Putnam. on Hierarchies and Systems of Notations

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On Hierarchies and Systems of NotationsAuthor(s): Hilary PutnamReviewed work(s):Source: Proceedings of the American Mathematical Society, Vol. 15, No. 1 (Feb., 1964), pp. 44-

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ON HIERARCHIES AND SYSTEMS OF NOTATIONS

HILARY PUTNAM

Introduction. By a "system of notations" we understand a (1-1)-mapping, M, from a set L (a segment of Cantor's second number

class) onto a family, F, of disjoint nonempty sets. The members of F

may be, for example, sets of expressions, e.g. such expressions as w,

xX2, co2, Eo, etc. Without loss of generality we may assume that the

sets in F are sets of natural numbers.We say "without loss of general-

ity" because expressions of the kind mentioned can always be re-

placed by numbers, using the device of godel-numbering.

In the most common systems of notations,' the number '1" is

used as a name for the ordinal 0, and 2,x s used to name the successorof the ordinal named by x. Thus 2, 22, 222, * * are names, respec-

tively, of 1, 2, 3, . . . . If M is a mapping of the kind described, we

think of the image of the ordinal a as the set of "notations" for a, and

we call this set Na. Thus we may write:

M: a- Na.

The reader may wonder why the set Na is allowed to contain more

than one number (i.e., more than one name or notation for the

ordinal a). The reason is that the possibility of having different namesfor the same ordinal, e.g. co and 2w, or so and weo, arises in all of the

common systems.

The most important restriction upon the mapping M is that it

should in some sense be inductively defined. This leads us to an ab-

stract "system of notations," D, which we present below. First we

need theDEFINITION. Let ... P, x, * be a formula of second-order

arithmetic containing one free variable P for a two-term relation be-

tween numbers, one free variable x for numbers, arbitrary bound

number variables, but no free or bound higher-order variables except

P. (The constants are to designate numbers and recursive functions

and predicates.) Then the mapping P<-I{xfI * *P, x, - } associ-

ates with every diadic relation P between natural numbers a set of

numbers, namely the set of all numbers x satisfying * . . P, x,

Such a mapping is here called an arithmetic operation.2

Receivedby the editorsOctober29, 1962.1 Cf. [C], [CK], K;R],P1.2 If ... P, x ... be allowed to contain second-orderbound variables, then the

operationmay be classifiedas Z, HI, 22, * **etc., accordingto the numberand qual-ity of these. (Cf. footnote3 below.)

44



ON HIERARCHIES AND SYSTEMSOF NOTATIONS 45

The following is the system D:

(1) If a=0, then Na,= { 1}.(2) If Np has already been defined for all <oa, then Nar+

= 12|IxCENa}.(3) If oa s a limit number and Ng has already been defined for all

f3<a, then Na-= [x9yx ? ay)], where 4 is an arithmetic operation

which leads from 2-term relations between natural numbers to sets of

natural numbers.In the definition of D, part (3), we have used x_ ay as an abbrevi-

ation for the 2-term relation (depending on a):

( 23)( 3y)(x E Np &y C N,, & <y <a).

We also impose the following:

Restrictions on the choice of 4.(1) The sets Na,defined above must be disjoint.

(2) There must be a limit number a such that NpF-5?5 for f3<a,and NA=-0 for a:<,B (we think of a as "the least ordinal for which

there is no notation in D").

D is, of course, not a single system but a schema for systems (deter-

mining a single system for each admissible choice of q5).In fact, it is

easily shown that all so far proposed systems are of theform D.Let CD=df.UNaV.We shall show that3 CD C2nll2. Ourshort proof

of this fact replaces the lengthy and intricate argument used by

Kreider and Rogers in [KR], since their main result is a special case of

ours. If we place certain further restrictions on X, we can give the

usual definition of the sets4 tn for nECD. We shall show that these

tnE22NII2. Even if we use a "jump operation" in lln[Il instead of

the "one-function-quantifier jump operation" (defined below), the

INare still in VnHl. Also the

1Nare still in VnC\ll even if we let

the operation 4 be in VC\ll), instead of restricting it to be arithmetic.

Thus we have a very strong closure property of the class Y1)ln :

we cannot get out of this class by iterating a jump operation through

all the ordinals for which we have names in a system D unless (i) we

are using a jump operation which is not in VOW)ll2 (i.e., which already

goes out of VnIIl in one jump) or (ii) we have already used an opera-

tion 4 which is not in 2nCll in defining D.

3A set definablein second order number theory is said to belong to the family

Zi, if it can be definedwith n function quantifiers, he firstof which is existential (insome prenexform), and is said to belong to the family l1 if it can be definedwith n

function quantifiersthe first of which is universal. ("function quantifier"=secondorderquantifier.)

'Cf. [K1;PI.



46 HILARY PUTNAM [February

The set gn are the complete sets of the various "hyperdegrees."Thus, our result may be expressed by saying that even if the series

of "hyperdegrees" is extended through D, we do not get any degreesfor three-function-quantifier sets. Similarly, we shall show that wecannot get analogues of hyperdegrees for all the analytic sets by usingan analytic jump operation and an analytic operation q5.

Classification of CD.

THEOREM . Let D be defined as above, and let CD= UNa. Then

CDC 12nI.2

PROOF. We have

nEzCD a3 a simple-ordering f) ( 3 a 2-place function g)(h){[ h is a

nonincreasingf-chain= (ax) (hl(x) h(x + 1))] & (y)(the

least element of f = y = (z)(g(y, z) =1@ z 1)) & (y) (z)

(1) (z is thef-successor of y= { w I g(z,w)-1 } 2w g(y, w)

-1}) & (y)(y is a limit element off {w I g(y, w) = 1}

= 4[4(( 3v)( au)(g(u, x) = 1 &g(v, z) 1 &(uf-precedes

v V u = v) &vf-precedes y))]) & (3x) (g(x, n) = 1)}.

Here we have used a well-ordering of all the integers in place of asegment of the second number class, and we have used the mapping

x<-+ {yj g(X, Y) = 1}

as our mapping M (obviously without loss of generality). The prefixis of the form a 2V and the reader can readily verify that the matrixinvolves only ordinary number quantifiers. In fact (1) becomes:

n C CD ( 3f) ( 3g) (h) [(x)f(x, x)=

1 & (y) (z) (f(y, z) 1 &f(z, y)=1=X y = z) & (x) (y) (f(x, y) = 1 V (y, x) 1) & (x) (y) (z)

(f(x, y) = 1 &f(y, z) = 1 =z>f(x,z) = 1) & ((x)f(h(x + 1),

hl(x)) = 1 ==> (3x)Jh(x) = J(x + 1)) & (y)((z)f(y, z) = 1

> (z) (g(y, z)-1 4 z 1)) & (y) (z) ((f(y, z) 1 & y 5 z

(2) & (w)(f(y, w) = 1 & w 0 y = f(z, w) = 1)) => (w)(g(z, w)

= 1 (3x)(w = 2x & g(y, x) 1))) & (y)((z)f(z, y)=1

& z 7 y :(W) (ZF w &f(z, w)=1 &w 5 y &f(w, y)

= 1)) =X (w)(g(y, w) = 1 X w C ?[X((U)(2V)(g(U, X)

= 1 &g(v,z) = 1 &f(u,v) 1&f(v, ) = 1&v y))]))

&(ax)g(x,n)= 1].



19641 ON HIERARCHIES AND SYSTEMS OF NOTATIONS 47

Contracting quantifiers,6 we have CDE22.To prove that CD llH we argue similarly, using instead of (1):

n C CD X (V simple orderingf) (g){ (h) (h is a nonincreasingf-chainX*( 3x)h(x)-=h(x + 1) & . . . (as in (1)) . .* &(y)(y is a

(3) limit element of f =X {w| g(y, w) = 1} = 4[ ... as in

(1) * *]) &(2 x)(y)g(x, y) # 1] X ( 3x)g(x, n) = 1} .

Note that the clause "( 3x) (y)g(x, y) 1" says that f reaches the least

ordinal for which there is no notation in D!

Bringing out the quantifier (h) (which comes out, of course, as an

existential quantifier), and contracting quantifiers, we have CDCH2l.

Thus CDC C4J'. Q.E.D.

THEOREM 2. If, in the definition of D, we allow q5to be a VnHl

operation, it remains the case that CDG22)ll'.

PROOF. In the proof that CD121 we note that the quantifier (h)

can be confined to the clause ((x)f(h(x+1), h(x))=1=X(2qx)h(x)

=h(x+1)). Then the function quantifiers in q can be moved ahead

of the number quantifiers by the devices of [K1] and finally brought

out. Although 4i occurs in a biconditional, the quantifiers can bebrought out in the order aV by splitting the biconditional into a

pair of conditionals and using the 3V form of q5when q4occurs in the

consequent and the V a form when q occurs in the antecedent. That

is, (w) (g(y, w) =-1'w [4* *]) becomes

(W) [(g(y, W) 1?'( i)(3)(i1) & ((6) ( X)%P2 g(y,) =-1)],

where wCck[ . ]4( 0a)(0)Ji(8)(3X)i2. Hence the original ex-

pression becomes

(w)(Oc) 8)() (A.)(g(y, w) 1 =4 56I)& (NX' g(y, W) 1)].

Bringing out the function quantifiers and contracting we have 3V in

front of the whole formula (except for the (h) which we confined).

Now we bring out (h) and contract again, and we have CDCEV. The

argument is similar for CDCll'. Note that, since we have to exploitthe fact that 4 can be written in both ways, we would not get a better

result by assuming that q5C21or qPCH'lhan we do by assuming that

q$C2VnCll.Generalizing this argument, we also see that if 4CCnllI,

n > 2, then CDGIV1C.1.

The sets ?. Suppose a set A is "one-function-quantifier" in a set

B, i.e., say xCA~(f)(ay)( ... B . .. ) (that the number quanti-

b Cf. [K1],also [KR, p. 346].



48 HILARY PUTNAM [February

fiers can be reduced to one when function quantifiersare present isshown in [KB1]).ince ( ay)( B * ) says that x belongs to a setwhich is recursivelyenumerablein B, f, we can write this as x E A

4=*(f)3y) Tf'B(g(x), g(x), y) for a suitable primitiverecursiveg. Thusthe set {x| (f)(ay)TfIB(x, x, y) } is a completene-function-quantifierform in B, i.e., every set whichis one-function-quantifiern B is recur-sive in this set (and if A is Hl' n B, A is even many-one reducible tothis set). This justifies calling the operation

B* {xjI (f)(y)TfB(x, x,y)

the completeone-function-quantifierump operation(or, more simply,the hyper-jumpoperation).We shall use D to iteratehis operation

transfinitely many times. First we impose the following.

Further restrictions on 4.(1) If a is a limit number, Na must not contain any power of 2.(2) There must exist recursive functions g, k such that if a is a

limit number and xENa, then g(x) ENpfor some limit number#j5a,and8 {k(x)} provides an order preservingcofinal mapping (see be-low) from U,,<pN,, into U7<aNy.

Let us write Ix I= a for xENa. Then (2) means that

(i) if y U,,<pN, then {k(x)}(y) is defined and {k(x)I(y)ENa(into property);

(ii) if y, zeCU.)<# , then IYIZ kIk(x)I)}(y)II {k(X)}(Z)I(order-preservingroperty);

(iii) for each yEU,<.x N., there is a zEU,,<pNy such that IYII5{k(x)} z)I (cofinalproperty).

Intuitively, the meaning of these restrictions is that it should bepossible to effectively distinguish notations for successor ordinalsfromthose for limit ordinalsand to associate an order-preserving o-

final recursive sequence from U,<pN, into U-,<aN, (for some effec-tively specified B)with each (notation fora) limit ordinala. We allowa = iin restriction (2), because otherwiseit-would not be possibletohave "recursivelyregular"ordinals.7

We now define the set !n (obtained by iterating the hyper-jumpoperation through D) as follows:

(1) t10

(2) If nED, then te ({xi (f)(ay)TfA"(x,x, y)}.(3) If nED, n is not a powerof 2, then

6 Ik(x)I is the symbol for the k(x)th partial recursive function in the standardenumeration.

7 Cf. [K:R;P]. [P] also defines the system C in a way which makes it clear thatthis system is of the form D.



I964] ON HIERARCHIES AND SYSTEMS OF NOTATIONS 49

II, {2x3Y 1 X E &{k(n)) (y) &y E Uy<1g(n)jyJ.

[Note that if In is a limit ordinal, (3) insures that S5m shall be recur-

sive in Dn for each m in the appropriate "fundamental sequence"(provided by { (n) } y) as y runs through the notations for ordinalsless than Ig(n) I).]

We see that each 'Dpcomes from Dn by the hyper-jump operation.And by an easy transfinite induction ,n is recursive in ,D. for

In| <Imi.

THEOREM 3. For each n E D, the set GVE2nC)I7.

PROOF. It suffices to write

x Cz ( f) (3g) (3g') (h) [ * entire matrix of (2)

& (w)g'(1, w) # 1 & (y) (z) (g(y, 2z) = 1 => (v) (g'(2z, v) = 1

X (a) ( Ey) Ta.P(v, V,y))) & (y)(z) (y is a limit element of

(4) f & g(y, z) = 1 =w {w Ig'(z, w) = 1} t= J(w, v) I g'({k(z)}

(v), w) = 1 & (aa)(ab)(f(a, b) & a 5 b & g(b, g(z)) = 1

&g(a,v) = 1}) & g'(n,x) = 1].

In (4) above "P" abbreviates "{vI '(z v)=1}."Here again we have used a well-ordering of all the integers instead

of a segment of the second number class. We have used the mapping

x -*{y g(x,y) = 1}

as our mapping a<-+N,; and we have used the mapping

y<* {zIg'(y,z) = 1}

as our mapping y"->Sy.Writing out the Ta Ppredicate in full and put-

ting g'(z, w) =1 and g'(z, w) = 1 for P(w) and P(w) whenever theseoccur in the full form of the predicate TahP(v, , y), we see that the

argument of Theorem 2 applies here. So, once again we confine (h) to

the single clause in which it occurs, split the biconditional involvingTarP into a pair of conditionals, and then bring out the function

quantifiers in the desired aV order. Thus t E2. The proof that

CJ E 12 is exactly similar, writing

XC DnX (f)(g)(g') { (h)(h is a nonincreasingf-chain = ( 3x)h(x)

(5)~~~ h(x + 1)) & ... as in (4) ... & ( 3x) (y) g(x, y) F61

9g'(x, x) = IX.

Finally, we note that the argument again generalizes to show that

if the operation



50 HILARY PUTNAM

B*-? (a) (ay) Ta,B(x, x, y)

is replaced by an operation which leads (uniformly) from B to a set

which is in 2G'nIH form relative to B, n 2, then the sets ,. will bein the class G'nII.

REMARK ADDED FEBRUARY 5, 1963. The method of this paper yields

an immediate proof that if B E CnI1' then the hyperjump of B also

E 12G2. Namely, it suffices to define

n E hyperjump (B)

<-v (3f)(3g)[(x)(f(x) = 1 Vf(x) = 0) & (x)(g(x) = 1 V g(x) = 0)

& (x) (f(x) = 1 X B(x)) & (x) (g(x) = 1

X (h) ( 3y) Thf (x, x, y)) &g(n) = 1]

X (f)(g) [(x)(f(x) = 1 V f(x) = 0) & (x)(g(x) = 1 V g(x) = 0)

& (x)(f(x) XI- B(x)) & (x) (g(x)1

X# (h) ay) Th (x, x, y)) *X> g(n)-1

Splitting the biconditionals into pairs of conditionals and proceedingwith the quantifiers as in the text, we have the desired result, whichwas proved by Addison and Kleene in A note on function quantifica-tion, Proc. Amer. Math. Soc. 8 (1957), 1002-1006, using an argumentinvolving the operation A.

REFERENCES

C. A. Church, The constructiveecondnumberclass, Bull. Amer. Math. Soc. 44(1938),224-232.

CK. A. Churchand S. C. Kleene, Formaldefinitionsn thetheoryof ordinalnum-bers,Fund. Math. 28 (1937), 11-2 1.

K1. S. C. Kleene, Arithmeticalpredicatesandfunction quantifiers,Trans. Amer.

Math. Soc. 79 (1955), 193-213.K2. , Introduction o metamathematics, an Nostrand, New York, 1952.KR. D. L. Kreiderand H. Rogers, Jr., Constructive ersionsof ordinalnumber

classes,Trans.Amer. Math. Soc. 100(1961), 325-369.P. H. Putnam, Uniquenessordinals in higherconstructiveumberclasses,Essays

on the foundationsof mathematics,pp. 190-206, MagnesPress,Jerusalem,1961.

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Putnam. on Hierarchies and Systems of Notations