- Email: [email protected]
A new system of proof-theoretic ordinal functions
Annals of Pure and Applied Logic 32 (1986) 195-207 North-Holland
195
A NEW SYSTEM OF PROOF-THEORETIC ORDINAL
FUNCTIONS
W. BUCHHOLZ Mathematisches lnstitut der Universitiit Miinchen, Theresienstrasse 39, D 8000 Miinchen 2, Fed. Rep. Germany Communicated by D. van Dalen Received 27 November 1984
In this paper we present a family of ordinal functions % (v ~< to), which seems to provide the so far simplest method for denoting large constructive ordinals. These functions are a simplified version of the 0-functions, but nevertheless have the same strength as those. This will be shown at the end of the paper (Theorem 3.7) by using proof-theoretic results from [1], [2], [5]. In Section 1 we define the functions % and prove their main properties. In Section 2 we define a primitive recursive notation system (OT, <) based on the functions % . This system has the great advantage that its ordering relation < is very simple and can be defined without reference to sets of coetficients or any similar concept. O T is introduced as a subset of a larger set T of terms, which plays an important role in Section 3. There we show that the statement PRWO0PoI2,o), which says that there exist no primitive recursive infinite descending sequences in ({x e OT: x < ~pog2,o}, <), is not provable in /-/I-CA0. This result is essentially used in Simpson [6] to establish the unprovability of a certain theorem of finite combinatorics. The proof of//1-CAo iz PRWO0P0f2~, ) is based on the following results from [1]:
I D v / V n 3kc~(k)=O
(v<~to)
where c~(k) e T, for all n, k e N; and every sequence (c~(k))k,N is primitive recursive. In Section 3 we will prove c~(k)eOT and (c~(k)q:O~c'~(k+ 1)
196
W. Buchholz
hierarchy (q0~)~
Inductive definition of 0*(o9 + 1) (i) 0 • O*(to + 1).
(ii) ~, r / • 0*(o9 + 1) ::> ~ + r / • O*(to + 1). (iii) a • O*(to + 1) & v ~ to ::~ OUR,, • 0*(to + 1). So by using only the functions a'~--~OaRo ( v = O , 1 , . . . , to) instead of (a, fl) ~--~ Oafl one obtains a system of ordinal notations which has almost the same strength as the full system O(to + 1). This suggests to define directly a family of ordinal functions ~ (v ~< to) corresponding to a ~ OaRo (v ~< to) such that the system of all ordinals representable in terms of O, +, I P o , . . . , ~P,o will be isomorphic to O*(to + 1). So we are led to the following definition of ~p,,a: ~p~a:= min{y: y , Q ( a ) } , where C~(a) denotes the set of all ordinals which can be generated from ordinals
Preliminaries. We are working in ZFC. The letters a, fl, y, 6, ~, T/, ~ always denote ordinals. 'On' denotes the class of all ordinals and 'Lim' the class of all limit ordinals. Each ordinal a is identified with the set of its predecessors so that a = (x • O n : x < a } and a
enumerates the class of
all infinite cardinals. We define
1
ie if
>o.
We denote by P the class of all additive principal numbers, i.e., P = {a • On:O < a A V~, T/< a (~ + r / • a)} = (to~: ~ • On}.
A new system of proof-theoretic ordinal functions
197
Definition of P(oO. (1) P ( 0 ) : = 0 . (2) F o r a¢ > 0 t h e r e are u n i q u e l y d e t e r m i n e d a¢o, . . . . , a¢,, • P with a~ = a~o + • . . + a¢,, a n d a~, ~<- • • ~< a¢o; w e set P(a¢) : = {a¢o, • . • , an}.
Definition. F o r
a~o, • • •, a¢,, • P we set a¢o # • • • # a¢, : = a~#(o) + • • - + a~#(n), where ar is a p e r m u t a t i o n of ( 0 , . . . , n) with a~#(o) ~>- • • t> a~#(n). 1.1. P r o p o s i t i o n .
(a) o: $ P ¢:~ P(o:) _~ o:.
(b) y • P ~ ( P o ~ ~_ yc~oL < y). (c) e(fl) ~_P(o: + fl) ~_e(o 0 U P(fl). (d) g2e • P, for all ~ • On.
Definition of sets of ordinals Co(o0 and ordinals ~pvol (v <~co) The definition p r o c e e d s by transfinite recursion on a~ simultaneously for all v ~< e0. Suppose that Cv(~) a n d ~p,,~ are defined for all ~ < a¢, v ~< o9. T h e n we set
q (~) := U c7(~),
~p,a~:= m i n { y : y ~ Co (aO},
n<¢o
where C~(tr) is defined by induction on n as follows
~ ( ~ ) := ~ ,
Cvn+l(~) :'- Cn(lT) U ( y : P ( y ) _ C~(a¢)} u {~,,~: tg • a~n co"(a:) ^ ~ • c.,(O ^ u ~ co}.
Remark. The condition " ~ e C , ( ~ ) " in the definition of C~+1(a¢) is included since it m a k e s the i m p o r t a n t p r o p e r t i e s of the functions ~po easier to prove. But it can be shown that by omitting this condition one does not change the sets Co(a 0. H e n c e Co(t 0 can be characterized as the least set X with:
(co ~ ~_x, (c2) v~, n e x (~ + n • x ) ,
(C3) V~ e X n a Vu-< o~ (~p=~ e X). In the following the letters u, v, w shall always denote ordinals ~
(a) W,,O= ~2,,.
(b) ~,,o: e P.
(c) ~ -< ~,,,#<~ + . (d) a'~C.(a')=_ Co(a) and ~p~,o~<~ ~p,fl. (e) r e c,,(~)¢,P(r) _= c,,(~). (f) ~, ,1 e c,,(o:) =>~ +,7 ~ c,,(,O. (g) ~ + ,7 e q,(o:) :>,1 e c-,,(a'). (h) a'o < a' and V~ (a,o ~< ~ < a" :::> ~ , C_~(a:o)) :::> C~(ffo) = Co(o:).
198
W. Buchholz
Proof. (a) By induction on n we get C~(0)= g'2o.
(b) Assume ~Pva~~ P. Then P(~potr) ~ ~p~tr __.Co(a 0 and thus ~a~ ~ Co(tr). Contradiction. (c) From f2~,~ Co(a 0 it follows that £2o ~< ~p~,a~. Obviously the cardinality of C~(a0 is less than O,o+1. Hence there exists 7 < ~ +1 with 7 ~ Co (tr) and therefore W~,te< f2o +1. (d) Trivial. (e) Using the fact that ~Pu~e P one proves Vy ~ C~(a0 (P(7) _~ C~(tr)) by induction on n. On the other side, if P(7)~-Co(a0, then P(7)~-C~(tr) for some n e ~ (since e ( 7 ) is finite and C~(tr) _~ C~÷l(a0) and thus 7 e Cv~+I(~) ~- C~(tr). (f) From ~, r/e Co(a;) we obtain P(~ + r/) ~_ P(~:) U P(r/) _~ Co(tr) and then n
(g) From ~ + 77 e Co(a;) we obtain P(r/) ~_ e ( ~ + r/) ~_ Co(tr) and then 77 (h) Suppose a~o< a~ and V~ (a~0~< ~ < c~--->~ ~ Co(a~o)). Then we get Co(a~o) ~_ C~(a0 by 1.2(d), and V 7 (Y ~ C~,(m)--->7 ~ Co(~o)) by induction on n. 1.3. L e m m a . a~ < fl and o~ ~ CO(a~) ==>~poo~ < ~pofl.
Proof. From the premise we conclude ~pvtr~<~,~,fl and ~p~,teeco(fl). Hence ~poa~< ~Pvfl, since ~p,~fl~ C~(fl). 1.4. Lemma. (a) Y = ~P,,,~i and ~i ~ C.,,(~i) for i = O, 1 ~ Uo = ul, ~o = ~1. (b) 7 e q , ( t r ) and £2,, < ~ y e P ~ a u , ~(7=~p,,~and e co( ) n (c) f2o < ~ , , ~ e c o ( o O a n d ~ C , , ( ~ ) ~ a~nCo(oc). ProoL (a) follows immediately from 1.2(c) and 1.3. (b) We have P ( 7 ) = { y } and 7ecn+l(tI')\Cnv(o~' ) for some n e ~ . Hence 3, = ~Pu~ with ~ e a~O C~(~) and ~ e Cu(~). (c) Let 7 : = ~/,u~. By (b) we obtain 7 = ~/,w~with ¢ e a~O Co(a0 n Cw(~ ). Now by (a) it follows that w = u and ~ = ¢ e a~n Co(a0 1.5. L e m m a . C_~(¢x) n f2,,+1 = ~,,a.
Proof. ~ ~_ C~(a0 n ~v+l holds by definition and 1.2(c). Now let y e Cv(a0 n K2~+1. We have to show that y < ~Pva~. 1. y < ~ : Then y < ~,~,a~holds by 1.2(c). 2. y e P: Then y = ~p,,~ with ~ < a~ and ~ e C,,(~) (1.4(b)). By 1.2(c) we have u ~
A new system of proof-theoretic ordinal functions
199
1.6. L e m m a
(a) ~p,,(or+ l ) = ~ min{yeP'ap'or< y}'
if oreco(or), I.ap,,or, otherwise. (b) o r e L i m :~ ~p~or=sup{~p~:~
e(v) := ~ Co, tea+l,
for v = O, for v > 0 ,
or*v:=
or, Do+or,
for v - O, for v > 0 .
1. We have 0 e Co(O) and ~pvO= K2~ = w °*~. 2~ Suppose or • Co(or) and Ipoc~= to ~*~. Then also or + 1 e Co (or + 1) and ~ v ( o r "~ 1) -- COt~*v+l -" O)(a~+l)*v by 1.6(a). 3. Suppose or e e(v) Iq Lim and V~ < or (~ e Co(~) ^ ~Pv~ = co~*~). Then by 1.6(b) we obtain ~poa~=sup{co~*°:~ 0 and thus cr • Co(or), since 0 • Co(0) ~_ C~(or). For DO < or < e(v) we have P(or) _ ~ and therefore by I.H. (induction hypothesis) ~ e Co(~) c_ Co(or) for all ~ • P(or). This yields tr e Co (or). 1.8. Lemma. (a) Co(or) ~_ e,~o+s. (b) e~,~+l <~ a~::> Co(e~o,+l ) = Co(or). Proof. (a) Using 1.7(b) and 1.2(c) one proves C~"(or) ___ea~+l by induction on n.
(b) follows from (a) and 1.2(h).
Definition of Guy. For every Y e Co(ca,+l) we define a finite set Guy =_On in such a way that, for each tr, Y • Cu(or)c~G,.Y =-o:. These sets will be used in Section 2 to define the set OT of ordinal notations. The definition of G.y proceeds by induction on min{n e N : y • C3(ea.+l)}:
(1)
200
(2) y = ~ p ~
W. Buchholz
with ~ • C o ( ~ ) :
G , , ~ : = /,t ~r : ,tU lG ,- , ~ [g,
ifu<~v,
if
v
1.9. Lemma. I f 7 • Co(eao+l), then y • Cu(o0 holds if, and only if, Guy ~_ ol. Proof. By induction on min{n • ~ : y • Cg(ea,+l)}: 1. y ~ P: By I.H. we have ~ • Cu(o:)C~Gu~ c_ tr, for every ~ • P(y). Hence P ( y ) _c C,,(¢r) ¢~ Guy c_ t~. By 1.2(e) we have y • Cu(a~) ¢=>P ( y ) ___Cu(t~). 2. y = V ~ with ~e • Co(~): 2.1. u ~< v: Then by I.H. we have ~ • Cu(tr)c~G,,~ c_ tr, and by 1.4(c), y • Cu (a~) ¢~ ~ • a~ A Cu (a~). From this we obtain y • Cu (~) ¢:> { ~:} LI Gu~ c a~. But Guy = {~} U Gu~. 2.2. v < u: In this case we have y • 12u _c C,,(tr) and G,,y = t~.
2. The notation system
(OT, < )
In this section we introduce a primitive recursive set O T of formal terms together with a primitive recursive ordering on O T such that ( O T , < ) is isomorphic to (Co(e~,+l), <). Let Do, D1, • . . , Do, be a sequence of formal symbols. Inductive definition o f a set T o f terms
(T1) Oe T. (T2) If a e T and v ~< ~o, then D~a e T; we call D~a a principal term. (T3) If ao, •. •, ak • T are principal terms and k I> 1, then (a0, • . . , ak) • T. In the following the letters a, b, c, d will always denote elements of T. For principal terms a we set: ( a ) : = a. Inductive definition o f a < b f o r a, b e T
(<1) b 4 : 0 = > 0 < b. (<2) u < v or ( u = v a n d a < b ) ~ D u a < D ~ b . (<3) L e t a = (ao, . . . , a,), b = (bo, . . . , bm), l <-m + n. T h e n a < b iff one of the following two cases holds: (i) n < m and ai = bi for i ~< n. (ii) B k < ~ m i n { n , m } ( a k < b k and ai=b~ for i < k ) . 2.1. Lemma. < is a linear ordering on T. Proof. Straightforward.
A new system of proof-theoretic ordinal functions
201
Abbreviations. L e t a • T and M, M ' c_ T: M<~M'
:¢:~ V x • M 3 y e M ' ( x < ~ y ) ,
M
:¢v
VxeM(x
:¢:~
::lx • M ( a <~x ).
a <~M
Inductive definition o f Gua c T f o r a • T
(G1) G , 0 : = ~. (G2) G,(ao, . . . , ak) := Guao t.J . . . U G.ak. ~{b}UG,,b, if u ---
(oT1) o • o r . (OT2) If ao, • . . , ak • O T (k >t 1) are principal terms with ak ~<" "~< ao, then (ao, . . . , ak) • OT. (OT3) If b • O T with Gob < b, then Dob • OT.
The elements of OT are called ordinal terms.
Proposition.
a • O T ~ G . a c OT.
Inductive definition o f an ordinal o ( a ) f o r a • T
(o.1) o(0) :=0. (0.2) o((ao, . . . , ak)):=O(ao) # . . . (0.3) o(Oob):=~poo(b).
# o(ak) (k >>-1).
2.2. Lemma. For a, c • O T we have: (a) o ( a ) • Co(ea,+I), (b) G~o(a) = {o(x):x • G,,a}, (c) a < c ::> o(a) < o(c). Proof. By induction on the length of a, simultaneously for (a), (b), (c): Let E :"- Eg2a,+l.
1. a = 0: trivial. 2. a = Dob: Then Gob < b and b • OT. (a) By I.H. we have o ( b ) e C o ( e ) and G o o ( b ) = { o ( x ) : x • G o b } ~ _ o ( b ) . From this we obtain o ( b ) • e N Co(o(b)) by 1.8, 1.9 and then o ( a ) = ~poo(b) • Co(e). (b) Since o ( b ) • Co(o(b)), we have Guo(a) = ~ {o(b)} LI G , o ( b ) ,
[1~,
if u ~
by I.H. we have G . o ( b ) = { o ( x ) : x • G~b}. Hence G ~ o ( a ) = { o ( x ) : x • G,,a}.
W. Buchholz
202
(c) We make a subsidiary induction on the length of c: (i) c = D~d with v < u : o ( a ) < Q o + l <~f2~ <~ ~Puo(d)=o(c). (ii) c = D ~ d with b < d : By the I.H. we get o ( b ) < o ( d ) and, as shown above, o ( b ) ~ Co(o(b)). This yields ~poo(b) < v/~o(d). (iii) c = ( C o , . . . , c,,,) with m i> 1 and a < Co: By the subsidiary I.H. we get o(a) <~O(Co) and thus o(a) < O(Co) # o(cO <~o(c). 3.
a = ( a o , • • • , a,,) w i t h
n >i 1 a n d a,, < -
• • ~< ao:
(a) By I.H. we have P(o(a))= {o(ao), ...,o(a,)}=_Co(e) and therefore o(a) ~ Co(e). (b) By I.H. we have G,,o(ai)= { o ( x ) : x eG,,(ai)} for i = 0 , . . . , n . Hence
G,,o(a) = 0 G~o(ai)= {o(x):x 6 0 Guai}= {o(x):x¢ G.a}. i=o
i=o
(c) Let c = (Co, • • •, c,,) with m t> 0. (i) n < m and a i - c i for i ~ n : o ( a ) = o ( c o ) # . . . # o ( c n ) < o ( c ) . (ii) k ~ < m i n { n , m } with ak
o(a) = O(Co) # " "
2.3.
Lemma.
(a)
Co(e~o+I)
# O(Ck-1) # o(ak) # ' ' "
-
# o(a,,) < o(c).
{o(x):x ~ OT}
(b) For every a ~ O T with a < DIO holds: o(a) = the ordertype o f ({x O T : x < a ) , <). (c) ~Poe~2o+l= the ordertype o f ({x ~ O T :x < D10}, <). Proof. Let e := e,%+1. (a) By induction on n we prove: a~ ¢ C~(e):~ :la ~ O T (re = o(a)). (Together with 2.2(a) this yields C o ( e ) = { o ( x ) : x ¢ O T } . ) for n = 0 the assertion is trivial. Let e: ~ C~+l(e)\C~(e). 1. c~ = O~o+ - • • + a~k with C~o, • • •, ock ¢ C~(e) and Crk ~<" • • ~< Cro: By I.H. there are ao, • • •, ak ~ O T with o(ai) = cri (i = O , . . . , k). By 2.1 and 2.2(c) we obtain a k ~ < " " < a o and thus a : = ( a o , . . . , ak) ~ OT. Now o(a) = o(ao) # ' " # O(ak) = 0{.
2. cr = ~ with ~ ¢ C~(e) f3 C~(~): By I.H. there exists b ~ O T with o(b) - ~. By 2.2(b) and 1.9 we obtain { o ( x ) : x ~ G~b} = G ~ ~_ ~ = o(b). Hence G~b < b by 2.1 and 2.2(c). It follows that D~b ~ O T and o(Dob) = o:. ( b ) , ( c ) By (a) and 2.2(c) the system ( { x ¢ O T : x < a } , < ) is isomorphic to (Co(e) f3 o(a), < ) , for each a ~ OT. By 1.5 we have Co(e) f3 O(310) = Co(E) (] if21 "- ~30E. This yields part (c). For a < D10 we have o(a) e Co(e) N/21 = ~poe and thus Co(e) rl o(a) -- o(a).
A new system of proof-theoretic ordinal functions
203
3. Unprovability of P R W O 0 P o ~ ) in H~-CAo Let or~< ~Poeu,+a. By PRWO(a~) we denote the statement that there are no primitive recursive infinite descending sequences in ({x e O T : o ( x ) < or}, <). Using a result from [1] we will prove the following theorem.
3.1. Theorem. IDv/PRWO(~Poeav+a) ( 0 < v ~< ¢o). Since ~Poe~+l< ~poK2,o, for all v < co, and since H]-CAo proves the same arithmetic sentences as (.Jv<,~ IDv, we get from 3.1: Corollary. /-/I-CAo ~tPRWO(aPog2,o). In Pohlers [5] it was shown that TI(v), i.e. the principle of transfinite induction up to 0eg÷10, is not provable in IDv, Theorem 3.1 improves this result (for v ~< w) in so far as PRWO(~/,oeav+a) is a/~2-sentence while the complexity of TI(v) is/711. Moreover PRWO(~Poeg+l) is a consequence of TI(v). Remark.
We repeat now some definitions from [1]. As before the letters a, b, c, d shall always denote elements of T. Definition o f a + b a n d a . n
a +0:=0+a:=a, (ao,
. . . , a,)
+
a.0:=0,
(bo,
. . . ,
bin) : = (ao, . • • , an,
a . ( n + l ) : = a
bo,
. . . ,
bin),
.n+a.
Proposition. (a + b) + c = a + (b + c). Definition o f To f o r v <~ o~
T~ := {0} tO {(Duoao,..., D,, a n ) : n >10, ao, . . . , a,, e T, Uo, . . . , Un <~ V}. Remark.
To~TI~.
Abbreviation.
. .~T,o=
T, and T , = {x e T : x
1 : = DoO.
We idenitfy [~ with the subset {0, 1, 1 + 1, 1 + 1 + 1 , . . . } of O T N To, Definition o f dora(a) a n d a[z] f o r a e T, z e dora(a)
([ l.O) dom(O) := O.
([ ].1) dom(1):=(O}; 110]:=0. ([ ].2) dom(D,,+O):= (D.+O)[z]:=z.
204
W. Buchholz
([ ].3) dom(Do,0):=~; (Do, O)[n]:=Dn+lO. ([ ].4) Let a = Dob with b :/=0: (i) d o m ( b ) = {0}: dom(a):= ~: a[n]:=(D~b[O]. (n + 1). (ii) dom(b) = T~ with v ~~1): dom(a):=dom(ak); a[z] : - - ( a o , . . . , ak-1) + a~[z]. Definition. 0[n] := 0, a[n]:=a[O], if dom(a) = (0}. 3.2. Lemma. (a) z e dom(a) ==>a [ z ] < a.
(b) z, z' e dom(a) = Tu and z < z' ==>a[z] < a[z']. (c) O # : a e T ~ d o m ( a ) e ( { O } , f~} O { T . : u < v } , dom(a).
and a[z]e T~ for all z e
Proof. Straightforward by induction on the length of a.
1.3. Lemma. a, z ~ O T and z ~ dom(a) ~ a[z] ~ OT. Before we are going to prove this lemma we want to give the Proof of Theorem 3.1. Let 0 < v ~< 09, n
c n: = DoD~ • • - D~0,
c~(k)'=c~[1][2]. . . [k].
~n [1, Corollary 4.0] we have shown: (1)
ID~/Vn 3kc~(k)=O.
:)he easily proves that c~ e O T Iq To; this can be done in PA (Peano Arithmetic). Since the proofs of 3.2 and 3.3 can also be formalized in PA, we obtain: (2)
P A F V n V k (cn(k) e o r ^ (c~(k),O--->c~(k + 1) < c~(k))).
9bviously the sequences (c~(k))k~ are primitive recursive, and by 1.3, 1.7 we lave o(c~) < ~Poe~+l. Together with (2) this yields:
(3)
PAFPRWO( o , +I) Vn 3kc (k)--0.
From (1) and (3) we obtain Theorem 3.1. For the proof of 3.3 we need the following definitions and lemmata.
A new system of proof-theoretic ordinal functions
205
Definition. G°a "= G~a U {0} b <~ a :¢:~b < a
and
Vu Vc (b < c < a ~ G~b <~G~c U G°z).
3.4. L e m m a . b <~z a, G~a < a, G . z < b ~ G.b < b. Proof. We have G . b < G.a U G°z < a. Assumption: b <. G.b. Then there exists a subterm d of b with minimal length such that b < G . d < a. By the minimality of d we have d = D,,c with G.c < b < c < a. Using b <:1=a and G . z < b we obtain G~b < G,,c U G°z < b. Contradiction.
3.5. Lemma. boa + bo <~ a + b and D,,bo <1, D,,b. Proof. 1. Suppose a + bo <~ c ~ u, we have
G.(D,,bo) =
{bo} U G.bo < {Co} U G.co U G°..z ~_ G,,c U G°z.
If v < u, then Gu(D,,bo) = fJ.
3.6. Lelnma. a • T and z e dom(a)=>a[z] <~za Proof. By induction on the length of a: By 3.2 we have a[z] < a. ~ Suppose a[z] <. c < a. We have to prove G,a[z] < G.c u G°z. 1. a = 1 or a = Dw+10: trivial. 2. a = D o , O:G.a[z]=G~D=+IO~ {0}. 3. a = D ~ b with d o m ( b ) = { 0 } : Then a [ z ] = ( D ~ b [ O ] ) . ( z + l) and G~a[z]= G.D~b[O]. B y I . H . and 3.5 we get D~b[0] <~o D~b = a. We also have Dob[0] < c < a and therefore G~D~b[O] <-. G,c U {0}. 4. a = Dob and dora(b) = Tw and v ~ w < o~: Then a[z] = Dob[x] with x := Dwb[1]. Suppose u ~
= {Co} U Guco U {b[1]} U G°b[1] < {Co} U G.co U G°I c Guc U G ° z 5. a = D ~ b and d o m ( b ) e { N } U { T w : w < v } " By I.H. we get b[z]<~=b and then a[z] = D~b[z] <~ D~b = a by 3.5. 6. a = ( a o , . . . , a k ) (k >~1): By I.H. we get ak[z] <~=ak and then a [ z ] = (ao, . . . , ak-1) + ak[z] ~ z (ao, . . . , ak-1) + ak = a by 3.5.
206
IV. Buchholz
Proof of Lemma 3.3. By induction on the length of a: 1 . a = ( a o , • • •, ak) • OT: Then ao, • . . , ak • O T and ak[Z] < ak <~ " " " <. ao. By I.H. we have ak[Z] • OT. Hence a[z] = (ao, . . . , ak-1) + ak[Z] • OT. 2. a = Dob • O T : Then b • O T and Gob < b. 2.1 dom(b) = {0}: By I.H. and 3.6 we obtain b[0] • O T a n d b[0] <3o b. From b[0] <30 b and Gob < b we get Gob[0] < b[0] by 3.4. Hence a[z] = (Dob[O]). (z +
1)•OT. 2.2. dora(b)= Tu with v<<.u b [ x ] e O T ) . By 3.6 we have b[1] <31 b. From this together with Gob < b and (3;,1
Dob[z]eOT. Finally we want to show that the ~p-functions have essentially the same strength as the O-functions. 3.7. Theorem. 0eg+10 = ~/)0EK2v+l (0 < V ~ (.0) Proof. By [2] and [5] we have 0ea,+10=lIDvl. The proof of 0eav+~0~
A
lID.[ = s u p { r k ( c ~ ) : k • ~ } ,
where c~= k O o O ~ " " O,,O,
(1)
and r k ( a ) = sup{rk(a[n])+ 1 :n e dom(a)}, for all a • To. By 3.2 and 3.3 we have: OeaeOTnTo
~
a[n]
(2)
From (2) and 2.2(c) we obtain by transfinite induction on a: rk(a) <~o(a),
(3)
for all a • O T O To.
From 1.2(d), 1.6(b), 1.7(b)we obtain: IPoe~+l = sup{o(ck) :k • ~}.
(4)
As already mentioned in the proof of 3.1 we have:
c~•OTn To. Now from (1), (3), (4), (5) it follows that
(5)
IIDvl <
Woe~+l.
A new system of proof-theoretic ordinal functions
207
Remark. The functions ~po (v ~< to) were first defined in an unpublished manuscript (1981) by the author. Later on this approach was extended by J/iger [4] and Schtitte [3].
References [1] W. Buchholz, An independence result for (HI-CA) + BI, Ann. Pure Appl. Logic, to appear. [2] W. Buchholz and W. Pohlers, Provable wellorderings of formal theories for transfmitely iterated inductive definitions. J. Symbolic Logic 43 (1978) 118-125. [3] W. Buehholz and K. Schiitte, Ein Ordinalzahlensystem ftir die beweistheoretische Abgrenzung der H~-Separation und Bar-Induktion, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Math.-Naturw. Klasse, 1983. [4] G. J~iger, p-inaccessible ordinals, collapsing functions, and a recursive notation system, Archiv f. math. Logik und Grundlagenf. 24 (1984) 49-62. [5] W. Pohlers, Ordinals connected with formal theories for transfinitely iterated inductive definitions, J. Symbolic Logic 43 (1978) 161-182. [6] S. Simpson, Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher B~iume, Arehiv f. math. Logik u. Grundlagenf. 25 (1985) 45-65.
195
A NEW SYSTEM OF PROOF-THEORETIC ORDINAL
FUNCTIONS
W. BUCHHOLZ Mathematisches lnstitut der Universitiit Miinchen, Theresienstrasse 39, D 8000 Miinchen 2, Fed. Rep. Germany Communicated by D. van Dalen Received 27 November 1984
In this paper we present a family of ordinal functions % (v ~< to), which seems to provide the so far simplest method for denoting large constructive ordinals. These functions are a simplified version of the 0-functions, but nevertheless have the same strength as those. This will be shown at the end of the paper (Theorem 3.7) by using proof-theoretic results from [1], [2], [5]. In Section 1 we define the functions % and prove their main properties. In Section 2 we define a primitive recursive notation system (OT, <) based on the functions % . This system has the great advantage that its ordering relation < is very simple and can be defined without reference to sets of coetficients or any similar concept. O T is introduced as a subset of a larger set T of terms, which plays an important role in Section 3. There we show that the statement PRWO0PoI2,o), which says that there exist no primitive recursive infinite descending sequences in ({x e OT: x < ~pog2,o}, <), is not provable in /-/I-CA0. This result is essentially used in Simpson [6] to establish the unprovability of a certain theorem of finite combinatorics. The proof of//1-CAo iz PRWO0P0f2~, ) is based on the following results from [1]:
I D v / V n 3kc~(k)=O
(v<~to)
where c~(k) e T, for all n, k e N; and every sequence (c~(k))k,N is primitive recursive. In Section 3 we will prove c~(k)eOT and (c~(k)q:O~c'~(k+ 1)
196
W. Buchholz
hierarchy (q0~)~
Inductive definition of 0*(o9 + 1) (i) 0 • O*(to + 1).
(ii) ~, r / • 0*(o9 + 1) ::> ~ + r / • O*(to + 1). (iii) a • O*(to + 1) & v ~ to ::~ OUR,, • 0*(to + 1). So by using only the functions a'~--~OaRo ( v = O , 1 , . . . , to) instead of (a, fl) ~--~ Oafl one obtains a system of ordinal notations which has almost the same strength as the full system O(to + 1). This suggests to define directly a family of ordinal functions ~ (v ~< to) corresponding to a ~ OaRo (v ~< to) such that the system of all ordinals representable in terms of O, +, I P o , . . . , ~P,o will be isomorphic to O*(to + 1). So we are led to the following definition of ~p,,a: ~p~a:= min{y: y , Q ( a ) } , where C~(a) denotes the set of all ordinals which can be generated from ordinals
Preliminaries. We are working in ZFC. The letters a, fl, y, 6, ~, T/, ~ always denote ordinals. 'On' denotes the class of all ordinals and 'Lim' the class of all limit ordinals. Each ordinal a is identified with the set of its predecessors so that a = (x • O n : x < a } and a
enumerates the class of
all infinite cardinals. We define
1
ie if
>o.
We denote by P the class of all additive principal numbers, i.e., P = {a • On:O < a A V~, T/< a (~ + r / • a)} = (to~: ~ • On}.
A new system of proof-theoretic ordinal functions
197
Definition of P(oO. (1) P ( 0 ) : = 0 . (2) F o r a¢ > 0 t h e r e are u n i q u e l y d e t e r m i n e d a¢o, . . . . , a¢,, • P with a~ = a~o + • . . + a¢,, a n d a~, ~<- • • ~< a¢o; w e set P(a¢) : = {a¢o, • . • , an}.
Definition. F o r
a~o, • • •, a¢,, • P we set a¢o # • • • # a¢, : = a~#(o) + • • - + a~#(n), where ar is a p e r m u t a t i o n of ( 0 , . . . , n) with a~#(o) ~>- • • t> a~#(n). 1.1. P r o p o s i t i o n .
(a) o: $ P ¢:~ P(o:) _~ o:.
(b) y • P ~ ( P o ~ ~_ yc~oL < y). (c) e(fl) ~_P(o: + fl) ~_e(o 0 U P(fl). (d) g2e • P, for all ~ • On.
Definition of sets of ordinals Co(o0 and ordinals ~pvol (v <~co) The definition p r o c e e d s by transfinite recursion on a~ simultaneously for all v ~< e0. Suppose that Cv(~) a n d ~p,,~ are defined for all ~ < a¢, v ~< o9. T h e n we set
q (~) := U c7(~),
~p,a~:= m i n { y : y ~ Co (aO},
n<¢o
where C~(tr) is defined by induction on n as follows
~ ( ~ ) := ~ ,
Cvn+l(~) :'- Cn(lT) U ( y : P ( y ) _ C~(a¢)} u {~,,~: tg • a~n co"(a:) ^ ~ • c.,(O ^ u ~ co}.
Remark. The condition " ~ e C , ( ~ ) " in the definition of C~+1(a¢) is included since it m a k e s the i m p o r t a n t p r o p e r t i e s of the functions ~po easier to prove. But it can be shown that by omitting this condition one does not change the sets Co(a 0. H e n c e Co(t 0 can be characterized as the least set X with:
(co ~ ~_x, (c2) v~, n e x (~ + n • x ) ,
(C3) V~ e X n a Vu-< o~ (~p=~ e X). In the following the letters u, v, w shall always denote ordinals ~
(a) W,,O= ~2,,.
(b) ~,,o: e P.
(c) ~ -< ~,,,#<~ + . (d) a'~
198
W. Buchholz
Proof. (a) By induction on n we get C~(0)= g'2o.
(b) Assume ~Pva~~ P. Then P(~potr) ~ ~p~tr __.Co(a 0 and thus ~a~ ~ Co(tr). Contradiction. (c) From f2~,~ Co(a 0 it follows that £2o ~< ~p~,a~. Obviously the cardinality of C~(a0 is less than O,o+1. Hence there exists 7 < ~ +1 with 7 ~ Co (tr) and therefore W~,te< f2o +1. (d) Trivial. (e) Using the fact that ~Pu~e P one proves Vy ~ C~(a0 (P(7) _~ C~(tr)) by induction on n. On the other side, if P(7)~-Co(a0, then P(7)~-C~(tr) for some n e ~ (since e ( 7 ) is finite and C~(tr) _~ C~÷l(a0) and thus 7 e Cv~+I(~) ~- C~(tr). (f) From ~, r/e Co(a;) we obtain P(~ + r/) ~_ P(~:) U P(r/) _~ Co(tr) and then n
(g) From ~ + 77 e Co(a;) we obtain P(r/) ~_ e ( ~ + r/) ~_ Co(tr) and then 77 (h) Suppose a~o< a~ and V~ (a~0~< ~ < c~--->~ ~ Co(a~o)). Then we get Co(a~o) ~_ C~(a0 by 1.2(d), and V 7 (Y ~ C~,(m)--->7 ~ Co(~o)) by induction on n. 1.3. L e m m a . a~ < fl and o~ ~ CO(a~) ==>~poo~ < ~pofl.
Proof. From the premise we conclude ~pvtr~<~,~,fl and ~p~,teeco(fl). Hence ~poa~< ~Pvfl, since ~p,~fl~ C~(fl). 1.4. Lemma. (a) Y = ~P,,,~i and ~i ~ C.,,(~i) for i = O, 1 ~ Uo = ul, ~o = ~1. (b) 7 e q , ( t r ) and £2,, < ~ y e P ~ a u , ~(7=~p,,~and e co( ) n (c) f2o < ~ , , ~ e c o ( o O a n d ~ C , , ( ~ ) ~ a~nCo(oc). ProoL (a) follows immediately from 1.2(c) and 1.3. (b) We have P ( 7 ) = { y } and 7ecn+l(tI')\Cnv(o~' ) for some n e ~ . Hence 3, = ~Pu~ with ~ e a~O C~(~) and ~ e Cu(~). (c) Let 7 : = ~/,u~. By (b) we obtain 7 = ~/,w~with ¢ e a~O Co(a0 n Cw(~ ). Now by (a) it follows that w = u and ~ = ¢ e a~n Co(a0 1.5. L e m m a . C_~(¢x) n f2,,+1 = ~,,a.
Proof. ~ ~_ C~(a0 n ~v+l holds by definition and 1.2(c). Now let y e Cv(a0 n K2~+1. We have to show that y < ~Pva~. 1. y < ~ : Then y < ~,~,a~holds by 1.2(c). 2. y e P: Then y = ~p,,~ with ~ < a~ and ~ e C,,(~) (1.4(b)). By 1.2(c) we have u ~
A new system of proof-theoretic ordinal functions
199
1.6. L e m m a
(a) ~p,,(or+ l ) = ~ min{yeP'ap'or< y}'
if oreco(or), I.ap,,or, otherwise. (b) o r e L i m :~ ~p~or=sup{~p~:~
e(v) := ~ Co, tea+l,
for v = O, for v > 0 ,
or*v:=
or, Do+or,
for v - O, for v > 0 .
1. We have 0 e Co(O) and ~pvO= K2~ = w °*~. 2~ Suppose or • Co(or) and Ipoc~= to ~*~. Then also or + 1 e Co (or + 1) and ~ v ( o r "~ 1) -- COt~*v+l -" O)(a~+l)*v by 1.6(a). 3. Suppose or e e(v) Iq Lim and V~ < or (~ e Co(~) ^ ~Pv~ = co~*~). Then by 1.6(b) we obtain ~poa~=sup{co~*°:~
(b) follows from (a) and 1.2(h).
Definition of Guy. For every Y e Co(ca,+l) we define a finite set Guy =_On in such a way that, for each tr, Y • Cu(or)c~G,.Y =-o:. These sets will be used in Section 2 to define the set OT of ordinal notations. The definition of G.y proceeds by induction on min{n e N : y • C3(ea.+l)}:
(1)
200
(2) y = ~ p ~
W. Buchholz
with ~ • C o ( ~ ) :
G , , ~ : = /,t ~r : ,tU lG ,- , ~ [g,
ifu<~v,
if
v
1.9. Lemma. I f 7 • Co(eao+l), then y • Cu(o0 holds if, and only if, Guy ~_ ol. Proof. By induction on min{n • ~ : y • Cg(ea,+l)}: 1. y ~ P: By I.H. we have ~ • Cu(o:)C~Gu~ c_ tr, for every ~ • P(y). Hence P ( y ) _c C,,(¢r) ¢~ Guy c_ t~. By 1.2(e) we have y • Cu(a~) ¢=>P ( y ) ___Cu(t~). 2. y = V ~ with ~e • Co(~): 2.1. u ~< v: Then by I.H. we have ~ • Cu(tr)c~G,,~ c_ tr, and by 1.4(c), y • Cu (a~) ¢~ ~ • a~ A Cu (a~). From this we obtain y • Cu (~) ¢:> { ~:} LI Gu~ c a~. But Guy = {~} U Gu~. 2.2. v < u: In this case we have y • 12u _c C,,(tr) and G,,y = t~.
2. The notation system
(OT, < )
In this section we introduce a primitive recursive set O T of formal terms together with a primitive recursive ordering on O T such that ( O T , < ) is isomorphic to (Co(e~,+l), <). Let Do, D1, • . . , Do, be a sequence of formal symbols. Inductive definition o f a set T o f terms
(T1) Oe T. (T2) If a e T and v ~< ~o, then D~a e T; we call D~a a principal term. (T3) If ao, •. •, ak • T are principal terms and k I> 1, then (a0, • . . , ak) • T. In the following the letters a, b, c, d will always denote elements of T. For principal terms a we set: ( a ) : = a. Inductive definition o f a < b f o r a, b e T
(<1) b 4 : 0 = > 0 < b. (<2) u < v or ( u = v a n d a < b ) ~ D u a < D ~ b . (<3) L e t a = (ao, . . . , a,), b = (bo, . . . , bm), l <-m + n. T h e n a < b iff one of the following two cases holds: (i) n < m and ai = bi for i ~< n. (ii) B k < ~ m i n { n , m } ( a k < b k and ai=b~ for i < k ) . 2.1. Lemma. < is a linear ordering on T. Proof. Straightforward.
A new system of proof-theoretic ordinal functions
201
Abbreviations. L e t a • T and M, M ' c_ T: M<~M'
:¢:~ V x • M 3 y e M ' ( x < ~ y ) ,
M
:¢v
VxeM(x
:¢:~
::lx • M ( a <~x ).
a <~M
Inductive definition o f Gua c T f o r a • T
(G1) G , 0 : = ~. (G2) G,(ao, . . . , ak) := Guao t.J . . . U G.ak. ~{b}UG,,b, if u ---
(oT1) o • o r . (OT2) If ao, • . . , ak • O T (k >t 1) are principal terms with ak ~<" "~< ao, then (ao, . . . , ak) • OT. (OT3) If b • O T with Gob < b, then Dob • OT.
The elements of OT are called ordinal terms.
Proposition.
a • O T ~ G . a c OT.
Inductive definition o f an ordinal o ( a ) f o r a • T
(o.1) o(0) :=0. (0.2) o((ao, . . . , ak)):=O(ao) # . . . (0.3) o(Oob):=~poo(b).
# o(ak) (k >>-1).
2.2. Lemma. For a, c • O T we have: (a) o ( a ) • Co(ea,+I), (b) G~o(a) = {o(x):x • G,,a}, (c) a < c ::> o(a) < o(c). Proof. By induction on the length of a, simultaneously for (a), (b), (c): Let E :"- Eg2a,+l.
1. a = 0: trivial. 2. a = Dob: Then Gob < b and b • OT. (a) By I.H. we have o ( b ) e C o ( e ) and G o o ( b ) = { o ( x ) : x • G o b } ~ _ o ( b ) . From this we obtain o ( b ) • e N Co(o(b)) by 1.8, 1.9 and then o ( a ) = ~poo(b) • Co(e). (b) Since o ( b ) • Co(o(b)), we have Guo(a) = ~ {o(b)} LI G , o ( b ) ,
[1~,
if u ~
by I.H. we have G . o ( b ) = { o ( x ) : x • G~b}. Hence G ~ o ( a ) = { o ( x ) : x • G,,a}.
W. Buchholz
202
(c) We make a subsidiary induction on the length of c: (i) c = D~d with v < u : o ( a ) < Q o + l <~f2~ <~ ~Puo(d)=o(c). (ii) c = D ~ d with b < d : By the I.H. we get o ( b ) < o ( d ) and, as shown above, o ( b ) ~ Co(o(b)). This yields ~poo(b) < v/~o(d). (iii) c = ( C o , . . . , c,,,) with m i> 1 and a < Co: By the subsidiary I.H. we get o(a) <~O(Co) and thus o(a) < O(Co) # o(cO <~o(c). 3.
a = ( a o , • • • , a,,) w i t h
n >i 1 a n d a,, < -
• • ~< ao:
(a) By I.H. we have P(o(a))= {o(ao), ...,o(a,)}=_Co(e) and therefore o(a) ~ Co(e). (b) By I.H. we have G,,o(ai)= { o ( x ) : x eG,,(ai)} for i = 0 , . . . , n . Hence
G,,o(a) = 0 G~o(ai)= {o(x):x 6 0 Guai}= {o(x):x¢ G.a}. i=o
i=o
(c) Let c = (Co, • • •, c,,) with m t> 0. (i) n < m and a i - c i for i ~ n : o ( a ) = o ( c o ) # . . . # o ( c n ) < o ( c ) . (ii) k ~ < m i n { n , m } with ak
o(a) = O(Co) # " "
2.3.
Lemma.
(a)
Co(e~o+I)
# O(Ck-1) # o(ak) # ' ' "
-
# o(a,,) < o(c).
{o(x):x ~ OT}
(b) For every a ~ O T with a < DIO holds: o(a) = the ordertype o f ({x O T : x < a ) , <). (c) ~Poe~2o+l= the ordertype o f ({x ~ O T :x < D10}, <). Proof. Let e := e,%+1. (a) By induction on n we prove: a~ ¢ C~(e):~ :la ~ O T (re = o(a)). (Together with 2.2(a) this yields C o ( e ) = { o ( x ) : x ¢ O T } . ) for n = 0 the assertion is trivial. Let e: ~ C~+l(e)\C~(e). 1. c~ = O~o+ - • • + a~k with C~o, • • •, ock ¢ C~(e) and Crk ~<" • • ~< Cro: By I.H. there are ao, • • •, ak ~ O T with o(ai) = cri (i = O , . . . , k). By 2.1 and 2.2(c) we obtain a k ~ < " " < a o and thus a : = ( a o , . . . , ak) ~ OT. Now o(a) = o(ao) # ' " # O(ak) = 0{.
2. cr = ~ with ~ ¢ C~(e) f3 C~(~): By I.H. there exists b ~ O T with o(b) - ~. By 2.2(b) and 1.9 we obtain { o ( x ) : x ~ G~b} = G ~ ~_ ~ = o(b). Hence G~b < b by 2.1 and 2.2(c). It follows that D~b ~ O T and o(Dob) = o:. ( b ) , ( c ) By (a) and 2.2(c) the system ( { x ¢ O T : x < a } , < ) is isomorphic to (Co(e) f3 o(a), < ) , for each a ~ OT. By 1.5 we have Co(e) f3 O(310) = Co(E) (] if21 "- ~30E. This yields part (c). For a < D10 we have o(a) e Co(e) N/21 = ~poe and thus Co(e) rl o(a) -- o(a).
A new system of proof-theoretic ordinal functions
203
3. Unprovability of P R W O 0 P o ~ ) in H~-CAo Let or~< ~Poeu,+a. By PRWO(a~) we denote the statement that there are no primitive recursive infinite descending sequences in ({x e O T : o ( x ) < or}, <). Using a result from [1] we will prove the following theorem.
3.1. Theorem. IDv/PRWO(~Poeav+a) ( 0 < v ~< ¢o). Since ~Poe~+l< ~poK2,o, for all v < co, and since H]-CAo proves the same arithmetic sentences as (.Jv<,~ IDv, we get from 3.1: Corollary. /-/I-CAo ~tPRWO(aPog2,o). In Pohlers [5] it was shown that TI(v), i.e. the principle of transfinite induction up to 0eg÷10, is not provable in IDv, Theorem 3.1 improves this result (for v ~< w) in so far as PRWO(~/,oeav+a) is a/~2-sentence while the complexity of TI(v) is/711. Moreover PRWO(~Poeg+l) is a consequence of TI(v). Remark.
We repeat now some definitions from [1]. As before the letters a, b, c, d shall always denote elements of T. Definition o f a + b a n d a . n
a +0:=0+a:=a, (ao,
. . . , a,)
+
a.0:=0,
(bo,
. . . ,
bin) : = (ao, . • • , an,
a . ( n + l ) : = a
bo,
. . . ,
bin),
.n+a.
Proposition. (a + b) + c = a + (b + c). Definition o f To f o r v <~ o~
T~ := {0} tO {(Duoao,..., D,, a n ) : n >10, ao, . . . , a,, e T, Uo, . . . , Un <~ V}. Remark.
To~TI~.
Abbreviation.
. .~T,o=
T, and T , = {x e T : x
1 : = DoO.
We idenitfy [~ with the subset {0, 1, 1 + 1, 1 + 1 + 1 , . . . } of O T N To, Definition o f dora(a) a n d a[z] f o r a e T, z e dora(a)
([ l.O) dom(O) := O.
([ ].1) dom(1):=(O}; 110]:=0. ([ ].2) dom(D,,+O):= (D.+O)[z]:=z.
204
W. Buchholz
([ ].3) dom(Do,0):=~; (Do, O)[n]:=Dn+lO. ([ ].4) Let a = Dob with b :/=0: (i) d o m ( b ) = {0}: dom(a):= ~: a[n]:=(D~b[O]. (n + 1). (ii) dom(b) = T~ with v ~~1): dom(a):=dom(ak); a[z] : - - ( a o , . . . , ak-1) + a~[z]. Definition. 0[n] := 0, a[n]:=a[O], if dom(a) = (0}. 3.2. Lemma. (a) z e dom(a) ==>a [ z ] < a.
(b) z, z' e dom(a) = Tu and z < z' ==>a[z] < a[z']. (c) O # : a e T ~ d o m ( a ) e ( { O } , f~} O { T . : u < v } , dom(a).
and a[z]e T~ for all z e
Proof. Straightforward by induction on the length of a.
1.3. Lemma. a, z ~ O T and z ~ dom(a) ~ a[z] ~ OT. Before we are going to prove this lemma we want to give the Proof of Theorem 3.1. Let 0 < v ~< 09, n
c n: = DoD~ • • - D~0,
c~(k)'=c~[1][2]. . . [k].
~n [1, Corollary 4.0] we have shown: (1)
ID~/Vn 3kc~(k)=O.
:)he easily proves that c~ e O T Iq To; this can be done in PA (Peano Arithmetic). Since the proofs of 3.2 and 3.3 can also be formalized in PA, we obtain: (2)
P A F V n V k (cn(k) e o r ^ (c~(k),O--->c~(k + 1) < c~(k))).
9bviously the sequences (c~(k))k~ are primitive recursive, and by 1.3, 1.7 we lave o(c~) < ~Poe~+l. Together with (2) this yields:
(3)
PAFPRWO( o , +I) Vn 3kc (k)--0.
From (1) and (3) we obtain Theorem 3.1. For the proof of 3.3 we need the following definitions and lemmata.
A new system of proof-theoretic ordinal functions
205
Definition. G°a "= G~a U {0} b <~ a :¢:~b < a
and
Vu Vc (b < c < a ~ G~b <~G~c U G°z).
3.4. L e m m a . b <~z a, G~a < a, G . z < b ~ G.b < b. Proof. We have G . b < G.a U G°z < a. Assumption: b <. G.b. Then there exists a subterm d of b with minimal length such that b < G . d < a. By the minimality of d we have d = D,,c with G.c < b < c < a. Using b <:1=a and G . z < b we obtain G~b < G,,c U G°z < b. Contradiction.
3.5. Lemma. bo
G.(D,,bo) =
{bo} U G.bo < {Co} U G.co U G°..z ~_ G,,c U G°z.
If v < u, then Gu(D,,bo) = fJ.
3.6. Lelnma. a • T and z e dom(a)=>a[z] <~za Proof. By induction on the length of a: By 3.2 we have a[z] < a. ~ Suppose a[z] <. c < a. We have to prove G,a[z] < G.c u G°z. 1. a = 1 or a = Dw+10: trivial. 2. a = D o , O:G.a[z]=G~D=+IO~ {0}. 3. a = D ~ b with d o m ( b ) = { 0 } : Then a [ z ] = ( D ~ b [ O ] ) . ( z + l) and G~a[z]= G.D~b[O]. B y I . H . and 3.5 we get D~b[0] <~o D~b = a. We also have Dob[0] < c < a and therefore G~D~b[O] <-. G,c U {0}. 4. a = Dob and dora(b) = Tw and v ~ w < o~: Then a[z] = Dob[x] with x := Dwb[1]. Suppose u ~
= {Co} U Guco U {b[1]} U G°b[1] < {Co} U G.co U G°I c Guc U G ° z 5. a = D ~ b and d o m ( b ) e { N } U { T w : w < v } " By I.H. we get b[z]<~=b and then a[z] = D~b[z] <~ D~b = a by 3.5. 6. a = ( a o , . . . , a k ) (k >~1): By I.H. we get ak[z] <~=ak and then a [ z ] = (ao, . . . , ak-1) + ak[z] ~ z (ao, . . . , ak-1) + ak = a by 3.5.
206
IV. Buchholz
Proof of Lemma 3.3. By induction on the length of a: 1 . a = ( a o , • • •, ak) • OT: Then ao, • . . , ak • O T and ak[Z] < ak <~ " " " <. ao. By I.H. we have ak[Z] • OT. Hence a[z] = (ao, . . . , ak-1) + ak[Z] • OT. 2. a = Dob • O T : Then b • O T and Gob < b. 2.1 dom(b) = {0}: By I.H. and 3.6 we obtain b[0] • O T a n d b[0] <3o b. From b[0] <30 b and Gob < b we get Gob[0] < b[0] by 3.4. Hence a[z] = (Dob[O]). (z +
1)•OT. 2.2. dora(b)= Tu with v<<.u
Dob[z]eOT. Finally we want to show that the ~p-functions have essentially the same strength as the O-functions. 3.7. Theorem. 0eg+10 = ~/)0EK2v+l (0 < V ~ (.0) Proof. By [2] and [5] we have 0ea,+10=lIDvl. The proof of 0eav+~0~
A
lID.[ = s u p { r k ( c ~ ) : k • ~ } ,
where c~= k O o O ~ " " O,,O,
(1)
and r k ( a ) = sup{rk(a[n])+ 1 :n e dom(a)}, for all a • To. By 3.2 and 3.3 we have: OeaeOTnTo
~
a[n]
(2)
From (2) and 2.2(c) we obtain by transfinite induction on a: rk(a) <~o(a),
(3)
for all a • O T O To.
From 1.2(d), 1.6(b), 1.7(b)we obtain: IPoe~+l = sup{o(ck) :k • ~}.
(4)
As already mentioned in the proof of 3.1 we have:
c~•OTn To. Now from (1), (3), (4), (5) it follows that
(5)
IIDvl <
Woe~+l.
A new system of proof-theoretic ordinal functions
207
Remark. The functions ~po (v ~< to) were first defined in an unpublished manuscript (1981) by the author. Later on this approach was extended by J/iger [4] and Schtitte [3].
References [1] W. Buchholz, An independence result for (HI-CA) + BI, Ann. Pure Appl. Logic, to appear. [2] W. Buchholz and W. Pohlers, Provable wellorderings of formal theories for transfmitely iterated inductive definitions. J. Symbolic Logic 43 (1978) 118-125. [3] W. Buehholz and K. Schiitte, Ein Ordinalzahlensystem ftir die beweistheoretische Abgrenzung der H~-Separation und Bar-Induktion, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Math.-Naturw. Klasse, 1983. [4] G. J~iger, p-inaccessible ordinals, collapsing functions, and a recursive notation system, Archiv f. math. Logik und Grundlagenf. 24 (1984) 49-62. [5] W. Pohlers, Ordinals connected with formal theories for transfinitely iterated inductive definitions, J. Symbolic Logic 43 (1978) 161-182. [6] S. Simpson, Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher B~iume, Arehiv f. math. Logik u. Grundlagenf. 25 (1985) 45-65.