2 of [15] also stated that IAn+j has a functional interpretation in Tn, a weaker version of Theorem 4 above. However, the proof I envisaged was incorrect.

6 Kreisel's method proves also the faithfulness of the interpretation of Z in the intentional T, but the translation is unnatural in this case, and it does not yield proofs of the faithfulness of the interpretations of subsystems we obtain in ?4 and [10]. The latter could presumably be proved by formalizing an interpretation by intensional effective operations such as that of [14].



480 CHARLES PARSONS

is used in translating T into Z by the interpretation by effective operations, and in proving the translation of the schema QFAC

Vbj ...b,3c, ...CkA(bl ...bjlC1 ...CO)

D> 3al ... akVbj ** *bjA(bj ... bl, alb, ... b, ... akbl ... bl).

This is straightforward but very lengthy, so that we do not give the proofs. But we have

LEMMA 4. (a) Tm is translated into IAn +1. (b) If m # 0, Tm* is translated into IR+1. (c) Let To be To* plus the functional Wfa = I7JcapfP (see [10]). Then To' is trans-

lated into IR! + FACo. LEMMA 5. If A is a formula of To, then QFAC is translated into a formula

provable in IRK + FACo. Let At be the translation of A '. The quantifier-free part of A -' is a formula of

To* and hence of To. Lemma 4 implies that if A -' is satisfiable in Tm (T,* with m 0 0, To) then At is provable in IA"+1 (IRm+1, IR1 + FACO). The argument that with QFAC A' is equivalent to A [6, p. 120], translates directly so that Lemma 5 implies that At = A is provable in IR! + FACo. Thus we have

THEOREM 6. If A is aformula of Zo and A-' is satisfiable in Tm, then A is provable in IA"+ . IfA-' is satisfiable in T*, then A is provable in IRt+1, and, by Theorem 2, in IRz + 1. If A-' is satisfiable in To, then A is provable in IR" + FACG.

Thus the interpretation of Theorems 4 and 5, and the interpretation of IRT + FACo in To (see [10]) are faithful.

The argument sketched does not work for the interpretation of IR. in To*. How- ever, if A' is satisfiable in T*, then a no-counterexample interpretation of A is provable in T* [8, p. 380]. The functionals are explicitly definable from primitive recursive functions and (rank 1) function arguments. From this one obtains an Herbrand alternation for A which is provable in To*, and since T* is a conservative extension of primitive recursive arithmetic [10, Lemma 5], also in PRA, and A is deducible from it in quantificational logic. Hence A is provable in Z0 and therefore in IRE.

Lemma 4 rests on the fact that if r is of rank ? m, then the predicates, Ore: e is the GMdel number of an effective operation of type a, Etef: e and f are the Godel numbers of extensionally equal effective operations of type '7,

are equivalent in Z0 to Im+ 1 formulae. If 'r = 0 we can define E~ef as e = f; if .7 = (a,) ... (ac,)0 as

VXi .. **XmYi .. *ens Ea,,(xj, yt) =, 3u3v[Tn(e, xi * @Xn3, U)

A Tn(fy y ... yn v) A Uu = Uv]}

and then define OQe as Elee. Then the crucial step is to show that R, corresponds to an effective operation,

that is to show for certain number p (if r = (a,) - * . (an)0)

(5.1) E,(a, a') A E(O)(.(e, e') = E[Sn3(,F, a, e, x), Sn3(F, a', e', x)].



ON f-QUANTIFiER INDUCTION 481

This is shown by induction on this formula, which reduces to the induction axiom on the consequent. p is chosen so that

{p}aeOx1 ... x. = {a}xl *x. {p}(a, e, Sx, xi* -xn) {e}(x, S3(p, a, e, x), xil- * xn)

(S as in [4, p. 342]). Thus for r of rank ? m, this is proved in IA" +1. If p is introduced by PR with value of type 7r and parameters a, * . ak of types

.1-. ek all of rank < m, then we use induction not on (5.1) but for some q on k

A Ef(ci, ct) D Ez[Snk+1(i, C1*i* C-k, x), Snk+1(q, cl - 'ck, x)] i=1

which, since the antecedent is now I1m+1. is a case of IRm+,. The proof of Lemma 5 involves formalizing the proof of the well-known theorem

of [9] that effective operations belong to a certain class of continuous operations. Although this proof is rather complex, it uses more than Z* only in one respect. Suppose e is a neighborhood function for a continuous (partial) functional, where ul... u?n code neighborhoods for its arguments. Then FACo is needed to show that for any y there is a neighborhood containing <ul .. ue, {e}u1... un -* 1> for all u1 ...* u* such that <u1 . *n> < y and {e}u1 ...u * # OA 0.

Instead of translating T into Z by taking the variables as ranging over effective operations, it is also possible to take the variables as ranging over continuous functionals which have recursive neighborhood functions and are defined for all arguments which themselves have such neighborhood functions (of lower ranks). The proof of the translation of QFAC is then much simpler (cf. [6, 4.141]), but the translation and the proof of Lemma 4 are more complex. Perhaps the overall argument is slightly simpler.

Some brief remarks should be made about the no-counterexample interpretation. Given a formula A of Z, let A, be the validity functional form (in the sense of [2]) of a prenex equivalent to A. Since A D A, is provable in predicate logic, if A is provable in a subsystem of Z then A, is too (if the function parameters are added). If A is provable in IA. +1, the A, is also, but then by Theorem 3 above it is provable in IRE+ 1 with the function parameters. Since the Godel interpretation of A, is a no- counterexample interpretation of A, it follows that the no-counterexample interpretation of IA"+1 for m # 0 can be carried out in Tm*. (Not if m = 0, since the recursions involve parameters of rank 1.)

It follows that the more direct no-counterexample interpretation (NCI) of IRm+i in Tm* is, for m # 0, not faithful. The same would be true for the interpretation in a second-order free variable system with ordinal recursion on standard orderings of type < Wm + i(W).7 It does appear that the NCI of IRm+m in Tm* is faithful for I1n+2 sentences.

The question remains open whether the NCI of IA"+1 in Tm* with in # 0 is faithful for all formulae. If in defining the NCI one replaces A, by the functional form of A (with the quantifiers where they stand), then this modified NCI is strongly non- faithful for the partial systems. In this sense, any instance of FAC has a NCI by

7 Cf. [15, Theorem 1].



482 CHARLES PARSONS

elementary functionals (i.e. in a proper subsystem of To). For example the functional form of an instance of FACI' is

Vx < aVzA[x,fx, z] v 3c{gc < a v A[gc, c90, hc]}

and it is satisfied by setting C= Xl x=gC, z= hc.

i<a

REFERENCES

[1] JUSTUS DILLER and KURT SCHWATE, Simultane Rekursionen in der Theorie der Funktionale endlicher Typen, Archiv fOr mathematische Logik und Grundlagenforschung, Vol. 14 (1971), pp. 69-74.

[2] BURTON DREBEN and JOHN DENTON, Herbrand-style consistency proofs, in Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 419-434.

[3] W. A. HOWARD, Functional interpretation of bar induction by bar recursion, Compositio Mathematica, vol. 20 (1968), pp. 107-124.

[4] S. C. KLEENE, Introduction to metamathematics, Van Nostrand, New York, 1952. [5] G. KREISEL, On the concepts of completeness and interpretation offormal systems, Funda-

menta Mathematicae, vol. 39 (1952), pp. 103-127. [6] , Interpretation of analysis by constructive functionals of finite types, Constructivity

in mathematics (A. Heyting, ed.), North-Holland, Amsterdam, 1959, pp. 101-129. [7] , Inessential extensions of Heyting's arithmetic by means offunctionals offinite type,

this JOURNAL, vol. 24 (1959), p. 284 (Abstract). [8] , A survey of proof theory, this JOURNAL, vol. 33 (1968), pp. 321-384. [9] G. KREISEL, D. LACOMBE and J. R. SHOENFIELD, Effective operations and partial recursive

functionals, Constructivity in mathematics (A. Heyting, ed.), North-Holland, Amsterdam, 1959, pp. 290-297.

[10] CHARLES PARSONS, On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 459-473.

[11] CLIFFORD SPECTOR, Provably recursive functionals of analysis, Proceedings of Symposia in Pure Mathematics, Vol. 5, 1962, pp. 1-27.

[12] W. W. TATr, Nested recursion, Mathematische Annalen, vol. 143 (1961), pp. 236-250. [13] , Intensional interpretations of functionals of finite type. I, this JOURNAL, vol. 32

(1967), pp. 197-215. [14] , Constructive reasoning, Logic, methodology, and philosophy of science. III

(B. van Rootselaar and J. F. Staal, Editors), North-Holland, Amsterdam, 1968, pp. 185-198.

[15] CHARLES PARSONS, Proof-theoretic analysis of restricted induction schemata, this JOURNAL vol. 36 (1971), p. 361 (Abstract).

[16] , On a number-theoretic choice schema. II, this JOURNAL, vol. 36 (1971), p. 587 (Abstract).

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