An Ordinal Analysis of Stability
An Ordinal Analysis of Stability Michael Rathjen
Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England
Abstract
This paper is the rst in a series of three which culminates in an ordinal analysis of 12 -comprehension. On the set-theoretic side 12-comprehension corresponds to Kripke-Platek set theory, KP, plus 1 -separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is -stable, i.e. L is a 1-elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 12 -comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on -inaccessible cardinals which is then employed for determining an upper bound for the proof{theoretic strength of the theory KPi + 8 9 is + -stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.
1 Introduction An ordinal analysis of 12-comprehension has been a main goal in proof theory.1 This paper is the rst in a series of three which culminates in an ordinal analysis of 12-comprehension. On the set-theoretic side 12-comprehension corresponds to Kripke-Platek set theory,2 KP, plus 1-separation, i.e. the schema of axioms 9z(z = fx 2 a : (x)g) for all formulas in which z does not occur free. The precise relationship is as follows (see [17], Theorem 5.4): Theorem 1.1 KP +1 separation and (12 ?CA)+ BI prove the same sentences of second order arithmetic, where BI stands for the scheme of bar induction. The ordinals such that L j= KP + 1-Separation are familiar from ordinal recursion theory. The results in this paper were obtained in 1995 when I was a Heisenberg Fellow of the German Science Foundation, Deutsche Forschungsgemeinschaft. 1 For more details see [15], [6], [9]. 2 It is crucial here that the In nity axiom is included.
1
De nition 1.2 An admissible ordinal is said to be nonprojectible if there is no total { recursive function mapping one{one into some < , where a function g : L ! L is called {recursive if it is de nable in L. The key to the `largeness' properties of nonprojectible ordinals is that for any nonprojectible ordinal , L is a limit of 1{elementary substructures, i.e. for every < there exists a < < such that L is a 1{elementary substructure of L, written L 1 L. De nition 1.3 Let < . is {stable if L 1 L. Ordinals satisfying L 1 L for some > have strong re ecting properties, e.g., Lemma 1.4 (cf. [18], 1.18) L 1 L+1 i is n{re ecting3 for all n. The last result makes it clear that an ordinal analysis of 12 comprehension will necessarily involve a proof{theoretic treatment of re ections beyond those surfacing in admissible proof theory. The objective of this paper is to give an ordinal analysis of not too complicated stability relations. My experiences in explaining the ordinal analysis of 12-comprehension have taught me that it is best approached in three steps. The rst step is given in this paper. It introduces an ordinal representation system based on -inaccessible cardinals which is then employed for determining an upper bound for the proof{theoretic strength of the theory KPi + 8 9 is + -stable. The second paper will build on this one and feature an ordinal analysis of parameter free 12-comprehension, whereas the third paper will give an ordinal analysis of full 12comprehension. It might be instructive to explain the increasing complexity of the ordinal analyses in the three papers in terms of their ordinal representation systems. In this paper the collapsing functions still collapse single points. In the second paper the collapsing functions collapse points from an interval but the interval doesn't contain ordinals with complicated re ection properties. In the last paper the collapsing functions collapse points from an interval which itself may contain ordinals < where is -stable. The results in this paper date from 1995. the most important forerunner of the techniques employed here was [13].
> 0 is said to be n {re ecting if L j= n {re ection. By n {re ection (n{re ection) we mean the scheme ! 9z [T ran(z ) ^ z 6= ; ^ z ]; where is n (n), and T ran(z ) expresses that z is a transitive set. 3
2
2
-indescribable
cardinals
It makes no sense to present an ordinal representation system without giving some kind of semantic interpretation. For ordinal representation systems in impredicative proof theory it is essential to understand the collapsing functions which they encapsulate. In this section we will indicate a model for the projection functions, employing rather sweeping large cardinal axioms, in that we shall presume the existence of certain cardinals, featuring a strong form of indescribability. Large cardinals have been used quite frequently in the de nition procedure of strong ordinal representation systems, and large cardinal notions have been an important source of inspiration. In the end, they can be dispensed with, but they add an intriguing twist to the relation between set theory and proof theory. The advantage of working in a strong set{theoretic context is that we can build models without getting buried under complexity considerations. Another objective of this section is to \ nd" a large cardinal analogue of the notion of being + -stable for all < . De nition 2.1 Let [ V= V be the cumulative hierarchy of sets, i.e.
2ON
V0 = ;; V +1 = fX : X V g; V =
[
! . Corollary 2.4 If is A--indescribable and < and < , then is A--indescribable. Proof : Suppose P1; : : : ; Pn V and hV+ ; Ai j= [P1; : : : ; Pn]. Let (u1; : : : ; un; v; w) := 9x2w \hVv+x ; Pi j= (u1; : : : ; un)": Then hV+ ; Ai j= [P1; : : : ; Pn ; ; fg] by Lemma 2.3. Thus there exist 0 < such that hV0 + ; Ai j= [P1 \ V0 ; : : : ; Pn \ V0 ; \ V0 ; fg \ V0 ]: ut This implies hV0 + ; Ai j= [P1 \ V0 ; : : : ; P0 \ V0 ] by Lemma 2.3.
Lemma 2.5 If and is A- -indescribable, then for all < there exists < such that is A- -indescribable. Proof : Let < . As is A- -indescribable by Corollary 2.4, we get hV+ ; Ai j= 9 \ is A- -indescribable"; and thus, by A- -indescribableness, there exists 0 such that < 0 < and hV0 + ; Ai j= 9 0 \ is A- -indescribable"; which implies that there exists < such that is A- -indescribable. ut To situate the notion of -inaccessibility with regard to consistency strength in the usual hierarchy of large cardinals, we recall the notion of a subtle cardinal. 4
De nition 2.6 A cardinal is said to be subtle if for any sequence hS : < i such that S and C closed and unbounded in , there are < both in C satisfying S \ = S : Since subtle cardinals are not covered in many of the standard texts dealing with large cardinals, we mention the following facts (see [8], x20): Remark 2.7 Let (!) denote the rst !-Erdos cardinal. (i) f < (!) : is subtleg is stationary in (!). (ii) \Subtlety" relativises to L, i.e. if is subtle, then L j= \ is subtle". Lemma 2.8 Assume that is a subtle cardinal and that A V . Then for every B closed and unbounded in there exists 2 B such that is A--indescribable. Proof : Assume that is subtle. Since is inaccessible, we may select a bijective mapping F : V ?! such that (1) CF = f < : F V maps V bijectively into g is closed and unbounded in . Now let B be closed and unbounded in . By the preceding we may assume B CF . In addition, we may assume that B consists only of cardinals. For a contradiction assume that there is no cardinal 2 B such that is A--indescribable. Let 2 B . Since is not A--indescribable, we can nd an Lset(P)-formula and a subset P V so that (note that + = ) (2) hV ; Ai j= [P; ] and (3) 8 < hV ; Ai j= : (P \ V ; ): For 2 B de ne (4) P = F 00P \ ( n !) [ f3n : n 2 F 00P \ !g [ f3n + 1 : hV ; Ai j= n(P ; )g [ f3n + 2 : hV ; Ai j= : n (P ; )g where h n : n 2 !i is an enumeration of the Lset (P)-formulas with two free variables. By subtlety of , we nd 0 < 1 both in B , such that P0 = P1 \ 0: (5) (5) yields (6) P0 = P1 \ V0 : hV ; Ai j= 1 (P1 ; 1) holds by (2). Therefore (5) viewed together with (4) implies (7) hV ; Ai j= 1 (P0 ; 0): Hence, using (6), hV ; Ai j= 1 (P1 \ V0 ; 0), contradicting (3). ut 5
Corollary 2.9 Let be a subtle cardinal. Then there exists a regular cardinal < such that
(8) (9)
8 < 9 < \ is -indescribable" 8 < 8 < [\ is -indescribable" ! < ]:
Proof : By Lemma 2.8 there exists < such that is -indescribable. Then is regular and satis es (10)
8 < 9 < \ is -indescribable"
by Lemma 2.5. We may assume that has been chosen minimal with property (10). If there existed a cardinal < and an ordinal < such that is -indescribable, then, by Corollary 2.4 and Lemma 2.5 we would get
8 < 9 < \ is -indescribable"; contradicting the choice of .
ut
Lemma 2.10 is 0-10-indescribable i is strongly inaccessible. Proof : See [4], Ch. 9, Theorem 1.3.
ut
6
3 Skolem Hulls and Collapsing Functions In this section we use the large cardinal notions of the previous section to develop collapsing functions. De nition 3.1 Let + denote the least cardinal > . Note that + is always a regular cardinal (on the basis of ZFC). The Veblen-function gures prominently in predicative proof theory (cf. [F 68], [21]). We are going to incorporate this function in our ordinal representation system. De nition 3.2 The Veblen{function ' := ' ( ) is de ned by trans nite recursion on by letting ' be the function that enumerates the class of ordinals
f! : (8 < )[' (! ) = ! ]g: Corollary 3.3 (i) '0 = ! . (ii) ; < ' =) + < ' . (iii) < =) ' < ' . (iv) < =) ' (' ) = ' . In the following we shall assume that is the least regular cardinal satisfying (11)
8 < 9 < \ is -indescribable":
then satis es (12)
8 < 8 < [\ is -indescribable" ! < ]:
According to Corollary 2.9, the existence of follows from the existence of a subtle cardinal. For < , let ( ) be the least cardinal such that is -indescribable; notice that ( ) < as well.
De nition 3.4 By recursion on we shall de ne sets of ordinals C ( ; ) (the th Skolem Hull generated from ordinals < ), a set < of re ection con gurations, projection instances, and re ection instances, and the ordinals for projection instances X 2 < . A projection instance X 2 < determines the collapse . Formally X is an expression built up from ordinals, symbols M; Pn and parentheses. By X 2 C ( ; ) we mean that X 2 !; is a regular ordinalg. Lemma 7.4 (Reduction Lemma) Let A = W(A)2J . Assume =2Reg, where := rk(A). Then: H ; :A ^ H ?; A =) H + ; ? Proof : Use induction on . For details see [2] , Lemma 3.14. ut Theorem 7.5 (Predicative cut elimination) Let H be closed under '. If H +! ? and [; + ! [ \ Reg = ; and 2H, then H ' ?: Proof : By main induction on and subsidiary induction on (cf. [2] , Theorem 3.16). ut
Corollary 7.6 H +1 ? ^ 2= Reg =) H ! ? : Lemma 7.7 (Bounding Lemma) Let 2Reg and 2H. If < and B 21(), then H ?; B =) H ?; B ( ;) :
Proof by induction on . Since < , B cannot be the principal formula of a re ection inference (RefH ). If B is not the principal formula of the last inference, the assertion follows by using the inductive assumption on its premisses and reapplying the same inference. Let B be the principal formula of the last inference, which then must be (9). B has the form (9x2L)F (x) with 0(){formula F (L0). Also,
H 0 ?; B; s2 L ^ F (s) for some 0 < and s2T () with jsj < . By the induction hypothesis, H 0 ?; B ( ;); s2 L ^ F (s) : Since jsj < ; , we have s2L s2L. Thus, applying (9), the assertion follows. 33
ut
8 Embeddings
The aim of this Section is to embedd P + 8 9 L 1 L+ into RS (OT ).
8.1 Basic embeddings
Regarding proofs, we will be drawing on [2] when the proof is almost literally the same. De nition 8.1 For ? = fA1; : : : ; Ang let
no(?) := !rk(A1 )# #!rk(An ): We de ne
no(?) ?; 0 (?) ? H[?] nono(?)
? :() for all operators H, H[?]
? :() for all operators H,
and
? :() for all operators H, H[?]
no(?)# ?:
Lemma 8.2 Let s t stand for the formula (8x2s)(x2t). (i) A; :A: (ii) s=2s: (iii) s s.
(iv) s2= t; s2t for s2T (jtj). (v) s 6= t; t = s:
Proof : [2], Lemma 2.4, Lemma 2.5.
ut
Lemma 8.3
[s1 6= t1]; : : : ; [sn 6= tn]; :A(s1; : : : ; sn); A(t1; : : : ; tn):
Proof : [2], Lemma 2.7.
ut
Corollary 8.4 (Equality and Extensionality)
s1 = 6 t1; : : : ; sn =6 tn; :A(s1; : : : ; sn); A(t1; : : : ; tn): Proof : [2], Theorem 2.9.
ut
Lemma 8.5 (Foundation)
(8x2L )[(8y2x)F (y) ! F (x)] ?! (8x2L )F (x): 34
Proof : Fix an operator H. Let A (8x2L )[(8y2x)F (y) ! F (x)]. First, we show, by induction on jsj, that if s2T ( ), then H[A; s]
(+) So assume that
H[A; t]
for all t2T (jsj). Using (_), this yields
H[A; s; t] for all t2T (jsj), and hence (1)
!rk(A) #! s +1 0 j j
!rk(A) #! t +1 0 j j
:A; F (s) :
:A; F (t)
!rk(A) #! t +1 +1 :A; t2 s ! F (t) 0 j j
!rk(A) #! s +2 :A; (8x2s)F (s) 0 Lemma 8.2(i), H[A; s] 0 :F (s); F (s);
H[A; s]
j j
via (8). Set := !rk(A)#!jsj + 2. By (1) and (^), H[A; s] 0+1 :A; (8y2s)F (y) ^ :F (s); F (s) : From the latter we obtain
therefore, using
+2 : A; s 2 L ^ [(8y 2s)F (y ) ^ :F (s)]; F (s) ; 0 and hence H[A; s] 0+3 :A; (9x2 L )[(8y2x)F (y) ^ :F (x)]; F (s) via (9). This shows (+). rk(A) s +1 Finally, (+) enables us to deduce H[A; s] !0 #! +1 :A; s2 L ! F (s) from which
H[A; s]
j j
the assertion follows by applying (8) and (_).
ut
Lemma 8.6 (In nity Axiom) If be a limit ordinal > !, then
(In nity Axiom)L; i.e.,
(9x2L)[z 6= ; ^ (8y2x)(9z2x)(y2z)]:
Proof : [2], Theorem 2.9.
ut
Lemma 8.7 (0{Separation) Let A[a; b1; : : : ; bn] be a 0{formula of LM . If 2Lim and s; t1; : : : ; tn2T (), then
(9y2L)[(8x2y)(x2s ^ A[s; t1; : : : ; tn]) ^ (8x2s)(A[x; t1; : : : ; tn] ! x2y)]: More concisely, we can express this by \ (0{separation)L ".
Proof : [2], Theorem 2.9.
ut
Lemma 8.8 (Pair and Union) Assume 2Lim and s; t2T (). (i) (9z 2L)(s2z ^ t2z): 35
(ii) (9z 2L)(8y2s)(8x2y)(x2z):
Proof : [2], Theorem 2.9.
Conventions. We shall write 9x and 8x instead of (9x2L ) and (8x2L ), respectively.
ut
De nition 8.9 The sequent calculus GML (\GML" stands for \Grundmengenlehre") is de ned as follows. The language of GML is LM . With the exception of 0{collection, GML has the same axiom schemes as KP. (However, it is understood that the axiom schemes are de ned with regard to LM . To be precise, GML comprises the axiom scheme of 0(LM ){ separation, whereas 0(LM ){collection is not an axiom scheme of GML.) In addition, GML has 9 axioms expressing closure of the set-theoretic universe under the rudimentary functions F0; : : : ; F8 of [3], VI. Lemma 1.11. The sequent calculus LK (\LK " stands for \Logischer Kalkul") has the same language as GML but no set-theoretic axioms. Lemma 8.10 Assume ! < is a limit ordinal. Let ?[~a ] = fA1[~a ]; : : : ; Ak[~a ]g be a set of LM {formulae, where ~a = a1 ; : : : ; an. If GML ` ?[~a ], then there exists m < ! such that, for all ~s = s1 ; : : : ; sn2T (),
H[?[~s ] ; ] L
!! +m L !(+1) ?[~s ] :
Here ?[~s ]L stands for fA1[~s ]L; : : : ; Ak [~s ]Lg.
Proof by induction on GML derivations. As to the axioms of GML, the claim follows easily from previous results of this Section. The inferences of GML are dealt with in the same manner as in [2], Theorem 3.12. ut
Lemma 8.11 Assume 0 < < . Let ?[~a ] = fA1[~a ]; : : : ; Ak [~a ]g be a set of LM {formulae, where ~a = a1; : : : ; an. If LK ` ?[~a ], then for all ~s = s1 ; : : : ; sn2T (),
?[~s ]L where ?[~s ]L stands for fA1[~s ]L; : : : ; Ak [~s ]Lg. Proof by induction on LK derivations.
ut
8.2 A variant
Instead of KPi + 8 9 L 1 L+ we are going to embedd a variant of it into the in nitary system. Lemma 8.12 There exists a sentence ModGML such that GML ` ModGML and for any transitive set M with M 6= ;, M is a model of GML i M j= ModGML . Proof : This follows from [3], VI, Lemma 1.11 and Lemma 1.6 or [18], Theorem 2.4. ut
Lemma 8.13 There is a 1-formula Sat1(a) such that for all transitive sets M N which
are models of GML the following holds:
i
h
M 1 N i 8a2M Sat1(a)N ! Sat1(a)M : 36
Proof : By [3], VI, Lemma 1.15 or [18], Lemma 2.4.
ut
Corollary 8.14 Let (M; N ) be the conjunction of the following formulas 1. M and N are transitive 2. 9u2M u2M ^ M 2N
3. M j= ModGML ^ N j= ModGML h
4. 8a2M Sat1(a)N ! Sat1(a)M
i
The theory KPi + 8 9 L 1 L+ is contained in the theory K which consists of KP plus the axiom (Stab), i.e.
(24)
h
8x 8 9M 9N 9 9f (M; N ) ^ x 2 M ^ i 2= M ^ ; f 2N ^ fun(f ) ^ dom(f ) = ^ 8 < f () = L+ :
Proof : To show that every set x is contained in an admissible set, pick M; N such that x2M and (M; N ). Then M and N are models of GML by Lemma 8.12. M 1 N holds by Lemma 8.13. Therefore, since M 2N , M is an admissible set. To verify 8 9 L 1 L+ in K , x an arbitray ordinal > 0. Pick M; N; ; f such that 2M , (M; N ), 2=M , ; f 2N and fun(f ) ^ dom(f ) = ^ 8 < f () = L+ : Let be the least ordinal which is not in M . Then and hence there exists a function g2N such that dom(g) = + and 8 < + g() = L . Notice that (L)M = L and L+ N . From M 1 N it therefore follows that L 1 L+ . ut Theorem 8.15 Let ?[~a ] = fA1[~a ]; : : : ; Ak[~a ]g be a set of L{formulae with ~a = a1; : : : ; an. When KP + 8 9 L 1 L+ ` ?[~a ], then there exists m < ! such that, for all ~s = s1; : : : ; sn2T , !m H[?[~s ] ] + m ?[~s ] : L
L
Proof : Compared to Lemma 8.10, there is only one new axiom to take care of. By (8.14) it suces to deal with (Stab). Let s; t be RS (OT )-terms. Put s;t := !max(jsj+1;jtj+1) and
s;t := (s;t ) + s;t. Let (s; t) be the formula h
i
9M 9N 9 9f (M; N ) ^ s 2 M ^ 2= M ^ ; f 2N ^fun(f ) ^dom(f ) = t ^8x 2 t f () = L+ : Then one can show (25)
H[s; t]
s;t !m s;t !m Ord(t) ! (s; t)
for some m. To be more precise, the witnesses for M and N are going to be L(s;t) and L(s;t )+s;t , respectively. In order to prove M 1 N one draws on Lemma 8.13. Let Sat1(a) be the formula 9uA(u; a), where A is 0. Let r; p be RS (OT )-terms satisfying jrj < (s;t )+ 37
s;t and jpj < (s;t). Starting from :A(r; p); A(r; p), one applies an inference (RefH ) with H := ((s;t ); jrj + 1-P0 ; s;t -P0 ; ;; 0) to get s;t !k s;t !k
H[s; t; r; p]
:A(r; p); (9z 2 L(s;t))(9x 2 z)[M r (z; x) ^ A(x; p)] H
for some k. From the latter one gets
H[s; t; r; p]
s;t !k s;t !k
0
0
:A(r; p); (9x 2 L(s;t))A(x; p)
for some k0. Using an inference (8) one gets
H[s; t; p]
s;t !k0 s;t !k0
(8x 2 L(s;t)+s;t ):A(x; p); (9x 2 L(s;t))A(x; p)
for some k0, i.e., s;t !k0 s;t !k0
H[s; t; p]
:Sat1(p)
L(s;t)+s;t
; Sat1(p)L(s;t) :
The latter yields (26)
H[s; t]
s;t !k1 s;t !k1
i
h
(8x 2 L(s;t)) Sat1(x)L(s;t)+s;t ! Sat1(x)L(s;t) :
for some k1. (26) is the main step to proving (25). Applying two inferences (8) to (25) yields
H
! (
8x 2 L)(8y 2 L)(Ord(y) ! (x; y)) : ut
9 Strengthening re ection
De nition 9.1 Let H be a re ection instance with interval [; + ]. A formula A is said to be in 1(F (H )) if it belongs to the smallest collection of formulae which contains F (H ) and is closed under quanti ers of the form (9x2L+ ). A formula A is said to be in _1 ;^(F (H )) if it belongs to the smallest collection of formulae which contains F (H ) and is closed under quanti ers of the form (9x2L+ ) and the connectives _ and ^. Lemma 9.2 Let H be a re ection instance with interval [; + ] and F (~t; L+ ) be a 1(F (H ))-formula. Assume H ; F (~t; L+ ) and k(~t); k(H ) H. Then H
#2 ~t max(;)+1 ; (9z 2 L )(9~y2z )[MH (z; ~y)
38
^ F (~y; z)] :
Proof : We proceed by induction on .
If F (~t; L+ ) is not the main formula of the last inference, then the assertion follows from the induction hypothesis by subsequently applying the same inference. Now suppose that F (~t; L+ ) is the main formula of the last inference. If F (~t; L+ ) is in F (H ), then the assertion follows by an inference (RefH ). Otherwise F (~t; L+ ) is of the form (9x2L+ )G(~t; s; L+ ) with jsj < + and the last inference was (9), hence H 0 0; G(~t; s; L+ ) for some 0 < and 0 ; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]. The induction hypothesis then yields
(27) H 0#2 0; (9z 2 L)(9~y2z)(9x2z)[MH~t (z; ~y) ^ MHs (z; x) ^ G(~y; x; z)] : From Lemma 8.11 we obtain (28)
:(9z 2 L )(9~y2z)(9x2z)[MH~t (z; ~y) ^ MHs (z; x) ^ G(~y; x; z)]; (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]: (Cut) applied to (27) and (28) then provides the desired result.
ut
Lemma 9.3 Let H be a re ection instance with interval [; + ] and G(~t; L+ ) be a _ ; ^ 1 (F (H ))-formula. Assume H ; G(~t; L+ ) and k(~t); k(H ) H. Then #!! (+1) ~t H max( ;+!(+1)) ; (9z 2 L )(9~y2z )[M (z; ~y) ^ G(~y ; z )] :
H
Proof : By pulling quanti ers (9x2L+ ) in front of the formula G(~t; L+ ) we can render it a 1(F (H ))-formula, say F (~t; L+ ). By Lemma 8.11 we then get
:G(~t; L+ ); F (~t; L+ ): Hence from H ; G(~t; L+ ) we obtain H ; F (~t; L+ ) 0
via (Cut), where 0 := max( ; no(F (~t; L+ ))#no(G(~t; L+ ))) + 1 and := max(; + ! ( + 1)). Using Lemma 9.2, we get (29) H # # ; (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)] : Again using Lemma 8.11, we have (30) (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ G(~y; z)]: Thus (Cut) applied to (29) and (30) yields 0
H
0
0
#!! (+1)
; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ G(~y; z)] : 39
ut
10 The Operators H In order to be able to remove critical cuts, i.e. cuts which were introduced by re ection inferences, we have to forgo arbitrary operators. We shall need operators H such that an H{controlled derivation that satis es certain extra conditions can be \collapsed" into a derivation with much smaller ordinal labels.
De nition 10.1 The operator H is de ned by \ H (X ) = fC ( ; ) : X C ( ; ) ^ < g Lemma 10.2 Let H be a re ection instance with projection instance X. (i) H is an operator. (ii) < 0 =) H (X ) H (X ). (iii) H is closed under ( ; 7! ' ) ;
Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England
Abstract
This paper is the rst in a series of three which culminates in an ordinal analysis of 12 -comprehension. On the set-theoretic side 12-comprehension corresponds to Kripke-Platek set theory, KP, plus 1 -separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is -stable, i.e. L is a 1-elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 12 -comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on -inaccessible cardinals which is then employed for determining an upper bound for the proof{theoretic strength of the theory KPi + 8 9 is + -stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.
1 Introduction An ordinal analysis of 12-comprehension has been a main goal in proof theory.1 This paper is the rst in a series of three which culminates in an ordinal analysis of 12-comprehension. On the set-theoretic side 12-comprehension corresponds to Kripke-Platek set theory,2 KP, plus 1-separation, i.e. the schema of axioms 9z(z = fx 2 a : (x)g) for all formulas in which z does not occur free. The precise relationship is as follows (see [17], Theorem 5.4): Theorem 1.1 KP +1 separation and (12 ?CA)+ BI prove the same sentences of second order arithmetic, where BI stands for the scheme of bar induction. The ordinals such that L j= KP + 1-Separation are familiar from ordinal recursion theory. The results in this paper were obtained in 1995 when I was a Heisenberg Fellow of the German Science Foundation, Deutsche Forschungsgemeinschaft. 1 For more details see [15], [6], [9]. 2 It is crucial here that the In nity axiom is included.
1
De nition 1.2 An admissible ordinal is said to be nonprojectible if there is no total { recursive function mapping one{one into some < , where a function g : L ! L is called {recursive if it is de nable in L. The key to the `largeness' properties of nonprojectible ordinals is that for any nonprojectible ordinal , L is a limit of 1{elementary substructures, i.e. for every < there exists a < < such that L is a 1{elementary substructure of L, written L 1 L. De nition 1.3 Let < . is {stable if L 1 L. Ordinals satisfying L 1 L for some > have strong re ecting properties, e.g., Lemma 1.4 (cf. [18], 1.18) L 1 L+1 i is n{re ecting3 for all n. The last result makes it clear that an ordinal analysis of 12 comprehension will necessarily involve a proof{theoretic treatment of re ections beyond those surfacing in admissible proof theory. The objective of this paper is to give an ordinal analysis of not too complicated stability relations. My experiences in explaining the ordinal analysis of 12-comprehension have taught me that it is best approached in three steps. The rst step is given in this paper. It introduces an ordinal representation system based on -inaccessible cardinals which is then employed for determining an upper bound for the proof{theoretic strength of the theory KPi + 8 9 is + -stable. The second paper will build on this one and feature an ordinal analysis of parameter free 12-comprehension, whereas the third paper will give an ordinal analysis of full 12comprehension. It might be instructive to explain the increasing complexity of the ordinal analyses in the three papers in terms of their ordinal representation systems. In this paper the collapsing functions still collapse single points. In the second paper the collapsing functions collapse points from an interval but the interval doesn't contain ordinals with complicated re ection properties. In the last paper the collapsing functions collapse points from an interval which itself may contain ordinals < where is -stable. The results in this paper date from 1995. the most important forerunner of the techniques employed here was [13].
> 0 is said to be n {re ecting if L j= n {re ection. By n {re ection (n{re ection) we mean the scheme ! 9z [T ran(z ) ^ z 6= ; ^ z ]; where is n (n), and T ran(z ) expresses that z is a transitive set. 3
2
2
-indescribable
cardinals
It makes no sense to present an ordinal representation system without giving some kind of semantic interpretation. For ordinal representation systems in impredicative proof theory it is essential to understand the collapsing functions which they encapsulate. In this section we will indicate a model for the projection functions, employing rather sweeping large cardinal axioms, in that we shall presume the existence of certain cardinals, featuring a strong form of indescribability. Large cardinals have been used quite frequently in the de nition procedure of strong ordinal representation systems, and large cardinal notions have been an important source of inspiration. In the end, they can be dispensed with, but they add an intriguing twist to the relation between set theory and proof theory. The advantage of working in a strong set{theoretic context is that we can build models without getting buried under complexity considerations. Another objective of this section is to \ nd" a large cardinal analogue of the notion of being + -stable for all < . De nition 2.1 Let [ V= V be the cumulative hierarchy of sets, i.e.
2ON
V0 = ;; V +1 = fX : X V g; V =
[
! . Corollary 2.4 If is A--indescribable and < and < , then is A--indescribable. Proof : Suppose P1; : : : ; Pn V and hV+ ; Ai j= [P1; : : : ; Pn]. Let (u1; : : : ; un; v; w) := 9x2w \hVv+x ; Pi j= (u1; : : : ; un)": Then hV+ ; Ai j= [P1; : : : ; Pn ; ; fg] by Lemma 2.3. Thus there exist 0 < such that hV0 + ; Ai j= [P1 \ V0 ; : : : ; Pn \ V0 ; \ V0 ; fg \ V0 ]: ut This implies hV0 + ; Ai j= [P1 \ V0 ; : : : ; P0 \ V0 ] by Lemma 2.3.
Lemma 2.5 If and is A- -indescribable, then for all < there exists < such that is A- -indescribable. Proof : Let < . As is A- -indescribable by Corollary 2.4, we get hV+ ; Ai j= 9 \ is A- -indescribable"; and thus, by A- -indescribableness, there exists 0 such that < 0 < and hV0 + ; Ai j= 9 0 \ is A- -indescribable"; which implies that there exists < such that is A- -indescribable. ut To situate the notion of -inaccessibility with regard to consistency strength in the usual hierarchy of large cardinals, we recall the notion of a subtle cardinal. 4
De nition 2.6 A cardinal is said to be subtle if for any sequence hS : < i such that S and C closed and unbounded in , there are < both in C satisfying S \ = S : Since subtle cardinals are not covered in many of the standard texts dealing with large cardinals, we mention the following facts (see [8], x20): Remark 2.7 Let (!) denote the rst !-Erdos cardinal. (i) f < (!) : is subtleg is stationary in (!). (ii) \Subtlety" relativises to L, i.e. if is subtle, then L j= \ is subtle". Lemma 2.8 Assume that is a subtle cardinal and that A V . Then for every B closed and unbounded in there exists 2 B such that is A--indescribable. Proof : Assume that is subtle. Since is inaccessible, we may select a bijective mapping F : V ?! such that (1) CF = f < : F V maps V bijectively into g is closed and unbounded in . Now let B be closed and unbounded in . By the preceding we may assume B CF . In addition, we may assume that B consists only of cardinals. For a contradiction assume that there is no cardinal 2 B such that is A--indescribable. Let 2 B . Since is not A--indescribable, we can nd an Lset(P)-formula and a subset P V so that (note that + = ) (2) hV ; Ai j= [P; ] and (3) 8 < hV ; Ai j= : (P \ V ; ): For 2 B de ne (4) P = F 00P \ ( n !) [ f3n : n 2 F 00P \ !g [ f3n + 1 : hV ; Ai j= n(P ; )g [ f3n + 2 : hV ; Ai j= : n (P ; )g where h n : n 2 !i is an enumeration of the Lset (P)-formulas with two free variables. By subtlety of , we nd 0 < 1 both in B , such that P0 = P1 \ 0: (5) (5) yields (6) P0 = P1 \ V0 : hV ; Ai j= 1 (P1 ; 1) holds by (2). Therefore (5) viewed together with (4) implies (7) hV ; Ai j= 1 (P0 ; 0): Hence, using (6), hV ; Ai j= 1 (P1 \ V0 ; 0), contradicting (3). ut 5
Corollary 2.9 Let be a subtle cardinal. Then there exists a regular cardinal < such that
(8) (9)
8 < 9 < \ is -indescribable" 8 < 8 < [\ is -indescribable" ! < ]:
Proof : By Lemma 2.8 there exists < such that is -indescribable. Then is regular and satis es (10)
8 < 9 < \ is -indescribable"
by Lemma 2.5. We may assume that has been chosen minimal with property (10). If there existed a cardinal < and an ordinal < such that is -indescribable, then, by Corollary 2.4 and Lemma 2.5 we would get
8 < 9 < \ is -indescribable"; contradicting the choice of .
ut
Lemma 2.10 is 0-10-indescribable i is strongly inaccessible. Proof : See [4], Ch. 9, Theorem 1.3.
ut
6
3 Skolem Hulls and Collapsing Functions In this section we use the large cardinal notions of the previous section to develop collapsing functions. De nition 3.1 Let + denote the least cardinal > . Note that + is always a regular cardinal (on the basis of ZFC). The Veblen-function gures prominently in predicative proof theory (cf. [F 68], [21]). We are going to incorporate this function in our ordinal representation system. De nition 3.2 The Veblen{function ' := ' ( ) is de ned by trans nite recursion on by letting ' be the function that enumerates the class of ordinals
f! : (8 < )[' (! ) = ! ]g: Corollary 3.3 (i) '0 = ! . (ii) ; < ' =) + < ' . (iii) < =) ' < ' . (iv) < =) ' (' ) = ' . In the following we shall assume that is the least regular cardinal satisfying (11)
8 < 9 < \ is -indescribable":
then satis es (12)
8 < 8 < [\ is -indescribable" ! < ]:
According to Corollary 2.9, the existence of follows from the existence of a subtle cardinal. For < , let ( ) be the least cardinal such that is -indescribable; notice that ( ) < as well.
De nition 3.4 By recursion on we shall de ne sets of ordinals C ( ; ) (the th Skolem Hull generated from ordinals < ), a set < of re ection con gurations, projection instances, and re ection instances, and the ordinals for projection instances X 2 < . A projection instance X 2 < determines the collapse . Formally X is an expression built up from ordinals, symbols M; Pn and parentheses. By X 2 C ( ; ) we mean that X 2 !; is a regular ordinalg. Lemma 7.4 (Reduction Lemma) Let A = W(A)2J . Assume =2Reg, where := rk(A). Then: H ; :A ^ H ?; A =) H + ; ? Proof : Use induction on . For details see [2] , Lemma 3.14. ut Theorem 7.5 (Predicative cut elimination) Let H be closed under '. If H +! ? and [; + ! [ \ Reg = ; and 2H, then H ' ?: Proof : By main induction on and subsidiary induction on (cf. [2] , Theorem 3.16). ut
Corollary 7.6 H +1 ? ^ 2= Reg =) H ! ? : Lemma 7.7 (Bounding Lemma) Let 2Reg and 2H. If < and B 21(), then H ?; B =) H ?; B ( ;) :
Proof by induction on . Since < , B cannot be the principal formula of a re ection inference (RefH ). If B is not the principal formula of the last inference, the assertion follows by using the inductive assumption on its premisses and reapplying the same inference. Let B be the principal formula of the last inference, which then must be (9). B has the form (9x2L)F (x) with 0(){formula F (L0). Also,
H 0 ?; B; s2 L ^ F (s) for some 0 < and s2T () with jsj < . By the induction hypothesis, H 0 ?; B ( ;); s2 L ^ F (s) : Since jsj < ; , we have s2L s2L. Thus, applying (9), the assertion follows. 33
ut
8 Embeddings
The aim of this Section is to embedd P + 8 9 L 1 L+ into RS (OT ).
8.1 Basic embeddings
Regarding proofs, we will be drawing on [2] when the proof is almost literally the same. De nition 8.1 For ? = fA1; : : : ; Ang let
no(?) := !rk(A1 )# #!rk(An ): We de ne
no(?) ?; 0 (?) ? H[?] nono(?)
? :() for all operators H, H[?]
? :() for all operators H,
and
? :() for all operators H, H[?]
no(?)# ?:
Lemma 8.2 Let s t stand for the formula (8x2s)(x2t). (i) A; :A: (ii) s=2s: (iii) s s.
(iv) s2= t; s2t for s2T (jtj). (v) s 6= t; t = s:
Proof : [2], Lemma 2.4, Lemma 2.5.
ut
Lemma 8.3
[s1 6= t1]; : : : ; [sn 6= tn]; :A(s1; : : : ; sn); A(t1; : : : ; tn):
Proof : [2], Lemma 2.7.
ut
Corollary 8.4 (Equality and Extensionality)
s1 = 6 t1; : : : ; sn =6 tn; :A(s1; : : : ; sn); A(t1; : : : ; tn): Proof : [2], Theorem 2.9.
ut
Lemma 8.5 (Foundation)
(8x2L )[(8y2x)F (y) ! F (x)] ?! (8x2L )F (x): 34
Proof : Fix an operator H. Let A (8x2L )[(8y2x)F (y) ! F (x)]. First, we show, by induction on jsj, that if s2T ( ), then H[A; s]
(+) So assume that
H[A; t]
for all t2T (jsj). Using (_), this yields
H[A; s; t] for all t2T (jsj), and hence (1)
!rk(A) #! s +1 0 j j
!rk(A) #! t +1 0 j j
:A; F (s) :
:A; F (t)
!rk(A) #! t +1 +1 :A; t2 s ! F (t) 0 j j
!rk(A) #! s +2 :A; (8x2s)F (s) 0 Lemma 8.2(i), H[A; s] 0 :F (s); F (s);
H[A; s]
j j
via (8). Set := !rk(A)#!jsj + 2. By (1) and (^), H[A; s] 0+1 :A; (8y2s)F (y) ^ :F (s); F (s) : From the latter we obtain
therefore, using
+2 : A; s 2 L ^ [(8y 2s)F (y ) ^ :F (s)]; F (s) ; 0 and hence H[A; s] 0+3 :A; (9x2 L )[(8y2x)F (y) ^ :F (x)]; F (s) via (9). This shows (+). rk(A) s +1 Finally, (+) enables us to deduce H[A; s] !0 #! +1 :A; s2 L ! F (s) from which
H[A; s]
j j
the assertion follows by applying (8) and (_).
ut
Lemma 8.6 (In nity Axiom) If be a limit ordinal > !, then
(In nity Axiom)L; i.e.,
(9x2L)[z 6= ; ^ (8y2x)(9z2x)(y2z)]:
Proof : [2], Theorem 2.9.
ut
Lemma 8.7 (0{Separation) Let A[a; b1; : : : ; bn] be a 0{formula of LM . If 2Lim and s; t1; : : : ; tn2T (), then
(9y2L)[(8x2y)(x2s ^ A[s; t1; : : : ; tn]) ^ (8x2s)(A[x; t1; : : : ; tn] ! x2y)]: More concisely, we can express this by \ (0{separation)L ".
Proof : [2], Theorem 2.9.
ut
Lemma 8.8 (Pair and Union) Assume 2Lim and s; t2T (). (i) (9z 2L)(s2z ^ t2z): 35
(ii) (9z 2L)(8y2s)(8x2y)(x2z):
Proof : [2], Theorem 2.9.
Conventions. We shall write 9x and 8x instead of (9x2L ) and (8x2L ), respectively.
ut
De nition 8.9 The sequent calculus GML (\GML" stands for \Grundmengenlehre") is de ned as follows. The language of GML is LM . With the exception of 0{collection, GML has the same axiom schemes as KP. (However, it is understood that the axiom schemes are de ned with regard to LM . To be precise, GML comprises the axiom scheme of 0(LM ){ separation, whereas 0(LM ){collection is not an axiom scheme of GML.) In addition, GML has 9 axioms expressing closure of the set-theoretic universe under the rudimentary functions F0; : : : ; F8 of [3], VI. Lemma 1.11. The sequent calculus LK (\LK " stands for \Logischer Kalkul") has the same language as GML but no set-theoretic axioms. Lemma 8.10 Assume ! < is a limit ordinal. Let ?[~a ] = fA1[~a ]; : : : ; Ak[~a ]g be a set of LM {formulae, where ~a = a1 ; : : : ; an. If GML ` ?[~a ], then there exists m < ! such that, for all ~s = s1 ; : : : ; sn2T (),
H[?[~s ] ; ] L
!! +m L !(+1) ?[~s ] :
Here ?[~s ]L stands for fA1[~s ]L; : : : ; Ak [~s ]Lg.
Proof by induction on GML derivations. As to the axioms of GML, the claim follows easily from previous results of this Section. The inferences of GML are dealt with in the same manner as in [2], Theorem 3.12. ut
Lemma 8.11 Assume 0 < < . Let ?[~a ] = fA1[~a ]; : : : ; Ak [~a ]g be a set of LM {formulae, where ~a = a1; : : : ; an. If LK ` ?[~a ], then for all ~s = s1 ; : : : ; sn2T (),
?[~s ]L where ?[~s ]L stands for fA1[~s ]L; : : : ; Ak [~s ]Lg. Proof by induction on LK derivations.
ut
8.2 A variant
Instead of KPi + 8 9 L 1 L+ we are going to embedd a variant of it into the in nitary system. Lemma 8.12 There exists a sentence ModGML such that GML ` ModGML and for any transitive set M with M 6= ;, M is a model of GML i M j= ModGML . Proof : This follows from [3], VI, Lemma 1.11 and Lemma 1.6 or [18], Theorem 2.4. ut
Lemma 8.13 There is a 1-formula Sat1(a) such that for all transitive sets M N which
are models of GML the following holds:
i
h
M 1 N i 8a2M Sat1(a)N ! Sat1(a)M : 36
Proof : By [3], VI, Lemma 1.15 or [18], Lemma 2.4.
ut
Corollary 8.14 Let (M; N ) be the conjunction of the following formulas 1. M and N are transitive 2. 9u2M u2M ^ M 2N
3. M j= ModGML ^ N j= ModGML h
4. 8a2M Sat1(a)N ! Sat1(a)M
i
The theory KPi + 8 9 L 1 L+ is contained in the theory K which consists of KP plus the axiom (Stab), i.e.
(24)
h
8x 8 9M 9N 9 9f (M; N ) ^ x 2 M ^ i 2= M ^ ; f 2N ^ fun(f ) ^ dom(f ) = ^ 8 < f () = L+ :
Proof : To show that every set x is contained in an admissible set, pick M; N such that x2M and (M; N ). Then M and N are models of GML by Lemma 8.12. M 1 N holds by Lemma 8.13. Therefore, since M 2N , M is an admissible set. To verify 8 9 L 1 L+ in K , x an arbitray ordinal > 0. Pick M; N; ; f such that 2M , (M; N ), 2=M , ; f 2N and fun(f ) ^ dom(f ) = ^ 8 < f () = L+ : Let be the least ordinal which is not in M . Then and hence there exists a function g2N such that dom(g) = + and 8 < + g() = L . Notice that (L)M = L and L+ N . From M 1 N it therefore follows that L 1 L+ . ut Theorem 8.15 Let ?[~a ] = fA1[~a ]; : : : ; Ak[~a ]g be a set of L{formulae with ~a = a1; : : : ; an. When KP + 8 9 L 1 L+ ` ?[~a ], then there exists m < ! such that, for all ~s = s1; : : : ; sn2T , !m H[?[~s ] ] + m ?[~s ] : L
L
Proof : Compared to Lemma 8.10, there is only one new axiom to take care of. By (8.14) it suces to deal with (Stab). Let s; t be RS (OT )-terms. Put s;t := !max(jsj+1;jtj+1) and
s;t := (s;t ) + s;t. Let (s; t) be the formula h
i
9M 9N 9 9f (M; N ) ^ s 2 M ^ 2= M ^ ; f 2N ^fun(f ) ^dom(f ) = t ^8x 2 t f () = L+ : Then one can show (25)
H[s; t]
s;t !m s;t !m Ord(t) ! (s; t)
for some m. To be more precise, the witnesses for M and N are going to be L(s;t) and L(s;t )+s;t , respectively. In order to prove M 1 N one draws on Lemma 8.13. Let Sat1(a) be the formula 9uA(u; a), where A is 0. Let r; p be RS (OT )-terms satisfying jrj < (s;t )+ 37
s;t and jpj < (s;t). Starting from :A(r; p); A(r; p), one applies an inference (RefH ) with H := ((s;t ); jrj + 1-P0 ; s;t -P0 ; ;; 0) to get s;t !k s;t !k
H[s; t; r; p]
:A(r; p); (9z 2 L(s;t))(9x 2 z)[M r (z; x) ^ A(x; p)] H
for some k. From the latter one gets
H[s; t; r; p]
s;t !k s;t !k
0
0
:A(r; p); (9x 2 L(s;t))A(x; p)
for some k0. Using an inference (8) one gets
H[s; t; p]
s;t !k0 s;t !k0
(8x 2 L(s;t)+s;t ):A(x; p); (9x 2 L(s;t))A(x; p)
for some k0, i.e., s;t !k0 s;t !k0
H[s; t; p]
:Sat1(p)
L(s;t)+s;t
; Sat1(p)L(s;t) :
The latter yields (26)
H[s; t]
s;t !k1 s;t !k1
i
h
(8x 2 L(s;t)) Sat1(x)L(s;t)+s;t ! Sat1(x)L(s;t) :
for some k1. (26) is the main step to proving (25). Applying two inferences (8) to (25) yields
H
! (
8x 2 L)(8y 2 L)(Ord(y) ! (x; y)) : ut
9 Strengthening re ection
De nition 9.1 Let H be a re ection instance with interval [; + ]. A formula A is said to be in 1(F (H )) if it belongs to the smallest collection of formulae which contains F (H ) and is closed under quanti ers of the form (9x2L+ ). A formula A is said to be in _1 ;^(F (H )) if it belongs to the smallest collection of formulae which contains F (H ) and is closed under quanti ers of the form (9x2L+ ) and the connectives _ and ^. Lemma 9.2 Let H be a re ection instance with interval [; + ] and F (~t; L+ ) be a 1(F (H ))-formula. Assume H ; F (~t; L+ ) and k(~t); k(H ) H. Then H
#2 ~t max(;)+1 ; (9z 2 L )(9~y2z )[MH (z; ~y)
38
^ F (~y; z)] :
Proof : We proceed by induction on .
If F (~t; L+ ) is not the main formula of the last inference, then the assertion follows from the induction hypothesis by subsequently applying the same inference. Now suppose that F (~t; L+ ) is the main formula of the last inference. If F (~t; L+ ) is in F (H ), then the assertion follows by an inference (RefH ). Otherwise F (~t; L+ ) is of the form (9x2L+ )G(~t; s; L+ ) with jsj < + and the last inference was (9), hence H 0 0; G(~t; s; L+ ) for some 0 < and 0 ; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]. The induction hypothesis then yields
(27) H 0#2 0; (9z 2 L)(9~y2z)(9x2z)[MH~t (z; ~y) ^ MHs (z; x) ^ G(~y; x; z)] : From Lemma 8.11 we obtain (28)
:(9z 2 L )(9~y2z)(9x2z)[MH~t (z; ~y) ^ MHs (z; x) ^ G(~y; x; z)]; (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]: (Cut) applied to (27) and (28) then provides the desired result.
ut
Lemma 9.3 Let H be a re ection instance with interval [; + ] and G(~t; L+ ) be a _ ; ^ 1 (F (H ))-formula. Assume H ; G(~t; L+ ) and k(~t); k(H ) H. Then #!! (+1) ~t H max( ;+!(+1)) ; (9z 2 L )(9~y2z )[M (z; ~y) ^ G(~y ; z )] :
H
Proof : By pulling quanti ers (9x2L+ ) in front of the formula G(~t; L+ ) we can render it a 1(F (H ))-formula, say F (~t; L+ ). By Lemma 8.11 we then get
:G(~t; L+ ); F (~t; L+ ): Hence from H ; G(~t; L+ ) we obtain H ; F (~t; L+ ) 0
via (Cut), where 0 := max( ; no(F (~t; L+ ))#no(G(~t; L+ ))) + 1 and := max(; + ! ( + 1)). Using Lemma 9.2, we get (29) H # # ; (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)] : Again using Lemma 8.11, we have (30) (9z 2 L )(9~y2z)[MH~t (z; ~y) ^ F (~y; z)]; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ G(~y; z)]: Thus (Cut) applied to (29) and (30) yields 0
H
0
0
#!! (+1)
; (9z 2 L)(9~y2z)[MH~t (z; ~y) ^ G(~y; z)] : 39
ut
10 The Operators H In order to be able to remove critical cuts, i.e. cuts which were introduced by re ection inferences, we have to forgo arbitrary operators. We shall need operators H such that an H{controlled derivation that satis es certain extra conditions can be \collapsed" into a derivation with much smaller ordinal labels.
De nition 10.1 The operator H is de ned by \ H (X ) = fC ( ; ) : X C ( ; ) ^ < g Lemma 10.2 Let H be a re ection instance with projection instance X. (i) H is an operator. (ii) < 0 =) H (X ) H (X ). (iii) H is closed under ( ; 7! ' ) ;
