On the primitive recursive functionals of the
finite types on ordinal numbers
Masamichi Wat6
1. Introbuction
In [2] 1955, S.C. Kleene extended the notion of recursive functions by considering the
functions of type l, i.e. the functions having function variables besides number variables-
Moreover in [3] 1959, he extended this notion by considering the functions of type n.
i.e. the functions having variables of types <n.
On the other hand, in 151 1962, G. Takeuti and A. Kino extended the notion of
recursive functions of types <__1 by using ordinal numbers instead of the natural numbers.
(AlsO in "On the recursive functions of ordinal numbers", J. Math. Soc. Japan, vol. 12
(1960) pp. 119-128, G. Takeuti did this, but in this paper we shall cite [5].)
In this paper, we discuss only about fprimitive recursive functions and predicates.
and are planning to study about general or partial recursive ones.
By a similar method of Kleene's one, we shall extend the notion of primitive recursive
functions in the sense of Takeuti and Kino [51 to the notion of ones of the finite types.
and prove the following.
First, we define the notion "ordinal function
is an extension of number theoretic
function )! " and the notion "( )-less primitive recursive function".
For a primitive
recursive function in the sense of Kleene [3], there exists a primitive recursivefunction
in our sense such that the latter is an extensio?e of the former, but the converse of
this does not hold. But if we restrict ourselves to ( '-less primitive recursive, the converse
a!so holds good.
Second, if we restrict variables to those of types <2, the primitive recursive func
tions in our sense are equivalent to the primitive recursive functions ile the sense of
Takeuli a7rd Kino [5].
2. Primitive recursive fuuctions
Let tn be any flxed and non-denumerable cardinal. In this paper, ordinal numbers,
functions etc. mean those of rrt-Ianguage. (Cf. [4], for the notion of in-1anguage.) For
each natural number j, objects and variables of type j are results obtained by replacing
1
"natural numbers" in the definition of objects or variables in [3] by "ordinal numbers".
And symbols used to express type-j variables, ordered sets of variables, etc. are like
those of [3].
The symbols '. Iq, max, j, gl, g2, pxx<., a.nd p" are analogues of [5].
Now, a function is called to be primitive recursive, if it can be defined by a finite
series of applications of the following schemata.
S 1.
(b) = ( '.
S2.
)(a,b,b)=1q(a,b).
S3.
(a,b,b)=j(a,b).
S4. p(oi,a,b)=(Y(a).
S5.
)(b)=c(X(b),b).
S6.
(a)=c(al)
S7. p(a,b)=;!xx<.X(x,b).
S8. p(a,b)=X(1x "(x,b),a,b).
S 9,
(c(j,b)=cYj(;,crj 2X((1!j,aj 2,b)).
where for S 6, al is a permutation of a, and for S 9, f 2.
A predicate is called to be primitive recursive, if its representing function is primitive
recursive.
Primitive recursiveness can be relativised with respect to
S O besides S 1-S 9 ; for example, when ipj(c,( , 2) e
SO.
r by using the following
r,
(c,b)= "j(c,AtXl(t,b),AcTX2(c,b)).
(When the variables c of cj(c) are ordinal variables only, S O is p(c,b)=cj(c).)
We have the following lemmas 1-2 as in [3].
Lemma 1. If
(a) is primitive recursive with a given description, then
)1(a, c)=p(a),
where cr, c are distinct variables, is primitive recursive with a description consisting of
applications of the same respective schemata as the given one.
Lemma 1* relativised of Lemma I holds good.
Lemma 2. For each j l, the function )(af, aj 1, b)=aj(Q:j 1) is primitive recursive.
Lemma 3. The following functions are primitive recursive.
( i) '(b)=0.
(ii) '!(a,b)=a'(=a+1).
Proof. ( i ) The functions pl(a,b,b)=1q(a, b) and , 2(a,b)=(t' are primitive recursive.
Then p3 a,b)= )1(a,p2(a,b), b)=1q (a,( ') is primitive recursive and so is p4(a. b)=,,..,x*<a
lq (x,( ') . And since p5(b)=a' is primitive recursive, p6(b)= 4( 5(b),b)=pxx< Iq (x, co) is
also primitive recursive. Then
)6(b)=0.
(ii) As above c(a,b)=px.<j(a,")Iq (a,x) is primitive recursive. But since for any
a, a+1<j(a,(o), !lxx<j(., )Iq(a, x)=a/.
2
Lemma 4. The following predicates are primitive recursive in. A,1
(in the case ( i ).
in A) .
( i ) A(a).
(ii) A(a)->B(a).
(iii) A(a)VB(a).
(iv) A(a)A (a).
Proof. Let p,X be representing functions of the predicates A,B, respectively.
( i ) A(a)
N( '(a))=0
lq(O,p(a))=0, where N is an analogue of [5].
(ii) A(a)-B(a)E;Iq( )(a),X(a))=1 N(Iq(p(a).X(a)))=0. But clearly sinc
lq are primitive recursive, N(Iq( )(a) , X(a))) is primitive recursive in.
N and
,X.
(iii) and (iv) follow immediately from ( i ) and (ii).
Lemma 5. The following predicates are primitive recursive.
(i) a<b.
(ii) a b. (For it is equivalent to lb<a,)
(iii) a=b. (For it is equivalent to a :bAb a.)
Lemma 6. The following functions are primitive recursive.
( i) (a,b,b)=max(a,b).
(ii)
b(a,b)=a.
Proof. ( i ) Since max(a, b)
j(a,b), thence max(a,b)=pxx<j(',b)'((x=aAb<a)V
(x=bAa b)). By Lemmas I and 5. R1(x,a,b,b)
x=a is primitive recursive. Similarly
R2(x,a,b,b) b<a, R3(x,a,b,b) E:x=b, and R4(x,a,b,b)
Therefore by Lemma 4, R5(x, a, b, b)E(x=aAb<a)V(x=bAa
a b are primitive recursive.
b) is primitive recursive.
Hence )(a, b,b)=max(a, b) is primitive recursive.
(ii) By ( i ) and Lemma 3 ( i ),
'! (a,b)=max(a,O) is primitive recursive. But
c (a, b) = a.
Lemma 7. The following predicates are primitive recursive in the predicate A.
( i ) (x).< A(x,a).
(ii) (Ex)x<aA(x,a).
Proof. ( i ) The following equivalence holds good.
(x)*<.A (x, a)Ea= OV(a> OAA (pxx<a IA (x, a) , a)) ( * )
For if a=0, then the left side of (*) is identically true for all a. Thence (*) holds good.
If a>0 and if for all x less than a A(x,a) is true, then pxx<a IA(x,a)=0 since there
does not exist x such that x<a and IA(x,a) . Therefore the right side of (*) is equi-
valent to A(O,a) . Thence the right side of (*) holds good. Finally if a>0 and there
exists an x less than a such that A(x, a) is false, then px.<a
that A (x a) rs false Therefore A(px.<a
A(x,a)=the least x such
A(x,a) , a) is false. So (*) is proved. On
3
the other hand, the right side of (*) is primitive recursive in A.
( ii ) It follows immediately from ( i ).
Lemma 8. The following functions are primitive recursive.
( i ) p(a,b)=gl(a).
(ii) ip(a,b)=g2(a).
Proof. ( i ) gl(a)=px*<a'(Ey)y<a'(j(x,y)=a) . And the right side of this equation
is primitive recursive by using Lemma 7.
( ii) It is proved similarly.
Now, we define max(al ' ""a ) be max(max(al ' ""a _1)'a ) . Then this function is
primitive recursive by induction on n using Lemma 6 ( i ).
Lemma 9. Let A1 ' ""A
A a) (i
be predicates such that Al(a)V"'VA (a) but not Ai(a)A
j) for all a. Then the followmg functron
) rs pnmrt ve recursrve m pl
' ,
and A1' "" A .
f P1(a) if A1(a),
9) (a) = I '
p (a) if A (a).
Proof,
(a)=pxx<*'*( l(a), ., (a))' ((Al(a)Ax=pl(a))V "' V(A (a)Ax=p (a))), thence by
SO, S7, Lemmas 1-6, p is primitive recursive in pl' ""
) , and Al' "" A,*.
Now, we define ordered n-tuple (n<a') of type-O variables as following.
<ao, "', a >=j(ao,j(al' ""j(a O) ))
It is clear that this function is primitive recursive. When a=<ao, "', a >, we put
(a)i=ai(i n). By S 8, Lemmas 8-9, there exists a primitive recursive function f such
that
{
f(a, b)=
if a=6(a),
gl(b)
f"( (a), g2(b)) if a> (a),
where 8 is an analogue of [5]. It is similar as [5] that
is primitive recursive. It is
clear that (b) =f(a, b) if a<( '. Thence the function p(a, b, b)=(a)b is primitive recursive.
Under these preparations, the analogous argument as [3] holds good. For example:
Theorem l. For each n
1, if p(a, 6", b) and e(a, b, Ir" 1) are primitive recursive, sO
is 9)1(a, b)= p (a, R " 1 6 (a, b, tn-1), b) .
Proof by similar means as in [3]. First replace in this theorem, O(a, b, f'e-1) and
)1(a, b) by e(a, b, c, T" 1) and pl(a, b, c), respectively. The resulting proposition, we prove
by induction on n, and, within that, induction on the length of a given primitive recursive
description of
)(a, o", b).
Case 7. a=(a, b) and p(a, (,", b)= )(a, b, o", b)=pzz<aX(z, b, o",
4
) where X is a pre-
viously introduced primitive recursive function. We need to show that
pl(a, b, b, c)= p (a, b, Ire (a, b, b, c, t), b)
where e(a, b, b, c, t) is primitive recursive, is primitive recursive. By Lemma 1; the de.scription of Xl(z, b, o, b, a)=X(z, b, (T,b) has the same length as that of X(z, b, 6, b). And by
S 6 and Lemma 1, since 6(a, b,b, c, r) is primitive recursive, so is el(z, b, b, a,c, T)=
e(a, b, b, c, f)' So, by the hypothesis of the induction on the length of dcscription,
X2(z, b, b, a, c)= Xl(z, b. Itel(z, b, b, a, c, f), b, a)
=X(z, b,
e (a,
, b, c, t), b)
is primitive recursive. Applyihg S 7, c(u, b, b, a, c)=/lzz<uX2(z, b, b, a, c) is also primitive
recursive. Using S 6 and Lemma 6 (ii), V(b,b, a, c)=a is primitive recursive. Thence by
S 5 and S 6,
pl(a, b, b, c)=ip( (b, b, a, c) , b, b, a, c)
= (a, b, b, a, c)
= pzz<aX2(z, b, b, a, c)
=kezz<aX (z, b, Ire (a, b, b, c, T), b)
=p(a, b, It6 (a, b, b, c, r), b)
is primitive recursive.
Case 8. We have
p(a, b, (1, b)=X( z92"(z, b, (T, b), a, b, c, b)
where x is previously introduced. Then we must prove that the following pl is primitive
recursive :
pl(a, b, b, c)= ) (a, b, ITe(a, b, b, c, 1:), b)
=X (1z9)"(z, b, Ire (a, b, b, c, 1:), b), a, b, Te (a, b, b, c, t), b).
By Lemma 1, we can express x((x, a, b, 6, b) as xl(( r, a, b, a, b, u) with the same length of
description as that of x, and by S 6 and Lemma l, e(u, b, b, c, r) as el (cv, a, b, b, u, c, r).
By the hypothesis of induction on the lehgth of description,
X2(a, a,
, b, u, c)=X1(cr, a, b, Irel(a, a, b, b, u, c, t), b, u)
=X ((x, a, b, re(u, b, b, c, r), b)
is primitive recursive, and thence applying S 8, so is
p2(a, b, b, u, c)=X2(1zp2"(z, b, b, u, c), a, b, b, u, c).
By S 6, p3(u, a, b, b, c)=p2(a, b,b, u, c) is also primitive recursive. Since v(a, b, b, c)=a is
primitive recursive, by S 5 so is pl(a, b, b, c)= )3( (a, b, b, c), a, b, b, c).
Now to see that
5
{pl(a, b, b, c)=p(a, b. Ite(a, b. ,b c, t), b),
we prove by induction on a, that for any fixed b, b, u, c, p2(a, b,b, u, c)=p(a, b, RTa(u, b, b.
c, r), b). Suppose that the equation holds for all z less than a.
p2(a, b, b, u, c)=X2(1z '2"(z, b, b, u, c), a, b, b, u, c)
=X(1z )2"(z, b, b, u, c) a, b, Ire (u, b, b, c, T), b),
so by the hypothesis of induction on a,
=X( zp"(z, b, IT6(u, b, b, c,
), b), a, b,
t6(u, b, b, c, t), b)
= p (a, b, Ara (u, b, b, c, T), b).
Q. E. D.
3. Takeuti- Kino vs. Kleene
As temporary use, we regard a as a system of arguments for a.
In order to examine the connection between number theoretic functions and ordinal
functions, we define simultaneously a predicate a'(aj) and an object oij for oij satisfying
the condition ( ,(oij). When j=0, we define Q)(a)sa<co, and a=a for a such as Q,(a).
When j>0, we define
(o(oi ) ((aj l) ((t,(a'
1-1)-aj(cYj-1)<a')A((rlj-1) (ct2j-1) ((o(oilj-1)Aa'(( t2j-1)
A(xlj-1=( j2 -1 _ (zj((rlj=1)=0!j(a!2j-1))),
and for a such as a'((Yj), the domain of oij is the set of all oij-1 such that ( '(o;j-1), and
we define oij(,x j 1)=0ij(oij-1) for all oij-1 such that ( '(oij-1).
Remark 1. In particular, if j=0 and a'(a)A(o(b), a=b if and only if a=b.
Remark 2. When (o(aj) , aj coincides with an object in the sense of Kleene [3].
If a)(aj) , we say that aj is an extension of c ' j. We write it as co(a) that c'(ccj)
holds for each element aj of a, and we define that a is the result obtained from a by
replacing each element aij of a by the corresponding crj. Then we say that a is an
extension of a.
Let ) be an ordinal function and
)! a number theoretic function in the sense of
Kleene [3]. When q)(a)= !( ) for all a such that co(a), we say that p is an extension
of
)"
Theorem 2. If
/ is a primitive recursive function in the sense of Kleene [3], then
there exists a primitive recursive function
) in our sense such that ) is an extension of
p', i.e. )(a)=p!(d) for a an extension of a.
Proof. We prove by using induction on the construction of
proof are numbered in according to the schemata of Kleene [3].
)'･ The cases of this
Cases 1-3. 6-7. They are clear from S 4, S 6, and Lemmas 3, 6.
Case 4. p!( )=c'(X!(5), ) . By the hypothesis of induction, there exists primitive
6
recursive function c and X such that if ( '(a,b) , then
" ( ,b)=cr(a, b), and if a'(b), then
X(b)=X/(b). So we put p(b)=c(X(b),b). Then if (t'(b), p(b)=c(X(b),b)=ip(X/(b),b)=
ip/ (X! (b), b) = p!(b).
Case 5. P'(O, b)=c!(b),
)1(a' , 6)=X/(a, p! (a, ). , ).
By the hypothesis of induction, there exist primitive recursive functions c and X such
that if ( '(b), then c(b)=ip'(b), and if ( '(a,b,b), then z(a,b,b)=X/(a,b,b). Since pl((r,a,b)
=a(a) is primitive recursive by S 4, so is p2(oi,a, b)=X(a,cr(a),b) by S 5. And since 6
stated in 2 is primitive recursive, so is ps((v, a,b)=X(6(a), ( (a(a)),b) by S 5. Thence by
Lemma 9,
, = if a=0,
e(( a b) fip(b)
( p3((v, a, b) otherwise,
is primitive recursive. Therefore
(a,b)=f, (1zp"(z,b),a,b) is primitive recursive.
Now we prove by induction on a, that if ( '(a,b), then p(a,b)= )/(a,b).
If a=0,
(O, b)= e (1z'O. O, b)=ip (b)=c'(b)= p!(O, b).
If 0<a<(o,
=X( (a),
"(
(a, b)=0( zp"(z, b), a, b)= 3( zpa(z, b), a, b)=X (6(a), {1zp"(z, b)} ( (a)), b)
(a),b),b)=x(a(a), )( (a),b),b)=x(6(a),p/( (a), ),b) (by the hypothesis of
induction on a) =x'( (a), p/( (a), b), b)=p!(a, b).
Case 8. p/(aj,b)=a:j(1(rj 2x!(aj ((j-2, b)). By the hypothesis of induction, there exists
a primitive recursive function x such that if ( ; (Qij, oij=2, b), then x (aj, aj-2, b)=x!(aj, ( j-2, b).
We put
)((xj,b)= ij( aj-2X ((xj, cvj 2, b)). If (') (aj, b), then x (otf,aj 2, b)=x'(aj, F , ) for
all ( j-2 such that a) ((xj-2). Therefore
where
a:j-2X (aj, aj 2, b) is an extension of crjl: xl(a:j, aj' 2, ).
( :j 2 means that ( j 2 ranges over the set of all ;j-2 such that (!'(aj-2), i.e. the set
of all number theoretic objects of type j-2, in contrast that loij-2 means that aj 2 ranges
over the set of all ordinal objects of type j-2 (in nl:-language). In other words, (';(10ij 2
X(al, aj-2, b)). Hence if ( '(( j, b), aj( aj 2X(oi', aj 2, b))=,x j(1( j-2X(oij cYj 2, b))=a j(1 7:
X'(Qij , aj-2, b)).
Q. E. D.
The converse of Theorem 2 does not hold good. Furthermore more generally:
Theorem 3. Let P/ be an arithmetical predicate in the sense of Kleene [1] or [2].
Then there exists a primitive recursive predicate P in our sense such that P is an extension of P!, i,e. P(a) Pl(a) for a an extension of . By Kleene r3], p. 7, VIII, a fortiori,
there exists a function (or a predicate) which is primitive recursive in our sense but is an
extension of a non-primitive recursive function (or predicate) in the sense of [3].
Proof. Let P!(a) be arithmetical in the sense of [2]. Then P!(a) is expressible in
one of the forms :
7
( )Rf( ,d), (E ) ( )R'( ,
,d), ( ) (E ) ( )R/( ,
(E )R'( , ), ( ) (E )R/( ,
,
,
), (Ex ) ( ) (Ez )R!( ,
,d),
,
, d),
where the R' for each is primitive recursive. Cf. [2], p. 315 (a) or ll], p. 285, Theorem
VII (d), p. 282, Cor., and p. 292, Theorem X.
For example, consider that P!(a) be expressible in the form (E ) ( ) RI( , , ). Then
by Theorem 2, there exists a predicate R primitive recursive in the new sense such that
if ( '(x,y,a), then R(x,y,a)
R!(x,y,d). Consider the predicate (Ex)*< (y)y< R(x,y,a).
Then this predicate is an extension of (E ) ( )R!( , ,d) and is prirnitive recursive in
the new sense by Lemma 7.
Remark 3. For the case that variables
of P! contain a variable of type 2 we do
not know whether or not Theorem 3 holds. By [3], p. 49. XLIX (6), the above proof is
not applicable for this case.
A fur!-ction which, in the list S 1-S 9, is obtained by using the schemata of Lemma
3
)(b)=0 and
,(a,b)=a/ instead of the schema S I p(b)=co, we call an (,,-less primitive
recursive function. Of course, the set of a'-less primitive recursive functions is a subset
of the set of primitive recursive functions. When we replace, in Lemmas l-9 except for
Lemma 3, "primitive recur-<ive" by "(o-less primitive recursive", the lemmas also hold
good. Hence.
Theorem 2/. When we replace, in Theorem 2, " a primitive recursive function p in
our sense " by "an (v-1ess primitive recursive function p in our sense", the theorem holds
gocd.
We stated in Theorem 3 that the converse of Theorem 2 does not hold, but the
converse of Theorem 2! holds good.
Theorem 3!. If p is an co-less primitive recursive function in our sense, then there
exists a primitive recursive function )! in the sense of Kleene [3] such that
sion of p', i.e.
is an exten-
(a)= !( ) for a an extension of d_
Proof. We prove it by induction on construction of p.
Cases 1-2 2Lnd 4. They are clear.
Case 3. Consider the number theoretic function j',
{
a+b(b+1) if a b,
f (a, b) a2 + b otherwise.
f is clearly primitive recursive in the sense of [3]. Now, j is an extension of f.
Case 5. By the hypothesis of induction, there exist primitive recursive functions xl
and c' in the sense of [3] such that when (o(b), X(b)=x'( ) and when (D(a,b), c(a,b)=
c'(a, ). So if ( ,(b), X(b)=x'( )<(L,, and thence c(X(b),b)=ip'(X/( ), ). Therefore if we
put
)!( )= 'r!(X!(6), ), ,r is primitive recursive in the sense of [3], and
8
' is an extension
of
)!.
Cases 6-7. Similarly.
Case 8. By the hypothesis of induction, there exists a primitive recursive function
x/ in the sense of [3] such that when ( '(oi, a, b), X (of, a, b)=x/( ,a, b). Since I Z( ) is
primitive recursive, if we put c'(x, a,b)=X!(
(x) ,a,b), ip' is also primitive recursive in
the sense of [3] by [3] p. 4 IV. Thence by similar method as [1] p. 231#G, if we put
p/(a,b)=ipl( "(a; ),a, ), p! is primitive recursive in the sense of [3]. On the other hand,
p (a b) c!( /(a b) a, b)=X!(
( f(a; b))
, a, b)=X!(A p'(L( ,b),a, b).
Now we prove by induction on a, that when (,'(a, b), , (a,b)= /(a,b). When a=0,
it is clear that lz o(z, b) is an extension of I )/o( , ). And since X is an extension of
xr, p(O, b)= f(0,6). When 0<a<( ,, by the hypothesis of induction on a, Iz )(e(z, b) is an
extension of
p!( ( ,b). And since x is an extension of x/, p(a, b)=p/(a, b).
Case 9. By the hypothesis of induction, there exists a primitive recursive function
Xf in the sense of [3] such that X is an extension of X!. So, when co((Yj,b),1aj-2X(cvj,
(rj-2, b) is an extension of
( j, j-2,6 ),
1 j-2x,( j, i j-2, ). Therefore if we put )!(e j, )=e j(1i j-2x/
" is primitive recursive in the sense of [3], and p is an extension of pf.
Q. E. D.
Remark 4 The exrstence of
! in this theorem is unique.
By Theorems 2 -3r, we have the
Theorem 4.
r is a primitive recursive function in the sense of Kleene [3], if and
only if there exists at least one co-less primitive recursive function
that
) is an extension of
in our sense such
!
Let ci(al ' "', a ) be the abbreviation of cr(<al ' "" a >). Then a(al , "', a ) is primitive
recursive. By this and Lemmas 1, 3, 6, 8, clearly,
Theorem 5. If
) is primitive recursive in the sense of Takeuti-Kino [51, it is
primitive recursive in our sense.
Clearly, the converse of this theorem does not hold, but if the maximum type of
variables is < 2, it holds good (since applications of S 9 do not occur). Namely,
Theorem 6. If a function
of variables of types<2 is primitive recursive in our
sense, it is primitive recursive in the sense of Takeuti-Kino [5].1)
By Theorems 5-6 we have the
Theorem 7. A function
of variables of types<2 is primitive recursive in our sense.
if and only if it is primitive recursive in the sense of Takeuti-Kino [5].
Nihon University
1)
Perhaps Schema (IX) of [5] should be written as followsl
1 , " ', an)=:g(hl ' " '
f(hil' ""him+
a ,h,,8,
, h,,
gi(hl ' ' hm' al' "" a? )' ""gl(hi , ..'
, _"i , ..',
where il' "'im+ is a permutation of 1, "',m+.
9 --
an))' m
m+
References
S．C．Kleene：Introduction to metamathematlcs，NQrth−Holland，1952．
1）
S．C，Kleene：Arit血metical predicates and function quanti丘ers，Tralls，Amer．Math，Soc，。
2）
vol．79（1955），312∼340。
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10