Reflection and forcing in E-recursion theory

Reflection and forcing in E-recursion

elements

of Tp let the restriction

91

theory

of p to sl, . . . , s, be defined by r = (T,, h,, gr)

where T, = {s ( s E T, & (3 s n)[s G &I}, h,(s) = h,(s)

if s E T,,

g,(s) = g,(s)

if s E T,.

3.12. Lemma. Suppose pII4(H,,, . . . , HSm,T) and let r be the restriction of p to s1, . . . ) s,. Then r I!-4(HS1, . . . , H,, T). Proof. Suppose rl)L4(HS1, . . . , H,). Let rl extend r so that rll+-4(HS1, . . . , H,). Extend rr to a condition r2 which duplicates (on a different part of T,) that part of p which was omitted to form r. Let m, be greater than any number mentioned in T,,. Define r, by (i)

if t E Trl, or if there is a u in T, so that u = u,,“(kI,. . . , k,),

(a) r E Tr, (b) rE T,

(3i sn)[u,s si], (Vi sn)[u,“(k,)& si], and t = u,“(kl + m,, . . . , k, + m,,). (ii)

(iii)

if tET,,,

(a) h,(t) = h,l(t) (b) h,(t) = hp(u,,“(kl,. (a) g,(t) = g,,(t) @) g,(t) = h,(u,“(k,,

. . , k,)

if tET,I, t=u,“(k,+mO ,..., and u,,“(kl, . . . , k,)E Tp.

k,+mJ

if t E T,, . . . , k,,))

if t E T,, t = u,“(k,+ m,, . . . , k, + m,) and uO”(kl,. . . , k,)E T,.

r2 is an element of P as both rl and p are in P. There is an automorphism

rr of

the partial order P and the forcing language generated by rr : uO”(kl,. . . , k,) f,

u,“(kl+

mO,

.

. . , k, + mo)

for the appropriate sequences from TV By standard arguments, if q is any element of P and I,!J(H,~,. . . , H,,) is a formula in 5?*, then qlkJ/(K,,

. . . , H,J

iff

ddl~$(H,~,,~,

. . . , Kd.

In paflicular, 74~) I~4(ff,+,~, . . . , H,&. BY construction, if k is less than n + 1, then rr(s,J = sk, so 7r(~)I!-4(H,~, . . . , H,,). But n(p) > r, and r,ll- -4(HS1, . . . , H,,). This contradiction proves the lemma. The first difficulty with a class notion of forcing is to show that the forcing relation for ranked sentences can be defined locally.

T.A. Slaman

92

The relation pII-+ for ranked sentences is recursive in p, r~‘.

3.13. Lemma. proof.

The

recursive Retagging

only

step

in the

is the unbounded Lemma

3.11

inductive

quantification implies

that

definition

of forcing

over P involved it suffices

which

in defining

to check

only

is not

E-

p IF - 4. The

those

conditions

extending p which name ordinals less than w . U (range b -{m} U{rk(-4))). This _ bounded quantification is E-recursive; then “p IF 4” is E-recursive by an effective transfinite

recursion.

The next lemma

implies

that I-.,[ T] is E-closed.

3.14. Lemma. Suppose plt-)((e, (I&, . . . , H,))(I, {h,(s) 1(3i s n)[s c si] & h,(s) #m} and w. Proof. The proof

proceeds

by induction

= y. Then y is recursive in 0, =

on y. By Lemma

3.12, assume

that p is

its own restriction to si, . . . , s, and so p sE 0,. The interesting case is when e=3” * 9. Define a sequence (A,, ) n
in q’, b(q, y(q), 0,)

recursive

W$JC q there

Rud(TC(T)) and HTC4)c are r,(q’, b) and yq(q’, b)

so that

r,(q’, b)Il-“b$ W~~~J’ and yq(q’, b) = 0, or r&q’, b)l~llbllr

= -r&q’, b).

This follows from two facts: first, forcing a ranked sentence is E-recursive by Lemma 3.13; second, since y(q’, b) must be less than y the inductive hypothesis holds. Let

Of course,

An+1

=

the Gandy

co4 .

sup[{AJ

selection

theorem

is also being

invoked

as before.

U {1/4(q’, b), y(q) 1q, q’ E P n &, &q~p&q’6r(q)&bET}].

The bounding principle (2.4) implies that A,+i, as a function of n, is E-recursive in w, p and 0,. p is E-recursive in 0, so f : n -+ A,, is E-recursive in 0, and w. Let A = lJ,<, A,,. The claim to be verified is that p Ikll(e, (Hsl, . . . , H,))(),
Refkction and forcing in E-recursion theory

93

condition extending p and let b be any element of r so that q IF b E W~~~=~~. It is sufficient to show that there is an r extending q so that r IkllbllT
If T is P-genetic over L,, then L,[ T] = E(T).

Proof. It has been remarked that L,[T]c_ E(T). Lemma 3.14 shows that it is impossible for any condition to force a computation using parameters from T to have height K. It also follows from Lemma 3.14 that if cy
then L,[T]

is X,-admissible.

Proof (A sketch). Assume L, is s;,-admissible. It is sufficient to verify &bounding in L,[ T]. The proof is exactly the same as in the previous lemma except that the E-recursive function ll*llTis replaced by a Xi-relation. &-bounding in Lx is used to construct a sequence (A,, ( n E w). 3.17. Lemma.

0, is not A, over L,[ T].

Proof. The E-recursive functions are closed under definition by effective transfinite recursion. The partial function which assigns the height of H, in T to those H, in the well-founded part of T is E-recursive in T. It is enough to show that the nonwell-founded part of T is not Z1 over L,[ T]. Suppose T E C, p is a condition in P and 4(x, y, z) is a A, formula which has only free variables x, y, z. Suppose pll-“there are infinitely many x E FTI so that 32 4(x, 7, 2)“. It is to be shown that there is a q and an H, so that qll-

T.A. Slaman

94

3z +(H,, T, z) and h,(t) # 00. This is if the subset of T picked out by 3z 4(x, T, z) is infinite, then it has an element in the well-founded part. Since the set {x E T 132 4(x, T, z)} is forced to be infinite there must be a t not appearing in Tp or r and a q extending p so that q 113~ c#J(H,,+r,z). Let r* be a term and r extend Otherwise, and

q so that rl!-+(H,, T, T*). If h,(t) # ~0 the claim is proven.

let 0 = rk(+(H,,

There is a retagging r’ of r which has h,(t) # ~0

T, T*)).

h,.(t) a y = max[{h,(s)

( t s s & h,(s) # co}U{w - p}]

so that Ret(o . /3, r, r’). r’ is defined by relabeling every nonempty node which is comparable to t and labeled UJ in r by y plus its height in T,. By the Retagging Lemma 3.11, r’Ik$(H,, r, T*). This completes the proof of the lemma. 3.18.

If p

km.

PbK,

(0,

(H

‘I’

E p

....H~n),T

={h,(S)

and (Si 1i c n) is a sequence =

SO that

(Vi s n)[ Si

E

T,], then

KOD

1 hp(S)$‘m

&

(3i

sn>[S

C

SiJj.1

Proof. Lemma 3.14 implies that ~~~~~~~~~~~~~~~~ K$. If K:~Q’..-~~+),~>KY”, then the z1 formula in 0, first satisfied at KID+ 1 in L would reflect below K&Hs.1.-7Hs_)‘T which is impossible. Thus, K~Hsi~~~‘Hsm”T~ KO. SUppOSe K p” > K ~H~l,-*H=?2T.There is a A0 formula 4(y, Hsl, . . . , I-l,, T) with only free variable y, a condition q so that q is in the generic and an ordinal 6 so that qlt-“(3y

E b+,Wlb#4y,

K,,

& (VS’ G S)(Vy E I&T])

. . . , H,,

-0

- C#J(Y, H,,, . . . , HSb,T)“.

By Lemma 3.10, the restriction r of q to sl, . . . , s, forces the same sentence. r is E-recursive in 0, and w ; by reflection below K:O the ordinal 6 + 1 must be recursive in 0,.

This is a contradiction.

As before, the above argument also establishes the relativized result for those (Y< K. In particular, KFb’UT= K:. The addition of T to L, does not change the reflection structure of L,. The preceding lemmata constitute a proof of Theorem 3.1 as follows. 3.1. Theorem. Let L, be countable and E-closed. There is a set of reals T so that (i) L,[T] = E(T). (ii) OT is not A, over E(T). (iii) If L, is admissible, then so is L,[T]. (iv) If (Va E L,)[K~< K], then (Va E Rud(TC(X)))[K:T< K]. (v) If there is a Z1 subset of K which is not E-recursively enumerable in L,, then the same is true in L,[T].

Reflection and forcing in E-recursion theory

9.5

Proof. Let T be the tree constructed by forcing with P. (i) is Corollary 3.15 (ii) is Lemma 3.17. (iii) is Lemma 3.16. Lemma 3.18 implies (iv). If there is a x1 subset of K which is not E-recursively enumerable in Lx, then (Va! < K)[K:
4. Lightface and boldface predicates Sacks’ Theorem 2.25 states that when X is a set of ordinals either for every element, a, in E(X) the convergence of {e},(a, X) is A,(a, X) over E(X) or there is an element p so that the universal _Z1predicate over E(X) is E-recursively enumerable using p. The crucial distinction was the condition: either for all a in E(X), Kx is not (a, X)-reflecting or there is an a so that for every b in E(X), K>b’X= Kx. In this section, a set of reals T is constructed so that the complete lightface _Z1 in E(T) predicate with only parameter T is E-recursively enumerable, however, the boldface x1 predicates on E(T) are not E-recursively enumerable. That is to say that K:= ~~ but for every a in E(T) there is a b in E(T) so that K:~< KT. In fact there will be a b in E(T) so that K;
KT.

4.1. Theorem. There is a set of reals T so that (i) K,T= KT. (ii)

tit

p

=

Kz.

For

any

(iii) For cdl a in E(T),

a

in

K$~-CK:~.

E(T),

K:“T<

KT.

96

T.A. Slaman

Condition (iii) is included for aesthetics,

it will simply follow from the existence

of an ordinal which is recursive in T but not countable 4.2.

Definition. If

K

is a limit ordinal, p EL,,

and (+ <

if whenever a A, formula 4(x) with only parameter

in E(T). K,

then cr is p-stable in L,

p is satisfied by some z in L,,

then it is satisfied by some z’ in L,. Fix L, so that K is countable, E-closed and inadmissible; there is an ordinal y which is uncountable in L, ; and there is a -y-stable in L, ordinal which is less than K. Let 07 be the least ordinal which is not countable in L, and let o be the least w;-stable in L,. L, is the ground model for the forcing construction used to prove 4.1. The set T is designed to make K:= CT and KT= K. For those 0 which are greater than or equal to (+ the forcing will not change the reflection structure, i.e. K F’ will equal K f and so be less than K. 4.3.

Lemma. u is the

limit

of

S = (6
Proof. The supremum of S is WY-stable in L, and is the limit of ordinals which do not bound a w;-stable ordinal.

ordinal. Hence, the limit of S is 0; the least o;-stable

The characterization of (+ is used to define the appropriate forcing using only the parameter 0’;.

in L,

variant of Steel

4.4. Definition. The partial order P: A condition p is a triple (T,, h,, g,) so that (i) T, is a finite tree on o. If s E T,, then s is a finite sequence of finite ordinals. Let In(s) be the length of s. (ii) h, : T, +

K

U{m} so that

(a) h,(( >) = 00. (b) If s, t E T, and S g t, then h,(s)> h,(t). (m>cf~ and 03) K.) (c) If s E T, and In(s) = 1, then either h,(s) = ~0 or h,(s) = 6 and there is a A,-formula 4(x) with only parameter w; so that L,+,l=3x 4(x) and L8f3x 4(x). (iii) gr, : T, -{s E T, ( In(s) = 1) + PC (the Cohen partial order). (iv) If p and q are in P, then q

1, then G(S) cpc g,(s). The class P of conditions is E-recursive in the parameter WT. 4.5. JMinition. Let G be P-generic over L,. If In(s)> 1 and there is a p in G so that s E T,, then let H, = lJ {g,(s) ) p E G & s E T,}. If In(s) s 1 let H, be a canoni-

97

Reflection and forcing in E-recursion theory

cal sequence number for s. Let FT be the set {H, I(3p E G)[s E T,]} and let --+ partially order FT defined by I-I, >T H, if s $ t. Let T = (FT, +-). It follows K;f>CT.

immediately proofs

The

facts in Section The

4.6.

the genericity

3 and so their

definition

However,

from

of the remaining

of forcing

descriptions has

the

inductive to define

Definition. p 114 for C#Jatomic

is defined

(iv)

pll-acb p II-7 = 7 p IFH, E FT p U-H+,, = “(n)“, pItI-&

(iv)

p IFH, T> H,

E(T)

and that

to the analogous

abbreviated. here

as in Section

for atomic

formulas.

3.

by and aEb.

if TEC. iff S-E TP and h,(s-)>O. p N-H, ) = 0

for all integers n, “(n)” is the sequence number for (n). iff s E T,, In(s) > 1, n is less than or equal to the length of g,(s) and the nth element of g,(s) is i (i.e. g,(s)(n)= iff pll-H,EF, and s$;.

Definition. If p and p* are conditions

4.7.

clauses forcing

if a,bEL_,

= i

(v)

J_.,[T]c

parallel

are somewhat

are needed

(i) (ii) (iii)

some changes

of G that

facts are roughly

in P, then

i).

Ret(y, p, p*) if

T, = T,*, gP = g,*.

(i) (ii)

(Vs E T,)[(h,(s)

< y v h,*(s)) = 0 j

h,(s) = h,+=(s)

& (h,(s) 2 Y 3 h,*(s) zz 711. This is the same as the original of p as in the previous section. as in Section 3 as well. 4.8.

Lemma. plk$

4.9.

Lemma.

Suppose ifi

definition

of when

The following

4 is ranked

and

Ret(o

p* is a y-absolute

two lemmata

. rk(4),

p, p*). Then

p*Ik$.

Suppose

pIk+(H,,,

. . . , H,“, T). Define p* by

(i) s E T,* if In(s) < 1 and s E T, or there is an i less than

s E si. (ii) If s E T,*, then h,*(s) = h(s). Then p* II-4(H,,, . . . , H,, T). 4.10.

Corollary.

parameter

0;.

retagging

have the same proofs

The relation

If also ln(s)a2,

p 1~4 for ranked

or equal

to n so that

then g,*(s) = g,(s).

formulas C#Iis E-recursive

in the

98

T.A. Slaman

Proof. The corollary follows from Lemma needed to define p EP E-recursively.

4.8 as before.

The parameter

4.11. Lemma. If p Iq\(e, Hsl, . . . , H,)llT = y and y 5 0, then y is E-recursive and 0, = {h,(s) 1i+,(s) # 03& s E T, & [In(s) s 1 v(3i G n)(s E Si)])* Proof. As in Section 4.12.

Corollary.

Lemma.

is

in w;

3.

E(T) = L,[ T].

Proof. Lemma 4.11 implies that no element of E(T) 4.13.

o;

can compute

K.

If p 3 V, then Kf'H%'~-.'H%'T=K~‘“P.

Proof. As in Section 3, use the lightface in p (p E-recursively computes w; and cr) definition of forcing and reflection (0, is defined in Lemma 4.11). 4.14. h-.

K; = (+.

Proof. It has already been remarked that K:~u. Suppose p I!-\l(e,T)\\= y. Then y is a solution of a A, in o; predicate and 0, = {h,(s) 1s E T, & In(s) = 1 & h,(s) #m}. By the definition of P, each element of 0, has a A, definition which uses only the parameter WT. Thus, 0, has such a definition since it is finite and y is A0 in 0;. But then y
KT=

KT=

K.

Proof. The proof of this lemma is very similar to the previous one. Suppose there is a A, formula 4(x, y), a condition p and an ordinal 6
so

By Lemma 4.9, p can be assumed to be definable by a A, formula with only parameter w;. By Corollary 4.10, the forcing relation for ranked sentences is recursive in the condition p and the GGdel number for the sentence. For

“b+,[Tlk3x 4(x, T) &J-,[Tl#3x 4(x, 7’)” the GGdel number is E-recursive pIt“L,+,[T]\3x~(x,

in 6. Thus,

T)&L,[T]~~x~(x,

T)”

is an E-recursive in w; predicate on K. The least solution to this predicate must be smaller than u since c is o;-stable in L,. 4.16. Lemma. o; is not countable in L,[T].

Reflection

Proof.

Suppose

and forcing in E-recursion

99

theory

T is a term in 6p*, the ramified language,

which names only

PI,,, . . . , H,, T} and p is a condition so that pII-“ is a function from w to WT.” By Lemma 4.9, it is safe to assume that each si is in T, and if s E TP, then either In(s) = 1 or there is an i less than or equal to n so that s G si. In fact, for each n there are only countably many extension q which truely decide the value of 7(n). These are the ones which have only nodes of length 1 and those s which are initial segments of some si. There are only countably many such conditions since each node of length n can be labelled with only one of countably many nonstability ordinals. Thus, P essentially has the countable chain condition with regard to 7. So for each rt, 7(n) is restricted to at most countably many values by an antichain which is an element of L, (recall: a< K). The definability of forcing a ranked sentence is also ranked so the range of T must be bounded in 07. 4.17.

Corollary.

For all a in E(T),

K>~-=CK:~.

Proof. w; is E-recursive in T since (+> WY. Theorem 2.21 implies that since o; is uncountable in E(T), K?=< K:~. The preceding constitutes 4.1.

a proof of Theorem

4.1.

Theorem. There is a set of reals T so that (i) K:= KT.

(ii) Let /3= ~0’. For any a in E(T), (iii) For all a in E(T), K~=
Let

T and

K

be from

K:‘*~
above.

T is easily

equivalent to a set of reals. (i) is Lemma 4.15;

seen

to be E-recursively

(ii) is the combination

4.13 and 4.14 and the condition of inadmissibility on L,‘s implying Corollary 4.17.

of Lemmata

K:<

K~;

(iii) is

The phenomenon of a-reflection beyond K; is tied to the homogeneity of E(a) beyond K: upto K: as to those phenomenon which are A, in a. In the above example L, and L, are exactly the same with regard to light-face phenomena. However, once the parameter u is introduced this total homogeneity disappears because of the Moschovakis and L, ends at K::.

5. Bounded existential

phenomenon

in LK. The homogeneity

between LKs

quantification

Suppose X is a set of ordinals, gc(X) is the greatest cardinal in E(X) and cf(gc(X)) is the cofinality of gc(X) in E(X). Theorem 2.13(i) says that if /3< cf(gc(X)), then the E-recursively enumerable predicates on E(X) are closed

100

T.A. Slaman

under (32 E p). On the other hand, if these predicates are closed under (3.~ E cf(gc(X))), then 2.13(ii) implies that they are closed under (32 EE(X)), full existential quantification. If the E-recursively enumerable predicates are augmented by any nontrivial bounded existential

quantification

the result is that all

x1 predicates on E(X) become recusively enumerable. The goal of this section is to find a set of reals T so that the structure of the x1 predicates

on E(T) is richer than in the well-ordered

case. The functions which

are generated by the schemes of E-recursion augmented by an additional one for selection from A are called the ES,-recursive functions. A formal definition follows; a detailed analysis is available in Hoole [5] where these notions were first formalized. 5.1. Definition. Let X be a set and A a subset of X. Define, by simultaneous transfinite recursion, a subset %& of Rud(TC((X, A))), a norm “A/~\\~ : S4TX + OR and associated H sets ‘AH:. (i) For each a in Rud(TC((x, A))) (1, a)EQ&;

‘~jl(l, a&

= 0;

and

‘AH:=

Rud(TC((X,

A))).

(ii) If (n, a) E ‘QQ, then (2”, a) E ‘M&;

SAll(2”, a>lls= “4lh a)llx+ 1;

‘AH,,+~ = {(e, a, b) 1b e W~~~Hz}.

(iii) If there is a z in A so that (n, (a, z))E~~&, then (7”, a)~ ‘~0~; ‘4(7”, a)llx = th e 1east (Y so that there is a z in A with (n, (a, Z))E ‘~0~ and (Y = ‘~ll(n, (a, z)&+ 1. ‘AH~+~ is defined in (ii). (iv) (3” * 5’, a) E ‘~0~ if (m, a) E ‘4JX, ‘A/(rn, a& = u and Wz:H,xc s4?x; ‘~l((3” . 5’, a& is the least limit ordinal h so that h > (+ and for each b in WzfaHZ, h>

SAllbllx

5.2. lemma.

If, for each a in E(X),

the partial E-recursive

(a, x) selects from A (see Definition 2.9), then

functions on E(X)

are exactly the partial ES,-recursive

functions. Proof. If (a, x) selects from A, then (iii) in the definition of ES,-recursive is E-recursive. The other clauses in the definition are exactly those of E-recursion. For example, the partial ES,,,-recursive functions are exactly the partial Erecursive functions by the Gandy selection theorem. In fact, if c is a counting of A, then the partial ES,-recursive functions are all partial E-recursive in the parameter c. As in the development of E-recursion, the usual notions can be defined for ES,-recursion theory. In particular, a predicate is ES,-recursively enumerable if it is the domain of a partial ES,-recursive function. 5.3. Theorem.

There are sets of reals A and T so that A c T and un ordinal

K so

Reflection and forcing in E-recursion

101

theory

that (i) E(T) = L,[ T]. (ii) E(T)

is inadmissible.

(iii) E(T) is closed under the ES,-recursive functions. (iv) 0 tl L, is ES,-recursive in T but if a E E(T), then 0 n L, is not E-recursive in a. (v) The complete Zi predicate on L, is not ES,-recursively parameter

enumerable

in any

from E(T).

The sets A and T have the desired property that something (6’n L,) is added when selection over A is allowed; but not everything which is X1 is added. Fix K to be any countable ordinal so that L, is not Xl-admissible but L, is E-closed relative to the predicate 6. For example, L, could be a countable version of the E-closure of w1 relative to 0. Three sets of reals A, TO and T, will be constructed by (Steel) forcing over L,. T,, will be a tree on Cohen reals as usual. A will be a subset of Fr so that there is a copy of A below each member of I, the set of immediate successors of the topmost node in TO. Each ordinal p less than K will be associated with a non-empty subset I0 of I; the members of A below the elements of Ia will have To-height cofinal in the ordinals recursive in /3relative to 0. Selection over A will thus include being able to compute 0 tl L, from TO. T1 will be T, together with a (generic) counting of A. The genericity of T, will imply that any E-computation relative to T1 can be duplicated by a Ocomputation in L,. But L, is E-closed relative to 6 and inadmissible so the complete E-recursively enumerable predicate relative to 0 is A, in L,. Thus, the E-recursively enumerable in T1 and so the ES,-recursively enumerable predicates in TO will not include the complete Z1 predicate on L,. 5.4. Definition.

The partial order P: A condition p in P is a triple (T,, h,, g,) so

that (i) T, is a finite tree on w. For all m,, if there is an ml so that (m,, ml)E T,, then (m,, 1) E TP (ii) h, : TP * K U(m) so that: (a) If s E T, and In(s) < 1, then h,(s) = ~0. In(s) is the length of the sequence s. (b) If s and t are in T, and s !$ t, then h,(s)> h,(t). (a>=~ and a>~.) (c) If s = (mo, ml) is of length 2 and an element of T,, then let & = 1)) = ~0, then /3,= 0. h,(s) must either be m or an E-closed h,((m,, 1)). If h,(&, ordinal which is the height of some notation (n, P,) in 0:; That is, h,(s) is m or an E-closed ordinal which is the height of an E-computation relative to 6 using parameter &. (iii) g, : T, +Pe (PC is the Cohen partial order o<*) so that (a) For all m, if (m, 1)~ T,, then g,((m, 1)) = (1). @I<,,,, ends in I0 if (I-&,,,, 1) has height p in the generic tree.)

102

T.A. Shman

(b) If s and t are in T,, have length 2 and have the same second coordinate which is not 1, then g,(s) = g,(t). (The elements of A are exactly those H, where In(s) = 2 and the second element of s is not 1.) (iv) If p = (T p, hp, g,) and q = (T,, h,, h) are in P, then p 3 q if: (a) T, c Tq, h, c h,. (b) If s E T,, then g,(s) +

G(S).

The definition of P even with the explicit reference

to 0 is still E-recursive.

To

see if p is a condition in P consists of checking several very local consistency requirements and the main requirement on h,, (ii)(c). However, if K = Ij(n, p,)()g, and K is E-closed, then the computation tree enumerating (n, p,) into SES is an element of I,,,,. If K is E-closed, then 6 nL, is an element of L,,, hence any computation using 6 tl L, which has height less than or equal to K is also an element of L,+1. It is any easy construction to show that the E-closed ordinals which are the heights of 0, computations relative to 6 are cofinal in the ordinals which are recursive in 0, relative to C?; none of the power of 6 is lost when this restriction is imposed. Let G be P-generic.

Let

TG =U{T,IPEG~

h=U

h,

PEG

and if there is a p in G so that s E T,, let H, =

U k,(s) t s E Tp & P E Gl.

Let and let CT0 order FrO by extension of sequences. The order must be built into the field in this way since the same real may appear many times in the tree. For example, Ho2) = HC2,2). Let A = {H, 1s E TG & In(s) = 2 & (Vm)(s#

(m, 1))).

In essence TO= (F,,,
and let +I order FT1 by extension of sequences. T1 reintroduces to TO the counting of A which is implicit in TG. By the genericity of G, h is a surjection from TO to K U(m) and if h(s) = 0, then It remains to formuTO below (H, ), . . . , H,) has height p. Thus, L,[T,]sE(T,). late the sequences of definitions and lemmata for TO and T1 which are parallel to those of the previous sections. First the definitions of forcing for atomic formulas.

Reflection

5.5.

Definition.

and forcing in E-recursion

I. p II0 4 if 4 is defined by if a,bEL,

(i) (ii)

plk,aEI) p Il-, 7 = 7

(iii)

p Il-0(H( ), H, 1 I, . . . , Hs>~ho

(iv) (v)

P 11, HW = (1) p Ilo H,(n) = i

(vi)

p It,(H(

II. p Il-1 4 if (i), (iii) (iv) (v)

or (b)

(iv)

103

theory

and aEb.

if rEC. if s-ET, & h,(s-)>O. for all integers n. if s E T,, (Vm)(s# (m, l)), n is less than or equal to the domain of g,(s) and g,(s)(n) = i.

), . . . , H,),>(H,

if ss t and h,(t-)>O.

), . . . , H,)

4 is atomic is defined by as above. if S-E T, and h,(s-)>O. for all n E w. if s E T,, In(s) # 2, is less than or equal to the domain of

(ii) p II-, (HT ), . . . , HZ) E FT, P IF1H?,,1) = (1) p IF, H;(n) = i

P It-, HTnh,,,&)

PI!-,(HT)

,...,

g,(s) and g,(s)(n) = i. if ho, ml>E TP; n is less than or equal to the domain of g,(( m,, ml)>

= G,d

H:)T,2(HT)

,...,

Ht)

and gP((mO, ml>) = i. iff sst and tET,

The definition of Ret(y, p, p*) is exactly the same as in the previous sections as is the proof of the next lemma. 5.6. L~IIIIWI. Suppose

4 is ranked and Ret(o . rk(4), p, p*). Then

p IF04 ifi p” IF0 4 Lemma

and

5.6 has the familiar

p II1

4 ifl p* It-i 4.

corollary

that the forcing

relation

for ranked

sentences is E-recursive. The difference between IF, and II1 appears in the next lemma. 5.7. Lemma.

Suppose p E P.

I. If P It,4Ws,, . . . , H,,.,ToIdefinePZ by (i) s E T,, if s E T,, and (a) (3 s n)[s E Si], or (b) s = (m, 1) and (3i s n)[(m)s Si], Or (C) s=(mo,m~),mI#l and (3i s n)(3js n)[(mo)z si & (3k)((k, ~JE sj)]. (If an element of A is mentioned in some sip then its image mentioned s must be included.) (ii) hPz = h,(s), g,;(s) = g,(s) if s E Tp;. Then ~zIl-4(H,~, . . . , H,, T,).

below

any other

T.A. Slaman

104

II. If p I!-1 C$(HT1, . . . ) Hz, (i) s E T,* if s E TP, and

T,), then define pT by

(a) (3i G n)[s E si], or (b) s = (m,, ml) and 3i s n[(mJ~ si]. (Now all nodes in A must be included since they are indexed by integers in T,.) (ii) h,;(s) = h,(s), gP$s) = g,(s) Then pT IF, 4(Hz1, . . . , HT,, T,).

if s E Tpr.

The proof of Lemma 5.6 is an automorphism argument as in Section 3. The conditions p; and pT contain exactly the nodes which would have to be fixed for an automorphism to preserve the forcing relations kc, and II-i respectively. As before, results.

the preceding lemmata can be used to prove the crucial bounding

Suppose p E P. I. If P lb k U&, . . . , H&)1&=

5.8.

Lemma.

0, = {h,(s)

y, then y is E-recursive

in 0,

1hpb) # M and (3i G n)[s c si] or (3m)(3i

[(m) E Si & S = (m, l)] or (3m,, [S=(mo,

ml)&

mJ(3i

ml= 1 &(mo)cSi

= {h,(s) I h,(s) # ~0 and [(3isn)[sGsi] (3m)(s

The feature

=(m,

s n)

s n)(3j s n)

&[(3k)((ky

ml)~~i)].

II. If P lkr Ilk (HT1, . . . , H~,))]lT, = y, then y is E-recursive O*, is defined by 0;

where

in 0:

relative to 0.

or

1) and (3i < n)[(m)E

Si])11.

which is unique to the proof of Lemma

5.7 is that during the

inductive step in proving II, given p as above pass to a condition pT (see Lemma 5.6) which is E-recursive in 0; relative to 0. Reference is made to 6 at each step in the recursion. There is now enough information 5.3. Theorem.

to prove Theorem

5.3.

There are sets of reals A and T and an ordinal

K so

that

(i) E(T) = L,[ T]. (ii) E(T) is inadmissible. (iii) E(T) is closed under the ES,-recursive functions. (iv) 0 0 L, is ES,-recursive in T but if a E E(T), then 0 nL, is not E-recursive in a. (v) The complete ;S1 predicate on L, is not ES,-recursively enumerable in any parameter from E(T). Proof. Let T= T,; A, To and T1 are constructed as above. (i) is a corollary of Lemma 5.7 I, since K cannot be the height of any E-computation using elements

Re@ction and forcing

from Rud(TC(T,)).

I., is inadmissible,

so L,[ TO] must be ES,-closed. allows T, to E-recursively

in E-recursiontheory

10.5

so (ii) holds. L,[ Ti] is E-closed

by 5.7 II,

The counting of A which is E-recursive

duplicate ES,-computation

in Tl

on TO. This proves (iii).

(iv) holds since the function /?+ E(P) is ES,-computable by construction. Then 6 n E(p) is E-recursive in E(P). Finally, the E-recursively enumerable predicates relative to 6 are A, over L, since L, is E-closed relative to 6 and inadmissible (see Theorem subset of

K

2.25).

But the complete co-E-recursively

cannot be E-recursively

enumerable

enumerable

in any a in E(T,)

relative to 6 Lemma 5.7 II.

This implies (v). These techniques can be iterated to build sets T, A,, A,, . . . so that adding on T is always a non-trivial addition and selection over A,+1 to ES&-recursion stays strictly within x1.

6. A question If no constraints

are placed on X, then the structure

of the E-recursively

enumerable predicates on E(X) and their closure properties seem also unconstrained. An interesting class of X’s is the one consisting of those X so that the Harrington characterization of ~~~ (as the least ordinal where the Moschovakis witnesses become available for a in L[X]) obtains for all a in Rud(TC(X)). If X is of this sort, X G 2” and for all a in X, K: E E(X), then E(X) is a weak version of E(2”). Is there a complete characterization of selection relative to this sort of X?

References [II R.O. Gandy, Generalized recursive functionals of finite type and hierarchies of functions, Paper given at the Symposium on Mathematical Logic held at Univ. of Clermont Ferrand (June 1962). [2] E.R. Griffor, Private correspondance. [3] E.R. Griffor and D. Normann, Effective cofinalities and admissibility in E-recursion, Preprint Series No. 5 (1982), Univ. of Oslo. [41 LA. Harrington, Contributions to recursion theory on higher types, Ph.D. Thesis, M.I.T. (1973). [5] M.R.R. Hoole, Recursion on Sets, Ph.D. Thesis, Univ. of Oxford (1982). 161 R.B. Jensen and C. Karp, Primitive recursive set function, AMS Proc. Symp. in Pure Math. 13, part 1, 143-172. [71 S.C. Kleene, Recursive functionals and quantifiers of finite types I, Trans. AMS 91 (1959) l-52. 181 S.C. Kleene, Recursive functionals and quantifiers of finite types II, Trans. AMS 108 (1963) 106-142. [91 Y.N. Moschovakis, Hyperanalytic predicates, Trans. AMS 138 (1967) 249-282. [lOI D. Normann, Set recursion, in: J.E. Fenstad, R.O. Gandy and G.E. Sacks, eds., Generalized Recursion Theory II (North-Holland, Amsterdam, 1978) 39-54. 1111 G.E. Sacks, The l-section of a type n object, in: J.E. Fenstad and P.G. Hinman, eds., Generalized Recursion Theory (North-Holland, Amsterdam, 1974) 81-93. 1121 G.E. Sacks, Countable admissible ordinals and hyperdegrees, Advances in Math. 20 (1976) 213-262.

106

T.A. Slaman

[13] G.E. Sacks, The k-section of a type n object, Amer. J. Math. 99 (1977) 901-917. [14] G.E. Sacks, Three aspects of recursive enumerability in higher types, in: F.R. Drake and S.S. Wainer, eds., Recursion Theory: its Generalisations and Applications (Cambridge Univ. Press, Cambridge, 1980) 184-214. [1.5] G.E. Sacks, On the limits of recursive enumerability, to appear. [16] G.E. Sacks and T.A. Slaman, Inadmissible forcing, to appear. [17] T.A. Slaman, Aspects of E-recursion, Ph.D. Thesis, Harvard Univ. (1981). [18] J. Steel, Subsystems of analysis and the axiom of determinacy, Ph.D. Thesis, Univ. of California, Berkeley (1977).