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Infinitely Long Terms of Transfinite Type
INFINITELY LONG TERMS OF TRANSFINITE TYPE W. W. TAIT Stanford University, Stanford, Calif., USA
1. Functionals of higher type were introduced into proof theory by K. Godel in [2], where he gives an interpretation of first order number theory in terms of the impredicative primitive recursive (p.r.) functionals of finite type. The aim of this work was to show that for the consistency of number theory, Gentzen's use of induction up to GO (with respect to p.r. properties) can be replaced by a quite different constructive - but like Gentzen's, non-finitist - principle, namely, the assumption of constructive functionals of finite type and of their closure under p.r. operations. However, another view of Godel's result is possible: Instead of assuming functionals of higher type, we may regard the definitional schemata for the p.r. functionals simply as rules of computation, i.e. for transforming symbols. On this view, Godel's result may be interpreted as a consistency proof relative to the quantifier-free theory of p.r. functionals (his system T), which in turn must be justified by a proof that all the constant numerical terms of the theory can be transformed by the rules of computation into unique numerals. This latter viewpoint is what I will discuss here. It has become especially interesting in virtue of Spector's [7] extension of Godel's interpretation to classical analysis by adding the general principle of bar recursion to the schema for primitive recursion. For, while on any reasonable conception of computability at higher types, the computable functionals are constructively closed under the p.r. operations, there is no known constructively valid interpretation of these operations together with bar recursion.') It appears 1) There is a constructively valid interpretation for bar recursion of lowest type (i.e. in Spector's notation, where c is a sequence of numbers or functions) using the
INFINITEL Y LONG TERMS OF TRANSFINITE TYPE
177
that the best hope for a constructive justification of bar recursion lies in an analysis of the computations of bar recursive functionals. However, here I will discuss only p.r. functionals, or rather a certain generalization of them. On another occasion I will apply the present ideas to the analysis of functionals involving bar recursion of lowest type. William Howard has recently extended Godel's interpretation to ramified analysis, as formulated by Schutte [6] and Feferman [I], using p.r. functionals oftransfinite type. In view of this, it is fitting that we formulate the results of this paper for the wider context of transfinite types, even though we will not discuss Howard's result here. Actually, Howard uses a more complex concept of transfinite types than is introduced here, but his result can be obtained using the present conception. Just as Lorenzen and Schutte greatly simplified the problem of cutelimination for formal proofs involving induction principles by effectively representing such proofs by infinite well-founded proof trees, 1) so a similar device will serve us here in analyzing the computations of functionals involving definition by recursion. Consider, for example, the (not necessarily numerical valued) functional cP defined by the primitive recursion: cp(n+ I, a, b) = b(cp(n, a, b), n). cp(O, a, b) = a, The p.r. functionals of finite type can be generated from such cp by means of A-abstraction and explicit definition (where the definiens may contain and the successor operation, of course). Now, write CPo = sab . a and CPn+ 1 = Xab . b(CPn(a, b), n). Then cP is represented by the "infinite term" (CPo, CPl" .. ) which takes the value CPn for the argument n. The close connection between this infinitary rule of term formation and the rule of infinite induction (used by Schutte in the construction of proof trees), i.e. A(O), A(I), ... != (x)A(x), is made clear if we consider the Godel interpretation of the latter:
°
(t/t)B(O, CPo, t/t), (t/t)B(l, CPl' t/t), •.. !=(Ecp) (x) (t/t)B(x, cp(x), t/t). continuous functionals of finite types. By the methods of this paper, however, we can expect to obtain something more, namely, a least ordinal", such that the functionals of finite type defined by recursion up to '" are closed under bar recursion of lowest type. 1) Lorenzen was first to give a constructive formulation of cut-elimination for infinite proofs, using transfinite induction. Schiitte showed how to determine exactly in some cases which ordinals are involved, thus restoring the precision of Gentzen's original results.
178
W. W. TAIT
For, using our infinitary rule of term formation, the solution for tp is simply qJ = (qJo, qJl' ... ). In dealing with transfinite types, we make two further uses of Lorenzen's and Schutte's idea. First, transfinite types are themselves to be infinite objects. Namely if a, and 'Ci are types for all i < «, o: .:s; CU, and the a i are all distinct, then the set {(a is 'C in i < a is a type. In fact, it is the type of functionals which are defined for objects of any type a., i < a, and whose value for an object of type a, is of type 'Ci' A particular case is when a = 1, in which case the type is simply denoted by (170' 'Co)' Also, we will use another kind of infinite term to piece together functionals of type {(ai' 'Ci)L -c c from functionals of types (ai' 'Ci), i < rx. Namely, if for eachi < rx,qJiisoftypeai,thenqJ = (qJo,qJl,oo.)isoftype{(ai,r)},and for each argument x of type aJi < rx), qJ(x) = qJi(X), We are dealing here only with functionals of a single argument. But it is well-known how to reduce functionals of several variables to functionals of one variable (but of higher type). In the next section we will set up a formal calculus of infinitely long terms, or rather, a "semi-formal" calculus in Schutte's sense, which will codify the constructions of functionals which we have been discussing. In section 3, we will prove by induction up to rx that every term t can be computed, i.e. can be reduced by certain conversion principles to an irreducible term t', where the ordinal a is given explicitly in terms of certain bounds on t. From the proof of computability, in fact, it is easy to give a definition of t' = f(t) where the functional f is defined by predicative transfinite recursion up to rx. "Predicative" here means that f is obtained by explicit definition and recursive definition up to a, where the latter is used only to define numerical valued functions. Thus the definition of f does not involve functionals of higher type (unlike the definitions of the p.r. functionals of finite type). 2. We will use the usual notations for countable ordinals and operations on ordinals, but these should be interpreted in terms of a suitable constructive system of ordinal notations. In fact, for the sake of definiteness, we will assume that all discussion of ordinals here refers to the p.r. wellordering of the natural numbers defined in Schutte [4]. All the ordinal functions which we will use are represented in that ordering by p.r. functions.
179
INFINITELY LONG TERMS OF TRANSFINITE TYPE
The types and their ranks are inductively defined by Tp
1.
Tp 2.
o is a type with
rank RO = O.
If (Ji and t , are types with R(Ji, Rx, < /3 for all i < ct. ~ OJ and if a, #- (J j for i #- j (i, j < ct.), then the set p = {((Ji' 'ri), /3}; < a is a type with Rp = /3.
When the rank of {((J;, 'r;), /3} i < a is not relevant in a given context, we usually abbreviate this type by {((J;, 'rJ}; < a' and for ct. = 1, by ((Jo, 'ro). It would not be satisfactory in Tp 2 to take for p simply {((Ji' 'ri)}; < a and define Rp to be the supremum of the R(Ji and R'r;, since this supremum is not computable from p. On the other hand, our present definition has the unpleasant feature that two types may be extensionally the same, and be distinct only in virtue of their ranks. However, we can easily avoid this difficulty by taking
{(ni'
to mean
pJ, /3}; < a = {((Ji'
'rJ, b}i <
a
In particular, the condition a, #- (Jj in Tp 2 should be interpreted in this way.') The x-terms (i.e. terms of type r) and their lengths are inductively defined as follows: Tm 1.
Each variable a' of type 'r is a r-term with length I a'
Tm 2.
0 is
Tm 3.
If s is a O-term, then so is Ss.
Tm4.
If p = {((Ji,'ri)}i
a O-term.
I0I =
I=
O.
O.
I Ss I = I s 1+1. «, Si is a 'rcterm with is a p-term.
1) D. Scott has suggested, quite reasonably, that we call the object defined by Tp 1 and 2 type notations and distinguish them from the corresponding extensional types. Thus, a i a j in Tp 2 would mean that the types corresponding to the notations a i and aj are distinct. Of course, in speaking about notations, we would have to restrict ourselves to sets {(ai' T i ) , {l} which can be represented by numbers, e.g. recursively enumerable sets. The present formulation allows that our types may be free choice sequences of a certain spread (see the remarks below about the constructive meaning of infinite terms), but it is too early 10 see whether this will be of any particular use.
*
180
w.
W. TAIT
Tm 5. If So, S1' ... are r-term, and I s.; I < 13 for n S 0, then (Si; f3)i <0) is a {(O, r), R,,+ l}-term. I (Si; 13) I = 13· Tm 6.
If r is a p-term, S is a a-term, and (a, r) e p, then (rs) is a r-term, I rs I = max (I r I, I s I) + 1.
Regarding the constructive meaning of this definition, there are two possible views. On the narrower view, infinite types and infinite terms are to be given by effective, and so finite, rules for their construction, and our theorems about terms are ultimately about these rules, and so really only concern finite objects. This is the view expressed by Schutte in connection with his work with infinite proof trees. E.g. see [5], p. 369. Another, more general, viewpoint is that infinite terms are essentially free choice sequences of a certain spread. The details of the construction of the spread are left to the reader; but it will be noted that the possibility of identifying the terms with the choice sequences of a spread depends essentially on the fact that each term has an associated length which exceeds the length of its subterms, and each type {(ai' 'i)' f3} has a rank which exceeds the ranks of the a, and 'i. lt is easy to see that each term has a unique type. In particular, if r is a p-term, s a a-term and rs is a r- and a ,'-term, then (a, r) s p and (a, r') e p, and so , = i', r', s' and t' will denote r-terrns, The rank Rt of a r-term t is simply R". When there is no need for greater explicitness, we will write {Ad", Si} and (s.) for the terms given by Tm 4 and 5. Also, for IX = 1, write Jeauos o for {Aau " siL < a. is intended to denote 0, and S the successor operation, so that the numerals are 0, T = SO, 2 = sT, etc. An occurrence of a variable b in a term is called bound if it is in a context Jeb . s; and otherwise, it is called free. Let t(a U ) be an arbitrary term, and s a a-term. Replace each variable with free occurrences in s in all of its bound occurrences in t(a U ) by a distinct new variable, and replace each part {AaO"', Si' f3} and (Si; 13) in t(a U ) by {Aau , • Si' IX} and (Si; IX), respectively, where IX = I S 1+ 13. Finally, replace each free occurrence of a" in the resulting expression by s. The result will be denoted by t(s). We can assume that the change of bound variables in t(a U ) is done in a unique way, so that t(s) is unique.
o
INFINITELY LONG TERMS OF TRANSFINITE TYPE LEMMA
I t(s) I
~
1:
If t(a")
is a t-term and s a a-term, then t(s) is a t-term with
1s I + I t(a) I·
The proof is by straightforward induction on I t(a) We will write t 1t2 ... t. for ( .. . (t1t2)" .tn ) . Now we can formulate the rules of conversion. 1. {Aa"', si(a"')} r": II. (s;)ii --+ s•. III. (r ;)st --+ (rit)S.
181
--+
I.
s.(r"n).
The relation r =1 s (r reduces to s) is inductively defined by 10. 2°. 3°. 4°. 5°.
If r --+ s then r =, s. If r =1 s then t(r) =1 t(s). If r, =1 s, for each i < a, then {Aa"', r;}i <
(l'
In 2°, t(r) and t(s) are obtained from a term t(a) in the manner specified above. It is clear that if r" =i s, then s is a a-term, and moreover, on the intended interpretation, r" and s denote the same functional. For i = I, II or III, if rs --+ t is an instance of rule i, then rs is said to be i-convertible ti-conv, or simply conv) into t with principal part r. If t contains no conv. subterms, then it is called irreducible or is said to be in normal form. 3. We prove in this section that every term can be reduced to a term in normal form. The non-trivial part of the proof consists in showing that we can reduce t to a term without 1- or III-conv subterms. Dt ~ a will mean that Rr < rt for every principal part r of a 1- or Ill-conv subterm of t. We can read Dt ~ o: as "the degree of t is ~ «", providing that we do not assume Dt to be an ordinal which is computable from t. LEMMA
2:
If Dt(a"),
Ds"
~
o: and Rs" <
rt,
then Dt(s)
~
«.
Let uv be a 1- or III-conv sub term of t(s). We must show that Ru < «, If uv is a subterm of s, this follows from Ds ~ rt. If u = s, it follows from Rs < «. If uv is III-conv and u is of the form sw (so that s is of the form (s.),
182
W. W. TAIT
u'(s)v'(s) where, if uv is f-conv (i = I or III), then u'(a)v'(a) is an z-conv subterm of t(a). But since Dt(a) ::s; a, it follows that Ru = Ru'(a) < a. LEMMA
3:
If Dr, Ds ::s; a, rs is a term, and Rs < a, s a and I t I ::s; Max (I rs I, 1s 1+ 1r I).
then there is a t
with rs =! t, Dt
Proof by induction on I r I. If rs is not 1- or III- cony, then t = rs suffices, since rs is the only possible cony subterm of rs which is neither a subterm of r nor of s.lf r = {Aa'''' r;(aO"')} and s is of type a n> then t = rn(s) suffices by Lemmas 1 and 2. If r = (r;; f3)u, then since I r n I < I r I, there is a r, with rns=1 tn> oi;« a and 1t n 1 s Max (I rns I, Is 1+lrnl),forn ~ O. Hence, t = (t .: y)u suffices, where y = Max (Max(f3, I s I) + 1, 1 s I + 13).
4: If Drs ::s; a and Rr a'r; ... rin ~ 0). LEMMA
~
a, then r is of the form (u i ) or else
In fact, r is of the form rOr l ••• rn> where r o is not of the form UV, and Rr., ~ Rr ~ a. Now in all cases other than r o = at and r o = (uJ with n = 0, r o is the principal part of a 1- or lII- cony sub term of rs. But this is impossible since Drs ::s; a. Let x(~) = 2"', and for y "# 0, let x(r) be the iah simultaneous solution 13 of x(~) = 13, for all y' < y. Then x(r) is a normal function of a.
1: If Dt ::s; a ::s; xf~)I'
THEOREM
and 1 t'
I
+
wY, then there is a t' such that t =
r or,
::S;a
The proof is by induction on )I, and within that, by induction on 1 t I. If I t I = 0, then Dt = 0, so that t' = t suffices. Using the fact that the length of a term exceeds the length of its subterms, and that x~Y) is normal in 13, we can set (Ss)' = Ss', paO"'. s;, f3}' = {AaO"'. s;, x~Y)} and (s;; 13)' = = (s~·l' X(Y») Let t • = rs Then there are r' and s' with Dr' , Ds' < , a r - r' p • s=jS', 1 r' I ::s; and I s'l ::s; Xm. If Rr ~ a+w Y, then r is of the form a'r ; ... r« or (uJ, by Lemma 5. Hence, r' is of the form atr~ ... r~ or (u;), and so r's' is not 1- or III-conY. Thus, t' = r's' satisfies the theorem. Assume now that Rr < a+w Y• We must consider three cases: Case 1. y = 0. I.e. Rr s; a+ 1, so that Rs < a. Then by Lemma 4 there is a t' with t=r's' t', Dt' ::s; a and 1 t' I ::s; 21sl+21rl < 2 1t l• Case 2. y = (j + 1. Then since Rr < a+w·· w, we have Rr < a+w· . k for sufficiently large k. Then Dr's' ::s; a + w· . k, and so by k iterated
Xrn
-I
,
183
INFINITELY LONG TERMS OF TRANSFINITE TYPE
applications of the inductive hypothesis for J, there is a t' with r's' Dt' ~ IX and (y)
< XI t
Case 3. 'Y
= lim 'Yn' Then since Rr < n
+ COYk.
IX
t' ,
I'
+ oi', there is a k with
Rr <
IX
+
Hence, Dr' s' ~ IX + co'", and so by the inductive hypothesis for 'Yk' there is a t' with r's' =1 t', Dt' ~ IX and
I t' I < -
(Yk)
Xmax (x\~I. x\~(l+
1
<
(y)
Xltl'
This completes the proof. THEOREM 2: If Dt ~ co~, then there is a t' in normalform with t =' t' and
I t 'I
~
X(.j lt l.
This follows from Theorem I and the following lemma. LEMMA
and I t'
5:
If
I s I t I·
Dt
=
0, then there is a t' in normal form with t = t'
The proof is by induction on I t I. If I t I = 0, then t is in normal form, so we can take t' = t. Set (Ss)' = Ss', {Aa<1'· Si' {J}' = {Aa<1'· s;, {J}, and (s.; 13)' = (s;; (J). Let t = totl ... t n where n > and to is not of the form uv. Since Dt = 0, it follows from Lemma 4 that to is a variable, or else it is of the form (Ui) with n = 1. In the first case, t' = t~t~ ... t~ suffices; in the second case, if t~ is a numeral then t' = u~, and if t~ is not a numeral, then t' = t~t~. This completes the proof that every term t can be reduced to a normal form t', From the proof, it is clear that if we restrict ourselves to terms of length < 13 and degree < IX, then t' = f(t) can be defined by predicative recursion up to X~~) . IX, since the doubie induction in the proof of Theorem 1, up to IX; and within that, up to i t I, can be transformed into an induction up to X~~) . IX. The essential unicity of the normal form for a given term follows from
°
m,
THEOREM
3:
If r -
s, r = t and t is in normal form, then s
=1
t.
184
W. W. TAIT
For, if s is also in normal form, then t can be obtained from s only by trivial uses of 2°, namely, where in going from u(v) to u(w) (where v =1 w), v does not actually occur in u(v), so that u(w) differs from u(v) only by some changes of bound variables. (See the definition of the substitution u(v).) Hence, in this sense, t is a mere notational variant of s. In particular, if r contains no free variables and is a O-term, then all its normal forms must be numerals, and hence, identical. The proof of Theorem 3 proceeds by induction on the "length" of a reduction of r to s, defined in a suitable way. I omit this proof, which is routine, long and unpleasant. 4. Thefinite types are obtained by restricting Tp 2 to the case CI. = 1, so that they are built up from by means of the composition (a, r). Similarly, the terms offinite type are obtained by restricting Tm 4 to the case CI. = 1, so that the terms given by this clause are of the form AaG • s. The impredicative p.r. functionals of finite type are obtained from the terms of finite type (in our sense) by replacing Tm 5 by the schema for primitive recursion given in § 1. But we saw in § 1 how to replace each
°
.2(1)'2
(with k exponents). In particular, let t be a constant (0, 0) - term, i.e, without free variables. Then for every n,j(tn) is a numeral. Then
INFINITELY LONG TERMS OF TRANSFINITE TYPE
185
References [I] S. Feferman, Systems of Predicative Analysis. To appear. [2] K. Godel, Dber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12 (1958) 280-287. [3] G. Kreisel, Inessential Extensions of Heyting's Arithmetic by Means of Functionals of Finite Type. Abstract. JSL 24 (1959) 284. [4] K. Schutte, Kennzeichnung von Ordnungszahlen durch rekursiv erklarte Funktionen. Math. Ann. 127 (1954) 15-32. [5] , BeweistheoretischeErfassung der unendlichen Induktion in der Zahlentheorie, Math. Ann. 122 (1951) 369-389. [6] , Predicative Well-orderings. These Proceedings, p. 280. [7] C. Spector, Provably Recursive Functionals of Analysis: A Consistency Proof of Analysis by an Extension of Principles Formulated in Current Intuitionistic Mathematics. Recursive Function Theory. Proc. of Symposia in Pure Mathematics, Vol V., Am. Math. Soc. (1962) 1-27.
Added in proof The work of Lorenzen to which we refer is "Algebraische und logistische Untersuchungen tiber freie Verbande", Journal of Symbolic Logic 16 (1951) 81-106. However, an earlier treatment of proof theory by means of a constructive theory of infinite proofs is given in P. Novikov, "On the consistency of certain logical calculus", Matematicesky sbovnik 12, no. 3 (1943) 353-369. In particular, a constructive consistency proof for arithmetic is given.
1. Functionals of higher type were introduced into proof theory by K. Godel in [2], where he gives an interpretation of first order number theory in terms of the impredicative primitive recursive (p.r.) functionals of finite type. The aim of this work was to show that for the consistency of number theory, Gentzen's use of induction up to GO (with respect to p.r. properties) can be replaced by a quite different constructive - but like Gentzen's, non-finitist - principle, namely, the assumption of constructive functionals of finite type and of their closure under p.r. operations. However, another view of Godel's result is possible: Instead of assuming functionals of higher type, we may regard the definitional schemata for the p.r. functionals simply as rules of computation, i.e. for transforming symbols. On this view, Godel's result may be interpreted as a consistency proof relative to the quantifier-free theory of p.r. functionals (his system T), which in turn must be justified by a proof that all the constant numerical terms of the theory can be transformed by the rules of computation into unique numerals. This latter viewpoint is what I will discuss here. It has become especially interesting in virtue of Spector's [7] extension of Godel's interpretation to classical analysis by adding the general principle of bar recursion to the schema for primitive recursion. For, while on any reasonable conception of computability at higher types, the computable functionals are constructively closed under the p.r. operations, there is no known constructively valid interpretation of these operations together with bar recursion.') It appears 1) There is a constructively valid interpretation for bar recursion of lowest type (i.e. in Spector's notation, where c is a sequence of numbers or functions) using the
INFINITEL Y LONG TERMS OF TRANSFINITE TYPE
177
that the best hope for a constructive justification of bar recursion lies in an analysis of the computations of bar recursive functionals. However, here I will discuss only p.r. functionals, or rather a certain generalization of them. On another occasion I will apply the present ideas to the analysis of functionals involving bar recursion of lowest type. William Howard has recently extended Godel's interpretation to ramified analysis, as formulated by Schutte [6] and Feferman [I], using p.r. functionals oftransfinite type. In view of this, it is fitting that we formulate the results of this paper for the wider context of transfinite types, even though we will not discuss Howard's result here. Actually, Howard uses a more complex concept of transfinite types than is introduced here, but his result can be obtained using the present conception. Just as Lorenzen and Schutte greatly simplified the problem of cutelimination for formal proofs involving induction principles by effectively representing such proofs by infinite well-founded proof trees, 1) so a similar device will serve us here in analyzing the computations of functionals involving definition by recursion. Consider, for example, the (not necessarily numerical valued) functional cP defined by the primitive recursion: cp(n+ I, a, b) = b(cp(n, a, b), n). cp(O, a, b) = a, The p.r. functionals of finite type can be generated from such cp by means of A-abstraction and explicit definition (where the definiens may contain and the successor operation, of course). Now, write CPo = sab . a and CPn+ 1 = Xab . b(CPn(a, b), n). Then cP is represented by the "infinite term" (CPo, CPl" .. ) which takes the value CPn for the argument n. The close connection between this infinitary rule of term formation and the rule of infinite induction (used by Schutte in the construction of proof trees), i.e. A(O), A(I), ... != (x)A(x), is made clear if we consider the Godel interpretation of the latter:
°
(t/t)B(O, CPo, t/t), (t/t)B(l, CPl' t/t), •.. !=(Ecp) (x) (t/t)B(x, cp(x), t/t). continuous functionals of finite types. By the methods of this paper, however, we can expect to obtain something more, namely, a least ordinal", such that the functionals of finite type defined by recursion up to '" are closed under bar recursion of lowest type. 1) Lorenzen was first to give a constructive formulation of cut-elimination for infinite proofs, using transfinite induction. Schiitte showed how to determine exactly in some cases which ordinals are involved, thus restoring the precision of Gentzen's original results.
178
W. W. TAIT
For, using our infinitary rule of term formation, the solution for tp is simply qJ = (qJo, qJl' ... ). In dealing with transfinite types, we make two further uses of Lorenzen's and Schutte's idea. First, transfinite types are themselves to be infinite objects. Namely if a, and 'Ci are types for all i < «, o: .:s; CU, and the a i are all distinct, then the set {(a is 'C in i < a is a type. In fact, it is the type of functionals which are defined for objects of any type a., i < a, and whose value for an object of type a, is of type 'Ci' A particular case is when a = 1, in which case the type is simply denoted by (170' 'Co)' Also, we will use another kind of infinite term to piece together functionals of type {(ai' 'Ci)L -c c from functionals of types (ai' 'Ci), i < rx. Namely, if for eachi < rx,qJiisoftypeai,thenqJ = (qJo,qJl,oo.)isoftype{(ai,r)},and for each argument x of type aJi < rx), qJ(x) = qJi(X), We are dealing here only with functionals of a single argument. But it is well-known how to reduce functionals of several variables to functionals of one variable (but of higher type). In the next section we will set up a formal calculus of infinitely long terms, or rather, a "semi-formal" calculus in Schutte's sense, which will codify the constructions of functionals which we have been discussing. In section 3, we will prove by induction up to rx that every term t can be computed, i.e. can be reduced by certain conversion principles to an irreducible term t', where the ordinal a is given explicitly in terms of certain bounds on t. From the proof of computability, in fact, it is easy to give a definition of t' = f(t) where the functional f is defined by predicative transfinite recursion up to rx. "Predicative" here means that f is obtained by explicit definition and recursive definition up to a, where the latter is used only to define numerical valued functions. Thus the definition of f does not involve functionals of higher type (unlike the definitions of the p.r. functionals of finite type). 2. We will use the usual notations for countable ordinals and operations on ordinals, but these should be interpreted in terms of a suitable constructive system of ordinal notations. In fact, for the sake of definiteness, we will assume that all discussion of ordinals here refers to the p.r. wellordering of the natural numbers defined in Schutte [4]. All the ordinal functions which we will use are represented in that ordering by p.r. functions.
179
INFINITELY LONG TERMS OF TRANSFINITE TYPE
The types and their ranks are inductively defined by Tp
1.
Tp 2.
o is a type with
rank RO = O.
If (Ji and t , are types with R(Ji, Rx, < /3 for all i < ct. ~ OJ and if a, #- (J j for i #- j (i, j < ct.), then the set p = {((Ji' 'ri), /3}; < a is a type with Rp = /3.
When the rank of {((J;, 'r;), /3} i < a is not relevant in a given context, we usually abbreviate this type by {((J;, 'rJ}; < a' and for ct. = 1, by ((Jo, 'ro). It would not be satisfactory in Tp 2 to take for p simply {((Ji' 'ri)}; < a and define Rp to be the supremum of the R(Ji and R'r;, since this supremum is not computable from p. On the other hand, our present definition has the unpleasant feature that two types may be extensionally the same, and be distinct only in virtue of their ranks. However, we can easily avoid this difficulty by taking
{(ni'
to mean
pJ, /3}; < a = {((Ji'
'rJ, b}i <
a
In particular, the condition a, #- (Jj in Tp 2 should be interpreted in this way.') The x-terms (i.e. terms of type r) and their lengths are inductively defined as follows: Tm 1.
Each variable a' of type 'r is a r-term with length I a'
Tm 2.
0 is
Tm 3.
If s is a O-term, then so is Ss.
Tm4.
If p = {((Ji,'ri)}i
a O-term.
I0I =
I=
O.
O.
I Ss I = I s 1+1. «, Si is a 'rcterm with is a p-term.
1) D. Scott has suggested, quite reasonably, that we call the object defined by Tp 1 and 2 type notations and distinguish them from the corresponding extensional types. Thus, a i a j in Tp 2 would mean that the types corresponding to the notations a i and aj are distinct. Of course, in speaking about notations, we would have to restrict ourselves to sets {(ai' T i ) , {l} which can be represented by numbers, e.g. recursively enumerable sets. The present formulation allows that our types may be free choice sequences of a certain spread (see the remarks below about the constructive meaning of infinite terms), but it is too early 10 see whether this will be of any particular use.
*
180
w.
W. TAIT
Tm 5. If So, S1' ... are r-term, and I s.; I < 13 for n S 0, then (Si; f3)i <0) is a {(O, r), R,,+ l}-term. I (Si; 13) I = 13· Tm 6.
If r is a p-term, S is a a-term, and (a, r) e p, then (rs) is a r-term, I rs I = max (I r I, I s I) + 1.
Regarding the constructive meaning of this definition, there are two possible views. On the narrower view, infinite types and infinite terms are to be given by effective, and so finite, rules for their construction, and our theorems about terms are ultimately about these rules, and so really only concern finite objects. This is the view expressed by Schutte in connection with his work with infinite proof trees. E.g. see [5], p. 369. Another, more general, viewpoint is that infinite terms are essentially free choice sequences of a certain spread. The details of the construction of the spread are left to the reader; but it will be noted that the possibility of identifying the terms with the choice sequences of a spread depends essentially on the fact that each term has an associated length which exceeds the length of its subterms, and each type {(ai' 'i)' f3} has a rank which exceeds the ranks of the a, and 'i. lt is easy to see that each term has a unique type. In particular, if r is a p-term, s a a-term and rs is a r- and a ,'-term, then (a, r) s p and (a, r') e p, and so , = i', r', s' and t' will denote r-terrns, The rank Rt of a r-term t is simply R". When there is no need for greater explicitness, we will write {Ad", Si} and (s.) for the terms given by Tm 4 and 5. Also, for IX = 1, write Jeauos o for {Aau " siL < a. is intended to denote 0, and S the successor operation, so that the numerals are 0, T = SO, 2 = sT, etc. An occurrence of a variable b in a term is called bound if it is in a context Jeb . s; and otherwise, it is called free. Let t(a U ) be an arbitrary term, and s a a-term. Replace each variable with free occurrences in s in all of its bound occurrences in t(a U ) by a distinct new variable, and replace each part {AaO"', Si' f3} and (Si; 13) in t(a U ) by {Aau , • Si' IX} and (Si; IX), respectively, where IX = I S 1+ 13. Finally, replace each free occurrence of a" in the resulting expression by s. The result will be denoted by t(s). We can assume that the change of bound variables in t(a U ) is done in a unique way, so that t(s) is unique.
o
INFINITELY LONG TERMS OF TRANSFINITE TYPE LEMMA
I t(s) I
~
1:
If t(a")
is a t-term and s a a-term, then t(s) is a t-term with
1s I + I t(a) I·
The proof is by straightforward induction on I t(a) We will write t 1t2 ... t. for ( .. . (t1t2)" .tn ) . Now we can formulate the rules of conversion. 1. {Aa"', si(a"')} r": II. (s;)ii --+ s•. III. (r ;)st --+ (rit)S.
181
--+
I.
s.(r"n).
The relation r =1 s (r reduces to s) is inductively defined by 10. 2°. 3°. 4°. 5°.
If r --+ s then r =, s. If r =1 s then t(r) =1 t(s). If r, =1 s, for each i < a, then {Aa"', r;}i <
(l'
In 2°, t(r) and t(s) are obtained from a term t(a) in the manner specified above. It is clear that if r" =i s, then s is a a-term, and moreover, on the intended interpretation, r" and s denote the same functional. For i = I, II or III, if rs --+ t is an instance of rule i, then rs is said to be i-convertible ti-conv, or simply conv) into t with principal part r. If t contains no conv. subterms, then it is called irreducible or is said to be in normal form. 3. We prove in this section that every term can be reduced to a term in normal form. The non-trivial part of the proof consists in showing that we can reduce t to a term without 1- or III-conv subterms. Dt ~ a will mean that Rr < rt for every principal part r of a 1- or Ill-conv subterm of t. We can read Dt ~ o: as "the degree of t is ~ «", providing that we do not assume Dt to be an ordinal which is computable from t. LEMMA
2:
If Dt(a"),
Ds"
~
o: and Rs" <
rt,
then Dt(s)
~
«.
Let uv be a 1- or III-conv sub term of t(s). We must show that Ru < «, If uv is a subterm of s, this follows from Ds ~ rt. If u = s, it follows from Rs < «. If uv is III-conv and u is of the form sw (so that s is of the form (s.),
182
W. W. TAIT
u'(s)v'(s) where, if uv is f-conv (i = I or III), then u'(a)v'(a) is an z-conv subterm of t(a). But since Dt(a) ::s; a, it follows that Ru = Ru'(a) < a. LEMMA
3:
If Dr, Ds ::s; a, rs is a term, and Rs < a, s a and I t I ::s; Max (I rs I, 1s 1+ 1r I).
then there is a t
with rs =! t, Dt
Proof by induction on I r I. If rs is not 1- or III- cony, then t = rs suffices, since rs is the only possible cony subterm of rs which is neither a subterm of r nor of s.lf r = {Aa'''' r;(aO"')} and s is of type a n> then t = rn(s) suffices by Lemmas 1 and 2. If r = (r;; f3)u, then since I r n I < I r I, there is a r, with rns=1 tn> oi;« a and 1t n 1 s Max (I rns I, Is 1+lrnl),forn ~ O. Hence, t = (t .: y)u suffices, where y = Max (Max(f3, I s I) + 1, 1 s I + 13).
4: If Drs ::s; a and Rr a'r; ... rin ~ 0). LEMMA
~
a, then r is of the form (u i ) or else
In fact, r is of the form rOr l ••• rn> where r o is not of the form UV, and Rr., ~ Rr ~ a. Now in all cases other than r o = at and r o = (uJ with n = 0, r o is the principal part of a 1- or lII- cony sub term of rs. But this is impossible since Drs ::s; a. Let x(~) = 2"', and for y "# 0, let x(r) be the iah simultaneous solution 13 of x(~) = 13, for all y' < y. Then x(r) is a normal function of a.
1: If Dt ::s; a ::s; xf~)I'
THEOREM
and 1 t'
I
+
wY, then there is a t' such that t =
r or,
::S;a
The proof is by induction on )I, and within that, by induction on 1 t I. If I t I = 0, then Dt = 0, so that t' = t suffices. Using the fact that the length of a term exceeds the length of its subterms, and that x~Y) is normal in 13, we can set (Ss)' = Ss', paO"'. s;, f3}' = {AaO"'. s;, x~Y)} and (s;; 13)' = = (s~·l' X(Y») Let t • = rs Then there are r' and s' with Dr' , Ds' < , a r - r' p • s=jS', 1 r' I ::s; and I s'l ::s; Xm. If Rr ~ a+w Y, then r is of the form a'r ; ... r« or (uJ, by Lemma 5. Hence, r' is of the form atr~ ... r~ or (u;), and so r's' is not 1- or III-conY. Thus, t' = r's' satisfies the theorem. Assume now that Rr < a+w Y• We must consider three cases: Case 1. y = 0. I.e. Rr s; a+ 1, so that Rs < a. Then by Lemma 4 there is a t' with t=r's' t', Dt' ::s; a and 1 t' I ::s; 21sl+21rl < 2 1t l• Case 2. y = (j + 1. Then since Rr < a+w·· w, we have Rr < a+w· . k for sufficiently large k. Then Dr's' ::s; a + w· . k, and so by k iterated
Xrn
-I
,
183
INFINITELY LONG TERMS OF TRANSFINITE TYPE
applications of the inductive hypothesis for J, there is a t' with r's' Dt' ~ IX and (y)
< XI t
Case 3. 'Y
= lim 'Yn' Then since Rr < n
+ COYk.
IX
t' ,
I'
+ oi', there is a k with
Rr <
IX
+
Hence, Dr' s' ~ IX + co'", and so by the inductive hypothesis for 'Yk' there is a t' with r's' =1 t', Dt' ~ IX and
I t' I < -
(Yk)
Xmax (x\~I. x\~(l+
1
<
(y)
Xltl'
This completes the proof. THEOREM 2: If Dt ~ co~, then there is a t' in normalform with t =' t' and
I t 'I
~
X(.j lt l.
This follows from Theorem I and the following lemma. LEMMA
and I t'
5:
If
I s I t I·
Dt
=
0, then there is a t' in normal form with t = t'
The proof is by induction on I t I. If I t I = 0, then t is in normal form, so we can take t' = t. Set (Ss)' = Ss', {Aa<1'· Si' {J}' = {Aa<1'· s;, {J}, and (s.; 13)' = (s;; (J). Let t = totl ... t n where n > and to is not of the form uv. Since Dt = 0, it follows from Lemma 4 that to is a variable, or else it is of the form (Ui) with n = 1. In the first case, t' = t~t~ ... t~ suffices; in the second case, if t~ is a numeral then t' = u~, and if t~ is not a numeral, then t' = t~t~. This completes the proof that every term t can be reduced to a normal form t', From the proof, it is clear that if we restrict ourselves to terms of length < 13 and degree < IX, then t' = f(t) can be defined by predicative recursion up to X~~) . IX, since the doubie induction in the proof of Theorem 1, up to IX; and within that, up to i t I, can be transformed into an induction up to X~~) . IX. The essential unicity of the normal form for a given term follows from
°
m,
THEOREM
3:
If r -
s, r = t and t is in normal form, then s
=1
t.
184
W. W. TAIT
For, if s is also in normal form, then t can be obtained from s only by trivial uses of 2°, namely, where in going from u(v) to u(w) (where v =1 w), v does not actually occur in u(v), so that u(w) differs from u(v) only by some changes of bound variables. (See the definition of the substitution u(v).) Hence, in this sense, t is a mere notational variant of s. In particular, if r contains no free variables and is a O-term, then all its normal forms must be numerals, and hence, identical. The proof of Theorem 3 proceeds by induction on the "length" of a reduction of r to s, defined in a suitable way. I omit this proof, which is routine, long and unpleasant. 4. Thefinite types are obtained by restricting Tp 2 to the case CI. = 1, so that they are built up from by means of the composition (a, r). Similarly, the terms offinite type are obtained by restricting Tm 4 to the case CI. = 1, so that the terms given by this clause are of the form AaG • s. The impredicative p.r. functionals of finite type are obtained from the terms of finite type (in our sense) by replacing Tm 5 by the schema for primitive recursion given in § 1. But we saw in § 1 how to replace each
°
.2(1)'2
(with k exponents). In particular, let t be a constant (0, 0) - term, i.e, without free variables. Then for every n,j(tn) is a numeral. Then
INFINITELY LONG TERMS OF TRANSFINITE TYPE
185
References [I] S. Feferman, Systems of Predicative Analysis. To appear. [2] K. Godel, Dber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12 (1958) 280-287. [3] G. Kreisel, Inessential Extensions of Heyting's Arithmetic by Means of Functionals of Finite Type. Abstract. JSL 24 (1959) 284. [4] K. Schutte, Kennzeichnung von Ordnungszahlen durch rekursiv erklarte Funktionen. Math. Ann. 127 (1954) 15-32. [5] , BeweistheoretischeErfassung der unendlichen Induktion in der Zahlentheorie, Math. Ann. 122 (1951) 369-389. [6] , Predicative Well-orderings. These Proceedings, p. 280. [7] C. Spector, Provably Recursive Functionals of Analysis: A Consistency Proof of Analysis by an Extension of Principles Formulated in Current Intuitionistic Mathematics. Recursive Function Theory. Proc. of Symposia in Pure Mathematics, Vol V., Am. Math. Soc. (1962) 1-27.
Added in proof The work of Lorenzen to which we refer is "Algebraische und logistische Untersuchungen tiber freie Verbande", Journal of Symbolic Logic 16 (1951) 81-106. However, an earlier treatment of proof theory by means of a constructive theory of infinite proofs is given in P. Novikov, "On the consistency of certain logical calculus", Matematicesky sbovnik 12, no. 3 (1943) 353-369. In particular, a constructive consistency proof for arithmetic is given.