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High α-Recursively Enumerable Degrees
J.E. Fenstad. R.O. Gandy, G.E. Sacks (Eds.) GENERALIZED RECURSION THEORY I I North-Holland Publishing Company (1978)
Q
HIGH a-RECURSIVELY ENUMERABLE DEGREES Wolfgang Maass Msthematisches Institut der Universitat Miinchen A degree g
is said to be high if &' = 0"
the jump of 2 and
where &'
is
0 is the degree of the empty set. Thus 0'
is a high degree but in ordinary recursion theory (ORT) there exist as well high recursively enumerable (r.e.)
degrees below
0'
according to a theorem of Sacks 1121. The proof of this result is a very nice application of the infinite injury priority method. It follows from the theorem of Sacks that the notion high is not trivial. Further results show that the notions high and low (
is low if
a' =
0' ) are in fact important for the study of
the fine structure of the r.e. degrees in ORT. The intuitive meaning is that 2 is high if 2 is near to if
2 is near to
0
0'
and 2 is low
in,the upper semilattice of the r.e.
degrees.
Therefore these notions are useful for the study of non-uniformity effects in this structure where one looks for theorems which hold in some regions of this semilattice but not everywhere (see e.g. Lachlan L43). In addition high degrees are interesting for technical reasons. Some results have been proved for high degrees and it is not yet known whether they are true for all r.e. degrees (see e.g. Cooper
tll).
Finally high degrees are a link between the structure of r.e. degrees and the structure of r.e. Martin
(see[l5]):
sets according to a theorem of
A degree contains a maximal r.e.
set if and
only if it is a high r.e. degree. In a-recursion theory for admissible ordinals 2 39
OL
the deeper
WOLFGANG MAASS
240
properties of r.e. degrees and r.e. sets are explored in a general setting and one tries to find out which assumptions are really needed in order to do certain constructions. We refer the reader to the survey papers by Lerman and Shore in this volume for more information. It turned out that in fact several priority arguments can be transferred to u-recursion theory (see e.g. Sacks-Simpson 1141, Shore 1163, Shore 1181). Other results of ORT have been proved for many admissible
u but 'it is still open whether they hold for all
admissible a( e.g. the existence of minimal pairs of a-r.e. degrees 161 ,[21]
and the existence of minimal a-degrees [171,[7]).
Lerman [5] closed the gap between provable existence and provable non-existence in the case of maximal cc-r.e. sets.
For some time one thought that the existence of high tx-r.e. degrees below 0 '
was as well completely settled by Shore c 2 0 3 ,
but an error was found in the proof of Theorem 2.3.
in [ 2 0 ]
* . The
problem was then open again except for z2-admissible u where the existence proof from ORT works and for
d
such that
0'
is the
only non-hyperregular a-r.e. degree where every a-r.e. degree below 0'
is low according to [201 (these are the types ( 1 ) and
( 4 ) in our characterization in $ 3 ).
We close the gap in this paper by proving that high a-r.e. degrees below 0'
exist if and only if u2cf u a w2p u
. This re-
sult was not expected and is different from the result in 1201. We think that the new result is a lucky circumstance for a-recursion theory since it was thought in C201 that the situation is somewhat trivial (every non-hyperregular cL-r.e. degree is high). Now it turns out that inadmissibility (in form of non-hyperregularity) influences the behaviour of the jump of an a-r.e. degree but is *I would like
thank R . A . Shore for informing me about this.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
241
not s o s t r o n g t h a t it overruns everything ( t h i s w i l l become even c l e a r e r i n our forthcoming paper C113 ). The plan o f t h i s paper i s as f o l l o w s : contains some basic d e f i n i t i o n s and f a c t s . In
we construct high ci-r.e.
> o2cfa 8 v2pa
degrees below
0'
f o r t h e case
. We give some motivation f o r t h e construction
s o t h a t t h i s chapter should be readable f o r anyone who has seen bef o r e an i n f i n i t e i n j u r y p r i o r i t y argument i n ORT (e.g.C23]).
construction r e f l e c t s s e v e r a l t y p i c a l f e a t u r e s of
The
a-recursion
theory and uses s t r a t e g i e s which would not work i n ORT. In
$&
0'
i n t h e case o 2 c f o
we prove t h a t t h e r e e x i s t no high w-r.e. 4
w2pa
degrees below
by using some basic p r o p e r t i e s o f
s t r o n g l y inadmissible s t r u c t u r e s . Along t h e way some first r e s u l t s a r e proved about a distinguished degree between which we w r i t e
O3I2.
A summary i s given i n
. Four types o f
0,'
and
0''
admissible o r d i n a l s have
t o be distinguished as f a r as t h e behaviour o f t h e jump o f r.e. grees i s concerned.
for
de-
242
WOLFGANG W S S
$0. P r e l i m i n a r i e s
p
Lowcase greek l e t t e r s a r e always o r d i n a l s , always l i m i t o r d i n a l s and
a i s always admissible i n t h i s paper.
W e consider only s t r u c t u r e s ';G = < Lp,B> r e g u l a r over D
s LR i s
Lp
, i.e.
zn% i f
' d x < (3 ( L r D
may c o n t a i n elements of
For
2,s
t h a t some
w r i t e s rrnp'p
p
ancfa
A set
D
ancflqa
An o r d i n a l
and
D
E.
$6-r.e.
{ K a Lo
I)-finite i f
&
Lo :
f x l < e , x > s U,"3
is
D
I K
C
D
.
zns
C
sets
i s E n % - i f and only i f
e
E (3 )
U,"
if
we w r i t e
for
(i.e.
D =
zn'&
which are given by some
n = 1
of
for
Wes
A ,D c L o
A srLD )
one says t h a t
i s %-reducible t o
A
i f t h e r e i s some index
D
e e (i such t h a t f o r
K c Lg
all
K
sn&i f t h e s e t i s xn$v . A s e t
i s called a (regular) 0-cardinal
for some
is
D
if
tame-
D3
.
.
anpLa,
~r3
For s e t s (written
.
such
zn$function
($-recursive)
K a Lo
d e f i n i t i o n . I n t h e s p e c i a l case (xi
(which
and one
instead of
W e f i x f o r t h e following u n i v e r s a l every s e t
dr
such t h a t some
anpa
b [ b is a (regular) cardinal]
L,.
En formula
d cofinally into
W e say t h a t a s e t
i s called
say t h a t a s e t
f o r the l e a s t
d s (3
s Lo i s c a l l e d
" p o s i t i v e neighborhoods" K C L,,
. We
6 ( i . e . p maps (3 1-1 i n t o b ). We w r i t e
into
I,% ( A ,$).
Lp )
is
B
L p as parameters) over t h e s t r u c t u r e %
f u n c t i o n maps
i n s t e a d of
L
B G LR and
i s d e f i n a b l e by some
for the least
projects
p
where
B
r\
one w r i t e s a n c f d X
1 c (3
A are
and
A
t)
3 H1 H2 8 L B ( E We
d
A
H1 si D
A
H 2 s Lo
-D
)
and KcLp-Ae3H,
H ~ r L ~ ( ( K , 1 , H l , H 2 > 6 WH~1*s D * H 2 C L R - D ) .
H I GH
The index
e
~ 1 -RECURS I VELY
can be communicated by w r i t i n g
One f u r t h e r defines t h a t
[XI
(written
A SwSD
L,,
6
says t h a t a degree
D
r e s t r i c t e d t o single-
i s defined by
A =$D
A asD
and t h e equivalence c l a s s e s a r e c a l l e d 6-deRrees
D **A
c
K
to
).
An equivalence r e l a t i o n
A E
.
A LZD
i s weakly $-reducible
A
i n t h e same way but with t h e s e t s tons
243
ENUMERABLE DEGRE ES
A
. One
has c e r t a i n p r o p e r t i e s i f t h e r e e x i s t a s e t
which has a l l t h e s e p r o p e r t i e s .
We study i n t h i s paper t h e
a-jump
operator
(see Shore c201
f o r a discussion o f t h e d e f i n i t i o n ) : A'
:= C
H1 H2
i s t h e jump o f a s e t write
instead o f
We
L, ( c
E
we^
). Since we have
A
4e
H2 c I.,-
A
i n a - r e c u r s i o n theory
A S L ,
WeL&
H1 5 A
A)
(we always
D + A'
ss D f
t h i s d e f i n i t i o n gives r i s e t o t h e d e f i n i t i o n of t h e u-jump
c w ~ 'f o r
operator
We w r i t e s t e a d of
(0')'
bility)
U2La =~
"1
for the
0
6
At
w-degree o f t h e empty s e t and
A
in-
0"
-
. Observe t h a t UlLU 0' and ( u s i n g the a d m i s s i 0" . Furthermore we have f o r r e g u l a r s e t s A t h a t . E
One says t h a t an u-r.e. otherwise
.
2
at-degrees
set
A
i s complete i f
A
0
;
0'
i s c a l l e d incomplete.
We o f t e n use without f u r t h e r mentioning t h e r e g u l a r s e t theorem
of Sacks which says t h a t every a - r . e .
regular
a-r.e.
For a s e t
set A
(see [13],
c L,
[22],[81
one w r i t e s
such t h a t a c o f i n a l function
f :
a-reducible t o
A
rcf A = u
A
. The s e t
, otherwise
A
rcf A
degree contains a
f o r proofs).
f o r the least
b + o e x i s t s which
6
ot
i s weakly
i s c a l l e d hyperregular i f
i s c a l l e d non-hyperremlar.
d
2 44
WOLFGANG MAASS
Observe t h a t we have f o r r e g u l a r ticular
r c f A = o l c f ' L a t A ' a , i n par-
A
i s non-hyperregular i f f
A
Hyperregularity i s
i s inadmissible.
a p r o p e r t y of de-
-contrary t o regularity-
g r e e s r a t h e r t h a n of s i n g l e r e p r e s e n t a t i v s : gree then
i s t h e same f o r every
rcf A
& i s an a-de-
if
A E &
.
Simpson proved i n h i s t h e s i s t 2 2 1 t h a t f o r any
r = rcf
have t h . a t
u - c a r d i n a l and
A
f o r some eC-r.e.
v2cfLu8
= c2cf u
.
A
yc
Q
we
is a regular
iff
The f o l l o w i n g Lemma combines of Shore c19]
i n b ) Simpson's r e s u l t w i t h Theorem 2.1.
. The proofs
z2 p r o j e c t i o n
of a ) and c ) a r e s t r a i g h t f o r w a r d ( c o n s i d e r a
from x
i n t o a2por f o r c ) ). Lemma 1 :
a)
0'
b)
There e x i s t s an incomplete non-hyperregular
either
i s a non-hyperregular
c)
such t h a t
*
o2cfLOLw = u2cfcr
w2pa < x
and
(we w r i t e
w2cfu
zn'&s e t
i n s t e a d of
Jn,p
d-r.e.
OL
. degree
<
iff
and t h e r e i s a
02cfLuw = o 2 c f
d
.
f o r every r e g u l a r a - c a r d i n a l
x
x .
F i n a l l y d e f i n e f o r any s t r u c t u r e
: = p 6 + ( %a(
u2cfa < u
< w2pu
or-cardinal K a o 2 p a such t h a t
We have
f,,P
or w2cfa
0 2 p a 6 w 2 c f o ~ < cu
regular
a-degree i f f
M L 6
b = e x i s t s such t h a t
M 6 Ln)
jn Lg ). PP
According t o J e n s e n ' s Uniformization Theorem C21 we have 9n*fi
pnla
= o n p (3
f o r every
n > 0
and every l i m i t o r d i n a l (3.
We w i l l o f t e n use without f u r t h e r mentioning t h e e q u a l i t i e s = unpa
for
n = 1,2
which a r e e a s i e r t o show because
uniformization is t r i v i a l f o r admissible a
.
z2 -
We r e f e r t h e r e a d e r t o D w l i n [2) f o r a l l d e t a i l s concerning constructibility.
2 45
H I G H a-RECURSIVELY ENUMERABLE DEGREES
C o n s t r u c t i o n o f h i g h a-r.e.
$1.
degrees
A t f i r s t we s k e t c h t h e c o n s t r u c t i o n of i n c o m p l e t e h i g h r . e .
sets i n ORT. The o r i g i n a l p r o o f i s due t o S a c k s c123. A d d i t i o n a l i d e a s o f L a c h l a n and S o a r e a r e u s e d i n t h e v e r y p e r s p i c u o u s v e r s i o n o f t h e c o n s t r u c t i o n as i t i s p r e s e n t e d i n S o a r e l 2 3 1 (we r e f e r t h e r e a d e r t o t h i s p a p e r f o r more m o t i v a t i o n and d e t a i l s c o n c e r n i n g t h e p r o o f i n ORT ). I n order t o b r i n g t h e requirement
A’
E
i n t h e r e a c h of
0’’
a r e c u r s i v e c o n s t r u c t i o n we a s s o c i a t e w i t h a f i x e d Z‘* set S
a r.e.
0”
*
< e , y > c BS where
3yvz
set
V y’
which i s d e f i n e d by
BS 5
y 3 z i+(e,y’,z)
r2 d e f i n i t i o n
is a fixed
+(- ,y,z)
Then w e have f o r e v e r y
e IQ.
I-S(e)
I
i s enough t o i n s u r e t h a t f o r a l l
e
6
= lim
Y+W lim
of
S
over
.
and i t
BS(
A(
w
L ,
.
l i m B( ) = l i m B ( < e , y > ) Y4w Y*Q Pe i s a r e q u i r e m e n t which i s h a r d t o s a t i s f y i f e 4 S since i n
.
t h i s c a s e we have t o p u t a l l b u t f i n i t e l y many e l e m e n t s o f EIy a u f
into
.
A
A c o n f l i c t a r i s e s b e c a u s e w e have t o s a t i s f y as w e l l f o r a l l
e ew
: i C = +e(A)
Ne
requirements ween
Ct(x)
Ne
and
where
C
0’
i s a f i x e d r.e.
s e t . The
a r e s a t i s f i e d by p r e s e r v i n g a d i s a g r e e m e n t b e t f o r a s u i t a b l e argument
+e,t(At,~)
x
and by
f o r c i n g t h e a p p e a r a n c e o f s u c h a d i s a g r e e m e n t ( r e s p e c t i v e l y o f an argument
x
such t h a t
+,(A,x)?
)
Ct(x)
and
w e l l a g r e e m e n t s between
of a n i n i t i a l segment possible
y
of
o
on t h e way o f p r e s e r v i n g as +e,t(At,~)
x
out
which i s c h o s e n as l o n g as
( r s w ) .(We h a v e w r i t t e n O e ( A , * )
which i s p a r t i a l l y r e c u r s i v e i n
for all
A
with index
f o r the function e.)
WOLFGANG MAASS
246
This strategy of preservation in order to get
'I
C se A
is on
first sight contrary to intuition. But if we preserve -as soon as it appears during the construction- agreement between Ct(x)
$ e,t (At,x)
for every
and
x a y we actually try to make C t p re-
C = C l w is not recursive we must then have y " a
cursive. Since
and therefore i+e(A,r) We write
=
C(r)
.
Ie for the injury set of
all elements x
that are put into e' < e
tive requirement Pel with computation in A
Ne which is the set of
A
-as demanded by some posi-
-
although they destroy a
which should be preserved in order to satisfy
. Then we can be a little more exact in our description of the
Ne
preservation strategy and say that although we try to make recursive we only succeed in making
Clr recursive in
r
before). But we can still get the wanted conclusion if I,
ment above. Since
I,
C $ 1,
is recursive in for every
e ao
ment where one shows that for every lim A(te,y,)
Y*Q
e'+ e
A
= lim
Y '*
yew3
B()
for every
e
Cry
(8
as
even
C # Ie for the argu-
is infinite since we need only
we can prove
Ie
f 6 A I e' < e h y c u l during the inductive argu6 o
'I
C se A
(we use here that
C
and
#
fte',y> c B I
e ).
Observe that in writing
C ( g ) etc. we have followed the usual
convention to identify a set with its characteristic function. The construction from ORT works as well for
z2 admissible
el
(Shore c.203). But there are several reasons why this construction does not work for the other admissible u
. We discuss five of
these problems in the following and we simultaneously try to motivate the new features of the subsequent construction for the case Q
>w2cfo
fi
w2pa
.
Assume s a t we succeed in constructing the set A
in such
247
H IGH a- RECURS IVELY ENUMERABLE DEGREES
a way t h a t
Ve
GO(
(A(e) = * B s ( e) )
( d e f i n e f o r any s e t means t h a t
M1
-
M : ~ ( e ):= and
M2 E L,
-
with M n ((el
-
M2
3
yeV Y a ye(- < e , y ) c A)
f o r some f i x e d p a r a p e t e r way q u e s t i o n s
c Stt
ItK
p
e cor
c a n ' t be tame-
P,e,Ye'
e2cfu < a
*
.
A')
6
{ye\ e
if
6
of w i t n e s s e s . S i n c e
K f
i s not
u
we need t h e
A'
r2 admissible
(see
we c a n h a r d l y e x p e c t t h a t t h i s bound e x i s t s f o r a l l a - f i n i t e
$2.)
such t h a t
K
e 6 S
t o questions about
r2 L,
if
i f we want t o r e d u c e i n t h e same
e x i s t e n c e of a bound f o r t h e s e t S c 0''
as b e f o r e
M~ = * M *
;
S 6, A '
that
3 ye(i
. But
LJ
x
BS
M, E: L = ) .
T h i s d o e s n ' t imply i n g e n e r a l t h a t
W e have o f c o u r s e f o r e v e r y
and
S
K G S
.
We overcome t h i s d i f f i c u l t y by u s i n g i n a p o s i t i v e way t h a t a i s not
t 2a d m i s s i b l e . F o r t h e s e s e t s a n d i n t h e case
-r.e.
%I
t h e r e e x i s t non-hype rre gula r
a
> w 2 c f a >, 0 2 p a
OL
t h e r e e x i s t e ve n
incomplete non-hyperregular
a-r.e.
But for n o n - h y p e r r e g u l a r
we c a n a v o i d t h e s e a r c h f o r w i t n e s s e s
Take a c o f i n a l f u n c t i o n
ye :
a-recursive i n
e
vx
Q
S
f*
[pf
I
rcf A
A
.
I
ref
for
p
z
A
b
y
A
i < e , z > c A)
*
which i m p l i e s t h a t f o r e v e r y a - f i n i t e
W e say
It
At
C
=-recursive
respectively
It
which i s weakly
CAI
(plxrcfAxK
Convention:
Q
Then we have
we have
K s S -
*
f : rcf A
r c f A 3 y z(y = f ( x )
6
f o r some f i x e d p a r a m e t e r K
A
sets a c c o r d i n g t o Shore C193.
'I
. i n t t and
"weakly c t - r e c u r s i v e i n "
4wfftt as u s u a l . But t h e r e i s a pro-
blem w i t h t h i s i n t e r p r e t a t i o n , seeL91.
11
For t h e considered
happen t h a t
0"
u where
d o e s n o t c o n t a i n a regular
t o r113 t h i s o c c u r s i f and o n l y i f
~ 2 p a it can
a > w2cfa
I,
L,
v3cfa < e3p u
set. According We w i l l con-
.
2 48
WOLFGANG MAASS
s t r u c t i n L111 a n
ci
where
cr 3 cf u
a3pu
<
w2pu
I
cr2cfor
G
a.
T h i s example i s t h e m o s t - d i f f i c u l t one w i t h r e s p e c t t o o u r cons t r u c t i o n of i n co mp l et e h i g h cc-r.e. contain a regular
t 2s e t
degrees since
and we have w 2pcl <
t h e d e f i n i t i o n of t h e tame
z2 p r o j e c t u m
does not
0"
t a 2 p u ( s e e C63 f o r
t v 2 p u ).
I n consequence o f t h e p r e c e d i n g t h e p l a n f o r o u r c o n s t r u c -
2
t i o n i s as f o l l o w s : We t a k e a f i x e d i n com ple te n o n - h y p e r r e g u l a r a-r.e.
set
and make s u r e t h a t
D
n o n - h y p e r r e g u l a r. A(e)
= * Bs(e)
requirement A(e)
. As
Pe
.
Further f o r
A(')
e > 0
=*
D
i n o r d e r t o make
A
we want t o have t h a t
b e f o r e we s e t up f o r e v e r y
e
s . u
a positive
which t r i e s t o s a t i s f y t h i s c o n d i t i o n c o n c e r n i n g
It i s c r u c i a l f o r t h e i n f i n i t e i n j u r y argument t h a t t h e s e t
o f t h o s e e l e m e n t s which s h o u l d be p u t i n t o
A
i n order t o s a t i s f y
a l l r e q u i r e m e n t s o u t o f a n i n i t i a l segment o f t h e p r i o r i t y l i s t i s
n o t t o o c o m p l i c at ed . According t o p o i n t 2 ) t h i s f o r c e s u s t o make o u r p r i o r i t y l i s t no l o n g e r t h a n o 2 p a sets BS n K
of
K L
be c a use o n l y f o r c i - f i n i t e
a-cardinality l e s s than v 2 p u
Lu is a-recursive.
it is guaranteed t h a t
It i s n o t e a s y t o work w i t h s u c h a
short p r i o r i t y list i n an i n f i n i t e i n j u r y construction since t h e o l - r e c u r s i v e a p p r o x i mat i o n t o t h i s l i s t i s v e r y weak i f 0 2 p a < e2cfu
. We i n t r o d u c e
a c l a u s e b ) i n t h e c o n s t r u c t i o n which makes
it p o s s i b l e t o c o n t r o l i n many s i t u a t i o n s t h o s e unwanted i n j u r i e s which a r e m e r e l y due t o bad g u e s s i n g o f p r i o r i t i e s . We w a n t t o p r o v e by i n d u c t i o n on t h e p r i o r i t y f o r every
e
case t h a t
p(e)
we have
=*
. There
that
i s a problem i n t h e
i s a l i m i t o r d i n a l since t h e i n d u c t i o n h y p o t h e s i s
d o e s n ' t imply t h e n t h a t p(i) < p(e))
A(e)
p(e)
u { A(i)
I p(i)
< p ( e ) ) =* U tB(i)
1
and we c a n ' t c o n t r o l t h e d e g r e e of t h e i n j u r y set
249
HIGH LRECURSIVELY ENUMERABLE DEGREES Ie
. We use the fact that this situation is only possible
a2cf OL > w
since o2cf a 5 u2p
OL
. a2cf a >
u
if
implies that there
are enough fixpoint stages in the construction so that it is in fact not necessary to determine the degree of the injury set
Ie
.
5) There is a problem with the preservation strategy of Sacks in the case that there are non-hyperregular injury sets 1, is non-hyper(which will occur in our construction since )'(A regular). If we want to preserve then agreements C(x) = be(A,x) for
x
part of
p
these computations may altogether use an unbounded
A
even if
8
< a
. Since this would endanger the positive
requirements of lower priority we have to be much more careful with preservations. For this sake we introduce "e-fixpoints" in the
. In the case
case e 2 c f o ~? w rcf
I)
o2cfu = u
we divide
o(
into
many blocks as in Shore El81 (doing the same thing in the
case a2cf OL
5
a2pa > w
would be troublesome because of limit
points in the priority list).
Theorem 1 :
Assume that u > e2cf ci k 02por
are u-r.e. sets such that
C (OD
and
then there exists an a-r.e. set A and
A' =u 0"
D
C
and
D
is non-hyperregular
such that
.
. If
D
6,
A
,C
#,A
The rest of this chapter is devoted to the proof of this theorem. After some preparations we will describe the construction
of the set A Lemmata 3 , 4 , 5
for the case v2cf a > that this set
A
w
. We will show in the
has the properties we want. The
construction for the case a2cfu =
o
is rather close to the con-
struction in OR1 and will be discussed briefly afterwards.
250
WOLFGANG MAASS
W e f i x f o r t h e following r e g u l a r a - r . e . that
and
C 4, D
(Dw)w<
is- non-hyperregular.
D
a r e i n the following fixed
~
sets
(C,, ,)
z u such
C,D
and
oL
a-recursive enumerations of
these s e t s .
t, L,
Take a
’-! such t h a t
A. formula Define the
set
Ot-r.e.
set
such t h a t
S cot
OL
I
5
.
a s follows :
oc
t ( l , y > & B : ~ 4( ( ( 3 = 0 h y e D ) v ( ( 3 > 0
LaI= Vy’
and f i x a
0”
6
S tr L a b 3 y v x Y((3,y,x)
(3
B s
S
A
y 3 x 1WP,gt,x)))
Then we have f o r (3 > 0 : f3
E
+ I~yIa B l =pd( L a k Vx’f’(p,J,x))
S
lfi6 s
4
Iyl
6
BI =
.
OL
Fix an a-recursive enumeration
of
(Bu)w.
and
u
f o r the
B
following. F o r any s e t
and any
x
6
L,
as an abbreviation f o r
M
fi
(fxl
A,
M
.
u
Lemma 2 : Assume t h a t n a l i t y l e s s than e 2 c f a s e t such t h a t
i s an
K
. Further assume t h a t
a define a
cro
for
Rg f
0
x
W
z2 function W ( X )n
LO
and we have
a-cardi-
6 K
. Then
of
W
. For given
f : K * a by
= ~ ( ~ LO ) n
u fW(X)\ x c
I n t h e following we w i l l w r i t e where
~~
A
is an oL-r.e.
i s regular.
Proof : Pix an enumeration f ( x ) :=/*v(w,
at-finite s e t of
i s r e g u l a r f o r every
~ w ( ~x )6 IK ]
<
.
La)
I
w i l l be t h e s e t of elements which have been put i n t o
before stage
u
we w i l l use i n t h e following
.
?here e x i s t s a bound
K3
x c W
i s a fixed
r\
0
.
for
e,r
x,
Lo c W
, L
d e f i n i t i o n of
HIGH *RECURSIVELY
For t h e considered enumerations at stage
"K
La
~r
3 H'
e
-
a-r.e. and
(Al)-
sets
and
A
t h e r e e x i s t s a computation o f
C"
with negative neighborhood
Case i ) :
at
P
W
H'
A
e9ff
and t h e i r
C
we w i l l say t h a t
(CT)-
u
L,(
25 1
ENUMERABLE DEGREES
.A
P
> r2cfa 8 02pu
if
H H
A
5
-
L,
.
As)
c2cfa >
and
for
"C Qe A"
.
(3
The next d e f i n i t i o n is t h e f i x p o i n t device which was mentioned i n point 5 ) of t h e motivation.
A i s an
e-fixpoint at s t a g e
f o r every
7
there i s a
<
(3 I A
: C)
such t h a t
+I
i s a stage
u < 'h such t h a t at stage
t i o n of
"C
se A"
borhood
H
and we have
for
-
lltf
-
H s La
AP
We say that t h i s e-fixpoint CAn A
+ Cfi
n
The " r e s t r a i n t function"
- C"
with negative neigh-
i s i n a c t i v e a t stage (3
.
A
.
and t h e r e
t h e r e e x i s t s a computa-
cr
c , L
C,,
r s z t < ;5
r :a
Q
+
if
w i l l play a similar
at
r o l e a s i n Soare C231 and i s defined by cases : There e x i s t s a stage
Case 1):
i n a c t i v e e-fixpoint at a l l s t a g e s i n Take t h e l e a s t such
u . Define
1
such t h a t some
0-1 (3
C a , @ ] :=
r(e,p)
ft
I
<
.
+PI
w s t
t o be t h e l e a s t
i s an
Q
X <
Q
which i s an i n a c t i v e e-fixpoint at a l l s t a g e s i n Lff,(l]. Case 2 ) :
p
Define
r(e,p)
at
otherwise. W e f i x an 1-1
onto u for
e c
see t h a t
y
t o be t h e union o f a l l e-fixpoints
.
X 2 L,
function
. Using the assumption
g
r (y
which maps cr2poc p a r t i a l l y
w i l l be t h e p r i o r i t y o f t h e requirements
g"(e)
a(
g
n dom g)
< a 2 p o ~ where
B ( < U ) :=
a2pu
4
u
{
it i s easy t o
u2cfor
i s or-finite and
I g"(e)
B('r) < iy
Pe,Ne
h,D
f
.
f o r every
252
WOLFGANG MAASS
a - r e c u r s i v e approximation f u n c t i o n g ' ( * )
We f u r t h e r need a n
( o f two a r g u m e n t s ) w i t h - a - r e c u r s i v e domain which h a s t h e p r o p e r t y that for all y < w2pa f o r all
x
E
y
n dom g
t h e r e e x i s t s an o r d i n a l and a l l t E rr
we have
such t h a t
'ty
gT(x)
g(x)
.
I n a d d i t i o n we want t o have t h a t
3 Wo V a > , uo(
(1)
g(x)J
(2)
V limits A
and t h a t
f*
<
ol(gl(x)
g"(*)
gQ(x)S )
and
* 3 e0 c 1 V o (r0G Q a
J
i s 1-1 for e v e r y
.
w 4 u
X + g"(x) 3 1)
Because o f t h e d i s t i n g u i s h e d r o l e o f t h e r e q u i r e m e n t f u r t h e r need t h a t
g(0)
2
0
and
gQ(0)
E
Po
for all
0
The d e f i n i t i o n o f an a p p r o x i m a t i o n f u n c t i o n
g'(.)
we
IY <
a
.
w i t h these
properties i s routine. Observe t h a t i n g e n e r a l we c a n ' t g e t t h e f o l l o w i n g p r o p e r t y which one would r e a l l y l i k e t o have :
vx
< (r2pc43uovz d
y
v0
5 uo( g O ( z )
g(z) )
E
( s e e t h e p o i n t s 2 ) and 3 ) ).
Construction : A t stage
u we c o n s i d e r e v e r y
f o r some
z ce2pa
If
E
i s n o t a l r e a d y a n element of
<(J,y'
at stage
.
'(3,~'
$+, Ab
such t h a t g Q ( z ) S P
we p u t
into
if
u
a)
( p , ~ )3
b)
qp,y, I z
r(gQ(zt),v) for all
( z + l ) n dom g r
.
End o f c o n s t r u c t i o n .
for all T
u
z ' e ( z + l ) n dom gQ
such t h a t n o t
and
( z + l ) n dom gQ 5
A
H I GH a- RECURS1 VELY ENUMERABLE DEGREES
For a n e g a t i v e requirement boundedly many s t a g e s
t h e r e a r e i n g e n e r a l un-
Ne
where some p o s i t i v e requirement
Q
t r u l y lower p r i o r i t y ( i . e . injure
25 3
g-'(e)
5
Pel
g - ' ( e l ) ) t h i n k s t h a t i t may
because of t h e weak approximation p r o p e r t y o f
Ne
of
g'(0)
.
The following Lemma shows t h a t i n some s p e c i a l s i t u a t i o n s t h e s e unwanted i n j u r i e s w i l l not occur because of c l a u s e b ) i n t h e construction. I n t h e following we always w r i t e
\dca i V x B
that
dom g (g"(x)
0
Lemma 3 : Assume t h a t
y n dom gQ = element
Q
,p,6>
dom g
+,d> < r(gv(z),e) 8'
a
y
n dom
g
and
, rr
and
y
z d
n dom g
such t h a t
A
then there e x i s t s a g(z')
=
Q
,
s, z
. If
at s t a g e
.
(3
5
such
r
.
= g(x))
< w2pa
i s put i n t o
for the least
rr
Q
such t h a t
z
Proof : According t o t h e c o n s t r u c t i o n t h e r e e x i s t s a such t h a t at s t a g e
gr(z')srJ T
f u r t h e r have
we have z' < z
. Since
c l a u s e b ) does not r e s t r a i n
( z l + l ) n dom g7 c ( z ' + l ) n dom gQ because
e r ( g r ( z ) , T )
not r e s t r a i n e d by c l a u s e a ) a t s t a g e t follows t h a t
( z l + l ) n dom g'
=
. Since
gyp ( ( z ' + l ) n dom g ) = gE ( ( z l + l ) n dom g ) have t h a t
z' e dom g
and
g ( z ' ) o g'(z')
2
and
8'
<(J,&>
. We
r&d> i s
r I U a Yr
( z l + l ) n dom g
an
C
and
it
and
. I n p a r t i c u l a r we
0
.
The following Lemma w i l l s o l v e t h e problem which w a s described i n p o i n t 3) of t h e motivation : g-l(e)
o f some n e g a t i v e requirement
have problems t o c o n t r o l i n j u r y s e t 1, 0
I n t h e c a s e where t h e p r i o r i t y
. Lemma 4
U
N~
i s a l i m i t o r d i n a l we
I g - ' ( i ) < g"(e)
3
and t h e
gives a s u f f i c i e n t condition f o r a stage
t h a t some computation which e x i s t s a t s t a g e
u w i l l not be de-
s t r o y e d l a t e r . I t i s important t h a t t h i s c o n d i t i o n can be expressed
254
WOLFGANG MAASS
by using j u s t
U f B(i) I g - ' ( i )
g - ' ( i ) < g"(e)
3
, not
3
g"(e)
<
. A fi-xpoint argument
U f A(i) I
i n Lemma 5 w i l l show t h a t
t h i s condition w i l l be met by an unbounded s e t o f stages. u2por
The p r o p e r t i e s o f imply t h a t f o r every )u' t h a t f o r every
3 53 1( X
3 A( A
z e
u2por
<
8n
t h e r e e x i s t s a stage
i s an i n a c t i v e g ( z ) - f i x p o i n t a t a l l s t a g e s
8n
dom gff =
x
sup f r ( $ ( z ) , u ) \ z
,
dom g
z
r ( g ( z ) , Z ) >/ r ( g ( z ) , u )
x < r(g(z),V)
= B('X)
r\ 0
dom g )
z
€
8
By Lemma 3
x
where
Q
)
r )
t 5 2
+
.
P'
.
.
x < o. with
such t h a t
x = f o r some
and no element
at stage
A
d
Q
.
%Q
with
x < r(g(z),
z ' e dom g
Uo)
and
. Therefore we have
Q
n
= B:t)
:
Q
A
d (zl+l)
8''
put i n t o
x
.
z' < z
>c d
. We consider two cases was not put i n t o at s t a g e u since x < r ( g ( z * ' ) , w ) n dom g . By o u r induction hypothesis we have
r(g(z"),c)
a
Uo
t h e r e i s some where
x c B(48) n
3
at stage
V
n dom g
Assume f o r a c o n t r a d i c t i o n t h a t some A
Q
x < U
at stage
A
Q
and f o r every stage t
and no element
i s put i n t o
n
i s put i n t o
x n dom g
€
Proof : Induction on i s put i n t o
such
ey
is a stage such t h a t
0 2 Yy'
BPY)
68 n
Then we have f o r every
f o r t h e l e a s t such
v8'
Lemma 4 : Assume t h a t
g(z')=
Ia
i s an i n a c t i v e g(z)-fixpoint at a l l s t a g e s
8n
im2cfoc
dom g
I n the following we w r i t e
4
and the assumption 02p Q
6
r ( g ( z t l ) , u o ) = r ( g q O ( z l t ) , Qo) a t stage
A
such t h a t
2.
0'
>"x
x
w i l l not be
either.
Qo
w a s not put i n t o
and
A
and not
a t stage
U
since there e x i s t s
U'GQ
( z l + l ) n dom g f f c ( z ' + l ) fi dom gQ'
.
:
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Since
( z t + l ) n dom g"
at s t a g e
s ( z ' + l ) n dom guO
3
x i s not put i n t o
r(g(z),r) I r(g(z),o)
f o r all
. Assume t h a t t h e r e a minimal s t a g e > . By t h e preceding no element is
Q
fro
such t h a t
I
r(g(z),co) < r(g(z),w)
w i l l be put i n t o r(g(s),Vo)
A
at some s t a g e
r(g(z),a)
4
where
t
w
5
y < r(g(z),.)
r c
i s d e f i n e d according t o c a s e 2 ) . Since
r(g(z),a) no element 5 wo :
y < r(g(z),uo)
w i l l be put i n t o
Otherwise assume t h a t
r(g(z),rO)
u1
A
> y
and
0' r(g(z),uo)
y
r
whereas
r ( g ( z ) , a o ) < cr at any s t a g e
i s t h e minimal such
i s defined according t o case 1 ) we have
= r(g(z),uo)
Q
i s t h e r e f o r e only p o s s i b l e i f
is defined according t o c a s e 1 ) of t h e d e f i n i t i o n o f
r
A
uo because o f c l a u s e b ) i n t h e c o n s t r u c t i o n .
It remains t o prove t h a t C '
255
c a n ' t be put i n t o
A
r(g
r
. Since
Q
l(z),o,)
at s t a g e
@ ,
as
it w a s shown i n t h e f i r s t p a r t of t h i s proof. Thus we have proved t h a t some X
i s an i n a c t i v e g ( z ) - f i x p o i n t at a l l s t a g e s i n
w0
[@,,OC) whereas t h e r e i s no i n a c t i v e g ( z ) - f i x p o i n t at s t a g e
Since we have
Txt ,C
definition of for all z LU Remark:
If
Q
stage
'L
au
r(g(z),r) a r ( g ( z ) , o )
u s a t i s f i e s t h e assumptions o f Lemma 4
x < sup f r ( g ( z ) , w )
ment
.
t h i s gives a contradiction t o the
and we have proved t h a t
.
Q
Iz
6 f
r\
dom g
. Therefore t h e s e s t a g e s
3
0
i s put i n t o
t h e n no e l e A
at any
play a r o l e i n t h i s proof
which i s similar t o t h e r o l e of " t r u e s t a g e s " ( s e e Soare1233) in t h e proof i n ORI. Lemma 5 :
a) b)
l c
S
~
=*
A
For every
e B a we have
and
.
Proof : For convenience we prove a ) and b) simultaneously by
256
MAASS
WOLFGANG
. Assume f o r t h e f o l l o w i n g t h a t
i n d u c t i o n on g"(e)
and t h a t a ) and b) a r e t r u e f o r a l l
el
g-'(e)
such t h a t
= z
g"(el)
.
< z
Observe t h a t t h i s a s s u m p t i o n d o e s i n g e n e r a l n o t imply t h a t
u
(A(e')
a3cfa
I g-'(e')
, which
z
5
u
=*
z1
<
f B ( e t ) I g-'(e')
< z J
from
A(et)
Lemma 2
i s regular :
=*
3
i f we have
i s o f c o u r s e p o s s i b l e ( s e e p o i n t 3 ) of t h e
m o t i v a t i o n ) . But we g e t t h e i n f o r m a t i o n t h a t g"(e')
z
<
B(e')
Since every
t h a t every
I
B(")
i s r e g u l a r we g e t
i s r e g u l a r as w e l l . Then
A(e')
u
implies that
I
U
g-l(e') < z 3
is regular. This
i s t h e o n l y fact which w e u s e from o u r i n d u c t i o n h y p o t h e s i s s o t h a t i n t h e case
elpa =
we d o n ' t need an i n d u c t i o n a t a l l ( t h i s i s
Q
r a t h e r s u r p r i s i n g i f compared w i t h t h e s i t u a t i o n i n ORT For
we write
z+l
J:=
f o r t h e set o f t h o s e s t a g e s
M
u
i s unbounded i n
M
~ A ( ~ g ' -) l (I e ' ) For
-
an <
13 z, n An =
dom g
A(' %
z
by u s i n g t h e r e g u l a r i t y o f
OL
.
I
a define
An
A(")n
pt > hn
:=
gz'(y)
6 i(
.
)
t 1
2 := n
H
sup
An
[An\ n is
B LJ
Z2 L,
3
. It
1,
.
X
some 'c
c
c(
IN
with
c(
We have
1
<
for the
i s constant i n
A
B('X)
An+'
a0
> rr*
a(
since the function
that
2e M
and <
U
define then
.
a - f i n i t e set of a l l
rn i s a n i n a c t i v e g ( z ' ) - f i x p o i n t
(Cr,a) := ft'\ r
dom g h )
f o l l o w s from p r o p e r t y ( 2 ) o f t h e a p p o x i -
m a t i n g f u n c t i o n and Lemma 3 We w r i t e
<
:=
an = B ( v ) ~ an
and t h e f a c t t h a t
)
A(")
B (((z+l) n
s e t s it i s e a s y t o s e e t h a t
are r e g u l a r u - r . e .
e x i s t s . For every g i v en
(V y
B:x)~
A
By u s i n g p r o p e r t y ( 1 ) o f g o (a A(")
0-
are s a t i s f i e d . W e w a r t t o prove
where t h e a s s u m p t i o n s o f Lemma 4 that
, seet231).
s
rt c
Q
3
kr0,a) f o r e v e r y
) z'
.
in
zt L z
Cr,a)
such
f o r some
Then we have t h a t r ( g ( z
r IN
hat
11
*)
a c c o r d i n g t o Lemma 4 1
HIGH &RECURSIVELY
where
ro i s t h e l e a s t element o f
show
sup C r ( g ( z ' ) , e ) I
i n order t o prove t h a t
z' E ( z + l )
dom g l
n
<
M
r e
. Therefore
M
i t i s enough t o
- IN)]
z ' € ((z+l) n dom g
A
sup { r ( g ( z * ) , v ) I
.
a
25 7
ENUMERABLE DEGREES
Q C
M
<
A
Thus assume f o r a c o n t r a d i c t i o n t h a t
Vc
-
< a 3 u e M ~ Z E' ( ( z + I ) n dom g
This implies t h a t f o r every
-
K P L,
C
3
c-)
K
Q
The p a r t
-
u c M 3 z ' e ( ( z + l ) n dom g
(sup K < r(g(zt),w)
IN)
we assume t h a t
It+''
s a t i s f y t h e r i g h t s i d e . By Lemma 4
. Therefore t h e r e i s at
r(g(zt),v) f i x p o i n t at
such t h a t
we have t h a t
which means t h a t
Q
t h e r e i s no stage a t stage
A
that
y
L
'c
Cc+l
and
2
X
C, n 1 = C A n
. Therefore t h e r e i s no
- C,
we have proved t h a t
y
s i n c e otherwise some
active g(e')-fixpoint i n
1
%
= C n A
.
1 y
k
and
At
A
By Lemma 4
t %
since
c
MIt
can be expressed
would be an in'12'
6
which shows t h a t
IN
a;
a;+l
:= p z > X;((the
C,
1; = C n A;
n
c OL
we define
A
A;+,
4
u
B(<') . But . Thus we have proved g 3 .
A
zt
Q
( z + l ) n dom
'c
.
< a
C S: A
by
same as i n t h e d e f i n i t i o n o f ( a t stage
-C
C sa B ( * I )
I n order t o prove a ) assume f o r a c o n t r a d i c t i o n t h a t
For
. Thus
K c L,
a-recursively i n B(
S := s u p f r ( g ( z ' ) , a ) I r e M
that
auch
e
The equivalence which w a s j u s t proved implies t h a t t h i s i s absurd because we have
I
1 i s put i n t o
c
X
, contradicting
[%,=)
Cw n
since
r
i s not an i n a c t i v e g ( z t ) -
such t h a t an element
au
is
r(g(zt),e)
some g ( z ' ) - f i x p o i n t
Q
sup K <
do
Q,z',K
defined according t o case 2 ) of t h e d e f i n i t i o n of
c IN
.
C,)
K c La-
A
of t h i s equivalence i s obvious from our
it+tt
assumptions. For a proof of
72'
.
IN)('E < r ( g ( z t ) , v ) )
L,
Q
An+,)
A
t h e r e e x i s t s a computation of
.
258
WOLFGANG M A S S
"C
-C
for
A"
6:
such t h a t
H
s
5
26 c
- C"
A'
:= sup
- A)) .
L,
We have again t h a t given
L,
1n
since the function
ci
n c r A;
. Now it
i s a g ( z ) - f i x p o i n t at
i s defined f o r some
definition of
c o
is
This implies t h a t
Q 6
f o r every
u
z2 d e f i n a b l e
F o r t h e p r o o f o f b) we choose
have
A(e)
-
-
r1 =
g e t h e r shows t h a t
A(e)
D = B(O) =4 A(')
S QUA'
C,
( u s e Lemma 4 )
A
Sz
hyperregular and
0
E
S n
v
x < rcf A
i s now very easy. We g e t
from Lemma 5
x < rcf A have
K
36 >
A
S n
x
rcf A
f ( x ) (7<&6> p e
OL
x
K
c A*
Q
La-
UZLa &A1 $2.)
this
S"
can be w r i t t e n a s a
i C
cqA
i s non-
D
Q
A) 0
6.
.
"K s L a -
n2 formula.
S1I
. CJ .
S )
such t h a t f o r a l l
Concerning t h e computation f o r
"K
and
x
Then we
we observe t h a t
Since we have
( s e e t h e first p a r t of t h e proof o f Theorem 2 b) i n
r2f a c t
It
which i s weakly
o(
f ( x ) (7C&d> E. A ) 6
[PI
.
i m p l i e s t h a t f o r every p c u
(we may assume without loss o f g e n e r a l i t y t h a t There i s a parameter
> S
. I n o r d e r t o show
f : rcf A +
). Lemma 5 b)
36 >
a.
.
=*
D r,A
7
.
.
1' 6 M
S <
such t h a t
i s non-hyperregular s i n c e
(A
A
B M
W
we f i x a c o f i n a l f u n c t i o n
a-recursive i n
and
= A(e) o;+l n w1 Further we 1 by t h e d e f i n i t i o n of S which t o -
v1
bq A
G1
A(e) n
The proof o f Theorem 1 and
and
according t o case 1 ) of t h e
M
which i s absurd s i n c e we have proved j u s t before t h a t
that
M
Q
f o r a l l these stages
r(g(z), I f ) =
follows from Lemma 4
H
c a n l t be t h e case t h a t
because t h i s c o n t r a d i c t s
r
f
2 6 >'trf it i s obvious t h a t A'
i f we s t a r t with some
r(g(a),u)
with negative neighborhood
can be expressed &-recursively i n
A'
.
259
HIGH a-RECURSIVELY ENUMERABLE DEGREES Case i i ) : a > w 2 c f a = The proof of Theorem 1
.
= w
U ~ D S
i s simpler i n t h i s c a s e s i n c e t h e
problem at l i m i t s of t h e p r i o r i t y l i s t d o e s n ' t occur ( s e e point 4 ) of t h e motivation). The c o n s t r u c t i o n i s c l o s e r t o t h e one i n ORT I 2 3 1 but we have t o be aware o f t h e o t h e r p o i n t s i n t h e motivation a n d t h e f a c t t h a t we c a n ' t use t h e r e g u l a r i t y of
as it i s done
A
i n ORT ( " t r u e s t a g e s " ) . According t o point 5 )
of t h e motivation we f i x a s t r i c t l y
increasing cofinal function recursive i n
B(O)
fw(X)J
3 %6 Q
If
:@
f : r c f D + a which i s weakly
with an index
. We d e f i n e t h e n
e
3 y 3 H ( E W
we go t o t h e l e a s t such
fQ(x) 1
e*c
r sw
minimal ( w i t h r e s p e c t t o a f i x e d c a n o n i c a l
4- o f Lot) say t h a t
such t h a t
$
f'(x)
and
E We,-
fi
A
A
H
5
a-
- A?)).
(01 I L,
and choose tx,$,$,
A 1 L,
well ordering
.
- A?)
c L,
We t h e n
i s t h e n e g a t i v e neighborhood of t h i s
computation. F u r t h e r we f i x a Z2 L, and
maps
g
approximation
3u
Vn<
w
that
g"(*)
all
cr
.
1-1 onto
w
v
g'(*)
m
,c
n
to
vc
function
. We have
ci
g
g
where (gr(m)
i s 1-1 f o r every
dom g ' ( * ) =
I
dom ,g
g(m))
OL I
. We
For all
x
e,
rr
a oc choose
l(e,u)
There i s a s t a g e neighborhood
at s t a g e
v
(Cw&
l(e,w)
~
require further
g(0)
C
go(0)
and
a
s rcf
D
and
w
Analogously as i n Soare C231 we d e f i n e f u n c t i o n s r e l a t i v e t o f i x e d enumerations
= u
i n t h i s case a very n i c e
and t h a t
u
such t h a t
=
for
0
1 and
' ,.
r
maximal such t h a t f o r
the following holds : %
cu
such t h a t
fr(x) 4
and t h e negative
$Iof t h i s computation s a t i s f i e s
t h e r e e x i s t s a computation o f
k
5
"C -.",A1'
L,
- A,
for
and
260
WOLFGANG MAASS
:= f T ( x )
llKx,r
satisfies
H C'
that
-
(we then w r i t e
A,
u ( e , x , a ) := p i ( ; s r I
w i t h negative neighborhood
C"
f o r t h e minimal such CI
H ey )
A
H )
that
)
Q
If we then have f o r t h i s
and
f o r t h e minimal such
.
y < u(e,x,a)
.
F i n a l l y we demand ( f o r any
H
we then demand in t h e case
is a successor stage t h a t no
at stage u-1
A
-
c L,
H s L,
and
S Q
For
- C,
w a s put i n t o
CG n f Q ( x ) = C,
l(e,o)
that
n fa(x)
l(e,o) < rcf D
. and
a l l t h e conditions in t h e d e f i n i t i o n a r e satisf i e d except t h e last one ( i . e . C? n f w ( x ) = C, n f G ( x ) ) we s a y
for
that
x = l(e,u) e
i s i n a c t i v e at s t a g e
r(e,cr) :=
Q
and define
x I l(e,u)I
.
sup Cu(e,x,e) I x < l ( e , u ) 3
.
I
sup Cu(e,x,w)
Otherwise we define r ( e , o ) :=
It is convenient t o choose t h e universal enumeration
i n such a way t h a t f o r all
r
Wo
.
so t h a t we have
=
(We)e
l(0,w) = r ( 0 , a ) = 0
Construction : A t stage
o
c Rg gv(.)
i n t o
we consider every
. If
A
<(3,y)
such t h a t
E
c ( l , r > i s not already an element o f
at stage
< ~ ,> r2 r(gQ(m), u )
w
A,
if
for all m 6 n
where
gQ(n) *
p
End o f construction.
The claims o f Theorem 1 following LemmK
.
follow as in case i ) from t h e
we put
.
H I G H +RECURSIVELY
n e o we have
Lemma 6 : For every , ( g ( n ) ) =*,(g(n))
a)
Proof: Induction on since
+
Wo =
that
u
that
A(
that
vU
Tn :=
i c ,uo I r
into
:=
8 e0
I
Vm
5
I :=
m
{
d
g(e) < n )
n (gv(m)
u-1
n 130.
6
Then t h e r e i s a stage Q
a el V m
6
=
D
m 6 n
A
Q
c Tn j
C s$~) A
<
t h e following
i s regular. Choose
i s i n a c t i v e at u
Qo
such
v0
and define
g(m))
Tn ( g(m)
v1 2
n = 0
and t h e induction hypothesis
.
I
i s put
)I
3
.
.
'
m e ( n + l ) - I and assume t h a t s u p ( l(g(m),o)
y
= y n Atn)
such t h a t
sup f r ( g ( m ) , w ) l
r e Tn3
= rcf D
which implies t h e c o n t r a d i c t i o n
. Thus we have shown t h a t
A(g(n)) =* ,(g(n)) b)
. Assume f o r
y tl A ( 4 n )
(by t h e d e f i n i t i o n of r c f D) C
0
such t h a t
Then we have
OL.
=
=
I ( r(g(m),r) = r(g(m),T))
Take f u r t h e r any
c Tn3
B
b) are t r i v i a l for
i s a successor stage and an element
at stage
A('n)
Define
Q
gQ(0)
W e get from t h e p r o p e r t i e s of
a)
V
. a ) and
n
and f o r a l l u
.
n > 0
and
.
b) i C ~ g , ' ~A)
26 1
ENUMERABLE DEGREES
sup f r(g(m),w) I
which i s used f o r t h e proof of
Q
as usual.
implies t h a t
sup t l ( g ( n ) , u ) I u L Tn)
= rcf D
which i s absurd according t o t h e preceding.
The proof of Theorem 1 i s now f i n i s h e d . W e have proved Theorem 1
i n order t o get t h e following c o r o l l a r y :
Corollary : Assume t h a t incomplete high a-r.e.
u 2 c f a )v2poc
degrees.
. Then t h e r e e x i s t
262
WOLFGANG M A S S
Proof of the corollary: The case Shore C203
. For the other admissible
non-hyperregular a-r.e. sets D
OL
OL
= o2cfo~ is proved in
there exist incomplete
if o2cfa >, U 2 p a
according to
Shore [ I 9 3 (see also E l l ] for another proof of this fact). Apply Theorem 1
$2.
D and an u-r.e. set C
to this set
0'
.
0312
The depree For those
6
ci
where incomplete non-hyperregular a-r.e.
degrees exist there exists a distinguished a-degree between 0' and
for which we write
0"
0312
. We will show in the following
and in t 1 1 3 that there is a close connection between O 3 I 2
and the
jump of non-hyperregular a-r.e. degrees. Lemma 7 :
rx
Assume
is such that incomplete non-hyperregular 0312
a-r.e. degrees exist. Then there is an a-degree 03/2 cg
a)
01
b)
0312 is the greatest
0, L, C)
set and
D
such that
01'
A2
0312
L,
degree
(i.e. 0312 contains a
for every A ~ L , set D
0312 is the greatest tame- z, L, degree (i.e. 0312 contains
a set
S
such that { K c L-1 K
D B,03/2
Remark:
*
If a is
A, Lu degree and
for the
OL
0312
0"
1
is
z, L
o1
and we have
D with this property )
for every set
swa
d) u 2 ' a
S
6,
I,
for the set U 2 L a ~ O fand t any 8 , admissible then 0 '
is the greatest
is the greatest tamez, L
degree. Thus
of the Lemma they meet together in the middle, one
coming from below, the other coming from above.
H I GH
8-
P r o o f : $:= tL,,C) set
missib1e.A S
missible
r,
0
i s A2L,
xl$,).
and
r2 L,)
(tame-
i s ina d-
i f and o n l y i f
Friedman [ 3 ] obse rve d t h a t f o r inad-
A l Lp
a greatest
(3
between
w i t h C c 0' r e g u l a r and cl-r.e.
S B L,
A 1 '& (tame-
is
263
RECURSIVELY ENUMERABLE DEGREES
P-degree e x i s t s which l i e s s t r i c t l y
and which i s a n u p p e r bound f o r t h e tame-
0'
d e g r e e s . T h i s r e s u l t c a n ' t be g e n e r a l i z e d t o a l l ina d-
Lo
m i s s i b l e s t r u c t u r e s c L ,D) even i f D i s r e g u l a r o v e r L o : b and regular i s The s t r u c t u r e f+ = < L L,C > w i t h C I 0' a:-r.e. i n a d m i s s i b l e (we h av e
nu
A,$-
is the greatest
w = o l cf s x L u < w l p S
degree
. But
. A cco r d i n g t o Lemma
01
=
u2pw
but
)
0'
% = < Lp,B> where
w e h ave o 2 p a < at f o r t h o s e
1
where i n c o m p l e t e n o n - h y p er r eg u l ar
d we have u - l p at
= ?:t
F r i ed m a n' s argument works as
w e l l f o r those inadmissible s t r u c t u r e s
0 1 p ~ t l )< (3
:w
w-r.e.
degrees e x i s t . Since
5 =
f o r the considered s t r u c t u r e
t h e r e i s no problem w i t h t h e a d d i t i o n a l assum ption i n t h i s c a s e . Take a r
I
Al&
and d e f i n e
C v M
:=
set if
S
c L,
set
out of t h e g r e a t e s t
2x I x
E C
that
S
i s (weakly) $ - r e c u r s i v e
a-recursive i n
in
03'2
2 contains a
I n o r d e r t o p r o v e c ) i t remains t o show t h a t $ s e t . I n t h e case e 2 c f a z -2pw
Theorem 1
i n C91
.
If we have u 2 c f o t
s t r o n g l y i n a d m i s s i b l e and tamegree
0
zl &
4
may o r may n o t e x i s t f o r t h e s e
f i n e s t r u c t u r e of
t h i s f o l l o w s from
0 2 p ac
then
&
s i t u a t i o n where i n co mp l et e n o n - h y p er r eg u l a r
x
&
r2p
01
a
is
s e t s which a r e n o t of de-
, de pe nding
& as it i s shown i n 52 o f [ g ]
we h a v e an & - c a r d i n a l
i f and o n l y
by u s i n g t h e c o r r e s -
.
E
M
. T h e r e f o r e we c a n prove
C v M
ponding p r o p e r t i e s of t h e &-degree
El
set
1 u f 2 x + 1 I x c M 3 . Then we have f o r e v e r y
a ) and b ) f o r t h e s o d e f i n e d oc-degree
tame-
%-degree
Al$
t o be t h e &- d eg r ee of t h e A 2 , L
03/2
i s (weakly)
S
M c ct
s u ch t h a t
a-r.e.
on t h e
. However
i n our
degrees e x i s t
rr2cfLaw
= o2cf g
264
WOLFGANG MAASS
. Therefore we can apply the construction of and get tame- ,F,% set of degree ; . Lemma 5 in Property d) follows from Theorem 2 in C91 . according to Lemma 1 L9]
4
The greatest 4, L ,
Remark :
z2L,
and the greatest tame-
degree can be determined for the other admissible
well. The results might be useful for the study of
0~
r2 L,
as
de-
grees. For u with u2cfa < u 2 p a = oc we have that the greatest
d2 L,
degree is equal to
degree is equal to
0'
places compared with
and the greatest tame-
0"
(thus these two degrees have switched their
z2admissible at ).
For the other ac with the property that
0'
is the only
non-hyperregular x-r.e. degree we have that v2cf OL and in this case there is a greatest ween
0'
and
A,
L
4
u2p u s
Q
degree strictly bet-
whereas the greatest tame- Z, degree is either
0"
equal to the greatest A , degree
is equal to
r2 ,L
(if e2cfLa(e2por) = o2cfs )
or
(otherwise) as one can see by using Lemma 1
0'
and arguments of $ 2 in C93
.
For all a which are not 1, admissible we have that the greatest
r
I.,
A 2 L,
degree
for every
2 has the property that U2L"
Swag
+b
w-degree a _ .
The following rather technical Lemma will be the heart of the proof of Theorem 2
'. It generalizes an observation of Shore
(Lemma 3.3 inc181) which also has important applications in (3recursion theory (see Lemma 3
, $2
in L91
1.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Lemma 8 :
265
$= c L p , B ,
Consider a structure
and a limit 9i rL ordinal I -r (I such that ulcf (3 < g and vlcf a 9 1,P 1,P (see $0. for definitions).
*
J
If Dc- L A is regular over L A and [ K {K
L21K 5 LA
6
Proof:
C,
XI% then
D3 is
D3 is Zl& as well.
The same trick as in Shore C181 is used. F i x a
y
definition
-
L iK
E
*
of the set f K c LA[ K c D 3
zl3
a cofinal
zl%
function p : vlcf X + I and a cofinal Z,% function .L %% & q : ulcf p + (3 Define a set M S ulcf a r d c f (J by
.
bM
:t)
VX
nl$
c Lp(y)
(X 6
D +
B'
q(d)*Lq(d)
t3
K ( x E. K A ~ ( K )
Then we have in fact M o L o and thus get a of f K L LxIK
K QL
x
A
- D 3 as follows : - D c) 3 ~ & ( < 8 , 6 >M€A
'
17).
zl d
definition
Lx
K 5 L1
K S L
P(y1
*
Ls(d) A
< Lq(~)~Ls(6)n B ) k C V x E K i a K ' ( x 6 K ' A V ( K ' ) ) J ) .
Theorem 2 : Assume that ff2cf ci * - 2 p u
and 2 is an in-
complete ci-r.e. degree. Then we have a)
@'
b)
= 0'
if 2 is hyperregular (Shore)
= 0 3 ' 2
if 2 i s non-hyperregular
Proof : a) i s contained in Shore f201 immediately from Lemma 8 : at-r.e. and regular, 2 := where
A
b)
6s
Q,
Choose D
6
2'
is a-r.e. and regular
Assume that A
6
*
.
. .
and
It follows
< L,,C > with C 6 0' regular and Z1 < Lu,A > :=
2 is al-r.e., incomplete, regular and
non-hyperregular. Then we have
o(
olcf<~*A'a
~lp<~a~~'a
according to Shore Cl8] (this fact follows immediately from Lemma 8
1- For
=
:= vlcf
function g X,
266
WOLFGANG MAASS
which a X
maps
3
Z2 formula
f s
vy
+9
. Take any s e t
1-1 onto 'Ic
OL
3
y
v z ~ < x , y , z ) over
Zl@(XyyyZ) H
[el
1 $ ( x Y ~ , z ))
f o r some f i x e d index
3
3
I n order t o get ( t h i s implies
A'
showing t h a t i s a 1-1
r2L,
is
:= f [ A 1 ]
. Since
where
f :
1
:= ~ l p < ~ * ~ ~ and ' o r D :=
with
C
E
a-r.e.
0'
= u2cfa
( t a k e a c o f i n a l Z2 L,
Q
We do t h i s by
+ ~lp*~,,~'-
t o the structure
.
?iThe assumptions of t h e Lemma a r e
sr
u l c f 9rb
u 2 p 0 ~= fl,ol
function
u2cf
6
q :o ~ c f+ a a ; f
,
= alp'LQyA'a
i s bounded f o r every
A"
i s r e g u l a r over
{K
e L~
IK
.
Q
OL
<
y1 %
E
XI
Therefore
EK
~y <
La
is 6
rJlcf'La*A>o( t h e r e f o r e
1
),
A" i s
(because
z1
LA I K s L2
which implies t h a t
-
(since
is
f-lCRg f
rl (La,A> A&
3 i s Zl'&
n2 L,
.
)
YQ
is then
q
because according t o Shore L18] we have
cofinal i n
p,'L-9A'
i s obviously
A'
and r e g u l a r ,
all satisfied i n t h i s situation :
u l c f SGOL
i s A, L ,
A'
r2L,
is
A'
Zl < L q y A > map. We apply Lemma 8
:= tL,,C>
Lemma 8
i n t 2 0 1 ) . We g e t t h e n
by Lemma 7 b )
03/2
$a
.ik
We have
A
( i t i s t h i s fact
we show t h a t
A'
O3I2
it i s enough t o show t h a t
Z 2 L,
(g(f) = y
from Lemma 7 d ) .
A'
dq
Z
S CWa A '
which i s a c t u a l l y proved i n Theorem 2.3.
03/'
Then we have
Ix3 s A'
x tt
. T h i s implies
e
.
L,
VyPE
which i s defined by
S
n
and
glcf'La*A'a
1.
according t o
t l
26 7
HIGH a - R E C U R S I V E L Y ENUMERABLE DEGREES
$ 3 . Summarp Two factors determine the results about the jump of u-r.e. degrees : the relative size of u2cfcc
and W2poc
and
the existence of incomplete non-hyperregular u-r.e. degrees. Therefore we distinguish four different types of admissible ordinals oc :
( I I a2cfor
%
w2pw
a n d there exist
"0 incomplete non-hyper-
regular u-r.e. degrees which are
(these are exactly those
(2) o2cfa
3 v2pa
I, admissible)
and there exist incomplete non-hyperregular
a-r.e. degrees (these are exactly those
&
rr2cfoc < 02por
OL
which satisfy
o(
> a-2cfa 8 u 2 p u )
and there exist incomplete non-hyperregular
a-r.e. degrees
&
u2cfa
0 - 2 ~ and ~ there exist "0 incomplete non-hyper-
regular at-r.e. degrees , For the types (2) and ( 3 ) there exists the distinguished degree 03/2
between 0 '
been described in Lemma 7 For
that
a
@'
.
0''
at of type ( 4 ) we have
or-r.e. degree For
and
with the properties that have
p' = 0'
for every incomplete
& (Shore [20]).
a of type ( 3 ) we have for incomplete a-r.e. degrees k = 0'
if
is hyperregular respectively 2' = O 3 l 2
is non-hyperregular according to Theorem 2 For
.
if
a of type ( 1 ) and (2) there exist incomplete o(-r.e.
degrees & such that for type ( 1 )
1.
a'
= 0''
according to $1. (see Shore c2OJ
WOLFGANG MASS
268
In particular we have thus shown the following : Corollary:
Assume that
is admissible. Then there exist
~~
high incomplete a-r.e. degrees if and only if U2cfa
a2pa
.
We will continue the study of type ( 1 ) and ( 2 ) in c111. It turns out that ( 2 ) is the most interesting type as far as results about the jump of u-r.e. degrees are concerned.
FtEFERENCES : C1’J S.B.
Cooper, Minimal pairs and high recursively enumerable
degrees, J.Symb.Logic 39 ( 1 9 7 4 ) , 655-660 L23
K.J. Devlin, Aspects of constructibility, Springer Lecture
Note 354 (1973) [3]
S.D. Friedman,
[4]
A.H. Lachlan, A recursively enumerable degree which w i l l not
(3-Recursion Theory, to appear
split over all lesser ones, Ann.Math.Logic 9 (1975), 307-365
151 M. Lemnan, Maximal a-r.e. sets, Trans.Am.Math.Soc.
188
(19741, 341-386 r.63
M.Lerman and G.E. Sacks, Some minimal pairs of &-recursively
enumerable degrees, Ann.Math.Logic [7]
W. Maass,
4 (19721, 415-422
On minimal pairs and minimal degrees in higher
recursion theory, Arch.math.Logik 1 8 ( 1 9 7 7 ) , 169-186
[81 W. Maass, The uniform regular set theorem in at-recursion theory, to appear in J.Symb.Logic W. Maass, Inadmissibility, tame RE sets and the admissible
[9]
collapse, to appear in Ann.Math.Logic I101
W.
Maass, Fine structure theory of the constructible
universe in oc-
and p-recursion theory, to appear in the
269
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Proceedings of "Definability in Set Theory" (Oberwolfach 19771, Springer Lecture Note [ll]
W. Maass, On
OL-
and
0-recursively enumerable degrees,
in preparation
[123
G.E. Sacks, Recursive enumerability and the jump operator,
Trans.Am.Math.Soc.
1133
G.E.
108 (1963), 223-239
Sacks, Post's problem, admissible ordinals and
regularity, Trans.Am.Math.Soc.
L143 Ann.
[15]
(1966), 1-23
G.E. Sacks and S.G. Simpson, The ct-finite injury method, Math.Logic 4 (1972), 323-367 J.R. Shoenfield, Degrees of Unsolvability, North Holland/
American Elsevier, Amsterdam/New York
116'1 R.A.
(1971)
Shore, Splitting an a-recursively enumerable set,
Trans.Am.Math.Soc. 204 (1975), 65-77
LIT] R.A. Shore,Minimal 4-degrees,Ann.Math.Logic ClSl
R.A.
4( 1972),393-414
Shore, The recursively enumerable u-degrees are dense,
Ann.Nath.Logic 9 (1976), 123-155
[191
R.A.
Shore, The irregular and non-hyperregular K-r.e.
degrees, Israel J.Math. 22, No.1,(1975),
28-41
f201 R.A. Shore, On the jump of an a-recursively enumerable set, Trans. Am.Math.Soc.
C211
217 (1976), 351-363
R.A. Shore, Some more minimal pairs of u-recursively
enumerable degrees, to appear
t223
S.G.
Simpson, Admissible ordinals and recursion theory,
Ph.d.dissertation, M.I.T.
[23]
(1971)
R.I. Soare, The infinite injury priority method, J.Symb.
Logic 41 (19761, 513-529
Q
HIGH a-RECURSIVELY ENUMERABLE DEGREES Wolfgang Maass Msthematisches Institut der Universitat Miinchen A degree g
is said to be high if &' = 0"
the jump of 2 and
where &'
is
0 is the degree of the empty set. Thus 0'
is a high degree but in ordinary recursion theory (ORT) there exist as well high recursively enumerable (r.e.)
degrees below
0'
according to a theorem of Sacks 1121. The proof of this result is a very nice application of the infinite injury priority method. It follows from the theorem of Sacks that the notion high is not trivial. Further results show that the notions high and low (
is low if
a' =
0' ) are in fact important for the study of
the fine structure of the r.e. degrees in ORT. The intuitive meaning is that 2 is high if 2 is near to if
2 is near to
0
0'
and 2 is low
in,the upper semilattice of the r.e.
degrees.
Therefore these notions are useful for the study of non-uniformity effects in this structure where one looks for theorems which hold in some regions of this semilattice but not everywhere (see e.g. Lachlan L43). In addition high degrees are interesting for technical reasons. Some results have been proved for high degrees and it is not yet known whether they are true for all r.e. degrees (see e.g. Cooper
tll).
Finally high degrees are a link between the structure of r.e. degrees and the structure of r.e. Martin
(see[l5]):
sets according to a theorem of
A degree contains a maximal r.e.
set if and
only if it is a high r.e. degree. In a-recursion theory for admissible ordinals 2 39
OL
the deeper
WOLFGANG MAASS
240
properties of r.e. degrees and r.e. sets are explored in a general setting and one tries to find out which assumptions are really needed in order to do certain constructions. We refer the reader to the survey papers by Lerman and Shore in this volume for more information. It turned out that in fact several priority arguments can be transferred to u-recursion theory (see e.g. Sacks-Simpson 1141, Shore 1163, Shore 1181). Other results of ORT have been proved for many admissible
u but 'it is still open whether they hold for all
admissible a( e.g. the existence of minimal pairs of a-r.e. degrees 161 ,[21]
and the existence of minimal a-degrees [171,[7]).
Lerman [5] closed the gap between provable existence and provable non-existence in the case of maximal cc-r.e. sets.
For some time one thought that the existence of high tx-r.e. degrees below 0 '
was as well completely settled by Shore c 2 0 3 ,
but an error was found in the proof of Theorem 2.3.
in [ 2 0 ]
* . The
problem was then open again except for z2-admissible u where the existence proof from ORT works and for
d
such that
0'
is the
only non-hyperregular a-r.e. degree where every a-r.e. degree below 0'
is low according to [201 (these are the types ( 1 ) and
( 4 ) in our characterization in $ 3 ).
We close the gap in this paper by proving that high a-r.e. degrees below 0'
exist if and only if u2cf u a w2p u
. This re-
sult was not expected and is different from the result in 1201. We think that the new result is a lucky circumstance for a-recursion theory since it was thought in C201 that the situation is somewhat trivial (every non-hyperregular cL-r.e. degree is high). Now it turns out that inadmissibility (in form of non-hyperregularity) influences the behaviour of the jump of an a-r.e. degree but is *I would like
thank R . A . Shore for informing me about this.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
241
not s o s t r o n g t h a t it overruns everything ( t h i s w i l l become even c l e a r e r i n our forthcoming paper C113 ). The plan o f t h i s paper i s as f o l l o w s : contains some basic d e f i n i t i o n s and f a c t s . In
we construct high ci-r.e.
> o2cfa 8 v2pa
degrees below
0'
f o r t h e case
. We give some motivation f o r t h e construction
s o t h a t t h i s chapter should be readable f o r anyone who has seen bef o r e an i n f i n i t e i n j u r y p r i o r i t y argument i n ORT (e.g.C23]).
construction r e f l e c t s s e v e r a l t y p i c a l f e a t u r e s of
The
a-recursion
theory and uses s t r a t e g i e s which would not work i n ORT. In
$&
0'
i n t h e case o 2 c f o
we prove t h a t t h e r e e x i s t no high w-r.e. 4
w2pa
degrees below
by using some basic p r o p e r t i e s o f
s t r o n g l y inadmissible s t r u c t u r e s . Along t h e way some first r e s u l t s a r e proved about a distinguished degree between which we w r i t e
O3I2.
A summary i s given i n
. Four types o f
0,'
and
0''
admissible o r d i n a l s have
t o be distinguished as f a r as t h e behaviour o f t h e jump o f r.e. grees i s concerned.
for
de-
242
WOLFGANG W S S
$0. P r e l i m i n a r i e s
p
Lowcase greek l e t t e r s a r e always o r d i n a l s , always l i m i t o r d i n a l s and
a i s always admissible i n t h i s paper.
W e consider only s t r u c t u r e s ';G = < Lp,B> r e g u l a r over D
s LR i s
Lp
, i.e.
zn% i f
' d x < (3 ( L r D
may c o n t a i n elements of
For
2,s
t h a t some
w r i t e s rrnp'p
p
ancfa
A set
D
ancflqa
An o r d i n a l
and
D
E.
$6-r.e.
{ K a Lo
I)-finite i f
&
Lo :
f x l < e , x > s U,"3
is
D
I K
C
D
.
zns
C
sets
i s E n % - i f and only i f
e
E (3 )
U,"
if
we w r i t e
for
(i.e.
D =
zn'&
which are given by some
n = 1
of
for
Wes
A ,D c L o
A srLD )
one says t h a t
i s %-reducible t o
A
i f t h e r e i s some index
D
e e (i such t h a t f o r
K c Lg
all
K
sn&i f t h e s e t i s xn$v . A s e t
i s called a (regular) 0-cardinal
for some
is
D
if
tame-
D3
.
.
anpLa,
~r3
For s e t s (written
.
such
zn$function
($-recursive)
K a Lo
d e f i n i t i o n . I n t h e s p e c i a l case (xi
(which
and one
instead of
W e f i x f o r t h e following u n i v e r s a l every s e t
dr
such t h a t some
anpa
b [ b is a (regular) cardinal]
L,.
En formula
d cofinally into
W e say t h a t a s e t
i s called
say t h a t a s e t
f o r the l e a s t
d s (3
s Lo i s c a l l e d
" p o s i t i v e neighborhoods" K C L,,
. We
6 ( i . e . p maps (3 1-1 i n t o b ). We w r i t e
into
I,% ( A ,$).
Lp )
is
B
L p as parameters) over t h e s t r u c t u r e %
f u n c t i o n maps
i n s t e a d of
L
B G LR and
i s d e f i n a b l e by some
for the least
projects
p
where
B
r\
one w r i t e s a n c f d X
1 c (3
A are
and
A
t)
3 H1 H2 8 L B (
d
A
H1 si D
A
H 2 s Lo
-D
)
and KcLp-Ae3H,
H ~ r L ~ ( ( K , 1 , H l , H 2 > 6 WH~1*s D * H 2 C L R - D ) .
H I GH
The index
e
~ 1 -RECURS I VELY
can be communicated by w r i t i n g
One f u r t h e r defines t h a t
[XI
(written
A SwSD
L,,
6
says t h a t a degree
D
r e s t r i c t e d t o single-
i s defined by
A =$D
A asD
and t h e equivalence c l a s s e s a r e c a l l e d 6-deRrees
D **A
c
K
to
).
An equivalence r e l a t i o n
A E
.
A LZD
i s weakly $-reducible
A
i n t h e same way but with t h e s e t s tons
243
ENUMERABLE DEGRE ES
A
. One
has c e r t a i n p r o p e r t i e s i f t h e r e e x i s t a s e t
which has a l l t h e s e p r o p e r t i e s .
We study i n t h i s paper t h e
a-jump
operator
(see Shore c201
f o r a discussion o f t h e d e f i n i t i o n ) : A'
:= C
H1 H2
i s t h e jump o f a s e t write
instead o f
We
L, (
E
we^
). Since we have
A
4e
H2 c I.,-
A
i n a - r e c u r s i o n theory
A S L ,
WeL&
H1 5 A
A)
(we always
D + A'
ss D f
t h i s d e f i n i t i o n gives r i s e t o t h e d e f i n i t i o n of t h e u-jump
c w ~ 'f o r
operator
We w r i t e s t e a d of
(0')'
bility)
U2La =~
"1
for the
0
6
At
w-degree o f t h e empty s e t and
A
in-
0"
-
. Observe t h a t UlLU 0' and ( u s i n g the a d m i s s i 0" . Furthermore we have f o r r e g u l a r s e t s A t h a t . E
One says t h a t an u-r.e. otherwise
.
2
at-degrees
set
A
i s complete i f
A
0
;
0'
i s c a l l e d incomplete.
We o f t e n use without f u r t h e r mentioning t h e r e g u l a r s e t theorem
of Sacks which says t h a t every a - r . e .
regular
a-r.e.
For a s e t
set A
(see [13],
c L,
[22],[81
one w r i t e s
such t h a t a c o f i n a l function
f :
a-reducible t o
A
rcf A = u
A
. The s e t
, otherwise
A
rcf A
degree contains a
f o r proofs).
f o r the least
b + o e x i s t s which
6
ot
i s weakly
i s c a l l e d hyperregular i f
i s c a l l e d non-hyperremlar.
d
2 44
WOLFGANG MAASS
Observe t h a t we have f o r r e g u l a r ticular
r c f A = o l c f ' L a t A ' a , i n par-
A
i s non-hyperregular i f f
A
Hyperregularity i s
i s inadmissible.
a p r o p e r t y of de-
-contrary t o regularity-
g r e e s r a t h e r t h a n of s i n g l e r e p r e s e n t a t i v s : gree then
i s t h e same f o r every
rcf A
& i s an a-de-
if
A E &
.
Simpson proved i n h i s t h e s i s t 2 2 1 t h a t f o r any
r = rcf
have t h . a t
u - c a r d i n a l and
A
f o r some eC-r.e.
v2cfLu8
= c2cf u
.
A
yc
Q
we
is a regular
iff
The f o l l o w i n g Lemma combines of Shore c19]
i n b ) Simpson's r e s u l t w i t h Theorem 2.1.
. The proofs
z2 p r o j e c t i o n
of a ) and c ) a r e s t r a i g h t f o r w a r d ( c o n s i d e r a
from x
i n t o a2por f o r c ) ). Lemma 1 :
a)
0'
b)
There e x i s t s an incomplete non-hyperregular
either
i s a non-hyperregular
c)
such t h a t
*
o2cfLOLw = u2cfcr
w2pa < x
and
(we w r i t e
w2cfu
zn'&s e t
i n s t e a d of
Jn,p
d-r.e.
OL
. degree
<
iff
and t h e r e i s a
02cfLuw = o 2 c f
d
.
f o r every r e g u l a r a - c a r d i n a l
x
x .
F i n a l l y d e f i n e f o r any s t r u c t u r e
: = p 6 + ( %a(
u2cfa < u
< w2pu
or-cardinal K a o 2 p a such t h a t
We have
f,,P
or w2cfa
0 2 p a 6 w 2 c f o ~ < cu
regular
a-degree i f f
M L 6
b =
M 6 Ln)
jn Lg ). PP
According t o J e n s e n ' s Uniformization Theorem C21 we have 9n*fi
pnla
= o n p (3
f o r every
n > 0
and every l i m i t o r d i n a l (3.
We w i l l o f t e n use without f u r t h e r mentioning t h e e q u a l i t i e s = unpa
for
n = 1,2
which a r e e a s i e r t o show because
uniformization is t r i v i a l f o r admissible a
.
z2 -
We r e f e r t h e r e a d e r t o D w l i n [2) f o r a l l d e t a i l s concerning constructibility.
2 45
H I G H a-RECURSIVELY ENUMERABLE DEGREES
C o n s t r u c t i o n o f h i g h a-r.e.
$1.
degrees
A t f i r s t we s k e t c h t h e c o n s t r u c t i o n of i n c o m p l e t e h i g h r . e .
sets i n ORT. The o r i g i n a l p r o o f i s due t o S a c k s c123. A d d i t i o n a l i d e a s o f L a c h l a n and S o a r e a r e u s e d i n t h e v e r y p e r s p i c u o u s v e r s i o n o f t h e c o n s t r u c t i o n as i t i s p r e s e n t e d i n S o a r e l 2 3 1 (we r e f e r t h e r e a d e r t o t h i s p a p e r f o r more m o t i v a t i o n and d e t a i l s c o n c e r n i n g t h e p r o o f i n ORT ). I n order t o b r i n g t h e requirement
A’
E
i n t h e r e a c h of
0’’
a r e c u r s i v e c o n s t r u c t i o n we a s s o c i a t e w i t h a f i x e d Z‘* set S
a r.e.
0”
*
< e , y > c BS where
3yvz
set
V y’
which i s d e f i n e d by
BS 5
y 3 z i+(e,y’,z)
r2 d e f i n i t i o n
is a fixed
+(- ,y,z)
Then w e have f o r e v e r y
e IQ.
I-S(e)
I
i s enough t o i n s u r e t h a t f o r a l l
e
6
= lim
Y+W lim
of
S
over
.
and i t
BS(
A(
w
L ,
.
l i m B(
.
t h i s c a s e we have t o p u t a l l b u t f i n i t e l y many e l e m e n t s o f E
into
.
A
A c o n f l i c t a r i s e s b e c a u s e w e have t o s a t i s f y as w e l l f o r a l l
e ew
: i C = +e(A)
Ne
requirements ween
Ct(x)
Ne
and
where
C
0’
i s a f i x e d r.e.
s e t . The
a r e s a t i s f i e d by p r e s e r v i n g a d i s a g r e e m e n t b e t f o r a s u i t a b l e argument
+e,t(At,~)
x
and by
f o r c i n g t h e a p p e a r a n c e o f s u c h a d i s a g r e e m e n t ( r e s p e c t i v e l y o f an argument
x
such t h a t
+,(A,x)?
)
Ct(x)
and
w e l l a g r e e m e n t s between
of a n i n i t i a l segment possible
y
of
o
on t h e way o f p r e s e r v i n g as +e,t(At,~)
x
out
which i s c h o s e n as l o n g as
( r s w ) .(We h a v e w r i t t e n O e ( A , * )
which i s p a r t i a l l y r e c u r s i v e i n
for all
A
with index
f o r the function e.)
WOLFGANG MAASS
246
This strategy of preservation in order to get
'I
C se A
is on
first sight contrary to intuition. But if we preserve -as soon as it appears during the construction- agreement between Ct(x)
$ e,t (At,x)
for every
and
x a y we actually try to make C t p re-
C = C l w is not recursive we must then have y " a
cursive. Since
and therefore i+e(A,r) We write
=
C(r)
.
Ie for the injury set of
all elements x
that are put into e' < e
tive requirement Pel with computation in A
Ne which is the set of
A
-as demanded by some posi-
-
although they destroy a
which should be preserved in order to satisfy
. Then we can be a little more exact in our description of the
Ne
preservation strategy and say that although we try to make recursive we only succeed in making
Clr recursive in
r
before). But we can still get the wanted conclusion if I,
ment above. Since
I,
C $ 1,
is recursive in for every
e ao
ment where one shows that for every lim A(te,y,)
Y*Q
e'+ e
A
= lim
Y '*
yew3
B(
for every
e
Cry
(8
as
even
C # Ie for the argu-
is infinite since we need only
we can prove
Ie
f
'I
C se A
(we use here that
C
and
#
fte',y> c B I
e ).
Observe that in writing
C ( g ) etc. we have followed the usual
convention to identify a set with its characteristic function. The construction from ORT works as well for
z2 admissible
el
(Shore c.203). But there are several reasons why this construction does not work for the other admissible u
. We discuss five of
these problems in the following and we simultaneously try to motivate the new features of the subsequent construction for the case Q
>w2cfo
fi
w2pa
.
Assume s a t we succeed in constructing the set A
in such
247
H IGH a- RECURS IVELY ENUMERABLE DEGREES
a way t h a t
Ve
GO(
(A(e) = * B s ( e) )
( d e f i n e f o r any s e t means t h a t
M1
-
M : ~ ( e ):= and
M2 E L,
-
with M n ((el
-
M2
3
yeV Y a ye(- < e , y ) c A)
f o r some f i x e d p a r a p e t e r way q u e s t i o n s
c Stt
ItK
p
e cor
c a n ' t be tame-
P,e,Ye'
e2cfu < a
*
.
A')
6
{ye\ e
if
6
of w i t n e s s e s . S i n c e
K f
i s not
u
we need t h e
A'
r2 admissible
(see
we c a n h a r d l y e x p e c t t h a t t h i s bound e x i s t s f o r a l l a - f i n i t e
$2.)
such t h a t
K
e 6 S
t o questions about
r2 L,
if
i f we want t o r e d u c e i n t h e same
e x i s t e n c e of a bound f o r t h e s e t S c 0''
as b e f o r e
M~ = * M *
;
S 6, A '
that
3 ye(i
. But
LJ
x
BS
M, E: L = ) .
T h i s d o e s n ' t imply i n g e n e r a l t h a t
W e have o f c o u r s e f o r e v e r y
and
S
K G S
.
We overcome t h i s d i f f i c u l t y by u s i n g i n a p o s i t i v e way t h a t a i s not
t 2a d m i s s i b l e . F o r t h e s e s e t s a n d i n t h e case
-r.e.
%I
t h e r e e x i s t non-hype rre gula r
a
> w 2 c f a >, 0 2 p a
OL
t h e r e e x i s t e ve n
incomplete non-hyperregular
a-r.e.
But for n o n - h y p e r r e g u l a r
we c a n a v o i d t h e s e a r c h f o r w i t n e s s e s
Take a c o f i n a l f u n c t i o n
ye :
a-recursive i n
e
vx
Q
S
f*
[pf
I
rcf A
A
.
I
ref
for
p
z
A
b
y
A
i < e , z > c A)
*
which i m p l i e s t h a t f o r e v e r y a - f i n i t e
W e say
It
At
C
=-recursive
respectively
It
which i s weakly
CAI
(plxrcfAxK
Convention:
Q
Then we have
we have
K s S -
*
f : rcf A
r c f A 3 y z(y = f ( x )
6
f o r some f i x e d p a r a m e t e r K
A
sets a c c o r d i n g t o Shore C193.
'I
. i n t t and
"weakly c t - r e c u r s i v e i n "
4wfftt as u s u a l . But t h e r e i s a pro-
blem w i t h t h i s i n t e r p r e t a t i o n , seeL91.
11
For t h e considered
happen t h a t
0"
u where
d o e s n o t c o n t a i n a regular
t o r113 t h i s o c c u r s i f and o n l y i f
~ 2 p a it can
a > w2cfa
I,
L,
v3cfa < e3p u
set. According We w i l l con-
.
2 48
WOLFGANG MAASS
s t r u c t i n L111 a n
ci
where
cr 3 cf u
a3pu
<
w2pu
I
cr2cfor
G
a.
T h i s example i s t h e m o s t - d i f f i c u l t one w i t h r e s p e c t t o o u r cons t r u c t i o n of i n co mp l et e h i g h cc-r.e. contain a regular
t 2s e t
degrees since
and we have w 2pcl <
t h e d e f i n i t i o n of t h e tame
z2 p r o j e c t u m
does not
0"
t a 2 p u ( s e e C63 f o r
t v 2 p u ).
I n consequence o f t h e p r e c e d i n g t h e p l a n f o r o u r c o n s t r u c -
2
t i o n i s as f o l l o w s : We t a k e a f i x e d i n com ple te n o n - h y p e r r e g u l a r a-r.e.
set
and make s u r e t h a t
D
n o n - h y p e r r e g u l a r. A(e)
= * Bs(e)
requirement A(e)
. As
Pe
.
Further f o r
A(')
e > 0
=*
D
i n o r d e r t o make
A
we want t o have t h a t
b e f o r e we s e t up f o r e v e r y
e
s . u
a positive
which t r i e s t o s a t i s f y t h i s c o n d i t i o n c o n c e r n i n g
It i s c r u c i a l f o r t h e i n f i n i t e i n j u r y argument t h a t t h e s e t
o f t h o s e e l e m e n t s which s h o u l d be p u t i n t o
A
i n order t o s a t i s f y
a l l r e q u i r e m e n t s o u t o f a n i n i t i a l segment o f t h e p r i o r i t y l i s t i s
n o t t o o c o m p l i c at ed . According t o p o i n t 2 ) t h i s f o r c e s u s t o make o u r p r i o r i t y l i s t no l o n g e r t h a n o 2 p a sets BS n K
of
K L
be c a use o n l y f o r c i - f i n i t e
a-cardinality l e s s than v 2 p u
Lu is a-recursive.
it is guaranteed t h a t
It i s n o t e a s y t o work w i t h s u c h a
short p r i o r i t y list i n an i n f i n i t e i n j u r y construction since t h e o l - r e c u r s i v e a p p r o x i mat i o n t o t h i s l i s t i s v e r y weak i f 0 2 p a < e2cfu
. We i n t r o d u c e
a c l a u s e b ) i n t h e c o n s t r u c t i o n which makes
it p o s s i b l e t o c o n t r o l i n many s i t u a t i o n s t h o s e unwanted i n j u r i e s which a r e m e r e l y due t o bad g u e s s i n g o f p r i o r i t i e s . We w a n t t o p r o v e by i n d u c t i o n on t h e p r i o r i t y f o r every
e
case t h a t
p(e)
we have
=*
. There
that
i s a problem i n t h e
i s a l i m i t o r d i n a l since t h e i n d u c t i o n h y p o t h e s i s
d o e s n ' t imply t h e n t h a t p(i) < p(e))
A(e)
p(e)
u { A(i)
I p(i)
< p ( e ) ) =* U tB(i)
1
and we c a n ' t c o n t r o l t h e d e g r e e of t h e i n j u r y set
249
HIGH LRECURSIVELY ENUMERABLE DEGREES Ie
. We use the fact that this situation is only possible
a2cf OL > w
since o2cf a 5 u2p
OL
. a2cf a >
u
if
implies that there
are enough fixpoint stages in the construction so that it is in fact not necessary to determine the degree of the injury set
Ie
.
5) There is a problem with the preservation strategy of Sacks in the case that there are non-hyperregular injury sets 1, is non-hyper(which will occur in our construction since )'(A regular). If we want to preserve then agreements C(x) = be(A,x) for
x
part of
p
these computations may altogether use an unbounded
A
even if
8
< a
. Since this would endanger the positive
requirements of lower priority we have to be much more careful with preservations. For this sake we introduce "e-fixpoints" in the
. In the case
case e 2 c f o ~? w rcf
I)
o2cfu = u
we divide
o(
into
many blocks as in Shore El81 (doing the same thing in the
case a2cf OL
5
a2pa > w
would be troublesome because of limit
points in the priority list).
Theorem 1 :
Assume that u > e2cf ci k 02por
are u-r.e. sets such that
C (OD
and
then there exists an a-r.e. set A and
A' =u 0"
D
C
and
D
is non-hyperregular
such that
.
. If
D
6,
A
,C
#,A
The rest of this chapter is devoted to the proof of this theorem. After some preparations we will describe the construction
of the set A Lemmata 3 , 4 , 5
for the case v2cf a > that this set
A
w
. We will show in the
has the properties we want. The
construction for the case a2cfu =
o
is rather close to the con-
struction in OR1 and will be discussed briefly afterwards.
250
WOLFGANG MAASS
W e f i x f o r t h e following r e g u l a r a - r . e . that
and
C 4, D
(Dw)w<
is- non-hyperregular.
D
a r e i n the following fixed
~
sets
(C,, ,)
z u such
C,D
and
oL
a-recursive enumerations of
these s e t s .
t, L,
Take a
’-! such t h a t
A. formula Define the
set
Ot-r.e.
set
such t h a t
S cot
OL
I
5
.
a s follows :
oc
t ( l , y > & B : ~ 4( ( ( 3 = 0 h y e D ) v ( ( 3 > 0
LaI= Vy’
and f i x a
0”
6
S tr L a b 3 y v x Y((3,y,x)
(3
B s
S
A
y 3 x 1WP,gt,x)))
Then we have f o r (3 > 0 : f3
E
+ I~yI
S
lfi6 s
4
Iyl
6
BI =
.
OL
Fix an a-recursive enumeration
of
(Bu)w.
and
u
f o r the
B
following. F o r any s e t
and any
x
6
L,
as an abbreviation f o r
M
fi
(fxl
A,
M
.
u
Lemma 2 : Assume t h a t n a l i t y l e s s than e 2 c f a s e t such t h a t
i s an
K
. Further assume t h a t
a define a
cro
for
Rg f
0
x
W
z2 function W ( X )n
LO
and we have
a-cardi-
6 K
. Then
of
W
. For given
f : K * a by
= ~ ( ~ LO ) n
u fW(X)\ x c
I n t h e following we w i l l w r i t e where
~~
A
is an oL-r.e.
i s regular.
Proof : Pix an enumeration f ( x ) :=/*v(w,
at-finite s e t of
i s r e g u l a r f o r every
~ w ( ~x )6 IK ]
<
.
La)
I
w i l l be t h e s e t of elements which have been put i n t o
before stage
u
we w i l l use i n t h e following
.
?here e x i s t s a bound
K3
x c W
i s a fixed
r\
0
.
for
e,r
x,
Lo c W
, L
d e f i n i t i o n of
HIGH *RECURSIVELY
For t h e considered enumerations at stage
"K
La
~r
3 H'
e
-
a-r.e. and
(Al)-
sets
and
A
t h e r e e x i s t s a computation o f
C"
with negative neighborhood
Case i ) :
at
P
W
H'
A
e9ff
and t h e i r
C
we w i l l say t h a t
(CT)-
u
L,(
25 1
ENUMERABLE DEGREES
.A
P
> r2cfa 8 02pu
if
H H
A
5
-
L,
.
As)
c2cfa >
and
for
"C Qe A"
.
(3
The next d e f i n i t i o n is t h e f i x p o i n t device which was mentioned i n point 5 ) of t h e motivation.
A i s an
e-fixpoint at s t a g e
f o r every
7
there i s a
<
(3 I A
: C)
such t h a t
+I
i s a stage
u < 'h such t h a t at stage
t i o n of
"C
se A"
borhood
H
and we have
for
-
lltf
-
H s La
AP
We say that t h i s e-fixpoint CAn A
+ Cfi
n
The " r e s t r a i n t function"
- C"
with negative neigh-
i s i n a c t i v e a t stage (3
.
A
.
and t h e r e
t h e r e e x i s t s a computa-
cr
c , L
C,,
r s z t < ;5
r :a
Q
+
if
w i l l play a similar
at
r o l e a s i n Soare C231 and i s defined by cases : There e x i s t s a stage
Case 1):
i n a c t i v e e-fixpoint at a l l s t a g e s i n Take t h e l e a s t such
u . Define
1
such t h a t some
0-1 (3
C a , @ ] :=
r(e,p)
ft
I
<
.
+PI
w s t
t o be t h e l e a s t
i s an
Q
X <
Q
which i s an i n a c t i v e e-fixpoint at a l l s t a g e s i n Lff,(l]. Case 2 ) :
p
Define
r(e,p)
at
otherwise. W e f i x an 1-1
onto u for
e c
see t h a t
y
t o be t h e union o f a l l e-fixpoints
.
X 2 L,
function
. Using the assumption
g
r (y
which maps cr2poc p a r t i a l l y
w i l l be t h e p r i o r i t y o f t h e requirements
g"(e)
a(
g
n dom g)
< a 2 p o ~ where
B ( < U ) :=
a2pu
4
u
{
it i s easy t o
u2cfor
i s or-finite and
I g"(e)
B('r) < iy
Pe,Ne
h,D
f
.
f o r every
252
WOLFGANG MAASS
a - r e c u r s i v e approximation f u n c t i o n g ' ( * )
We f u r t h e r need a n
( o f two a r g u m e n t s ) w i t h - a - r e c u r s i v e domain which h a s t h e p r o p e r t y that for all y < w2pa f o r all
x
E
y
n dom g
t h e r e e x i s t s an o r d i n a l and a l l t E rr
we have
such t h a t
'ty
gT(x)
g(x)
.
I n a d d i t i o n we want t o have t h a t
3 Wo V a > , uo(
(1)
g(x)J
(2)
V limits A
and t h a t
f*
<
ol(gl(x)
g"(*)
gQ(x)S )
and
* 3 e0 c 1 V o (r0G Q a
J
i s 1-1 for e v e r y
.
w 4 u
X + g"(x) 3 1)
Because o f t h e d i s t i n g u i s h e d r o l e o f t h e r e q u i r e m e n t f u r t h e r need t h a t
g(0)
2
0
and
gQ(0)
E
Po
for all
0
The d e f i n i t i o n o f an a p p r o x i m a t i o n f u n c t i o n
g'(.)
we
IY <
a
.
w i t h these
properties i s routine. Observe t h a t i n g e n e r a l we c a n ' t g e t t h e f o l l o w i n g p r o p e r t y which one would r e a l l y l i k e t o have :
vx
< (r2pc43uovz d
y
v0
5 uo( g O ( z )
g(z) )
E
( s e e t h e p o i n t s 2 ) and 3 ) ).
Construction : A t stage
u we c o n s i d e r e v e r y
f o r some
z ce2pa
If
E
i s n o t a l r e a d y a n element of
<(J,y'
at stage
.
'(3,~'
$+, Ab
such t h a t g Q ( z ) S P
we p u t
into
if
u
a)
( p , ~ )3
b)
qp,y, I z
r(gQ(zt),v) for all
( z + l ) n dom g r
.
End o f c o n s t r u c t i o n .
for all T
u
z ' e ( z + l ) n dom gQ
such t h a t n o t
and
( z + l ) n dom gQ 5
A
H I GH a- RECURS1 VELY ENUMERABLE DEGREES
For a n e g a t i v e requirement boundedly many s t a g e s
t h e r e a r e i n g e n e r a l un-
Ne
where some p o s i t i v e requirement
Q
t r u l y lower p r i o r i t y ( i . e . injure
25 3
g-'(e)
5
Pel
g - ' ( e l ) ) t h i n k s t h a t i t may
because of t h e weak approximation p r o p e r t y o f
Ne
of
g'(0)
.
The following Lemma shows t h a t i n some s p e c i a l s i t u a t i o n s t h e s e unwanted i n j u r i e s w i l l not occur because of c l a u s e b ) i n t h e construction. I n t h e following we always w r i t e
\dca i V x B
that
dom g (g"(x)
0
Lemma 3 : Assume t h a t
y n dom gQ = element
Q
,p,6>
dom g
+,d> < r(gv(z),e) 8'
a
y
n dom
g
and
, rr
and
y
z d
n dom g
such t h a t
A
then there e x i s t s a g(z')
=
Q
,
s, z
. If
at s t a g e
.
(3
5
such
r
.
= g(x))
< w2pa
i s put i n t o
for the least
rr
Q
such t h a t
z
Proof : According t o t h e c o n s t r u c t i o n t h e r e e x i s t s a such t h a t at s t a g e
gr(z')srJ T
f u r t h e r have
we have z' < z
. Since
c l a u s e b ) does not r e s t r a i n
( z l + l ) n dom g7 c ( z ' + l ) n dom gQ because
not r e s t r a i n e d by c l a u s e a ) a t s t a g e t follows t h a t
( z l + l ) n dom g'
=
. Since
gyp ( ( z ' + l ) n dom g ) = gE ( ( z l + l ) n dom g ) have t h a t
z' e dom g
and
g ( z ' ) o g'(z')
2
and
8'
<(J,&>
. We
r&d> i s
r I U a Yr
( z l + l ) n dom g
an
C
and
it
and
. I n p a r t i c u l a r we
0
.
The following Lemma w i l l s o l v e t h e problem which w a s described i n p o i n t 3) of t h e motivation : g-l(e)
o f some n e g a t i v e requirement
have problems t o c o n t r o l i n j u r y s e t 1, 0
I n t h e c a s e where t h e p r i o r i t y
. Lemma 4
U
N~
i s a l i m i t o r d i n a l we
I g - ' ( i ) < g"(e)
3
and t h e
gives a s u f f i c i e n t condition f o r a stage
t h a t some computation which e x i s t s a t s t a g e
u w i l l not be de-
s t r o y e d l a t e r . I t i s important t h a t t h i s c o n d i t i o n can be expressed
254
WOLFGANG MAASS
by using j u s t
U f B(i) I g - ' ( i )
g - ' ( i ) < g"(e)
3
, not
3
g"(e)
<
. A fi-xpoint argument
U f A(i) I
i n Lemma 5 w i l l show t h a t
t h i s condition w i l l be met by an unbounded s e t o f stages. u2por
The p r o p e r t i e s o f imply t h a t f o r every )u' t h a t f o r every
3 53 1( X
3 A( A
z e
u2por
<
8n
t h e r e e x i s t s a stage
i s an i n a c t i v e g ( z ) - f i x p o i n t a t a l l s t a g e s
8n
dom gff =
x
sup f r ( $ ( z ) , u ) \ z
,
dom g
z
r ( g ( z ) , Z ) >/ r ( g ( z ) , u )
x < r(g(z),V)
= B('X)
r\ 0
dom g )
z
€
8
By Lemma 3
x
where
Q
)
r )
t 5 2
+
.
P'
.
.
x < o. with
such t h a t
x =
and no element
at stage
A
d
Q
.
%Q
with
x < r(g(z),
z ' e dom g
Uo)
and
. Therefore we have
Q
n
= B:t)
:
Q
A
d (zl+l)
8''
put i n t o
x
.
z' < z
>c d
. We consider two cases was not put i n t o at s t a g e u since x < r ( g ( z * ' ) , w ) n dom g . By o u r induction hypothesis we have
r(g(z"),c)
a
Uo
t h e r e i s some where
x c B(48) n
3
at stage
V
n dom g
Assume f o r a c o n t r a d i c t i o n t h a t some A
Q
x < U
at stage
A
Q
and f o r every stage t
and no element
i s put i n t o
n
i s put i n t o
x n dom g
€
Proof : Induction on i s put i n t o
such
ey
is a stage such t h a t
0 2 Yy'
BPY)
68 n
Then we have f o r every
f o r t h e l e a s t such
v8'
Lemma 4 : Assume t h a t
g(z')=
Ia
i s an i n a c t i v e g(z)-fixpoint at a l l s t a g e s
8n
im2cfoc
dom g
I n the following we w r i t e
4
and the assumption 02p Q
6
r ( g ( z t l ) , u o ) = r ( g q O ( z l t ) , Qo) a t stage
A
such t h a t
2.
0'
>"x
x
w i l l not be
either.
Qo
w a s not put i n t o
and
A
and not
a t stage
U
since there e x i s t s
U'GQ
( z l + l ) n dom g f f c ( z ' + l ) fi dom gQ'
.
:
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Since
( z t + l ) n dom g"
at s t a g e
s ( z ' + l ) n dom guO
3
x i s not put i n t o
r(g(z),r) I r(g(z),o)
f o r all
. Assume t h a t t h e r e a minimal s t a g e > . By t h e preceding no element is
Q
fro
such t h a t
I
r(g(z),co) < r(g(z),w)
w i l l be put i n t o r(g(s),Vo)
A
at some s t a g e
r(g(z),a)
4
where
t
w
5
y < r(g(z),.)
r c
i s d e f i n e d according t o c a s e 2 ) . Since
r(g(z),a) no element 5 wo :
y < r(g(z),uo)
w i l l be put i n t o
Otherwise assume t h a t
r(g(z),rO)
u1
A
> y
and
0' r(g(z),uo)
y
r
whereas
r ( g ( z ) , a o ) < cr at any s t a g e
i s t h e minimal such
i s defined according t o case 1 ) we have
= r(g(z),uo)
Q
i s t h e r e f o r e only p o s s i b l e i f
is defined according t o c a s e 1 ) of t h e d e f i n i t i o n o f
r
A
uo because o f c l a u s e b ) i n t h e c o n s t r u c t i o n .
It remains t o prove t h a t C '
255
c a n ' t be put i n t o
A
r(g
r
. Since
Q
l(z),o,)
at s t a g e
@ ,
as
it w a s shown i n t h e f i r s t p a r t of t h i s proof. Thus we have proved t h a t some X
i s an i n a c t i v e g ( z ) - f i x p o i n t at a l l s t a g e s i n
w0
[@,,OC) whereas t h e r e i s no i n a c t i v e g ( z ) - f i x p o i n t at s t a g e
Since we have
Txt ,C
definition of for all z LU Remark:
If
Q
stage
'L
au
r(g(z),r) a r ( g ( z ) , o )
u s a t i s f i e s t h e assumptions o f Lemma 4
x < sup f r ( g ( z ) , w )
ment
.
t h i s gives a contradiction t o the
and we have proved t h a t
.
Q
Iz
6 f
r\
dom g
. Therefore t h e s e s t a g e s
3
0
i s put i n t o
t h e n no e l e A
at any
play a r o l e i n t h i s proof
which i s similar t o t h e r o l e of " t r u e s t a g e s " ( s e e Soare1233) in t h e proof i n ORI. Lemma 5 :
a) b)
l c
S
~
=*
A
For every
e B a we have
and
.
Proof : For convenience we prove a ) and b) simultaneously by
256
MAASS
WOLFGANG
. Assume f o r t h e f o l l o w i n g t h a t
i n d u c t i o n on g"(e)
and t h a t a ) and b) a r e t r u e f o r a l l
el
g-'(e)
such t h a t
= z
g"(el)
.
< z
Observe t h a t t h i s a s s u m p t i o n d o e s i n g e n e r a l n o t imply t h a t
u
(A(e')
a3cfa
I g-'(e')
, which
z
5
u
=*
z1
<
f B ( e t ) I g-'(e')
< z J
from
A(et)
Lemma 2
i s regular :
=*
3
i f we have
i s o f c o u r s e p o s s i b l e ( s e e p o i n t 3 ) of t h e
m o t i v a t i o n ) . But we g e t t h e i n f o r m a t i o n t h a t g"(e')
z
<
B(e')
Since every
t h a t every
I
B(")
i s r e g u l a r we g e t
i s r e g u l a r as w e l l . Then
A(e')
u
implies that
I
U
g-l(e') < z 3
is regular. This
i s t h e o n l y fact which w e u s e from o u r i n d u c t i o n h y p o t h e s i s s o t h a t i n t h e case
elpa =
we d o n ' t need an i n d u c t i o n a t a l l ( t h i s i s
Q
r a t h e r s u r p r i s i n g i f compared w i t h t h e s i t u a t i o n i n ORT For
we write
z+l
J:=
f o r t h e set o f t h o s e s t a g e s
M
u
i s unbounded i n
M
~ A ( ~ g ' -) l (I e ' ) For
-
an <
13 z, n An =
dom g
A(' %
z
by u s i n g t h e r e g u l a r i t y o f
OL
.
I
a define
An
A(")n
pt > hn
:=
gz'(y)
6 i(
.
)
t 1
2 := n
H
sup
An
[An\ n is
B LJ
Z2 L,
3
. It
1,
.
X
some 'c
c
c(
IN
with
c(
We have
1
<
for the
i s constant i n
A
B('X)
An+'
a0
> rr*
a(
since the function
that
2e M
and <
U
define then
.
a - f i n i t e set of a l l
rn i s a n i n a c t i v e g ( z ' ) - f i x p o i n t
(Cr,a) := ft'\ r
dom g h )
f o l l o w s from p r o p e r t y ( 2 ) o f t h e a p p o x i -
m a t i n g f u n c t i o n and Lemma 3 We w r i t e
<
:=
an = B ( v ) ~ an
and t h e f a c t t h a t
)
A(")
B (((z+l) n
s e t s it i s e a s y t o s e e t h a t
are r e g u l a r u - r . e .
e x i s t s . For every g i v en
(V y
B:x)~
A
By u s i n g p r o p e r t y ( 1 ) o f g o (a A(")
0-
are s a t i s f i e d . W e w a r t t o prove
where t h e a s s u m p t i o n s o f Lemma 4 that
, seet231).
s
rt c
Q
3
kr0,a) f o r e v e r y
) z'
.
in
zt L z
Cr,a)
such
f o r some
Then we have t h a t r ( g ( z
r IN
hat
11
*)
a c c o r d i n g t o Lemma 4 1
HIGH &RECURSIVELY
where
ro i s t h e l e a s t element o f
show
sup C r ( g ( z ' ) , e ) I
i n order t o prove t h a t
z' E ( z + l )
dom g l
n
<
M
r e
. Therefore
M
i t i s enough t o
- IN)]
z ' € ((z+l) n dom g
A
sup { r ( g ( z * ) , v ) I
.
a
25 7
ENUMERABLE DEGREES
Q C
M
<
A
Thus assume f o r a c o n t r a d i c t i o n t h a t
Vc
-
< a 3 u e M ~ Z E' ( ( z + I ) n dom g
This implies t h a t f o r every
-
K P L,
C
3
c-)
K
Q
The p a r t
-
u c M 3 z ' e ( ( z + l ) n dom g
(sup K < r(g(zt),w)
IN)
we assume t h a t
It+''
s a t i s f y t h e r i g h t s i d e . By Lemma 4
. Therefore t h e r e i s at
r(g(zt),v) f i x p o i n t at
such t h a t
we have t h a t
which means t h a t
Q
t h e r e i s no stage a t stage
A
that
y
L
'c
Cc+l
and
2
X
C, n 1 = C A n
. Therefore t h e r e i s no
- C,
we have proved t h a t
y
s i n c e otherwise some
active g(e')-fixpoint i n
1
%
= C n A
.
1 y
k
and
At
A
By Lemma 4
t %
since
c
MIt
can be expressed
would be an in'12'
6
which shows t h a t
IN
a;
a;+l
:= p z > X;((the
C,
1; = C n A;
n
c OL
we define
A
A;+,
4
u
B(<') . But . Thus we have proved g 3 .
A
zt
Q
( z + l ) n dom
'c
.
< a
C S: A
by
same as i n t h e d e f i n i t i o n o f ( a t stage
-C
C sa B ( * I )
I n order t o prove a ) assume f o r a c o n t r a d i c t i o n t h a t
For
. Thus
K c L,
a-recursively i n B(
S := s u p f r ( g ( z ' ) , a ) I r e M
that
auch
e
The equivalence which w a s j u s t proved implies t h a t t h i s i s absurd because we have
I
1 i s put i n t o
c
X
, contradicting
[%,=)
Cw n
since
r
i s not an i n a c t i v e g ( z t ) -
such t h a t an element
au
is
r(g(zt),e)
some g ( z ' ) - f i x p o i n t
Q
sup K <
do
Q,z',K
defined according t o case 2 ) of t h e d e f i n i t i o n of
c IN
.
C,)
K c La-
A
of t h i s equivalence i s obvious from our
it+tt
assumptions. For a proof of
72'
.
IN)('E < r ( g ( z t ) , v ) )
L,
Q
An+,)
A
t h e r e e x i s t s a computation of
.
258
WOLFGANG M A S S
"C
-C
for
A"
6:
such t h a t
H
s
5
26 c
- C"
A'
:= sup
- A)) .
L,
We have again t h a t given
L,
1n
since the function
ci
n c r A;
. Now it
i s a g ( z ) - f i x p o i n t at
i s defined f o r some
definition of
c o
is
This implies t h a t
Q 6
f o r every
u
z2 d e f i n a b l e
F o r t h e p r o o f o f b) we choose
have
A(e)
-
-
r1 =
g e t h e r shows t h a t
A(e)
D = B(O) =4 A(')
S QUA'
C,
( u s e Lemma 4 )
A
Sz
hyperregular and
0
E
S n
v
x < rcf A
i s now very easy. We g e t
from Lemma 5
x < rcf A have
K
36 >
A
S n
x
rcf A
f ( x ) (7<&6> p e
OL
x
K
c A*
Q
La-
UZLa &A1 $2.)
this
S"
can be w r i t t e n a s a
i C
cqA
i s non-
D
Q
A) 0
6.
.
"K s L a -
n2 formula.
S1I
. CJ .
S )
such t h a t f o r a l l
Concerning t h e computation f o r
"K
and
x
Then we
we observe t h a t
Since we have
( s e e t h e first p a r t of t h e proof o f Theorem 2 b) i n
r2f a c t
It
which i s weakly
o(
f ( x ) (7C&d> E. A ) 6
[PI
.
i m p l i e s t h a t f o r every p c u
(we may assume without loss o f g e n e r a l i t y t h a t There i s a parameter
> S
. I n o r d e r t o show
f : rcf A +
). Lemma 5 b)
36 >
a.
.
=*
D r,A
7
.
.
1' 6 M
S <
such t h a t
i s non-hyperregular s i n c e
(A
A
B M
W
we f i x a c o f i n a l f u n c t i o n
a-recursive i n
and
= A(e) o;+l n w1 Further we 1 by t h e d e f i n i t i o n of S which t o -
v1
bq A
G1
A(e) n
The proof o f Theorem 1 and
and
according t o case 1 ) of t h e
M
which i s absurd s i n c e we have proved j u s t before t h a t
that
M
Q
f o r a l l these stages
r(g(z), I f ) =
follows from Lemma 4
H
c a n l t be t h e case t h a t
because t h i s c o n t r a d i c t s
r
f
2 6 >'trf it i s obvious t h a t A'
i f we s t a r t with some
r(g(a),u)
with negative neighborhood
can be expressed &-recursively i n
A'
.
259
HIGH a-RECURSIVELY ENUMERABLE DEGREES Case i i ) : a > w 2 c f a = The proof of Theorem 1
.
= w
U ~ D S
i s simpler i n t h i s c a s e s i n c e t h e
problem at l i m i t s of t h e p r i o r i t y l i s t d o e s n ' t occur ( s e e point 4 ) of t h e motivation). The c o n s t r u c t i o n i s c l o s e r t o t h e one i n ORT I 2 3 1 but we have t o be aware o f t h e o t h e r p o i n t s i n t h e motivation a n d t h e f a c t t h a t we c a n ' t use t h e r e g u l a r i t y of
as it i s done
A
i n ORT ( " t r u e s t a g e s " ) . According t o point 5 )
of t h e motivation we f i x a s t r i c t l y
increasing cofinal function recursive i n
B(O)
fw(X)J
3 %6 Q
If
:@
f : r c f D + a which i s weakly
with an index
. We d e f i n e t h e n
e
3 y 3 H (
we go t o t h e l e a s t such
fQ(x) 1
e*c
r sw
minimal ( w i t h r e s p e c t t o a f i x e d c a n o n i c a l
4- o f Lot) say t h a t
such t h a t
$
f'(x)
and
E We,-
fi
A
A
H
5
a-
- A?)).
(01 I L,
and choose tx,$,$,
A 1 L,
well ordering
.
- A?)
c L,
We t h e n
i s t h e n e g a t i v e neighborhood of t h i s
computation. F u r t h e r we f i x a Z2 L, and
maps
g
approximation
3u
Vn<
w
that
g"(*)
all
cr
.
1-1 onto
w
v
g'(*)
m
,c
n
to
vc
function
. We have
ci
g
g
where (gr(m)
i s 1-1 f o r every
dom g ' ( * ) =
I
dom ,g
g(m))
OL I
. We
For all
x
e,
rr
a oc choose
l(e,u)
There i s a s t a g e neighborhood
at s t a g e
v
(Cw&
l(e,w)
~
require further
g(0)
C
go(0)
and
a
s rcf
D
and
w
Analogously as i n Soare C231 we d e f i n e f u n c t i o n s r e l a t i v e t o f i x e d enumerations
= u
i n t h i s case a very n i c e
and t h a t
u
such t h a t
=
for
0
1 and
' ,.
r
maximal such t h a t f o r
the following holds : %
cu
such t h a t
fr(x) 4
and t h e negative
$Iof t h i s computation s a t i s f i e s
t h e r e e x i s t s a computation o f
k
5
"C -.",A1'
L,
- A,
for
and
260
WOLFGANG MAASS
:= f T ( x )
llKx,r
satisfies
H C'
that
-
(we then w r i t e
A,
u ( e , x , a ) := p i ( ; s r I
w i t h negative neighborhood
C"
f o r t h e minimal such CI
H ey )
A
H )
that
)
Q
If we then have f o r t h i s
and
f o r t h e minimal such
.
y < u(e,x,a)
.
F i n a l l y we demand ( f o r any
H
we then demand in t h e case
is a successor stage t h a t no
at stage u-1
A
-
c L,
H s L,
and
S Q
For
- C,
w a s put i n t o
CG n f Q ( x ) = C,
l(e,o)
that
n fa(x)
l(e,o) < rcf D
. and
a l l t h e conditions in t h e d e f i n i t i o n a r e satisf i e d except t h e last one ( i . e . C? n f w ( x ) = C, n f G ( x ) ) we s a y
for
that
x = l(e,u) e
i s i n a c t i v e at s t a g e
r(e,cr) :=
Q
and define
x I l(e,u)I
.
sup Cu(e,x,e) I x < l ( e , u ) 3
.
I
sup Cu(e,x,w)
Otherwise we define r ( e , o ) :=
It is convenient t o choose t h e universal enumeration
i n such a way t h a t f o r all
r
Wo
.
so t h a t we have
=
(We)e
l(0,w) = r ( 0 , a ) = 0
Construction : A t stage
o
c Rg gv(.)
we consider every
. If
A
<(3,y)
such t h a t
E
c ( l , r > i s not already an element o f
at stage
< ~ ,> r2 r(gQ(m), u )
w
A,
if
for all m 6 n
where
gQ(n) *
p
End o f construction.
The claims o f Theorem 1 following LemmK
.
follow as in case i ) from t h e
we put
.
H I G H +RECURSIVELY
n e o we have
Lemma 6 : For every , ( g ( n ) ) =*,(g(n))
a)
Proof: Induction on since
+
Wo =
that
u
that
A(
that
vU
Tn :=
i c ,uo I r
into
:=
8 e0
I
Vm
5
I :=
m
{
d
g(e) < n )
n (gv(m)
u-1
n 130.
6
Then t h e r e i s a stage Q
a el V m
6
=
D
m 6 n
A
Q
c Tn j
C s$~) A
<
t h e following
i s regular. Choose
i s i n a c t i v e at u
Qo
such
v0
and define
g(m))
Tn ( g(m)
v1 2
n = 0
and t h e induction hypothesis
.
I
i s put
)I
3
.
.
'
m e ( n + l ) - I and assume t h a t s u p ( l(g(m),o)
y
= y n Atn)
such t h a t
sup f r ( g ( m ) , w ) l
r e Tn3
= rcf D
which implies t h e c o n t r a d i c t i o n
. Thus we have shown t h a t
A(g(n)) =* ,(g(n)) b)
. Assume f o r
y tl A ( 4 n )
(by t h e d e f i n i t i o n of r c f D) C
0
such t h a t
Then we have
OL.
=
=
I ( r(g(m),r) = r(g(m),T))
Take f u r t h e r any
c Tn3
B
b) are t r i v i a l for
i s a successor stage and an element
at stage
A('n)
Define
Q
gQ(0)
W e get from t h e p r o p e r t i e s of
a)
V
. a ) and
n
and f o r a l l u
.
n > 0
and
.
b) i C ~ g , ' ~A)
26 1
ENUMERABLE DEGREES
sup f r(g(m),w) I
which i s used f o r t h e proof of
Q
as usual.
implies t h a t
sup t l ( g ( n ) , u ) I u L Tn)
= rcf D
which i s absurd according t o t h e preceding.
The proof of Theorem 1 i s now f i n i s h e d . W e have proved Theorem 1
i n order t o get t h e following c o r o l l a r y :
Corollary : Assume t h a t incomplete high a-r.e.
u 2 c f a )v2poc
degrees.
. Then t h e r e e x i s t
262
WOLFGANG M A S S
Proof of the corollary: The case Shore C203
. For the other admissible
non-hyperregular a-r.e. sets D
OL
OL
= o2cfo~ is proved in
there exist incomplete
if o2cfa >, U 2 p a
according to
Shore [ I 9 3 (see also E l l ] for another proof of this fact). Apply Theorem 1
$2.
D and an u-r.e. set C
to this set
0'
.
0312
The depree For those
6
ci
where incomplete non-hyperregular a-r.e.
degrees exist there exists a distinguished a-degree between 0' and
for which we write
0"
0312
. We will show in the following
and in t 1 1 3 that there is a close connection between O 3 I 2
and the
jump of non-hyperregular a-r.e. degrees. Lemma 7 :
rx
Assume
is such that incomplete non-hyperregular 0312
a-r.e. degrees exist. Then there is an a-degree 03/2 cg
a)
01
b)
0312 is the greatest
0, L, C)
set and
D
such that
01'
A2
0312
L,
degree
(i.e. 0312 contains a
for every A ~ L , set D
0312 is the greatest tame- z, L, degree (i.e. 0312 contains
a set
S
such that { K c L-1 K
D B,03/2
Remark:
*
If a is
A, Lu degree and
for the
OL
0312
0"
1
is
z, L
o1
and we have
D with this property )
for every set
swa
d) u 2 ' a
S
6,
I,
for the set U 2 L a ~ O fand t any 8 , admissible then 0 '
is the greatest
is the greatest tamez, L
degree. Thus
of the Lemma they meet together in the middle, one
coming from below, the other coming from above.
H I GH
8-
P r o o f : $:= tL,,C) set
missib1e.A S
missible
r,
0
i s A2L,
xl$,).
and
r2 L,)
(tame-
i s ina d-
i f and o n l y i f
Friedman [ 3 ] obse rve d t h a t f o r inad-
A l Lp
a greatest
(3
between
w i t h C c 0' r e g u l a r and cl-r.e.
S B L,
A 1 '& (tame-
is
263
RECURSIVELY ENUMERABLE DEGREES
P-degree e x i s t s which l i e s s t r i c t l y
and which i s a n u p p e r bound f o r t h e tame-
0'
d e g r e e s . T h i s r e s u l t c a n ' t be g e n e r a l i z e d t o a l l ina d-
Lo
m i s s i b l e s t r u c t u r e s c L ,D) even i f D i s r e g u l a r o v e r L o : b and regular i s The s t r u c t u r e f+ = < L L,C > w i t h C I 0' a:-r.e. i n a d m i s s i b l e (we h av e
nu
A,$-
is the greatest
w = o l cf s x L u < w l p S
degree
. But
. A cco r d i n g t o Lemma
01
=
u2pw
but
)
0'
% = < Lp,B> where
w e h ave o 2 p a < at f o r t h o s e
1
where i n c o m p l e t e n o n - h y p er r eg u l ar
d we have u - l p at
= ?:t
F r i ed m a n' s argument works as
w e l l f o r those inadmissible s t r u c t u r e s
0 1 p ~ t l )< (3
:w
w-r.e.
degrees e x i s t . Since
5 =
f o r the considered s t r u c t u r e
t h e r e i s no problem w i t h t h e a d d i t i o n a l assum ption i n t h i s c a s e . Take a r
I
Al&
and d e f i n e
C v M
:=
set if
S
c L,
set
out of t h e g r e a t e s t
2x I x
E C
that
S
i s (weakly) $ - r e c u r s i v e
a-recursive i n
in
03'2
2 contains a
I n o r d e r t o p r o v e c ) i t remains t o show t h a t $ s e t . I n t h e case e 2 c f a z -2pw
Theorem 1
i n C91
.
If we have u 2 c f o t
s t r o n g l y i n a d m i s s i b l e and tamegree
0
zl &
4
may o r may n o t e x i s t f o r t h e s e
f i n e s t r u c t u r e of
t h i s f o l l o w s from
0 2 p ac
then
&
s i t u a t i o n where i n co mp l et e n o n - h y p er r eg u l a r
x
&
r2p
01
a
is
s e t s which a r e n o t of de-
, de pe nding
& as it i s shown i n 52 o f [ g ]
we h a v e an & - c a r d i n a l
i f and o n l y
by u s i n g t h e c o r r e s -
.
E
M
. T h e r e f o r e we c a n prove
C v M
ponding p r o p e r t i e s of t h e &-degree
El
set
1 u f 2 x + 1 I x c M 3 . Then we have f o r e v e r y
a ) and b ) f o r t h e s o d e f i n e d oc-degree
tame-
%-degree
Al$
t o be t h e &- d eg r ee of t h e A 2 , L
03/2
i s (weakly)
S
M c ct
s u ch t h a t
a-r.e.
on t h e
. However
i n our
degrees e x i s t
rr2cfLaw
= o2cf g
264
WOLFGANG MAASS
. Therefore we can apply the construction of and get tame- ,F,% set of degree ; . Lemma 5 in Property d) follows from Theorem 2 in C91 . according to Lemma 1 L9]
4
The greatest 4, L ,
Remark :
z2L,
and the greatest tame-
degree can be determined for the other admissible
well. The results might be useful for the study of
0~
r2 L,
as
de-
grees. For u with u2cfa < u 2 p a = oc we have that the greatest
d2 L,
degree is equal to
degree is equal to
0'
places compared with
and the greatest tame-
0"
(thus these two degrees have switched their
z2admissible at ).
For the other ac with the property that
0'
is the only
non-hyperregular x-r.e. degree we have that v2cf OL and in this case there is a greatest ween
0'
and
A,
L
4
u2p u s
Q
degree strictly bet-
whereas the greatest tame- Z, degree is either
0"
equal to the greatest A , degree
is equal to
r2 ,L
(if e2cfLa(e2por) = o2cfs )
or
(otherwise) as one can see by using Lemma 1
0'
and arguments of $ 2 in C93
.
For all a which are not 1, admissible we have that the greatest
r
I.,
A 2 L,
degree
for every
2 has the property that U2L"
Swag
+b
w-degree a _ .
The following rather technical Lemma will be the heart of the proof of Theorem 2
'. It generalizes an observation of Shore
(Lemma 3.3 inc181) which also has important applications in (3recursion theory (see Lemma 3
, $2
in L91
1.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Lemma 8 :
265
$= c L p , B ,
Consider a structure
and a limit 9i rL ordinal I -r (I such that ulcf (3 < g and vlcf a 9 1,P 1,P (see $0. for definitions).
*
J
If Dc- L A is regular over L A and [ K {K
L21K 5 LA
6
Proof:
C,
XI% then
D3 is
D3 is Zl& as well.
The same trick as in Shore C181 is used. F i x a
y
definition
-
L iK
E
*
of the set f K c LA[ K c D 3
zl3
a cofinal
zl%
function p : vlcf X + I and a cofinal Z,% function .L %% & q : ulcf p + (3 Define a set M S ulcf a r d c f (J by
.
:t)
VX
nl$
c Lp(y)
(X 6
D +
B'
q(d)*Lq(d)
t3
K ( x E. K A ~ ( K )
Then we have in fact M o L o and thus get a of f K L LxIK
K QL
x
A
- D 3 as follows : - D c) 3 ~ & ( < 8 , 6 >M€A
'
17).
zl d
definition
Lx
K 5 L1
K S L
P(y1
*
Ls(d) A
< Lq(~)~Ls(6)n B ) k C V x E K i a K ' ( x 6 K ' A V ( K ' ) ) J ) .
Theorem 2 : Assume that ff2cf ci * - 2 p u
and 2 is an in-
complete ci-r.e. degree. Then we have a)
@'
b)
= 0'
if 2 is hyperregular (Shore)
= 0 3 ' 2
if 2 i s non-hyperregular
Proof : a) i s contained in Shore f201 immediately from Lemma 8 : at-r.e. and regular, 2 := where
A
b)
6s
Q,
Choose D
6
2'
is a-r.e. and regular
Assume that A
6
*
.
. .
and
It follows
< L,,C > with C 6 0' regular and Z1 < Lu,A > :=
2 is al-r.e., incomplete, regular and
non-hyperregular. Then we have
o(
olcf<~*A'a
~lp<~a~~'a
according to Shore Cl8] (this fact follows immediately from Lemma 8
1- For
=
:= vlcf
function g X,
266
WOLFGANG MAASS
which a X
maps
3
Z2 formula
f s
vy
+9
. Take any s e t
1-1 onto 'Ic
OL
3
y
v z ~ < x , y , z ) over
Zl@(XyyyZ) H
[el
1 $ ( x Y ~ , z ))
f o r some f i x e d index
3
3
I n order t o get ( t h i s implies
A'
showing t h a t i s a 1-1
r2L,
is
:= f [ A 1 ]
. Since
where
f :
1
:= ~ l p < ~ * ~ ~ and ' o r D :=
with
C
E
a-r.e.
0'
= u2cfa
( t a k e a c o f i n a l Z2 L,
Q
We do t h i s by
+ ~lp*~,,~'-
t o the structure
.
?iThe assumptions of t h e Lemma a r e
sr
u l c f 9rb
u 2 p 0 ~= fl,ol
function
u2cf
6
q :o ~ c f+ a a ; f
,
= alp'LQyA'a
i s bounded f o r every
A"
i s r e g u l a r over
{K
e L~
IK
.
Q
OL
<
y1 %
E
XI
Therefore
EK
~y <
La
is 6
rJlcf'La*A>o( t h e r e f o r e
1
),
A" i s
(because
z1
LA I K s L2
which implies t h a t
-
(since
is
f-lCRg f
rl (La,A> A&
3 i s Zl'&
n2 L,
.
)
YQ
is then
q
because according t o Shore L18] we have
cofinal i n
p,'L-9A'
i s obviously
A'
and r e g u l a r ,
all satisfied i n t h i s situation :
u l c f SGOL
i s A, L ,
A'
r2L,
is
A'
Zl < L q y A > map. We apply Lemma 8
:= tL,,C>
Lemma 8
i n t 2 0 1 ) . We g e t t h e n
by Lemma 7 b )
03/2
$a
.ik
We have
A
( i t i s t h i s fact
we show t h a t
A'
O3I2
it i s enough t o show t h a t
Z 2 L,
(g(f) = y
from Lemma 7 d ) .
A'
dq
Z
S CWa A '
which i s a c t u a l l y proved i n Theorem 2.3.
03/'
Then we have
Ix3 s A'
x tt
. T h i s implies
e
.
L,
VyPE
which i s defined by
S
n
and
glcf'La*A'a
1.
according t o
t l
26 7
HIGH a - R E C U R S I V E L Y ENUMERABLE DEGREES
$ 3 . Summarp Two factors determine the results about the jump of u-r.e. degrees : the relative size of u2cfcc
and W2poc
and
the existence of incomplete non-hyperregular u-r.e. degrees. Therefore we distinguish four different types of admissible ordinals oc :
( I I a2cfor
%
w2pw
a n d there exist
"0 incomplete non-hyper-
regular u-r.e. degrees which are
(these are exactly those
(2) o2cfa
3 v2pa
I, admissible)
and there exist incomplete non-hyperregular
a-r.e. degrees (these are exactly those
&
rr2cfoc < 02por
OL
which satisfy
o(
> a-2cfa 8 u 2 p u )
and there exist incomplete non-hyperregular
a-r.e. degrees
&
u2cfa
0 - 2 ~ and ~ there exist "0 incomplete non-hyper-
regular at-r.e. degrees , For the types (2) and ( 3 ) there exists the distinguished degree 03/2
between 0 '
been described in Lemma 7 For
that
a
@'
.
0''
at of type ( 4 ) we have
or-r.e. degree For
and
with the properties that have
p' = 0'
for every incomplete
& (Shore [20]).
a of type ( 3 ) we have for incomplete a-r.e. degrees k = 0'
if
is hyperregular respectively 2' = O 3 l 2
is non-hyperregular according to Theorem 2 For
.
if
a of type ( 1 ) and (2) there exist incomplete o(-r.e.
degrees & such that for type ( 1 )
1.
a'
= 0''
according to $1. (see Shore c2OJ
WOLFGANG MASS
268
In particular we have thus shown the following : Corollary:
Assume that
is admissible. Then there exist
~~
high incomplete a-r.e. degrees if and only if U2cfa
a2pa
.
We will continue the study of type ( 1 ) and ( 2 ) in c111. It turns out that ( 2 ) is the most interesting type as far as results about the jump of u-r.e. degrees are concerned.
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