εn-Arithmetic

&"-ARITHMETIC J. P. CLEAVE Brisrol University, W K

and

H. E. ROSE Leeds University, U K

In [ l ] Grzegorczyk defined the class of functions 8" (for each n 2 0 ) by the following procedure: The initial functions are

U,(x)

=x

+ 1,

Ul(X,Y) = x,

U2(X,Y) = Y

and

f,(X3Y)>

which is given by f,(X,Y) =Y

+1,

fl(X,Y) =x

+Y

2

f 2 k Y ) =(x

fk+3(0,Y) =fk+2(Y fk+3(.

+ l).(Y + 11,

+ 1,Y + 11,

+ 1,Y)=fk+3(x,fk+3(x,Y)),

and the class is closed under the operations of substitution (replacing a free variable by a function, another variable or a constant) and limited recursion given by $ ( O , X ~ , . . . , x m ) = ~ ( ~ . .1. > x9 m ) $(y 1 , ~ 1 , . . . , x m ) = q ( y , x l, . . . , ~ ~ , $ ( ~ , ~ l , . . . , x m ) ) $(y,X1,...,xm) < r(~,xI,...,xm),

+

where the functions p , q and r have already been defined. It is shown in [ l ] that f , ( x , y ) is monotonic increasing in x,y and n and that 8"containsfi(x, y ) if i
298

J. P. CLEAVE AND H. E. ROSE

In this paper we shall discuss some metamathematical properties of €"-arithmetic. We begin by giving a new characterisation of €" functions by showing that the function fn(x,y ) may be replaced by the Ackermann function. Using some of the results of this work we show that the addition of a certain class of limited doubly recursive functions to 6"does not extend the system and this is sufficient to establish that the valuation function for equations of &"-arithmetic is an 8"" function provided n 2 2 . Applying this to Rose [3] we get our second main result: If n 2 2 the consistency of €"-arithmetic is derivable in 8"' '-arithmetic. Finally, adapting Kreisel [4], we prove that, again if n 2 2, &"-arithmetic is categorical with respect to recursive models. These results are analogous to those in Wang [5] where it is shown that the consistency of a certain partial system of elementary number theory (obtained by restricting the number of quantifiers involved) may be established in a similar but slightly more complicated system (that is, by allowing one more quantifier in the proofs); whereas extending €"-arithmetic to &"+'-arithmetic increases the complexity of the functions involved I ) . We note that all the results quoted or proved in this paper about &" functions and €"-arithmetic are valid for n 2 2; that is, @-arithmetic (and not d3-arithmetic- which by analogy with the g3 functions, the elementary functions, would be called Elementary Arithmetic -) is the smallest system in the Grzegorczyk hierarchy having these properties. So @-arithmetic, rather than b3-arithmetic, is the system which should be singled out by giving a special name to it; we suggest Primary Arithmetic.

1. The Ackermann function

l) In [6]Cleave has given an w 2 chain of classes of functions Emcomputed in such a way that the lengthof thecomputationof a functionf(x)of Ea+lconsideredasafunctionof x must itself belong to Ea and it can be shown that Em. = gr+l ( u = 1,2, ...). Hence the position of a function in the Grzegorczyk hierarchy is a measure of its complexity.

&"-ARITHMETIC

299

1. The class of functions 9" is identical to the class 8". THEOREM (Problem 7 of [I].) For n=O, 1 and 2 the result is obvious. To prove the result for n > 2 we first show LEMMA2. There is a function G,(x,y)belonging to 8" with the property Gfl(x,u>>9" (X,v). Then g,,(x, y ) is defined in 8"by g,(O,y)= 1 ,

gn(x3Y)
g,(x+ l,Y)=gfl-,(g,(X,Y),Y),

Conversely we show 3. There is a function F,(x, y ) belonging to S n with the properLEMMA ty F A X , Y ) > f " ( X > Y>. by the following procedure. Define first Thenf,(x, y) is defined in 9" hfl(x,Y ) by

h ( 0 , Y >=

4 ' 2

h"( x

+ 1,

~)=fn-l(hn(x,

Y>, hn(x, Y ) ) ,

h,(x, Y ) < Ffl(X, Y ) * (This last inequality follows as h,(x, y)
f,,(o, y>

=ffl-l

+ 1, Y + I ) ,

(Y

fn(X

+ 1, Y ) = h n ( x + I . f n ( x , Y ) ) ,

Y ) < F n ( X , Y>*

f n i ~ ,

Theorem 1 follows easily from these lemmas. To prove lemma 2 we show that G.(x., Y > =fn(x, Y )

'

Sfl(X7

Y).

By induction we get first and so, as

f"(X

+ 1, Y > > f n -

f"(% Y ) >Y ,

f"(X

1 (ffl(X3

Y),ffl(X? Y>>

+ 1, Y ) > f , , - l (f&

Y>Y Y ) .

Using the defining equations of gn(x,y ) the result follows. Proof of lemma 3. We need first three further lemmas. LEMMA 4. For n> 1, g,(x.y, z ) 2 g f l ( X ,g,(Y, z)). This is derived by induction on n. For n = 2 both sides of the inequality are equal to x . y * zand for n= 3 both sides are equal to z * ' ~ Assuming . the result for n = k we get immediately, since g,,(x,y ) > x . y for n>2, gk (gk (x, Y ) , ). 2 gk (x>g k ( Y 3

'1)

'

(1)

300

J. P. CLEAVE A N D H. E. ROSE

Now, as g, (1, Y ) = Y ,

'),')

g k + l ( X ' Y , ' ) = gk((gk...gk('9')2'.'

where gk occurs x.y- 1 times and gk+ I (x,gk+ 1 ( Y , ')) = = gk[gk... gk[gk(...gk(',')

... Z ) , g k ( * . . g k ( ' , ' ) " .

".

7I)'

g k ( . . - gk('5')

... ')I

where g k occurs (x- l ) . ( y - 1 ) times, and the result follows using (1). We define now g,"(x, y ) by g,"(x,y)=y,

s,""(X,Y)=Sn(X,Snm(X,y))

and we get from lemma 4 LEMMA 5. (4 g,"(x, Y ) S S n( X " , Y ) , (b) yr(x, y ) belongs to P' if, n >2. LEMMA 6. For n > 2 , g , , . l ( x . ( m + l ) , y ) > g , " ( x , y ) . PROOF.We have g,"

(x,y ) 5 g n (x: Y ) by lemma 5 5 ~ n ( ~ n + l ( m , x ) , Y a) s g n + l ( m , x ) 2 x m

< g n (gn+1 ( m ,gn+ 1 ( x , Y ) ) , gn+ 1 =gn+l(m

+ 1Afl+I(X>Y))

I gn+ ( x - ( m

+ l),y )

(

~

Y)) 3

by lemma 4.

To prove lemma 3 we show now that F n ( x , y ) = gn2x((2y+2

+ 1>",Y)'

f,(X,Y).

We derive this by induction on n and .x,the result holds for n = 3 since f3

( x , y ) < y 2 2 x and

gix((2y+2

+ 1)3, y ) > y 2 3 ' y ' 2 x .

+

As inductive hypothesis we assume the result for n = k and for all x _ < y 2, then gk+1((2Y+ +2l ) k + l , y ) > g : y + 2 ( ( 2 y + 2 l ) k , y ) by lemma 6 by hypothesis f k (Y + L') by a simple > fk(y + l?Y+ '1 inductive argument

+

'

=f

2 j

k

+ 1 (O, y ) .

301

&"-ARITHMETIC

4 (0, f X ( X , y

_I_

l)),4 (0,f X ( X , y

1)

I), ...

Y

4 (0,O)

*

(We may assume thatf(x, y ) is monotonic in both x and y, if it is not C : = o f ( i , j ) . )This sequence has the property we can replace it by

x;=o

302

J . P. CLEAVE AND H. E. ROSE

that ail the information required to calculate any one of its terms occurs in the succeeding terms and the first equation in the schema of the theorem, to calculate 4(t, u) we require 4(t, u' 1) and + ( t i 1, 4(t, u'- 1)) but this occurs as one of 4(t' I , 0), 4 ( t I,~I), ..., + ( t I~, f ( t , u~ I)). We will now give a formal expression R ( x , y , z ) belonging to 8"corresponding to the relation z = 4 ( x , y ) using this sequence. Let p u be the a-th prime number, (m), the exponent of p , in the prime factorisation of m, A: and E: the usual bounded universal and existential quantifiers, g(a)=xP=o(f"Li(x,y - I ) + 1) - so g(a) is the number of terms in the last a+ I lines of the sequence above and

F ( x , Y ) = (Pq(x))f(X'Y)(g ( X I + 1) then R ( x , JI, z ) E E ~ ' x ' Y ' [ A ~ ( o ) [ (=mp) (i 0 , i ) ] & A ; [ ( m ) q ( j ) + = l p ( j 4- 1, o)& A p + l ) L ( e ( j ) + z )[ ( m ) g ( j ) + k + Z

=(

m ) q ( j ) ~ ~ + ( m ) g ( j , + k + ~ + l= I l (&m~) q ( x ) I *

In this expression + ( t , u) corresponds to ( t ~ ) ~ ( ~ ~ ~ )I + fn>2, , , + ~R. ( x , y , z ) is a relation of 6"and now 4 (x,y ) is defined in 8"by

4 ( x , Y) = P Z [ Z r f ( x ,

Y ) &R(x, Y , .)I.

(See [I] for a discussion of the least number operator p in

c?"".)

2. The consistency of &"-arithmetic In this section we shall show, for n 2 2 and using the proof procedures of &"+'-arithmetic, that &"-arithmetic is consistent. (It will follow also that the proof of the consistency of 6'- and &'-arithmetic can be carried out in b3-arithmetic.) The proof will mirror exactly that given in [3] where five conditions are laid down on systems of recursive arithmetic R, and R, such that the consistency of R, is derivable in R,. We will give these conditions now and show that they can be adapted, where necessary, to our case. CONDITION 1 . R, contains primitive recursive arithmetic. This condition can be weakened considerably without affecting the proof. All that is really required in this condition is that R , contains a function th(y) which is the Godel number of the y-th theorem of R, and it is clear from 131 that th(y) is a function of 6' (a pairing function is an essential requirement for the definition of th(y); this is contained in 6'

&?"'ARITHMETIC

303

and no other function used in the definition increases faster). So condition 1 may be replaced by: R, contains @-arithmetic. CONDITION 2. R, may be given a finite codification by introducing function variables for functions of some finite number of argument places. CONDITION 3. The axioms of R, are finite in number and, upon arbitrary substitution of numerals for numerical variables and functions for function variables, are verifiable. CONDITION 4. The axiom schemata of R,,are those of primitive recursive arithmetic and rules of substitution for function variables. Ternary recursive arithmetic 191 is a codification of primitive recursive arithmetic satisfying conditions 2, 3 and 4. In this formulation functions of one, two and three argument places are given by recursion and composition (i.e. functions are obtained by substitution of a particular function for a function variable in a schema of composition or recursion) and terms are obtained by substitution of functions, in the sense above, for numerical variables. The set of terms in this system is the set of primitive recursive functions; this is established in [9] and the proof uses only the functions and methods of b2-arithmetic. Hence, for n 2 2 , a similar codification of &"-arithmetic can be given. In fact a simpler formulation can be given in the following manner. w(a, b) is the pairing function given in [9] as are the functions m,a and m2a where m,w(a,, a 2 ) = a i , ( i = I , 2 ) and w(mla, m2a)=a. The functions of this codification are w(a, b), min (a, 6) the minimum of a and b and a finite number of one variable functions 4 i ( u )( i = O , ..., k), including mla, m2a andf,(m,a, m2a), and the function definition schemata will be Composition Cfg(a)=f(g(a)) and Limited recursion R fg ( a ) where R fg (0) = 0 and R f g ( a + 1)=min ( f ( a + I), g(Rfg(a))). (It is clear that this form of limited recursion is equivalent to the original one @(O)=O, 4(a+ I)=g(@(u)), #(a)
304

J. P. CLEAVE AND H. E. ROSE

A valuation function v,(a, b) for the single variable functions of 8"arithmetic as codified above is given by the following definition: v,(a, b) is the value of the a-th function at argument b. u n ( i , b) = 4 i ( b ) for

+ +

i = 0, ..., k

k 1 , b ) = W(U,(m,n,b),~,(m,a,b)) v, (4a + k + 2, b) = min (un (m,a, b), u, (m,a, b)) vn(4a + k + 3 , b ) = U , ( ~ , U , U , ( ~ , U , ~ ) ) vn(4a + k + 4,O) = 0 U, ( 4 + ~ k + 4, b + 1) = U, ( m , ~ ,U, ( 4 i ~k + 4, b)) U,,(~U

where

vn(4u + k + 4 , b ) I u n ( m , a , b ) .

LEMMA9. If n 2 2 then o,(a, b)
so ~,(4a

w(a,b)<(a

+ b + 1), s 4 w ( u , b ) + 1 .

+ k + 1, b) = w ( u , ( ~ , u ,b),u,(mza, b)) w(fn+l(m,a,b),fn+,(m,a,b))

by hypothesis

< ( f n + 1 (mla, b ) + f n + 1 (mza,b ) + 1)' by (2) I f n + l ((mla + m2u + I)', b) using properties of (4a + 1, b ) by ( 2 )
I

fn+

fn+l(a,b)

1

Case 3. This follows similarly as min(a, b ) l w(a, b). Case 4. vn(4a + k

+ 3, b) <

fn+,

( m l a ,f n + l (m,a, b)) by hypothesis

I f n + , ( m , a + m2a + 1, b ) using definition of

5 fn+

(4a

+ k + 3, b )

fn+

by (2).

1(a,

b)

(2)

&"-ARITHMETIC

305

Case 5 is immediate. LEMMA 10. If n 2 2 then u,(a, h) is an 8"" function. Using lemmas 7 and 9 this is established in an exactly similar manner to theorem 8. (Note 1. This is an improvement on the result in [l], but our method will not yield anything when n=O or 1 asfp(a, b) is not an b2function. 2. It follows from lemma 10 that for every 4 in b", n 2 2 , if for all m t,4(Ocrn)) then kn+,4(x). We have shown now that all the conditions of [3] are satisfied with R, as &"-arithmetic and R, as 6"+'-arithmetic, provided n 2 2 , and so THEOREM 11. kn+ Consistency of &"-arithmetic, where the symbol kn+ refers to derivability in &"+'-arithmetic.

3. Undecidability and categoricity with respect to recursive models It will be shown in this section that for n 2 2 , 8"-arithmetic is undecidable (in fact, creative) and categorical with respect to recursive models, that is, has no recursive non-standard models. Both these results derive from the fact that for n 2 2 certain recursively inseparable sets can be represented in 6-arithmetic. ln preparation for this we examine first certain syntactical properties of &"-arithmetic. For any n, let P ( y ) and Q(y) be formulae (i.e. equations) of 8"arithmetic each with one free variable y . Let W , ( y ) denote P ( y ) & Az[iQ(z)], W,(Y) denote Q(y)&Az[l P(z)l, Xl(x) denoteE:[W~(y)l and X 2 ( x ) denote E ; [W2( y ) ] ; these are &"-formulae. (Note E : [ f ( n ) = g (n)] = o( 1L(1 2 I f ( n ) , g (n)l))= 0, so the bounded quantifiers belong to 8' as 1 1 I x , v ~ = ( I - ( x l y ) ) ' - ( y - x ) . ) LEMMA 11. t,i(Xt(z)&X2(t)). We have

n:=

l ( W i ( x ) & W 2 ( ~ ) ) -l P ( x ) v E , " [ Q ( b ) ]v ~ Q ( Y ) V J Y [ P ( ~ ) I and this formulae holds always because in the disjunction either P ( x ) or Q(y) occurs with its negation and the result follows. Now let ) 3 TI (m,x, x , Y ) 4 2 (x,Y ) = TI (m2x, x9 Y ) 41 (x.Y

306

J . P . CLEAVE AND H. E. ROSE

where T, (e, x,y ) is the usual Kleene TI predicate defined in [lo]. Kleene remarks, in chapter XI of [lo], that TI is a predicate of b3-arithmetic; in fact it is an g2 predicate (to show this we have to replace the pairing function using prime numbers by our w function (an g2 function) and note that, although the iteration of M' is not an 6' function, this does not affect the result as we do not require a Godel number to be attached to a unbounded number of formulae). It follows that d1 and 42 are also 8'functions. Hence there are formulae R ( x , y), S(x, y ) in &"-arithmetic for n 2 2 each with only two free variables such that for allj, k

Now let R(O(j),y ) be the P ( y ) and S (O"), y ) be the Q ( y ) for lemma 11 and let W 1(O"), y ) , W2(O"), y ) , X,( O ( j ) . y ) and X , (O"), y ) be defined as in that lemma. Then for n 2 2 k,

1(XI (O"',

2) &

x,(O?

We shall also require

x,(O"', F" x,

t-,

(O(j),

t ) & t < z --t t ) & t < z+

(4)

t)).

x,(O(j), z) x,(O"', z)

(5)

which can easily be shown to hold. Next let HI and H2 denote the informal counterparts of W, and W,, that is H , ( j , Y ) = 4, ( j , Y ) [ 4i 2 (jl H2 ( j , Y ) = 42 ( j , Y ) 1-l 41 ( j , 41 .

x

.>I

So we have for n 2 2 by (3) and as X , ( O ( j ) , O ( k ) ) and X 2 ( 0 ( j ) O(k))contain , no free variables (j) o(k)) El; [Hl ( j , 41 Ffl XI (0 (6) ~2[ H (~j , cf kfl x2 (0( j ) OW)).

.>I

+-+

Now it can easily be shown that LEMMA 12. The sets Ill and Il,, where

n, = . f [ ( E y ) H , ( x , ~ ) ] are effectively inseparable.

and

n, = ~ [ ( E Y ) ~ ( x , Y ) ]

&"-ARITHMETIC

307

We come now to the main results of this section; the properties of &"-arithmetics we require are (4), ( 5 ) and lemma 12. Let 9" denote the set of (Godel numbers of) &"-formulae with at most one free variable which are derivable in &"-arithmetic. We shall call an 6"-formula F (x) refutable if there exsists a provable counterexample, that is, if there exists a number j such that k,,i F ( O ( j ) ) . Let 9" denote the set of Godel numbers of refutable one variable formulae in &'-arithmetic. THEOREM 13. If n 2 2, 9" and 9,are effectively inseparable *). PROOF.Let XI = j

(it

[(Ek)(kflXI (0

1,= j [k"

>

x,( O ( j ) , x)]

1

>>I

(k)

0

.

We shall show first, in four steps, that C, and 1, are effectively inseparable. .Zl nC, =0 (the empty set). (4

For if n z ~ C nC, , then I-,, X,(O"', O(k)) for some k since mEC,.Further k n i X , (O("), x) since mEC,. Hence k n iX , (O'"), O'kj) but now, by (6), we have a contradiction and so (a) follows.

Let mel7,. Then ( E y ) H , ( m , y ) and so H,(m, k ) for some k , hence

E:[H,(M,x)]. So by (6), k?,X , (O'"), O'k)) and thus M E Z , .

Let mEn,. Then H,(m, k ) for some k and so E:[H,(m,x)] and (6) gives /-,, X,(O("), O(k)). But from (4) we have I-flX , (O'"), O(k))-+ iXI (O'"), x), so kniXI(O("),x) and hence r n ~ C , . (d) It is well known that any two disjoint recursively enumerable supersets of two effectively inseparable sets are themselves effectively inseparable (chapter V of [ll]). From (a) - (d) and lemma 12 we conclude that C, and C, are effectively inseparable.

* This result, for full primitive recursive arithmetic, was proved by G . Kreisel in his review of [3] (see Math. Rev. 25, p. 746).

308

J. P. CLEAVE A N D H. E. ROSE

Now it is clear that there exists a (primitive) recursive function g such that g(m)is the Godel number of iX1(O("), x). So mE

z,

--

( E j )(bn i 7 X , (o'"', 0"'))

9 ( m )E Bfl

and

C,) is many-one reducible to (%fl,9,,), so again using chapter V Hence (Il, of [Ll], Bfland 3,are effectively inseparable as Cland C, are. COROLLARY 14. For n 2 2 , F nand Bnare creative. This follows from theorem 13 using [ll] again. In [4] Kreisel proved that a free variable primitive recursive arithmetic satisfying certain conditions is categorical with respect to recursive models (pp. 173-4); a similar method is given in [12]. We shall not reproduce the proof here but using (4), ( 5 ) and lemma 12 it can easily be checked that, for n 2 2, &"-arithmetic satisfies Kreisel's conditions, hence THEOREM 15. For n 2 2 , &"-arithmetic is categorical with respect to recursive models.

References [ I ] A. Grzegorczyk, Some classes of recursive functions, Rozprawy Matemat. 4 ( 1953). [2] R. L. Goodstein, Logic-free formalisations of recursive arithmetic, Math. Scand. 2 (1954). [3] H. E. Rose, On the consistency and undecidability of recursive arithmetic, Zeitschr. f. Math. Logik 7 (1961). [4] G. Kreisel, Mathematical significance of consistency proofs, J. Symbolic Logic 23 (1958). (51 H. Wang, A survey of mathematical logic (North-Holland Publ. Co., Amsterdam, 1962). [6] J. P. Cleave, A hierarchy of primitive recursive functions, Zeitschr. f. Math. Logik 9 (1963). [7] A. Lachlan, Multiple recursion, Zeitschr. f. Math. Logik 8 (1962). [8] P. Axt, Enumeration and the Grzegorczyk hierarchy, Zeitschr. f. Math. Logik 9 (1963). [9] H. E. Rose, Ternary recursive arithmetic, Math. Scand. 10 (1962). [lo] S. C. Kleene, Introduction to metamathematics (North-Holland Publ. Co., Amsterdam, 1952). [ I l l R. Smullyan, Theory of formal systems, Ann. Math. Studies 47 (1961). 1121 A. Mostowski, On recursive models of formalised arithmetic, Bull. Acad. Pol. Sc. 5 (1957).