IN
TH£ORY
a-RECURSIC~
by
RI CEAP.D
j':,..
SHORE
(1968)
DEGREE OF
DOC~OR
OF
PH2:LOSOPHY
at the
MASSACHUSETTS INSTITUTE OF
~ECHNO~OGY
Septeri1ber, 1972
S i g rl a 'C u ~ e 0 ~f
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st -:ll. , 1,.::'7,2
Department of Mathematics, Au
Certified.
...
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•
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...
Thesis Supervisor
Ch£tirm.2-r!;)
T~!e p(3.1~t rn8n~: [:"1
c· n Gr acI Lt 2_ -:":·s
r: ':r-"}~::!.. t t :-:: e
~S t.: ~J.(~1
~,~
"',
1,,·
r...,,}
Richard, f'i.. Shore
Subrr.itted to the !)epartrnE:rlt of
thematiC's jn
in partial fulfillment of the requirements for
t, 1972
~ne
de~ree
of Doctor of Philosophy.
ABSTRACT
The pri.ority method as a.pplieci to Y'(:::Cl,U"S:2.C{l I)n
l1i~::sibJ.::ordinals is studied and developed by ~ereralizin~ to ~-recu~­
sion theor::7 several classtc theorer~ls of ordil 1 ;:1.ry y'eeUt'si.crl
theory 'dhcse ppoofs r'eqlJire diff':;rent tY}JCS of' ~)riority 2..':'P;U-·
ITlents.
The siDplest form of the priority cethoi is the Fri
I'·'IuCr})~lj.. l{
ty~)e
of' fj.rlj_te i11 jur Jl ar
l
t
gt.t!1iCl1t
S
9
i~t~;C1~-ll'1:L
i~.
burg1~j,~3;0
been f;ener;:Llized to a--recur::;iDD tneory by :}. Sac~=s a.nd S.
S:!.;';lpson.
In cha~)t2rs I and II ~'l,-::': shoier hO~,I tbe aSSU?T,rJt:'OL of
E~-ad~issibilit
can be used to extend t~eir ~ethodG to ~ne
r)J:)()rJel~~~~es~
Ir:
tl"le f'o ll()f:;tir},~;
Theorem:
If
a
lS
L 2 -admi s sible there i. S D ni nirrlal 0. -::1:::
e
e '" 0 n,r"l~'
) +- c ' "a - r . l..
:::.
C1."""
{)l>""'~ ~ ~ '.:
~. r~ ~
)J
f_ iJ 'S.... e LJ •
'.-< v 1 • \..• .;...
......... ',~ L i.- .L '-.-. 1 ,
.. ". ,.
the \~xlstenc(: of nj_nir:1al a-degree::> are <;11so establ::Ls[)ec;cjlxt
without the usc: of ~J. priori ty 2.ri:~umerlt. In chaote2 2:1 '(·;t::
x
a p,r i or i 't '::7 2. r' [; tlr:"le~J~C t/.~j~ ~~:1 3. fOl"C j_ns~ c c,' ~ 3 ~~ ]~U_ c~, i Cl~ t; c ;),t (J ·~.t (:; t. ~ e
a----;;~;--.;;
- c;; l~ d.1. ::;,r-...~..... \ r a
'- ·l n
J..
,- ....,
1.11 i
r· ..l-
,'....\./- \-,
('l
A?'
\.,;
.C
4
Theo:r'er:L
paraT)1 e
exist
Lj
If CJ.
lS
En-admissible then
Ln
~) u t: .~ e t s of
a.
Indeed, we
unifor~lv
i 1) J~ e
sets
.t\
a:
~-l
far
Inc }"12.. D "',_: e IJ I }~ I
21 (1 ttl j_ :3 ~:
II '0
~~~T (~
L r:; t:
C
t) '1' O\I e t
rJ e a
_~i r2~
S ;-:: 1 i~~~_j
~ t~': 'Z 1.11 a.r
!.}£~~._ ~i~~~ e () r (~r~l
O~ -1~ •
z:. ::' ;~ t
.r~ () ~r'l
~=. ~~~r e}~;1
E;~ .tl cl
::t
2.nd such th::.:. t
In chanter 1'1
for all admissible
of the
VIe
a:
.~:..
3-
carE:
s U ::~ {"l t: lie. t~
<0.
EJ roo .f c~ t~ t,.: J. ~3 i.:,! -: f: C:' e 1~~ ) f~)
C C 0,1'.' S e
1 nf i nit e in j u r ~v D r ~. 0 r it Y aI' p:- Lt T~ e n t
<a. c.
r (:: ~ 1,~ it") (: c ~J_ ;.:~ -'"7 (~l \~)~'._~> .:t ·CL f~~~~ ,'::\ .t"l
for d.ll r:.c :L::; s i 1 (;' 0 I .~ n .,,;,:, S .
1
.
3
ArnonE;, those:"tho have officia.lly been ny te3.c.hsY's I
1\lould Ii ke to
than~\:
r:lave conveyed
~'lho
tho'3e
attituJes as well as formal naterial:
insi~~ht
and
Prof. Burton Jreben.,
ny first ITH.=:nto-r allc]. continuinG: advisor in mathernatie;al
logic, Prof. hartley
Jr., who introduced me
~ogers,
~ost
whc taugnt me set theory as it should be ta
and COD\rictlon and, of course, Pro'f. Gerald Sacks, '/\iho
taught me idodf:?l theory and set
t~leory
theory -' ",) u t mer'e importantJ.,! has
ShOH'1.
as :ie12. f;.S r'2CUTS ion
me
1:0:";
beautiful
and fruitful are the interconnections anong them.
also like
tO~}lank
I \'Jould
Prof. Sacks in his capG.city of thes.is
advisor for encouragins me to work on
and for
rlcJj
a-recursion theory
the Questions dealt with in
s u ~~ _~; (; s t: 1.1i ~:~:: Trian ~l
thts thesis.
Of those from
w~om
I have le2rned
infor~ally,
lowe
3peclal thanks to Stephen SiLlJscn for many lessons learned
but e spes ially for my
theory.
often
FOL' tl1ei.r
half-ba~ed
fi~s
t
i
liS tru.e
';Jil1in:~nes;3
t iC!-j:in
a-recur::;j. on
to listen and react ta my
ideas I would like to thank Lee Harrington
anel <-John naclntyre.
Important debts
~re
also owed to Professors Joseph
for 1:.1s
"
L,f
clear
an(~.
s
Ie presentations of difficult theorems of
To the latter I
r e C i..lr s J. 0 n t 11 e 0 r v in [ 11 ]
indebted for his work on
pleas~re
much
L
[lJ from which I have
a~
gai~ed
and insight.
For financial SUpDort I must thanJ::: the Depa::etn2nt
of
~Iather:1atic.s
at :/t.I.'l'. and the National Science ?ounc:a-
tiona
Finally I
~AJoulc.1
::;'ilce to thank my v;ife., Haomi,. and
my cat f'OI' the!!' conste.n
de
SUDport and COr1D8.n:i.onshJ.p.
1;1-
, it is after my cat that all the important results
in this
t~esis
are named.
5
TABLE OF CONTENTS
2
ACKNOT,'ILEDG;"1ENTS
3
5
IHTRODUCTIOi',y
CHAPTER I:
7
~INIMAL
13
a-DEGREES
1. Definitions and
Bas~.c
Lemmas
15
2. I·linimal Dezrees VTi thout Priorities
23
.. 0"
.
1G
'8J.SSlbl.e
ra Ina
3 • L.'\' 2- Ad'
2_vQ
4. The Construction
32
5. The Priority Argument
39
6. 0 pen
1. The
Q'J. est ion s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . It 1~
Forcin~
48
Relation
2. The Cons true t ion
51
3. The Priority Argument
57
11
"t •
P'.,...,eser~~;'10"
..l ' .• _
v ~ 1 ( '.)
)... n _. ".rlmi··,...ir~l~f-n
\.\.. . ,., i::> ..:; _ U -l.. ...... v.Y • • • • • • • • • • • • • • • hL
-'.J...
65
1. 'llhe Cons true t ien . . . . . . . . . . . . . . . . . • . . . . . . . . . 68
2. The Priority
Argument~
~~
73
TABLE: OF
(cont:tnued)
CHAPTER IV:
rrI-IE DENSITY rrHEORETI
1. The General Plan
83
84
2. The Key Functions and Their
Approximations
87
3. The Requirements and Construction
90
4. The Proof -- An Outline
92
BIBLIOGRAPHY
a
BIOGRAPHY
98
,~
.... 0
7
Introduction
In generalizing or aXionatizing a field of
mathe~~tics
one tries to capture in a more abstract form the basic
concepts of the subject.
Of course, one keeps in sind the
essential properties and fundamental theorems that
like to preserve.
would
o~e
The first major test for the correctness
or everl appropriateness of the
then alsG seneralize the deeper
atte~pt
is whether one can
theore~s
and
l~!Dortant
methods of proof that are characteristic of the original
field.
Thus for recursion theory one atteoDts to reform-
ulate in a new setting the basic notions of cocputation J
recursive and recursively enumerable.
In
for~ulating s~ch
an abstraction Olle is gUided, of course, by one's intuition
about computation but one also has several basic facts and
relatjonships th2t lnust be preserved.
Thus one nakes sure
that basic theoress such as the recursion theorem
a~d
enu-
meration theorem are carried over.
When it comes
ti~e
to apply the above test to an attenp-
ted generalization of recursion theory we feel that the key
technique to be handled is the priority method.
pondin~ly,
Corres-
the crucial theorens are the ones that depend
on the priority method in ordinary
~0cursion
theory.
Cur
reason is that the priority method is the most distinctive
and characteristic technique cf recursion theory, while t}le
~\!hich
theorems for
it
needed
.lS
are :L.n veneral the deepest
and most difficult in the field.
In this thesis we concern ourselves with the generalization of recursion theory to recursi0D on
For
ordinals.
the history of this subject
adm~ssible
an~
general
esp~-
background we refer the reader to [2J, [7J, [8J and
cially to [13J and its bi.blio:.:-raphy.
Af.i far
2_8
the Pl>l.CJ:'>j_t:l
method is concerned the first success in this field comes
in [10J, where G. Sacks and S. Simpson ~eneralize the
Friedburg--l'luchnik solution to Pc,st' s problem to every
admissible
t~is
any
They do such
C~.
one problem.
theore~
about
rv:()~1E~,
They develop
r.e.
hovrever, thS.H
~echniques t~at
,;'.l~3t
~ill
L1 solve
tand~e
sets which in ordinary re:ursion
theory calls for a simple finite injury priority
argume~lt.
Later, in [3J, M. Lerman developed an alternative arprcac!l
to the finite
inj~ry
nethod.
While Sacks and Simpson
draw heavily on model theoretic notions
is more
pu~el~T
recurs ion : '12oret ic .
in [10J depend on sequences of
r t arne
t .1e
approach
Tl1us cruc ial cases
L1 -submodels
.L. . 2--proJ"ec
t "lon +-'
",c..lces over
papers significantJy viniicate
Lcr~~n's
•
+1
lDS
",ear...
a-recursion
generalization of ordinary recursion theory,
Poth of these
~heory
as a
~articularly
insofar as recursive enumerability is concerned.
In chapters I and II of this thesis we use the assumpticn of
Zn -admissibilitv to extend the methods of [10J to
.>
handle constructions of
sets.
E
sets rather than just
n
In particular) assusin;
l:dnil~lEJ_l
to construct a
in the connlete
Cl,-,dezree \'Th1ch is
C'i-r. e. set.
construct ',:i thout pr:Lori ties
ott)
for
from
E
8
3
L':/ - a d r~: iss 1. b 1 e
a C rl
r2
0'
to
(belov:
Th u s the i Tn rJ r 0 v e ~,1 e n t
0, •
corresponds to Sacks' use of a
a r:dniIEal degree recu:t'si ve
in ordinary recursion theory [6J.
rather than
ex -de0~ree for ev-::ry
illlprovem:::nt to produce a minir:lal
adm~ssible.
hO~'T
c.;-cie[~ree
r:-iinimal
3,
.J
to
ex-recursive
'Je also in:licEJ te
priori t;;r a:egument to construe t
in
E
1
Perhaps a
~o~e
complex type of priority argu-
ment is called for.
Ie chapter II
\'Ie
.;i ve' another e;{aD'ple of this method
but vlith tHO additional tFists.
ment ;,vhich is uniform in
Ct
?iT'st,
VJe
present an argu-
ra.. ther than one
~'lhic:h
spli ts
into cases according to the internal nature of
u.
we mix a forcing construction into the priority
a~gurnent.
We then prove that for any
(uniformly) a pair of
L
n
1: -adJ:lissi.ble
n
subsets of
o.
a.
Second,
there is
whicl-. are
incoLparable.
Clearly the next step in the attack on priority methods
In
ry,-reCU.I'Slon
cated types of
theory
TJa.S
nrior~tv
ar
to g:eneral:tze the !"1ore ccmpltpts.
The major other form
of prior-i t/ argurnent
metllod 0:' 2acks
lS,
of course, the inf:Lni te injury
..Ln. [16] Lerman and S<::c 1cs use an
infinite injury al'guuent to construct sinimal pairs of
a-r.e. degrees for many but not all
Zl-admissible
a.
We attack the infinite injury method by beginning with
an intermediate type of argument -- finite injuries but
unbounded preservations.
In particular we prove (chap-
ter III) the Splitting Theorem for all
El-admissi~le
a.
Our method seerrlS to be sufficiently [:eneral to handle
other
eYa~nples
of this type of argument such as ernbedding
partial ordering
a-r.e. degree.
into the
a-r.e. degrees below any fixed
Finally in chapter IV we sketch a proof
of the Density Theoreu for all
~l-admissibles.
In addi-
tioD to the techniques of chanter I I I we develop the machinery needed to handle a full infinite injury priority
argument.
success
fo~
Though we, of course, view this as an important
a-recursion theory we feel that further exam-
pIes of the infinite injury Method must be investigated
before we will be in a position to claim a general method
comp~rable
to the ones prOVided in [10J and chapter III
for the two types of finite injury arguments.
We hope
to pursue this course in later work.
In closing, we would like to make a few remarks about
other important tests for
c-recursion theory.
In addition
to carrytn r : over:' basic tc1e3.s a. ?ood €:eneT'8.11z2.tion must
.., ..,
..L L
offer 30!:,,(cthins nel'l as vTell.
of investigat ion
\tIi
It shouJ.d offer ne,,'; areas
thln t:le su.1Jj ec t
i tsel f
and snculd pro--
mote nevr interactions ':/i ttl other fields of mathematics.
to have, success in both of these areas.
the first
~'le
As an example of
cite the ne'!." probleLls presented by DOn-re[;u-
larity and non-hyperregularity &nd refer to [13J for the
beginninp;3 of in-lportant worle on these questions.
feel that
01..1.1'"
lemna IV. 2.1 l'IiII
allo~'r
al.30
further progres s
in study of the non-regular and non-hyperregular
degrees.
~:;e
a-r.e.
As far as the second criterion is concerned,
\1e \'Jou1d cite on tile one hand the ir:'"(portant
of model 2nd
argunents in
set~·thecretic mc~hocls
·:t-recursion theory.
contri~ution~)
to all '::'ork on p1"1 ori ty
Indeed,
on~~
could
~'lell
claim that the subject is becoming part of set theory
rather
than recursion theory.
the hope -:ha t
the rnethocls of
On the ot:'1er
CJ. -recurs ion
~v1nd} '1','2
have
t he'J=:"'Y 1.J1J..1 pro'le
useful in set theory and perhaps also in sodel theory.
As
8.
ITdnOI"
eXCl:1ple
whose results
ability
theor~T
3ee~
of' this
..,,78
vTould cite our clls.pter II
to be 2 part of set theory or defin-
rather t:lan recursicn theory, although a
priority arcu:'lent is cru,cial to the construction.
f;.
porc
important exarlple is furnished by Si;npson f s report '1_'-13 ]
of
"fdO:'~ kin
pros res ~) (; 11.
U
:31
the methods of
a-recursion
12
admissible
i~itial
se
s of
a cant :1n",1.ed :1n terac t ion b 0t~Ieen
L.
?inally,
~1e
expect
Ci.-,·recur's i()n theory and
Jensen's set,-theoretic :,vorlc [1J on t:18 fine structure of
the constructible universe.
Chapter I
Minimal a-Degrees
In this chapter we considRr the problem of constructing
minimal a-degrees for admisslble ordinals
Ct
result wi.II consist of constructing, -:for every
OUI'
main
L,,",-admi.ssible
Co
a , a set recursive in the complete a-r.e. set which is of
minimal a-degree though some other cases will also be covered.
vIe ass'L11!l€ here and throughout this thesis an acquaintence
~tlith
the basic concepts of a-recursion theory such as the vario'l'.s
notions of a-reducibility and regularity.
All the necessary
inf'ormation can be found, for example, in the introductory
sections (1,2 and the beginning of 6) of [10].
For more
general in:formation on a-recursion theory we refer the reader
to [ 2 ], [ 8] and [ 9] •
The first results on minimal degrees is that of Spector
[14].
in
On
He proved the existence of a minlinal degree recursive
.for ordinary recursion theory.
Unfortunately, his
proof is not suited to the needs of a-recursion theory [5].
Instead we rnlst turn to the later improvements introduced
by Sacks [6] to construct a minimal degree recursive in
Of •
Though the most important change introduced by Sacks was the
use of a priority argu.ment, his b,::tsic construction was also
less re3trictive than that of Spector.
It1i'laS this extra
freedom that MacIntyre [4] exploited to construct miniJnal
a-degrees for all counta.ble admissibles
U.
His construction,
however, proceeded in w-many steps by using a counting of
a
and so produced a set of minimal degree whieh in general was
not even constructible.
On the other hand a minor combi-
natorial twist (see section 1.7) allows one to carry out the
argunlent for all
section 2).
recursive in
~3-a~missible
ordinal3 (we do this in
The set constructed is, as in Spector's argu.ment,
0".
In section 4 we reintroduce the priority
technique to make our construction recursive in
a
0'
is
~2-admissible
Of
•
Now If
we can essentially work recursively in
without too much difficulty (seetion
3).
Thus we can
adopt the techniques of [10] to prove that the priority argument succeeds (section 5) and so
degree as well as recursive in
ou:~
0' .
set is indeed a minimal
Definitions and the basic lenu'nas
1.
A set
and evers
B
is of minimal a-degree iff
C a-recursive in
B a-recursive in it.
B
is not a-recursive
B
is either ,a-recursive or has
Our general plan is to construct
B
in
stages and to see to it that the second condition is taken care
of in one of two ways corresponding to the two possible values
for the degree of
Steps will, of course, have to be taken
C.
to assu:l"e ourselves that
is not recursive but the assoc:'ated
B
requirements will be much less troublesome.
This section
~ontains
some det"'initions and the basic lenunas that 'will be used to handle
the more crucial minimality requirements.
1.1
Sequences:
An a-finite sequence
disjoint a-finite sets
called the length of
:finite :sequences
runction with
and
!!.!.!. =
(lth 0-) •
One should think of an
(0"0 lJ {lth
and
CL-
cr*,., in the obvious way:
(J
+ i3
0*0
13
r
€
,.o} , 01 U [lth
0'*1
and
(J
0
by
respectively to the ordinal lth
1
a
we call
extends
a
and
In this section only
(0"
'T
,.
:)
-
'f'
or
,. c
incompatible if
p
j
(J
+ ~
J
13
€
1"1}>
correspond to extending the
representing function associated with
say that
a
as being B·n initial segment of a representing
0
"., written
and
of
0
In particular
0
<0'0' vI>
0 as the zeros and 0 1 as the ones of the
We define concatenation of two a·-finite sequences
fUnction.
a
is a pair
union is an in:itial segment of
1~hose
0
cr
and
a,-fini te seq.uenc2s exclusi.vely.
1
cr)
assigning the values
(1
.
O"i -c '1"i
.if
cr ~~
In general we
'f'
and
:J
i = 0,1
,. !S. cr
will be used to denote
-1.,.
-I.."')
1 2
Trees:
8
A tree is a partial a-recursive function
T
from
a-finite sequences to a-finite sequences such that
1)
If one of
T (0')
of
l i { 0*0)
T(O*O)
J
and
In this case
T(cr*l)
will also denote by
If
2)
T(T)
= 0,1 and T(o)
i
every
~ ~
for
which we
T(cr)l •
J
is a limit ordinal then
lth cr
(or union) of
T(cr)O
and
'1' ( 0*0)
T(cr)
are proper incompatlble extensionH of
T(cr*l)
for
is defined then so are all
T(cr*l)
J
a ,
mo~e
is the limit
T(a)
T(cr)i =
precisely
T(T)
is defined iff
U
T(T)i
'fCO
de~ined
is
for
1" C O .
As examples
of the many obvious but useful properties of trees
we note the following:
3)
If
T(o) ~ T('f)
4)
cr
and
~ 1"
T (if)
is defined then so is
and
T (0)
.
If
and
(J
'1"
are incompatible and
T (0)
,
T( '1")
a""~
.L .....
defined then they are incompatible.
We say that an a-finite sequence
a
C1
(necessarily unique) such that
is said to be on
T
is
on T if there is
,----
'1"
=
T(a)
We call
Bf3.
1.3
(~f3
<B n
> y)
A set
B c a
if there are unboundedly many initial
T
segments of its representing function on
(vy)
'T.
«B n f3 , B n t3>
f3 , B n f3>
is on
a beginning of
B
i.e.
~r)
•
and denote it by
Note that any set that is on some tree is regular.
Reduction Procedures:
In order to handle the minimality
requirements we d.efine an Ct-r.e. approximation to
[€]B:
(Recall that
Clearly if
=y
[e]B(x)
R€
is
e-th a-r.e. set) •
If
is a beginning of
.
Conversely if
there is a beginning of
indeed if
,
B
B
is regular and
B
,
.-
If
=y
[E]If(x)
and
then
( E] '\' (x) = y
such that
= y as well.
=y
[E]B(x)
,
a::> ff'
then
(e]O(x)
Using this
approximation to
[€]B
we define the key notion of splitting
and prove the lemma that is associated with satisfying a
minimality requirement bJi" making
,.
and
(j
distinc·t
and
Yl.
and
[ E ] If (x)
and
If.
split
=
Y2
Lemma 1.4:
beginning
for
e
.
B
t3
Then if
°,
iff
'T' ::>
and there aloe
p
[E]O(X)
x
=
Yl
This is clearlJi" an a-1'.e. relation of
Bca
is on a tI'ee
T
and
no pair of sequences on
T
split
Suppose
of
€
such that for some
Y2
.
for
p
recur sive:
( €] B
B
[E]B
cr
for som.e
Bt)
is a representing function it is
a-recursive.
Proof:
extends
[€]B(x)
To compute
[E]If(X)
such that
is defined.
and so there is one extending
,.
Be.
[€]
B
• 0
on T which
[E]'T{X) =
[E]B(x)
On the other hand, since
extending
1.8 defj.ned must give this same answer.
a-recursive so is
,.
By our remarks
B, ,. , such that
above there 1s beginning of
there is no splitting,any
look for any
Bt:)
for which
[E]'f(X)
Since this search is
We now turn to the second way of satisfying the minimality
requirement - making sure that
B
ts a-recursive in
1.5
is a splitting tree for
E
if whenever
are defined they split
T(O)
T
T(o*l)
Lenuna 1 .. 6:
I f a hyperregular set
a splitting tree for
B
B ~ a [€ J
then
E"
and
is regular (it:is on T) and
B
B
to calculate initial segments of'
T(O)
•
is on
B
T
J
~rperregular
~ ca
[€]B
and so to show how
B
from
[€ ]B
step by step:
been calculated to be a beginning of
has
and
is a representing function
[e]B
it suffices [7] to prove that
say
€
T (0,",0)
•
Since
Proof:
for
[e]B.
B.
ira
advance one more step all we have to do is decide which of
ancl
T( 0*0)
T( 0*1)
is a beginning of
splitting tl"'ee for
distinct
Yl
Y2
[e]T(a*l) (x) = Y2.
such that
[e]B
Since
theI'e is exactly one value for
of
.for which
B
for
i
=
° or 1
.,..
[E] '(x)
and
Since
x
and
[€]T(a*O) (x) = Yl and
is a representing flnlction
[€] 1" (x)
is defined.
T(a*i)
is a
T
we can a-recursively find an
€
and
B"
for any
1"
a beginning
Thus
is then beginning of
B. 0
This lemma can in fact be proven without the assumption
that
B
is hyperregular [4] but the proof is more difficult
and less intuitive.
In any case our final set
B will be
hyperregular ..
Now for som.e operations on trees that will indicate how
1t-le
intend to expl.oi t lemr.las 1.4 and 1.6.
1
1.7
The Splitting rrrees:
,-
on
Sp(TJE,-r) (.0)
is computed as follo1\Ts:
has been computed as
For
the tree
T
T(p) , look for the least computation
showing that some pair of a-finite sequences
ni ~ T(p*i)
set
split
for
T(p)
Sp(T,€,,-)(a*i) =
they are undefined).
set
Sp(11 ,e,'T) (a)
If
1".
:=
Sp (T, €, 1")
~i
1 on T
When this pair is found
E.
00'
with
"'1
(Of course if there is no such pair
Finally if
Ith
CJ
is a Ii-mit ordinal we
Sp(T,e,T)(cr) = U Sp(T,€,~)(p) •
p=.cr
This definition has one oddity that should be remarked on -
the requirement that
~
'1i
T(p*i) •
Though not needed in ordinary
minimal degree arguments it is inclu(led here to make sure that
when the operation is iterated into the transfinite the resulting
trees do not shrink down to a single path (see 1.9 and 1.10 for
further details).
This twist was introduced independently 'by
MacIntyre [5] to handle the> case of
1.8
The Full Tree:
€..
Thus, if for every
B to be on some
Clearly
€
no extension on
Sp(T,€,,.)
Sp(T,e,-r)
,.
of
€
on
T
which is on some given
Sp(T,€~,.)
extending
(recall that
T)
•
TO
'f0
and
,.1
,.1
T
may have
and so we would seem to be stuck.
Sp(T, €,,.)
and
is a splitting tree for
we would satisfy every mini.mality
What we do in this case is fall back on
on
L ..
This however majr not be possible -
B
no proper extension on
a regular cardinal of·
we could arrange for our final set
requirement by lel1'Jna 1.6.
some begimling
a
lenh~a
1.4.
then there are no
If
has
7
110' '11
respectively which split
,.
for
are the immedj~ate extensiorls of
T
which spl.it
T
There may, hOi;,fevex, be
PI' P2
on
T
for
and so lemma 1.4 would not apply directly.
e
possibilities.
( €]
p
PI
(
x) ~ [E]
~ ~O.
If there is a split
P2
( x)
then
remarks in 1.3
On the other
beginning of
and that
B
[€]B
B
lies on
E
.
In ei.ther case we will take
The desired tree is
is on
Fu (T,
is a subtree of
(Tt
p..
T; it lies above
a
the
whj~ch
B
p
=
,.0
or
will hopefully
:Fu (T, p) , the full subtree of
p, which is defined by
=
'jl
have successfully handled the minimality
"VIe
as indIcated and form a new tree on
T( PI)
for then by our
then lemma 1.4 \'1ill tell us that (modulo
B
requirement :for
above
T
is not total and so does not concern us.
hyperregularity of B)
lie.
~O
if there is no such split and we make
~and
and
is unde f in e d for all
In this case we try to assure ourselves that
is a beginning of
,.1
~l
extending
( E] P ( x)
t'~'lO
There are
P
T
p) (0') = T (PI ~~(j)
where
if every sequence on
if every sequence on
T'
Tf
extends
p) •
1.9 Intersecting Trees:
~
Since our constnlction will take place
infinitely many steps we must provide a method to carry us
through the limit stages (at successor stages we will essentially
Fu).
just use
Sp
or
function
RI
(recursive intersection) whieh operates on a-finite
sequences of trees
iT \) 1,,~\
•
v "-l.. .
RI (T } (.0) =
\)
If
(say
RI (T } (u)
To that end we deflne an a-recursive
UT
(¢)
if they are compatible.
\) <).. \)
has been calculated and is on every
',)
(,-) . - a)
T
\)
\
'I)
then \ve set
rr,
..I..
'\)
'I!
\)
(T \) *i)
assuming they are compatible.
ordinal
U
(~)
~
(J
Since the
is a limit
sequ~nce
is a-finite we can a-recursively eompute all of
this union must be
of
')
Ith
if they are in fact defined and if they are compatible
v
'J
RI {T } (,.)
'rco'
{T,,1,,('X.
T
Of course if
a.
length less than
o~
RI (T V-1
Thus
by the admissiblity
a
is a partial a-recursive function.
It
may however be empty and even if not it may fail to be a tree
since
RI {T } (0'*0)
"
being defined.
can be defined without
The following jJnportant lemma obviates
thes~
difficulties in our construction.
Lemma 1.10:
If
is an a-fin:Lte sequence of non
{T }
'J V(A
\) < A , T Ll
empty trees such that for every
T
y
by
v
< '"
an application of
HI {T\}},,(A ' then
is
and indeed l::f
every
T
Sp
or
Proof:
Tv
ls a non empty tree
T
'V
\)
•
T"
(,. -*1)
\)
v
on
T" (¢)
Thus
= U T\)(¢).
('r *1) i = 0,1
\J
If then
are
'J
By
v
<
\)f
is a nested
(J::::
dE~fined
al"e nested sequences.
We have two possibilities
p
T a r e nested s i.e. if
RI £T\)}(¢)
and
\)
Ir\J'
su:ffices to show that for any
for som.e
for limit
is on
is a subtree of
RI {T}
we claim.
y
Assume the lemma for.all limit ordinals less than
sequence and
on
and every T
Fu
RI {TV}V(A
It is then clear that the
then
is gotten from
HI (T) and has a proper extension on
'V
then it has one on RI (T ) •
\)
v
cr
~p
T,,('T"'V)
is
for every
'J
,
induction it
< A T') (T \} *1) C
- T'rr'l(\~ '\)-t-"l*i) •
Tv +1 :; Sp (T.
:J € -' p)
or T~ +1
. ~
In either case it is clear that
= :Fu(rr
~
, p)
'))
'-
trl' v·1
pc RI
-
(¢)
and so
is on
0
T
above
v
case we recall from the definition of
= Tv ('f)
~
p.
In the fir~jt
that since
Sp
'-
fTl
.. \)+1
(1" \)+1 )
T '1 ('f '1*i) ::> T l ''I JA-i) • In the second
v+
\)T
v\ v
case it is immediate from the definition of FU that
we have that
T +l( 'f+1*i)
'V
'\J
and so
= T'-' (,. '\) *i) .
RI {T } (o*i)
v
We remark that
=
Sp
Thus th~
T (1" -*i)
'J
V
are compatible
U T ('f *i) . 0
v v
J
Fu
and
RI
not only give recursive
trees but that G5del numbers for these trees can be found
uniformly from the arguments.
2
2.
Minimal degrees without priorities
In this section we give the basic min:Lmal degree argument
without priorities as in
[4,5] where MacIntyre proved that every
countable ad.rnissible and every regular cardinal of
minimal degrees.
L
have
The method actually can be seen to give the
result for every 'E -admissible "as well as for a few ordinals
3
which are not even L2 -admissible.
examples.
2.1.
The construction:
procedures of' length
our set
in
B
Stage 0:
Stage
0'
~
extension
T +
where
i
o 1 and
= 0 or
1.8 and let
Stage 1:
(Ta}a<A.
of every
At each stage
TO
be the identi t~y map and let
Since
e5
of length
To
~
U k( 0)
()<cr
t3 0
be
¢.
is total we can choose an
of length at least
U ken)
which is
T'~~5
Rk (5).
If
t3 cr = t3.
let
Sp(T
5
J
k(o) , t3)
Otherw'ise let
is total let
T 6+1
=
Fu(T cr , f3i)
is chosen as indicated by the discussion o:f
1
t3 cr = t3 i •
In either case
is a.-finite then by lennna 1.10
~
A
=
U T (¢)
cr <"- cr
=
T O+1
is total.
If
and
Let
and
and a t')tal
lies:
B
vIe eonstruct
we \'1il1 have
(j
~(J
= 5+1:
of
k: y onto) a.
given by
of
Let
incompatible with
it be
<a
redlction
a
13 cr
on which
To
Say vre have a list of the
many stages.
y
an initial segment
tree
y
We also include some relevant
is a total subtree
T~(¢)
;'-
The only d.ifficulty arising i.n the construction is the
-) II
t::' .. !
possibility that at some limit stage the sequence
might not be a-finite.
A reasonably straightfor\'lard
however" will sIlovl that if
~3-admissible
then
is the identity and
k
fT(j}cr<A
is always a-f:Lnite.
we can in a brute force \'lay require that
v=
cardinal of some model of
of size less tha,n
o:r
a
of'
La.
as
a
consider any
=
a. •
Cc a
Now
f
such that
and more
La
is a
in which it is also
whose cardinality in the model is less
cf(a) = U-' C
Since
V = L
a
To see that this last example is correct
a
is the last cardinal of
subset of
V = L
such that the last cardinal of
the cofinality of
invert
or a regular
tl'
well as all acl1nlssibles
regular cardinal of some model of
a
Alternatively
and that every subset
is a regular cardinal of some model of
than
be
is
a
Examples of ordinals with this second property include
generally all
As
b
(~) y
cal~ulation-,
in the model ts in fact a member
y
all countable admissibles
a*
(l'cr} cr<A
=
C <a
is bounded below
La
fCC]
~{e can thus
a C- L a •
to obtain C itself as
and so in fact a member of
fCC]
is a. bounded
L=
an a-finite set.
2.2
c~~
The minimality of
B:
Assuming the construction of 2,1
be carried out we
Claim:
B
is of rr.ininla.l degree.
Proof:
B
is not a-recursive nor evel''1l a-r.e. since 1-;e
made sure at step
6
that
~.
there ts an a-finite map
assures us that
on the a-finite set
a, say by
B I Rk (6)
for each
6 < Y
(of
course
t
=
key]
k(5) •
=
B
T6+1 ::: Sp ('1'5
J
a) •
Consider now any a-reduction procedure
is clearly on every
k( 5)
~ f3 0+1)
and
<k
If on the other hand
Fu (T 5- ' f3 0)' then the discussion of 1.8 assures us
is
that (modulo the hyperregulari t;yr of -B) if
function it is recursive.
to make
and so if
is a representing func'c ion
[E]B
B ~a [E]B.
then lemma 1.6 tells us that
T 5+1
Tv' v
is a repre:3enti~g
[E]B
Though we have not taken any steps
hyperregular vIe could ee4si1y do so as is done in [5].
B
Alternativel,y we can again refer to a proof of le!l1~a 1.4 that
does not use hyperregularity [4]. 0
2.3
Some cou.nterexamples:
Our first example shows that the
man~l
construction. of 2.1 does not handle as
might at first believe.
1'Iore specifically we indicate h.ow to
construct exampl.es of adrnissible ordinals
is a regular cardinal of
have the property
let
=
u
{rt}" ~ U (002 1
.A-
Ai"
U
i<w
x = L6
Working in
= U A n w2
(A.
L~ •
i)
~.
tf3}
,
=
Ai +1
~"2 )
= a but
a
~
does not
be the
cofinality
and
w1
Define sets as follo'T;ls
lU3
A
n
S[ (u A.l.
n
,
ltJ
2)
+1
U A~]
..L
of
w
is an initial segment of
2
is the transitive collapse of A
has cofinality
w
since
1J)2
= . U Ai n
then
w2
•
J..(t.fj
is a regular cardinal of
11)2
1-
u
L
a
sueh that
L, we let
which is o:C
w2
a
is clearly an elementary submodel of
(WI < 6 < w2)
Since
U
A
and
cardinality
1.1)2
cf(a)
be a Sk.olem-function for
S
A = w
O
1
A
Land
(*)
first admiss:ible after
ordina,l s as one
is not a member of
L (). .
L
6
, however, the sequence
and so of course not in
on the other hand is adlnissible and of cofinall.ty
by the elementariness of
It is our desired example.
A •
Our second example touches on the que8tion of hY::9crregu1.arity
and exhibit s various ordinal s 1,'lhich do not h9.ve non-·h:yperregular
minimal degrees.
Claim:
La.
(Assume
cf(a) =
and
is the last cardinal o!
If
V == L).
then any nQn-hyperregu1ar set
tV
i.e. every a-r.e. set is a-recursive in
Proof:
f
Since
B
is
B •
non-hyperregula~there
is a function
< !J)a B whose domaln is an initial segment of a
and whose
-
we can clearly take
=
hand, cf(a)
As
a
range is unbounded in
dam f
Let
= w.
wI.
Finally, since
a.
L, this subset is constructlble in
dam f
~
C
be any
o~
On the otner
implies that some countable subset of
t')
is sent unboundedly into
v=
is the last cardinal
such that
f
is complete
B
an-d
w
a)
dom f
1
La
a-r. e. set., whj&ch we can B,ssume to be
regular without any loss of generality [9].
For some limited
formula
We define a functi.on
j: w ~
qJ(x,y), y
ill
Vy < fen)
[~x
e La
C
is
Using
j
as a
B.
C (::.:) La 1== 3:x w(x,y)
j(n)
as follows:
since
in
€
~(x,y)
regula~
Given a set
~x €
(=>
while
paramete~
is the least
V
=
L
Lf(m)
:ts a
y
then
~r €
101
Mn C
~(x,y)]
.
j
it is easy to see that
and an
.
exists
C
m and the task of deciding if
:f(n) > U M
In
such that
assures us that
just compute finitely many values of
such that
m
until an
f
n c
ill
n
< j (n)
and if not
Mn c
such that
= ¢
~
CJ
tp (x,
y)
¢
J
is reached
No,., we just check Cl'.-recursi vely
x
=
.
~.'
j..p
,.L
......~
.i.l.
there
yes
2
This argument is essentially just a simplification of one in
[13] where Simpson proves that any non-hype:cregular a-r. e.
degree is complete for
va.2~lous
at s .
Though he uses a special
representati.on theorem for a-r.e. sets whieh is not tr1J.e for
arbitrary subsets of
a, it is not really needed.
We note"
for example, that the argu.m.ent given here also works for
and
that
a's
such that
~w
is the last cardinal of
La
and sJ.ch
cf(a) = w •
Finally we would like
to point out that
no a-r.e. degree can be minimal.
by Cor III.3.4
2
'E2-Admissible Ordinals
3..
An ord2nal
!:2
over
L(l,
is L2 -admissible if there is no function
which maps a proper in:'tial segment. of a onto
a
an unbounded subset of
Though we will be using
a,.
2:
2
functions and so L -aCL'1lisibili ty ratt}er heavily, V.re would like
2
to disguise this fact and make things look as much like the
case as possible. We will then be able to ca.rry out most
l
of our argu.l11ents formally as in the case of ~l -functions ano.
L
Thus we follow Jensen [1] ion making a' reduction
admissibility.
from
2:
to
2
structure
i:
by adding on an extra predicate
l
and working in the expanded
La
specific we J.et
be a complete
A
1:]. -set,
to t':1e
A
To b(~
<La.'A>.
e. g.
A = {<n, \»
I
r\ E
or for the more model-theoretically oriented \ve could let
L
satisfaction for
1
note the following facts:
La,
in a more obvious way..
3.1:
1: 2
admissible
code
Prop.
<=>
admissi.ble
For such
Be
2)
over
is
1)
over
2
over
<La,A> ,
over
La
3)
<=>
is regular and hyperregular.
a
we also have
a
is
B
La
iff
.
a-A
Now
is
recursively enumerable, i. e.
is recursively enumerable in
Similarly
B
<La,A>
B
A
1:
A
iff
B
is
1
iff
is a-A recursive, i. e.
is a-recursive in
A
B
~l
fj2
.
On the other ha.nd
bounded, iff
:recursive
notion of
is
A
<La' A>, iff'
i:
c(,
Ii\)}
a-~
B
B c a
is a-A finite, i"e. a-A
is a-flnite.
Thus for example the
cardinal coincides with that of a-cardinal.
2
Except for the second equivalence of (1), which can be found i.n
(7,8], the proofs of all these facts are
quantifier manipulations using the
so are omitted.
to
1:
out in
<La' A>
2
afuaissibility of
a
and
A remark on the roles these facts will play
(2) tells us that adjo:Lning
is, however, useful.
reduce
~~
c:..
by straightforw3.rd
A
really does
2: , while (1) assures us that vIe can still ca'l"'ry
l
most of the familiar operations of recursion
Finally (3), though essientially a
l
restatement of the L: -admissibj.lity of a, embodies the form
2
in which this property will be most heavily used.
theory on
L: -admissibles.
The basic recursion theoretic facts such as G5del numJering,
the enmneration theorem and the recursion theorem are of course
true in any ad..missible structure, but
<La,A>
is much closer
to an admissible ord.inal than an arbitrary admissible set.
We give
examples of the analogy and the reduction of
tltlO
that is introduced. First considt:;r the basic lemrn.a
1
of the Sacks-Simpson priority argument [lO] in this generalized
to
I:
setting.
Lel1'.ma 3 . 2:
[Iv
I ') < p}
Let
~
for some
p
be a regular a-A cardinal and let
<~
be a unifornLly a-A-r. e. sequenc.e
of a-A-finite sets each of a-A-cardinality less than
Then
U
I
v<p
')
Proof:
is
a-A finite and of a-A ca.rdinality
Since
'He
N.
<
•
~
'will later need the proof of this lemma
Let
as well as the statement we repeat the proof from (10].
g
be a
1-1
a-A recursive enumeration of
U
\;<p
x {') 1 ..
I
'V
~
If
the domain
o~
g
admissibility).
g
is
is less than
~
So say
c dom g
a-A recursive and
:finite and of
a-A
where
g[ ~J n I
J
is
\j
N we are of course done (by
<La,A>
is admissible.}
cardinality
On the other hand, U J
'J
~.
~
this lemma into the language of
{Iv
I v < p}
Let
for some
o:f a-cardinality <
U I
Then
3.3
v
~
<
g[~] =
~~
to
~2
~l
we translate
La
be a sequence of a-finite seta
2 over La .
is a-finite and of a-ce.rdinality less than ~.
~
which is uniformly
The a-A projection:
to be the least ordinal
function
a-A
be a regular a-cardinal and let
~
p
is
• 0
To ilDlstrate the reduction of
3.2~:
g[~]
Since
P~"A
such that there is a partial
a , Pa,A
1 (La,A) • For l:2-adJnissible
enjoys most of the p:roperties generally associated
with
and will act as its analog.
1
a*
1
Pa,A < a
f: p onto> a
'5.:
Following Jensen 'we def:tne
p
set bounded below
pI
a,A
reduction procedures of
Of course
in
of
which is
it is the last
2:
Thus for example, if
a-A (and so a) cardinal, any a-A r.e.
is a-A finite, and we can index the
<La' A>
Pa1 , A
by a
1
Pu,A
list recursively
is just the usual
~
~2
projectl1,m
a •
All such facts as
3~27
course be proven directly in
•
a-A-fecursi.ve union and so ,..,re ha-.:.re
contradicted the regularity of
Lemma
Clearly
g[N] •
U J
v<P 'J '
x {\}}, and so is of a-A cardinality <
is an
v
and consider
and those about
1
Pa,A could of
Lo:, though the manipulations
become more involved.
The advantage of worktng in
<La,A>
is that there is no need to reprove them - they all become
For example we can essentially m:Lm:tc the Sacks-
obvious.
Simpson solution of Posts problem in a-recursion theory [10]
to get an analogous :L"'1comparability resul t for
One word of' warning is in order". however
t
~2
sets (eh. II).
Though recurs:i.on
theoretic and combinatorial arguments tend to carryover to
<La' A>
without any changes, model theoretlc ones do not.
main obstacles arise because a
~l
(La,A) -,Skolem hull of
The
8,
transitive set is not necessarily transitive, while one taken
relative to
is always transitive.
La
In
particula~the
notion of an a-stable ordinal (i. e. one 1"lhi.ch is a
of
1:
1
su'omodel
La ) does not prove useful in this setting and will be
replaced.
Remark 3.4:
Indeed
everJ~hing
throughout
1:01"
one goes from.
any
Ll
Nothing in this section i.s specific to
remains correct when
n > 2.
to
1:
2
2
~s
replaced by
:E
2
n
The only real break occurs when
•
4-.1:
assume
We
4.
The Construction
is
L
a
admissible and work in
2
the sense that our construction will be
recursive and our
a.G·A
priority arguments will be done accordingly in
the other hand,$ we are trying to build a set
<La,A>.
~J'hich
Or:
1s of minimal
and so '\'1e are basically interested in reduction
degree in
procedures
in
La
on our notation:
He
in
rather than
<La/A>
and
all reduction procedures
will be assumed to be ones from
La
so we compromise
[E]
and
r~e.
sets
and all trees are
a-recursive functions.
At the end of each stage
of our construction we will
cr
tT~}i<f
have an a-A-finite sequence of' trees
for some
f
, - (J
such that each
of
Sp
or
Fu
T~+l
is obtained from
and each
T~
have an a-finite sequence
~a' ~ ~o
desired
for all
by an application
RI (T~1i<A •
is
such that
v'lhich is on every
f3 cr
crt < a •
T~
B
will of course be our
=
set of minimal degree.
from the straightfoT'\'lard one of
<'
cr~
The construction will differ
~2
in that the trees will no
longer be required to be total functions since such decisions
are not a-A recursive.
As a result
strictly increasing function.
can no longer be a
fa
The Lllpact of the priority
argument in the next section will be that though
f
O'
is not
monotone it eventually stays above each point of' interest, and
so for each
i
there is eventually a
of' the requirement of priority
i.
T.1
1tlhich takes care
Unfort1mately, we cannot
in all cases recursively specj_fy 'the priori.ties in adv8.nce, but
cr
must content ourselves, as in section 5 of [10], vlith an ap-
proximation procedure which eventually orders the
This approximation will be given by an a-A recursive
correctly..
~ction
e
priori~ies
k(a,e)
at stage
which tells us what requirement has priority
a.
The precise definition of
on the nature of
k(a,e)
and will be given in §5.
a
depends
For the p'.lrposes
of' understanding the construction it will suffice to note that
k(a,€)
will have the following properties:
1)
5 <a
There is a
k(a,e), as a function of
initial segment of
o.
called the doma:in of
k
such that
is eventually constant on ever.f
€,
(0 should be thought of as the fir.. a.l
order ty-pe of" the priorities).
He let
k(e) = lim k(a,e)
(J-?a
for
<
€
2)
k(e)
For everJ
Y<a
there is an
e < 6
such that
=5 •
3)
then
6 •
,.
If'
,.
is the least
e
(~O" ~
such that
e)(k(o,€)=k(e»)
is zero or a successor.
The first tlvO properties are to make the prj_ori ty argu.r!lent work,
while the last is .just technically convenient for the construction .
At first readj.ng it is probably best to think of
identity function as it will in fact be '\"lhen
k
as the
1
Pa,A = ex, and to
ignore the complications introduced by changes in the va.lue
4.2
We begin the construction at stage
o = identity and
TO
13,) = .0
\
a
W'e continue
o
9-E
by setting
follows:
Stage cr
y + 1.
=:
when one sets
x
Keeping in mind the convention that
equal to the
and there Is no such
least
<
€
f
such that
V
<
to the least· i
of
~y
<t
y
'i1
+1
then
x
<t
y
with some property
t, we let
=
k(cr,E) I- k(V,E)
be the
"(1
cr equal
such that there is no proper extension
and set
f
on
Case 1:
f
(J
=:
+1 .
,..,
TY one 0 f
'T")
R (
k
(J,
has a
11)
=:
~i Ch ,
~ (J
We let
TY1. for
<
i
AO.
·1 y
~y' 18 necessar1
say
WI1
~Oy
=:
fl"t'
v
=:
and set
n+1
fa
\) + 1 •
~lere
any sequence such as
extension on
T~
the definition of
!3
~y
< 11. By lemma 1.10 f O' :Ls again a successor,
of the form In (T~,t3) then
Now if TY
\)+1
Case 2:
for some
•
.'
T'!
1
f cr '
By 1.2.1 it has two incompatible
n
proper extensions on
incompatible with
By the minimalif:r of
TY.
proper extension on
say
le~st
~
13 y
has one on
f
f3 y '
rr ,
v
t3 y on
T~+l.
Since this 'Vrould contradict
TY
-- Sp(T Y , k(Y,\J) , 8)
\>+ 1
')
these are the only trees we ever put
we must have
as
in at successor places.
successor of
which is on both and has a proper
We now let
T~
t3(J
be the immediate
dictated by 1.8 and set
.0'
Y
Ti = Ti
Note thg,t all the informe"tion needed to carry out these
steps is a-A-recursive.
Essentially we ask only if certain
partial a-recursive functions are defined a.t specified points,
and except for these questions which are trivially recursive
in
and computing the a-A recursive function
A
we proceed
k
a-recursively.
Let
stage X:
TO"
Since
constant as
< 11.
= J.L8({V I f y < e} is unbounded in A)
can change at stage
i
definition of
i
i1
assures us that for
"l
cr
~
only if
cr
A.
We now set
lL~
Similarly
f,
=
.\
i <
fei
'1
k(a,i)
f3\ = u f3 0'
(j<).
.
f3 a
( (J
(J.L9
< -n)
<\)
exists for eveTy
(3:\)
< e) (lim k(cr,v) I k(:\,'J)).
T~l = lim T~J.
are nested and so we can set
i < fA
for
0"-+)..
can happen when the changes, in
f)..
<
T)..
other hand if
f
If
f,,.
we set
fA
is a limit ordinal we set
m).
lo.p
J.)..
=
Sp ( 'I')..
of course:J but
f)..
=
'v + 1
(which
are u..l1bounded in
k(cr,\J)
,,
11,
is eventually
(j-)A
there are two possibilities for
or ",hen
v
rr
-i
~
1\
By induction the
,
< i, the
J
k ( A.., \) )
')
" t3,:\)
TA.
:fA.
.
A-
On the
= RI {r.[1~1.
J. ~ J. <.l.)~
-f'
Note that since our construction is a-A recursive everything
~ d b y some sage
t ·lS u- A
genera-lie.
In particular
This is the
f'" lnle
•t
iT~}i<f is a-finite
A
c~J.cial
.p'
• t
-:z 1 -. i\ "
an d so a-J.lnl"
e ('0
... rop • ./".;;
aDd so
T~
is a tree.
A
use of the L2 -admissibility of
a
in the
construction .
The construction is no\v complete.
we will make the choice of
k
explicit and show that the priority
argument succeeds :i.e. for every
bounded.
Before
vI8
In the next section
E
<
dom k
to'
I
!
do this ho-:,.. . ever "hOC vrllI show
.p
. . . a < €}
that this
is
suffices to establish the minimality of
Lemma 4 . 3:
For every
for which
Proof:
=U
y
{a
< dom k there is a last stage
€
Moreover if
Since
I fa <
+ l}
E
I fa < €
{a
B.
+ l}
and then look at
~ f
y
of
U {a
I fa < € + I} < \) < y
the possibility that
limit stage
I fa <
{a
B
4.2 . 1
and
4.2.2
ken)
= e and also the least
By
~ T)(k(a,~)
such ordinal." f
If
=
and
~
€
we can find an
f Y
>
y
and so the
T ~ ,.,
'Tl
+ 1.
'T,
\ve
claim that
that
cannot be least.
T
~
Since
is a limit ordinal greater than
assures us that
f
y-<
E
-
Consider
a
such that
'1'1
is the least
and so by lemma 4 .~) there is a first
+1
=
..
a
fa
O'
In
€.
such that
T
such that
and so
then
Finally we have
is not a-r.e
'f'
cr
y - v + 1
0
•
B
k(~,~) = k(~»
implies, by our chaie e of
in
fy =
We in fact show that
R€.
>
>
y
is not a-recursive.
any
a
f
part of the construction tells us that
Lemma 4.4:
(va
would contradict the definition
must be unbound.ed in
again a contradiction and so
Proof:
shows us that
is a limit ordinal and
+ l}
An i.nspectj_on
€.
a contradiction.
y
€
fy -
constru~tion
y + 1 and so f y < €
On the other hand if
t hi s cas e
To prove the
f v •
I
of the successor stage of the
Y+l
€'
is bounded vIe can let
lemma it clearly suffices to show that
f
<
€
is a successor.
ty I
,.
f Y :::
cr
is a successor:
then
fa
11 +- I}
=
,,+ 1
is unbounded
On the other hand 4.2.3
:f
,.
< +1
't"l
a,nd
cr
is the first st.age >
in vrhich
'T
f 0'
in case 1 of the construction at stage
t3(J
that
was incompatible with
t3cr' cannot equal
= ." + 1
cr
and so we made sure
and so
R
e
·,,;e mnst have been
B, which extends
• 0
Re
Note however that since our construction is a-A-recursj*ve
so is
Thus by Proposition 3.1
B.
which is hyperregular" and so
B
1s c:l-recursive in
is hyperregular.
B
In particUlar
we can employ Lemma 1.6 once we choose our trees.
The trees
we need are obviously the limits of
O'-7"U.
as
~he
~
A
The limits exist because of the priority argument and the fact
> i for all 0' > 'T th2D T(ji = Ti'T for all
We set T. = lim
and note tha.t B is ()n every T.1
that if
fa
~
t3 a
T~
is on
= Ti
Lemma 4.5:
for every
If
Proof:
Let
we can take
a >
and so
TO' 1 = T
n+
J
TO
~
TO'
,.,
=
T
~
= T"n
t3 0) = Sp ( T'n,
'
E ,
(again
b:~l
4.3
By len~a
be as in lemma 4.2.
'T
and
1
n+~
two possibilities for
Sp (T ~ , k ( 0, ,.,)
Now for the final le:rmla ..
to be the last stage at which
'T
since
is recursive in it.
B
and
n
•
~
is a representing function then it
[€]B
is either recursive or
> ~
cr
>
cr
f cr == "'". +1
4.3).
The re are
it can be
t3 a)
or
In(fJ~~, ~ 0) = In(T
'I
In the first case (which can occur either if'
cr
:I
,..,
t3 0.) ..
is a limit
ordinal or if it is a successor and we were on case 1 at stage a),
B
is recursive in
,. ]B
LE:
must be a successor and
and so 1.8 tells us that
by 1 eIIhTIla 1. 6 •
1'le
In the second case
must have been in case 2 at stage
[€]B
is recursive" 0
(J
Thus modulo the proof of the priori.ty argument in 95
Vie
have proven our main result.
Theorem 4.6:
If
a
is
~2
admissible there is a minimal
a-degree recursive in the complete a-r.e. set. 0
5.
The priority argQment
In this section we prove
Theorem 5 . 1:
For each
< dam
E
k
the set
IE =
try i
fa <E}
is bou.nded.
The proof splits into three cases as in [10].
we will use a different
For each case
to assign priorities and then prove
k
that we stop returning to each initial segment of priorities.
Before we split into cases,
however~
there is one basic fact
about the construction which we need in every case.
Let
Lemma 5.2:
0' E
['rj~'J),
then
fa
=
"1
k(o.,y)
f'
"'0
>
E
for at most two
Proof:
E,
and suppose that for all
< 'J < a
and that
a's
It is clear that
for all
f
'.
= k\Tl,Y)
y <
J
[ n~ \»)
i.n
can be a limit ordinal
fa
€
A
I f y < >"'+1) is unbounded on a. This ~'lould imply
however that ty I f
<)..} is also unbounded in ( J . Thus our
y
assumption that fa L € for cr E [,.,,,v) assures us that
only if
{y
a
(,;,'.1)
fa :/
€
for
let
cr
be the least
E
if
cr
€
are two possibilities for
SP(T~, k{cr,y) , fjo.)
is a limit ordinal..
E
en,\))
rnO'
.I.E
Since
such that
.•
f 0
In(T~ , f' cr)
>
does not change in this interva.l.
return to
€
after
a
f
and before
for
€
~
E
cr =
and
€
= Y+ 1
There
E •
[~,\))
,
T~
Thus in the first case a
v
",·ould require some
~6+1 (0 < e < v)
but not on
If
to have a proper extension on T~ = T~
cr
~5
Since t-"A uR +J. ~ A~v this is
-Y+ 1 = In(T
.\. . - y' 6 G
..
'J'
(-I
impossible and we do not return to
T~
tively assume
to
cr
< e <
stage
or
fa > e
In this case
see that we can not return to
Proof
and that we first return
T~ = In(T~+l'~o).
and so
6
o.
Alterna-
\i.
for
so we must be in case 2 of the construction
and
6
'before
SP(T~,k(cr,y) ,t3 oJ
=
at a later stage
€
e
5.1:
Now just as above we
bef-ore stage
1"
'J.
0
<a
Let
f
P~,A, onto a
Let
k(cr,€) = f(e)
case 1.
a-A recursive map from
e
Pa,A
a~
be a partial
if there is a computation with GBdel number less than, cr
which shows that
f(e}
is defined and a G5del number for the
empty functi.on otherwise.
If we take domain
k
pl
a, A ' k
has all the properties required of it in seetion ll·.2~ The
=
only one which is perhaps not inunediately obvious is the
It :follows, however, directly from the fact that any
first.
P~,A is a-A finite (3.3).
a-A r.e. subset of
this case
(~)
has the additional property
k
at most once for each
Note that in
k(a,e)
changes
1
e < Pa,A •
The theorem is now an immediate consequence of' an in-
duction on
after
€
i€
that
€.
Ie
€+
be the first infinite
a-A cardinal
(it is, of course, regular) and inductively assume
< €+
which takes each
of
Let
above it.
At a successor stage we consider the map
a
such that
By 5.2
and
fa =
(*)
E
to the least element
this map is at worst three-
to-one on its domain and there are at most three
fa
=
E
domain.
which are above all elements of
Thus
IE
cr's
with
and so not in its
Finally, at limit levels,
leIlh"1la 3.2 carries the i.nduction s.long for us.
0
l~ 1
"
1
Case 2: ' Pa,A = a,
and there is no last a-A eardlnal"
This is the simplest case.
(j ,
The domain of
€.
k
k( 0', €}
We let
is now
for all
E
and it trivially has
a
(*)
the properties required in 4.2 as well as
there is no last cardinal vie 'can argue
1..
:=
~
exa~tly
Since
as in case
0
P~,A = a
Case 3:
N.
and there is a last a-A cardinal
As usual this is the most
inter~sting
case.
Indeed
it is the only one in which we deviate from [10].
Since
L1
<La' A>
is admissi.ble it is
partial function
all
h
1 formulas of
sequence (5, } " 60
.
6).,
in
6, .
.-
to define a
which is a Skolem function for
<La/A> [1].
2:
eas~r
U
h[~]
1
Pa,A - a
,
vJ'e use
°,+1 -
U
1:1
hE
to define a
6, + 1]
and
this sequence is unbounded
C<'X.
a (an easy consequence o:f 3" 3) " We now think of the
=
U
Since
priorities as being arranged in blocks
[0, 6,+1)
and
within each block as arranged in an N-list by using canon i ca11 y
Let
Ch osen
f ' "• ~
H
maps
h(x)
We approxirnate
(6,1
6~+1 = U hcr [ 0 ~ + 1]
be the least map
f.
vr+l "
if there is a computation < cr
hO (x) :;-:: h(x)
showing that
onto>
is defined and
hO(X) = 0
by sequences
(oC}: ~g
and
f: N ~
0~ =
U 5~
C<'X. ~
0'
6,+1 •
otherwise .
= hO" [~]
Finally let
,
,
fO"
We would now naturally defille our approximation to the priority
listing by
"There
that this would not satisfy 4.2.3.
we perturb
+n ,
"1
<
~)
except
To accomodate this requirement
k ( a + 1 , ~ .. \) + "'1) = fa+ 1 ( ,.., )
slightly and set
k
= ~ • v
€
')
and
k(\, N· \)+.,.,) = fA(,..,)
if this,value is also
\)'
f~+l( Tl) + 1
and
k
that
(j--»
otherwise.
We must now check the other requirements.
The computation of
values of
v
A
This alteration clearly assures us
satisfies 4.2.3.
1)
lim fa(n}
is entirely determined by the
k
except for the above perturbation
not affect the requirements other than 4.2.:3.
wh~ch
does
t5~},,<y
Thus if
reaches a constant value so does k(o,e) for E < N · Y •
(1
Once 5( = Cc for , ~ y all we r.eed to correctly compute
are the computations of
of course values of the
5~+1
compute
Finally if
6(1+1
C
,
such that
(j
>
,
~
•
6(, + 1
,
6, ~
+ \)) =
+
Thus at stage
(J
are all correct for
e~+l
<
E
~'+l
1 , k
«(J ,
in
a
~ •\)+"
=
~Y+l
+1
we
~
5(+1
., + \",)
- T
- 6
- -~~. "J+'!!
" 1'+-1
then by definition
for each
and so for some
=
e
there is a
v < N f
e(\)
=
e
correctly '\V'e will have
Indeed we can say
E.
C<l
more~r
fOT
every
e • 0
To finish this :final case vle induct on
J
and so by
is also correct. 0
Now once we have computed
k( (j
h
t&C 1 is unbounded
As
They are
€
correctly and of course never change again.
alone we see that
2)
X
function
~l-Skolem
~Y+l.·
definition occur by
for
h(x)
we prove that
~.
\) + 'T1.
Setting
is bounded below
an~
6V + 2
~
b) J~.v+~ < ~+
• v + 11.
By induction
calculation for
< .~+
u_T
6\)+2
there is an
~·\)+~+l
Moreover as
x ~ 6\)+1
By lemma 5.2 for some
above
h(x).
L
l
since' h
is a
f
O'
~.\)+~
h(x) ~ I N. v+ •
n
there are exactly n a's
=
~
• \)
+~+1
'rhis can be
<
6\)+2 = h[ 0\)+1 + 1].
limit ordinal less than
~~
for the cardlnality of
J
domain of
J
~.
\;+il
<Lo,A>
as
6\)+2 = h[ ()\J+l + 1] "
the funetion
.., . \,rrn
(ax)(vy
g
is
y <
~
recursive.
0
J
l:l
J~.~+il
At stages
clearly implies
fact empty.
a
~.
I N• + £
~.y
v n
c 6
-
y+l
o\) +..J..
•
The
is certainly
Thus a bound on
Skolern fU..l'lction
= g[y]
y
which
g
g[ yJ c ~\)+2 •
and so
~
y)(g(y) < x)
€
a-A
and
thing proceeds trivially:
<
~
For
is defined from parameters <
on some ordinal < 0V+l
v
.
J~
is given as the value of the
g( y]
Thus
An examination of the proof of lem.ma 3.2
Moreover the sentence
true in
.
As for the rest ".. e must
•
is of course some
g
•
\)..L
we can again argue as in case 1
~. \)+f1
shows that when applied to
enumerates
6 +.,
Skolem function they must all occur
below the limit ordinal
look more closely.
sho~
Jib
.
S
oun d e d ln
sentence with parameters
~l
to
5~2
such that
<2
n
,.3uch that
written as a
and so using our
I\.\\"\. \) c
- a
v+l
above we can employ lemma
k
that
Assume we have succeeded for
h
is bounded below
of the induction every6~+2
for every
and so
J~.y
'"
v <y ,
is in
6~
1.
Open Questions
The basic problem is of course to construct a minimal degree
for arbitrary 2:1 -admissibles. In partlcular '\Ale view ~~w
(of L) as a good test case for new approaches. It is not
hard to convince oneself that something different is actually
needed:
consider the functions
B(x)
if
I
Lx F 3:
[n]B
<
n
given by
cardinals
o othervrise •
In any procedure like that of section
splitting tree until one reaches stage
4 Tn will be a
~
n
at which point
we must return to it and make it the full subtree of'
Thus there will never be a stage at l'lhich all of
\'1111 have reached their limits.
T.
1.
Tn - 1 •
i < to
Thus there will never be a
chance to properly attack requirements a.fter
IJ).
On the
other ha.nd cotu1terexam.ples seem even less likely.
2.
In connection
~ji th
the second example of 2.3 we ask:
what admissible ordinals
a-degrees?
a
For
are there non-hyperregulax' minimal
Indeed doing much of anything with non-hyperregu.lar
sets is something of a challenge.
See (13] where some important
first steps have been taken.
3.
When can we embed
arbitrary finite lattices (or even
distributive ones) as initial segments of the a-degrees of
unsolvability?
Chc.pter II
G. Sacks and S. Simoson ha~e asked [10, §7 Question
5J if it is obvious that there ar€, for each
Cf.,
L
L )
(over
n
subsets of
a
ble.
If one understands '2
there
2,1"'e
L~
n
-adY:1issible
which are
a
in
:Ls
0'1
\,...'
~
and
I II,
n
reductton nroceclures ':':1icn 1Jut out
ex
when one feeds in
B
unqualified
!1
-'
.,
then the answer is an
In this sense
yes~·.
generalization of
('1
B
PI::,
in '.
fl',:.-recursi ve
in'(
n
(replace
is a dirEct
II
by
in the definition) and so amenable to the methods of
and
A
1.5.
Indeed one simply choeses a complete
a.nd minic s the cons truc tion of [10 J
0. -- A-
r . e. set s
Band
in the other.
1::111 i
I:-'7l e:::1
E
produc e
1,::·0
1.3
n-l
t~'!o
neither of which is
By the renJ.rks on
ti.'Ei.nslation (1.3) this
i ate <1 y g i vet h e des ire d res u 1 t
for t his d 8 f' j, n 1-
tion of
There is, however, the more obvious and natural notion
of
are
The
.,. /\
-n
in '<
L
and
n
3. n Sit] e r>
B
to be consider
formulas of
IT
d
n
:i nth i s
C cu3 e
definition is easily seen to
bility than the
for~er
(L a ,C)
i s s t 111 ;; yes ~
no longer quite so obvious.
for
~
is
n
in
C
iff there
which define
f
but the
Dr 0 0
3.
f
is
First of all, this later
ve a much
n > 2.
stron~er
reduci-
Consider, for eX22ple,
ani'" Hi th cor-:nlc:te
11:
A
and
B.
Cle2:elv
}J £ D,
to the second de -f'::Lni tion.
~n
ever, any thins
A
in
should also note that as
n~
n
in"
(A)
n
and
2.C
T.,!:! ttl the fir s t
is
~n
c:L.::f'ini tio:':, hO'..J -
and so
4 ~n(0),
B
20rd inc
t
B
~n(A).
(One
the second notion of
is not transitive vrhile the first clea:rl J is).
r
The second definition, moreover, is really model-
theoretic in nature and one can ga5n no direct hold on
it via purely recursion-theoretic constructions.
son for this is that individual fnets about
via the SeCODe} definition of
is
13
in general, on unboundedly much of
C.
very nature the first definition uses
about
C
for any sinsle decision.
a construction inside
much
informatio~~e
L
a
~le
depend,
Of course by its
on~y
a-finitely much
Since at any stage of
a-finitely
would seem to be at a loss as far as
is concer'ned.
It
~'lould
could hope to handle would be reduction
procedures which were
L /L .
n
a.
The solution to this difficulty,
by Prof.
C"
we car only handle
the strong notion of
seem that all
derived
3
in
tIn
The rea-
~ili~ch
was suggested
Sacks, is to mix a forcing construction into the
priority argument to produce sets which are sufficiently
generic so that the
titlO
notions of reduc=_bility coincide.
More specifically, one hones to make the truth of
sentences about
C
depend on only
L
n
a-finitely much cf
via a forcing re12.. t ion H.hic h is it self
Thts ":0'.110.
C
1.'~ ,.."{
then give us a method of handling the
rotion of
stron~
directly in a construction inside
T
oW
a. .
carry out this suggestion below to prove the theorem.
We also answer question 3 of [10J in this chanter
carrying out our construction of
sets uniformly.
integers
k
Thus for each
and
1
a. , Godel numbers for
incomparable.
which are,
rn
!J n -- inc ornparab
1e
.
n
fo~
by
L
n
we show that there are
every
subsets of
a
2: -admissible
n
which are
~-
n
(N. Leman and S. Simpson have informed us
that they have also found a unlf'ol"In solut:tcn for the case
n,= 1.)
4
1.
The Forclnrr
~elation
He assume that the reader has a gen2ral familiD.rity
with various standard
forcin~
descrlbe the partlcular systen
but
~ive
procedures.
~'Ji th
no proofs for the standard
'.[hie::l
therefore
~e
'liTe
are concern.2d
The basic
thec~e~s.
techniques and lemmas for the subject in various recursiontheoretic settings can be found for examnle in
1.1
Defini t ions:
?he gener2.1 f:"'arllev.7ork
will be working is that of Cohen style
~Iore
models of "VJeal: f:ct theories.
be fore ins over
L 's
G
sions (see I.l).
is that the
for
ex
ex
\'ii
thin
c=~ass
intend
specifically
~:Jhich
'de
~de
viill
Our forcing
E ·-adnisslble.
n
inelu--
j.nterp:cet::.1tion, of course,
seq~ence lS
a-finite
1•
forcing over
-finite sequences ordered
~he
rL '--,q
an initial segment of
the represent inn; :''llnct::Lon for our Q"e:neric precLtcate.
far 'as lansu2se is concerned ours
~'lill
r"S
contatn, i,n addi-
tion to the usual logical symbols, bounded quantifiers,
£, =, constant symbols
predicate symbol
G
a
for each
and a unary
for the generic predicate.
~e
will,
however, in our construction only consider formulas
in strict pren2x nor0al forri, i.e.
bounded quanti.fiers
C'l¢
we
~'lill
follo~'T
ones in which all
all unbounded ones.
T:'lUS
by
med.n the forT1ula r;otten by crlanzing ;':.11 quan-
fore
on the
60
relation
unrestrjct
nu~t:er
For
¢, forcins eqhals truth.
formulas
ifr every constant symbol in
cr\~ ~
and
L,~~
-'- v
6
~
a
J 1
The inducti7estep is as usual:
C3a
~ 10.)(0\1- ()(§:.))
belongs to
representin~ ~unction.
at\- :jjxQ(x)
iff
ifJ~
and
0) (T >¥ -3 X'1' (x) ) .
(1fT J
6
nrec isel;1 ~
0 ...'"'0
is internreted as having
G
~hen
as an initial segment of its
a
der~d-led b~/ induct10n
GIl- Q ls
T/ith thls c.efinition lt is easy to
prove all the standard facts about forcin;; in tlJe usual
inducti ve ;'lay.
¢
Cn
> 1)
En
is a
\l- ¢
relation while for
Thus in every case, for formulas in
is
(J
(J
h- iP
the
is
rn
One should also note that for
formulas
=a
n
Z n-,IV
it
IT n-- l '
These basic facts will enable us to use
~.
n
rn-admissibility of
a
to carry out the forcing part
of our construction.
1.2
r n -genericitv:
r,
iff .for every
(La' G
>F 9
Band
C
not:i.on of
ditions.
that if
we call a subset
<J
'J
sentence
n
<=> (3 a
~
-
r. n _r)"A-ne""'ic
.)..':",_ . .
.1
n!J.
n
.....
,-1,
f
G) ( (j 'J- rp ) •
'Fa
'I ' -
in!! via
'i'
OUI'
of
En-g._.eneric
a
j
language
By making our
L:
n
sets
\,r111 be able to handle the strong.,.,
.,J
a-finite facts, i.e.
As an examnle of the
is
of
G
•
then
un-generlc
po~er
forcins con-
this gives us we note
is
fJ
n
G
strong sense iff it is also true in the weak sense.
in the
We
must
l~rove
and
1[;
x
G
of
and
<=>
Say there are
(x) (x €
8 <=> <La
<Lo.,G) rL ~)(x)).
i
B <=>
is
b.
11
in
x, x
G
En
e
0(x)).
formulas
n
,G') 1=
j
(30 ~ G)(ot~
sentences is
[
Then b:'T the
we have that for all
x
En
B
such that
B
.J,.
'f;
the if Dart:
¢
2:
¢ (x)
Ln
-~enerici+;'v
'
,-
B <=> (30
C
0)(0 I~ ¢(x»)
Since forcin~ for
this is precisely the statenent that
in the v!eak sense.
Finally we remark that by the usual lemmas about forcing (in particular the \Iforcinr; equals truth;; lemma)
being
I _
n l
a,~
rn-generic is equivalent to there being, for each
formula
,¢.
¢,
a .
a c G
such that
al~
or
¢
Indeed these are the requirements that we will
try to satisfy in our construction.
1.3
Before proceeding we should comment on the restriction
to strict
prene~
formulas.
This corresncnds to yet another
variation in tl1e meaning: of
(or
;!
11
n
in'1 ) .
Though we will for the sake of sjrnplicity prinarily concern
ourselves
~!Ti th
strict
En
formulas, we will indicate in
section 4 how to add on additional requirements to the
construction and handle
E
n
formulas in which there are
no restrictions on the positions of the bounded quantifiers.
!W
".,
--",'
2.
i
I
The Construction
We first glve an jntuitive oicture of what we are
trying to do 2nd then rri ve the actual details of the con·In
struc~tion.
sections, 13
bot~1
ally syT'1T1etr-ic roles and
~;,Je
C
cmd
generall~l
'Hill
attentions to the
3
necessary we shall
dlstin~uish bet~een
2.1
n
assume
and
each
L
sure that no
'.'1 hen
the
n-l
En
a
is
sentence.
~he
towards our first
~oal
goals.
tNO
co~ple~ent
To beg in,
'/! e
'Ar :i.ll
assi~n
The
0
of
C
rod e r
priorities
Once this has been done we nroceed
n,
by, at each stage
current guess at an initial segment of
r n- 1
nota·-
second will be to make
sentence defines the
to these reauire2ents.
C
forcing conditions contained
sentences in some way and so
n
and
rn-admissible.
G i s in t e r pre t e cl a s ' ) .
E
~~e~
Throughout the rest of this chapt0r
first will be to decide by
in
3
The construction will have
The idea:
confine our
side of the construction.
tions by subscripting.
we fix
Tllill play virt;"l"
B
extendin~
our
to force the
formula of highest priority not yet decided or its
nega t.ion ~
tension
By
al~ia.y s
sentences
extenslon.
~t3
the asual J8!:lmas about
exi::; t s .
A
~n
the~e
~oreover
is a
6n
for(:in,~ -'
such an ex-"
as forcing for
L:
n·-l
U
1
. n-l
function givinz such an
~or
in the usual fashion.
have a potential witness
X
X
define
C.
into
£:
for~ula
6
ye will
s
E:
our current a0proximation to
we put
rn
eac~
B
forces
-"\i
-
thus making sure that
~'Je
Of course in both parts
necessary negative information about
then
(X-l"G)
does not
·.·r:i.ll protect the
B
only against later
encroachments by reouirements of lower priority.
The problem with making this construction independent
of
a
arises, of course, with the specification of
ority ordering.
1.Te
pri-
In chanter I as in [lOJ various orderings
depending on internal characteristics of
in advance.
t~e
a
are specified
will here uniformly generate (v:La approx-
imations, of course) a priority
llstin~
that, for each
turns out to be essentially the same as the one
~iven
a,
in
I. 5 .1. The idea ls to :niriic the proof for case 3 in I. 5
(there is a last
a parameterless
a-cardinal and
P~
\'1e first use
to generate blocks
Then within eRch block we act as if we
were indeed in the situation of case
0
a).
r. n
of requirements.
block is
=
3,
that is, if the
we consider the reauirements fer formulas in
=:
h[o]
ordered in a
d
type list.
The proofs will then
follow 1.5 and [10J.
2.2
The priority listing:
Slnce
one can easily get a parameterless
a
is
r. -admissible
n
E -Skolem function
n
[1 ~ .
h
in
T
~~.
,
z:;
.,..1
1
...J...
(')).
\.
-:.
•
U
l:<A
If thpre is a last
1':11~~
blocks
11-
a-cardinal arld
P
=
a
then these
a
eventually coineide wi th those of I. 5.3.
If there is no last cardinal then the
bounded in
.
lng
.
.,... ,- ]
..L.~).
a last
(L
') )
'
8~
sit ua t 1 (, n
tion
a
\":ill be un-
and l'lill incluje e·ach
Finallvv if
•
on
< a
, CI.
~nto
which projects
be as 1 n I . 5 • 1
\"1 iII
a-c:ardinal 'thus mlmic-·
then there will be
a
0 f' c:) u r s e in the con s t rue --
(1 ) .
but this is Tdhat
ca.n only approxinate the
1'le
"'.jill happen "tn the lirr,it li
•
UsinG the notation of 1.5.1 (3) ~:je let
1-1 map froT:1
u
onto
least ordinal for
.
\" l •
by stage
en
e•
for some
and
y
=
enc;
s <
E'
hnf~(Ef).
~'Jhere
W
D
,\
hnrnCe::)
:Ls the
S
theI'~
ls such a map constructed
(requirement
~)
at stage
R
y
a-card :tnal i ty
if'
n' I
to have
S)
B = hnriCs)
~
~',E'
and there is no pair
<
and
~oreover for fixed
c;
£'
end
l·-itb either
<
E:
E
<
'?
ment
8n
'?
RQ
We now define
higher priority than
(o;+l)n x
be the lec1st
n<Cl
' t es t'ne
approxlma
~
be the ant,~i
.. r.:
n.
at stage
proxirr.at:ton to
o~
associated
B
wi~~
has
hi7he~
~:;.uch
enr;
that
a reauire-
priority
t~an
I"'"
'v •
2.3
rrhe reqt}lrer:lents:
'iTe be
n by making a.n
Ci
-re:;urs :1ve
will be two types of requirements
If
'E::
~
En _ 1
sentence,then at
sta~A
cr n , our current initial segment of
that
or
is a
¢.
(for both
<P E: •
then
..P
\...
S
If
¢
£
fled at stage
C) •
rerplires
n
B, forces
is a
reouires that
say that any
Band
does not define
require~ent
R
He
of the first type is satis-
s
n
It ls a. stann + 1
dard fact about forcing that at any stage
we can
satisfy any unsatisfied requirenent of this type by choosing
an •
a proper extension of
Moreover, by the remarks at the
end of 1.1, this operation is
~.
(and so relatively harm-
n
less).
Satisfying the second type of requirenent is More dif-
ficul t but follar,;!s the
U3\J2.1
DC. ttern
of the
Priedl:>:.~:r}.;_·
!·1uchnik theorem.
~
We first split
into
~any
a
disjoint recursive subsets
will come
fro~
Z
stap;e
quantifiE:!' ','Ihich
COLl.rSe
j~f
..-:nB \ Ir:.J
j.3
in
n
"'"
'1"
E:
we will have a
for every
from
requir~~ent
n + 1, l.r
does not define
¢E::
At each stage
E:
specific potential witness
He can
The idea. is that
< a.
E::
potential witnesses to the fact that
C
unbounded, nqirwise
~
<
a.
of the second type at
E:
n (·'1' vi
Cx_£.'
~,
E:
~
~ ,-1.
':1
'--
"~ii
tnos""
.
,~
I
<
4
,.)
f'oY>
~
.J-
into
the first
c.
1s also
since we have nut a bound on the witness for
the init
1 existential
~
Of course if
then
H
2.4
t:
ntlfler.
is n0t of either of these forrs
E:
and '/Ie never cone ern ourseJ..ve s
5.s n.ull
The construction:
sta~e
At each
a-finite sequences
approxi~,ate
for
C.
Band
one sten
L:
n
~1ich
~D
:1. t .
Ne will have an
are intended to
of the I'epresent
segm,.~nt s
rnhese sequ.::nc2s 'cTill be changed froY:l s':ep to
tLeir length.
increas
be
initial
n
~J1 th
the
~)ince
the next will he
~n
of
DrOCf-?SS
~;oin~
from
the sets we enumerate
~ill
•
We begin qt stage
o
by setting, for both
and
f"\
V
X
each
< a.
E:
E:
equal to the least elements of
n +
At stage
1
we let
fled requrie0ent of hiChsst pl'tori ty
1rJi th
be the unsatis-
B.
1:'!e n0 1;.J
satisf~T
some changes in the
causes.
_.
.A-
nI
S
:·1.ore prec isel:"
D
;"p
as dictated
~33.t-
is associated
by 2
.3
2nd IEake
to reflect the injuries this
is a
if
require~ent
of the
extension of
exT1)c
E:
.
set
nO~:J
"
i:\B
first type we let
priority t"lan
can
\'Jhi~h He
fro be specific let us assume that
lsfy.
for
Fer
o~
for all
of hirsher
E:
;-'
lower nric:rity
"nan
.
eoua.l to the leFl.st ele 1nent of
'.
'7
1.J
S
above
ste~';
and
:tno.l1y
all
€
< a.
If
1\,-.;
~
to
("1
V •
.
for
is cd."' tne second
Formally
':Ie
t~TDf.~
then we a.dd
set
Everything else remains unchanged.
Had
T?
J
S
been"'"
.
0.
requireaent we would of course have proceeded dually.
At lir:11t sta.g:es
we sinply set, for both
an}
3.
~he
Priority Argunent
In this section we will show that, for
act to satisfy
require~ents
with
h1~he~
e~ch
we
B~
priority than
only boundedly often. T',TS C2D then e8.S j.ly conclud.e
S
,
and
constructed above ~2ve the detv
t:la t the set s
R
....
sired properties.
3.1
1]:111e first point to consider ::'8 our apprcxiL-:ti.tion for
the
6~.
U
l:;~y
0~
Let
= a.
be the least
y
it
ordi~al
suc~
t~at
ca~i be elthe:c a l.i;:lit or successor ordInal.
In either case the
implies tliat:> for
eaC~l
such tha.t
for all
such a stage has been
stase
3.
v <
~eQched
orn~al"d
lie in any block <
every
~
y
and
'J
all
and
< v
n' \)
11 >
(..J )
o .,..
_
-
~
fOI"
(J .,..
Once
all
s
< 'V.
~j1inally
for every
x
~)riorities
Hhcse
He note that :"'or
n v , E < 0V+1
there is a stage
such th2.t
< 0 \)..l..
+1
V
":>
':>
requj.rer"ent~3
ctre fixed.
s < 8v
T) ,
we can, of course, find a
nv < 8 v +1 ' by which
later stage
Prom such
v < y, there is a st3.£:e
<
s
and
n
>
-
nE
4
r:I:hus vIe c a.n al ~Ja'y s as s uyne any J.ni t ial s egr::ent 0 f 'Che pr>i--
ority listing to be fixed from some
now prove the main
T~~m8
~-':'=.:.:.:...
__ )~.
:-~~_~....
sta~e
onw~rd.
We can
lerr~rna.
-.~o~.
..
.
0a._c~
'- _ 1!
v < Y,
~
_
<
g ..)
_
,
~.te~p
....
_
~
_
l·~
~.'
~.J.--.
hClj·.r..
,d.
~
on the stages at
we act to
~hich
s~tisf~
requirc~ents
of
higler prior1ty t an
Proof:
We proceed by induction to prove an even
stron~er
result:
at 1,{hich
He
than
1) JV,E
J
the set of stages after
nV,E
act to sati::.fy requirements of hi;ller priority
2\ J
< E+ •
, is bounded belo,;; <5
p
l'hf veE)
v+l
'llhe proof is siI:dlar to 1.5.1.
lemrYla 1.5.2
/
Corresponding to
have the fact thc"l.t once
VIe
V,E
h2,ve stopped
~'Je
acting to satisfy all requiremerts of higher priority than
R '
S we act at most once more to satisfy
R
a successor (2) is
iIT~ediQte by
acting to satisfy an
at vThich
hull.
• +-
I->
h
:i
a
S~~olem
s a
also continues to hold
For
lS
llO
induction.
L
n
~
~
ioreover our
,~
fact and so the stage
occurs lies inside the appropriate
lv
As
::>
Thus for
S
fu.nction,
'v'le
L:
Skolem
have that (1)
by the definition of
0\>+1.
a limit ordinal lemma 1.3.2' (for
E
n
L -admisr.
sibles) carries along the induction for (2) since our construction is
~n.
To see that (1) remains true we argue
more closely from the proof of 1.3.2' as we did from that
of 1.3.2 in the last paragraph of l.5e
Of cour2e;1 at
stages of the induction corresponding to
trivially as in I . 5 :
that
c
J\>+l , J --
for limit
v
s
u V
J V,E: c °V+l
-
+l < °\)+2
and
s::: 0
Vle
proceed
ror every
:::
J\>+l,O
:::
O.
The a.rgument
is equally trivial ..
to ou!"
21 via
so~e
simple lemmas.
E < B
V
such that
Proof:
If
y
y
o y = U h[e \) +lJ - a..
D
=
hf \) (E).
is a limit ordinal the result is
obvious, so assume
tells us that
=
S
=
v + 1.
By the definition of
'Thus t,he
don h A 0v + 1
is not
",'
.. i •
parameters
Lemma
Vie
3 .l~:
Proof:
By lemma
~1ave
that
L -generic ~
n
Band' Care
Consider any
L:
and
E
3.3 there a.re
\)
< y
It
Of course,
Thus
immeGiately
a-finite.
and so over ;ts tran-
is, however, definable over
sitive collapse
y,
sentence
~
n-J.
<
ev
¢ Qp
By 1 errl.llla
•
such that
hfv(E)
= S.
3.2 we can go to a stage after which we never act
A.t
to satisfy a requirenent of higher priority than
such a stage we
if it is not yet satis-
must satisfy
Moreover, the procedure for Doving the
fied.
so protecting this requirenent assures us that
remain satisfied.
can now conclude the
s
~S
and
will
is in fact decided by
Thus each
an initial segment of each set
~e
x ,
cons~ructed.
D~oof.
Proof
--
-
L
LeL,
he any
1>13
variable and let
'J
and
s
For the sa:{e of definiteness
1!Ihen
define
is
G
the s t a C. e F; i ve n for"
from tbe
n
given by lemma 3 . 3 .
be
3.S
~:Je
sho;/J
inte~preted
\)
cO~-lstruct5""on
formdJ.a \,:1 th one free
\'
i.>
l-
and
[:
/ L
ljJ
K"
t· , ...J/'-
r;,~ (\"i\.
v
n
o.-~
is
Zn -1
,
p~7 '/)
--
as
n'
n
if
Rn
~
be
':! e see
n
Say
•
,
n·
>
vThere
be
C?_
1'[:-:.. t
n e s s fer the in i t i 2.1
and so
represent
nr
,
there
anG
rk(a) such that
an initial seg~ent of
is not satisfied at stage
not yet in
3. 2 •
does no"C
then by assunptlon
an cl 1 e t a
an
p
p
is constant for
that
Quant ifi:2r.
1s a stace
~)
Let
b Y 1 e mE; 3.
Let its value be
~
t~lat
D.
We now see that
n + 1 (i.e.
vAS
is
C) 't're can and indeed must tilen satisfy it.
To do this we put
into
C --
a contradiction.
Finally ve note that our construction does not denend
on
a
nor does it mention any infinite parameters.
we have in fact establi
procised at the beginning of
Thus
the theorem with the uniformity
t~e
chapter.
4.
As
r
...:..,
Pres ervin:-~
0 i1 t
sed in 1. 3
n
n O~,J i nd i
i:; E.
Gat e
1'1o:-r
but not recessarily in
s~rict pre~ex for~.
4.1
~?tructure
If the
E~(B)
then every
v
T
.l~:
n
ad cL:'ng
used in
are
a.
~(~Tnaj ..
ns
stri~t
formula is equivalent to cne in
as an additional
('
E
T
.LI
.fCJT'h ~
of:
the
r t n20 r e L
;12red to
can be s
prenex
0U
•
sirQ5-1a.r to those
[lO,§6] and III.4 to make sure that
t\'JO
destr~y
We achieve this recult by
uire:-;lents to our cOLstruction
vario~s
sets
We treat each st ict
hyperregul~r.
mUl.-3. vlith
a
does not
p~e~icate
for'-
free varJ..ELbles as a function and try
to
preserve computations on cortain initial segments on which
it is tot
Of course, this is handled via the forcing
relation and so the construction remains
priori ty art·;urnent
conputat
( 2, C
for each '\ralne ~
1 ts
evei.Tt:;ually keep
~:Je
ran;e~!8uD.ject.
(I
obvjou31~l
the first
n in. t h 1 S con t
the rune tlon itself
This
By the
preser~,tln~
t ua 11 y a for c 5_ n ~s c c' n d i t i
T~nus
~n'
bEcome;:"
suffIces to make
Lin
t~ X t )
and
¢o, preser-
that it suffices to consider, for a function
tJ
vations on initial segments up to the cofinality (in
of
ev
(where
=
(3
hf \) (€:)
we add on requ.irements
rn
a (strict prenex)
Let
cr ll (S)
be the
'0
). ~ p
y~,s
Also let
,'rhi-ch
all
does not force
a=
n n
hif (s)
v
R pQ of our neVI kind if
rro
for
such that
Q'
f..)
C"
>.J
La-cofinality of
11 •
If
.
£:
¢
a
)
this
(-<
"
AV
is
B
formula witn two free variables.
stage
value.
< G\) )
for some
L
b~T
as COillDuted
0
'::
y < cfT)e en)
be the least
fer
v
to have a specific
Q
}-.
..1 Tl-r1!(s)
.. \> .
we say that we can satisfy an
'\TTl
,}\).>
this stage an extension of
£
< cfT)(e n )
and
V
(J
il
VIe
vrhich forces
can find a.t
(v n
,
Oq",\)
~.J
to have a specific value.
)
S
)
These new requirements fit natu-
rally into the priority ordering.
The construction pro-
ceeds as before by always trying to satisfy the requirement
movi~g
of highest priority possible and then appropriately
corresponding to requirements of lower priority.
the
4.3
r~ehe
proof:
have only to indicate how the new re-
quirements are handled in the proof of the priority lemnla.
Everything else in
§3 remains the same.
foJlows as indicated in
with the required
Lemma
l~. 1{ :
once the
r n -admissibility
priority lemma is establishea the
<L a ,B)
~oreover,
4.1.
of
We therefore conclude
le~~a.
For each
v < y,
E
<
e
v
there is a
bou~d
on the star;2S at -;orh1ch VIe act to satisf;! requirements of
higher priority than
Rhf (c).
\)
First note that we can adjust
Proof:
Cf1\;,E: (8
sary to make sure that
mind the proof of' 3.2,
e
that for each
U{~J"\I,t:-c-,!s' <
K£ -
immediat~2.
of 3. 2
{ T
U
v ,c.,...
f
If
have
"'r(:~
~ £
T
R
....Tr'"
the set
& R,
E:
<
£
f'
n~.
(c"
v '-
For
2.
bound
£
& R
hf (£:)
v
.
I~eePinG
is of the nev.]" typ~}
I
< <5 v +1..l.
T1
aT
Any
on
is of an old type}
for
s' < £:
and
.
TIle conslde:t'
a member of
T
y < cf 8
one corresponding element, T,
to force
T
1'n f v ( E t) (v
Co\)
rh,
i~
l\
n.
is a
En
K£ - n
~
En -admissibilitv
of
.
a
v
£:
< 6
n mao from
~
<
for which
£
with
v <
pn
a
W x
T
cf 8"v
~he
bound is a member of the
1\
£:
- T) •
a-finite.
onto the
Thu.s by the
'
En
yr
£:
it is
its range is bounded.
-
the existence of a bound is itself a
and so
Thus
at which we acted
corresponding to them.
eler:1ents of
< 6
£ '
cree v )
subset of
We nO·tl ha.v8 a total 1-1
T1
there can be at most
of
H
unbounded in
there are
W
v
of
0-
S
to have a value.
,)
,2
We rirst handle the set
Since
is
a successor this fact is
E
1s necessarily permanent by the definition of
for each
in
is a limi t ordinal then by the aI'guill('rlts
s
n
v
ev
<
E
) = cf(e ).
if neces-
£
see that it suffj.c es to Drove
T'!1e
6 v +l .
bounded below
()
< y,
v
nv
Moreover,
fact in parameters
)'
"1'1
2,kolem huJ 1
c
of
_. n
In pa.rticu1ar,
not in
Now we deal with the elements of
vIe
clain that the assoclated
8 \) .
If cf(e V
) <
8
''0
Yv,s'
T
8
is a regular
v
are unbounded in
,S
into
D
9.
by sendin0;
s
Y
a-+
€'
is the least such member of
~
n
and
a
3 < 8
a subset of
s x
is in f
2. C t
a -. f
~
"'-
j
-
contrad~cted
'J
lS
the
~he
.
The correspondence is
to.
::1
'tTi
th
and the
As
j. n:i. teo
P"
f\S
of this bound is a
0v
T
are in a natural 1-1 correSpODc.ence
E:
n
tells us that
and
Since this nan
~o
Thus
0
\)
8
be the. bound just established.
v
L
L
s
Gvo
elements of
subset is
KE:. -
the
segr;lent of
i1he:re
n
regularity of
Let
segnent of
~inal
E -admissible we have
is
a-cardinal and the
maD frOf:1 trJis final
~l
On the
','Ie see then tha t
P
\)v·
involved nust cover a
vre can cIeri-ne
7,J
are bounded below
then thi.s is iIi1Inedlate"
other hand, suppose
J.s
_.
n
rn
r n -admissibilitv,. of
1.s bounded.
a
As before the existence
fact depending on parameters below
and so is indeed attained below
6 V +'.
.i.
Chapter III
The Splitting
eoren
In ordinary recursion theory one can distinGuish
various types of priority arguments.
between
fin~_ te
The najor split is
andinfini tein,i ury constructions but finer
distinction3 can and should be
Thus for example one
d~awn.
should mark the difference tJet-'reer, the arguments for the
Friedburl,,;->,Iuchnik solution of ?ost' s problem [10 J
sp l 1·t~·
vln~ theorem
r~
LO,
l~l
~j.
and Sac:(;3'
In the former there 1.s an a
ETiori recurslve bound on the preservat10ns initiated for
a given requirement and so on tlle injuries it inflicts on
lesser requircnents.
There are, however, no such bounds
a va i 1 a b 19 1 nth e proof
0
f
t h ,2 1 a t t e r
t:1 e o:e e til.
As long as one renalns ','Ii thiE the confines
o~·
o:;':-'GJ.nary
recursion theory these distinctions are relatively unimpor-
tanto
All that one ever needs in the proofs is that the
injuries to each initial seFment of requirenents are bounded.
Since the union over an initial segoent of
w
of finite
sets is finite, the proofs go through without regard to the
more subtle questions of the existence of recursive bounds
on the inj ury set 3.
The 81 tuG. ~~ion changes draLa t ically
\;Jnen
one enters the realm of recursion tllGOry generalized to all
admissible ordinals
r~l1e
0,.
an ini t :.0.1 se gr.;ent of
a
problen: is that the union
of set s each
neI~al
Q
O'-ler'
f ;/ihich iso, -1'1ni t e
tIle cell. etlon itself
must be given as an.
C'k-flnite u.nion of
one to be sure that the
proor of the
UniOY1
is
?riedburg-Muchn~k
a-finj_te sets for
a-·fini te.
theorem, towever, one is
not so lucky as to have everything presented on a silver
platter.
of the
In the
Indeed} a cruciD.l point
theore~
S2.c>~s-;:)1;".pson
proof
a-recursion theory [10J is their lemma
in
2.3 (see I.3) which enables them to handle unions which
are only
Unfortuna.tely, their lef:1l.-rna only a?plies
a-r. e.
. . bound on the size
to collections with a un-,- f arm §:. Rrlorl
~
of the .i1erILers.
It is tl181'efore admirab1y sui ted to pric1r--
i ty argurle:nts of the
Friedbure;-i\'~uchnik
ty'pe bu t does not
seem to suffice for ones like the splitting theorem
lack appropriate bounds.
~~ich
Indeed to date the only priority
arguments that have been carried out for all admissibles
have been of the Friedburg-T'ruchnl.k type.
1~:re
In thls ehapter
exhibi t a construction that enables one to handle nrior--
ity arguments of the second type.
a
In particular we Drove
strong form of the splitting theoren for all admissible
ordinals.
TheoreF1.:
be a non-recursive
a-r.e. set,'S
A,B
B.
<~
C
i\
be a regular
Let
and
a-r.e. set.
8
and such that
D
Then there are regular
such that
D
a-r.e. set and
is nat
AVB
=:
C, AI\, B :::: ¢,
a-recursive in
A
or
Note:
We have been informed
tI. Lerman that C. Teo
Chong has independently proven that there are incoLipa.rable
a-r. e. degrees
belo~'l
every
stronger result is a s
1
c,:-r. e. degree.
C.
.L .....
Though an even
corollary of our theorem (Cor.
3.3) the same type of priority argument is needed even for
the weaker result.
~e
have net seen his proof but have
been giv2n to understand that his methods are rather dif-
ferent from ours.
1.
The Construction
The intuitive picture:
1.1
Ve ber1n by noting that since
a-r.e. degree contains a regular set we can assume
every
~·.;ithout
loss of generality that
c
and
d
be cne--one
C
and
D
respectively.
approximations to
DO =
n
is regular.
.i./
I:Ie
let
Cl,-·recursiv2 func;tions which enumerate
As usual,
and
('
v
denote the associated
\'Te
IU
CO =
by
c (1)
and
1<0
LJ
r1(lO)
\
u
•
1<0
The general plan of the construction calls for pu.tting
c(o)
into exactly one of
vlill assure us that
A , 3, -c~
<
We will
also
v
-
0v .
to keep out of
as soc iated
A.
or
n
0.
AU13
=
B
v~riGus
......
(?
0-
This
and that
elements that we wish
for the sake of requ.ir'2r.lcnts
the co:yli t i on that
These
at stage
=
C, A/"tE
h~ve
~
and
't:j_ th
and
fA
~equ1rements ~ill
D
not be recurs J. ve 1n
be ordered ty a priority
sys tern that \;il1 be develoDed along '.11 th the construe tioD °
At any stage
we will have an approximation to the final
a
priority listing lirhich we
\'7illu~)e
c(o)
B.
is put into
\<Th1chever
~.',rill
A
or
pre servc~ the
to determine whether
Essentially we will choose
r~?qu 1renent s
of tlH: highest
possible priority.
In line with Sacks' original approach to this theorem
[6J
\'Ie
try to insure that
"":/,"": ":" 4"
1.
/
..
j,.:to.
A
\
S )
by the
rounciabcut
in~tiaJ
on
~1he
sez~ents
as
lon~
ey seeE to agree Nith
2S t
[')
idea is that if for'" SO;-'l"?
E:
=
[s l'~
j
D,
then
TiTe
J.
'lIould
eventually be preserving the first available comrutation
each
D
basi s
0
would then be able to
a-recursively -- a contradiction.
hO~'iever) ,;::e
f such c on3 idel"2ttions,
that there is
SO:llC
of
C01~j.t·DJu~atl·on,~~
_
~
_ _
'-'
_
~
x.
priori
~alue
bound on the
~!'._cJA\/X').
c.
•
On the
can only- ar p:ue
for Hhich uc preserve
x' s
cannot as s =1.3n any unj.form
~'le
to this bound and so find ourselves heir
to the problerns discussed in the introduction to this
chapter.
Our strategy for handlins these difficulties is to
first arrange the requirerLents
then to consider
requirement.
[S
"
lJR
r~ ~~
1\
[s J1->.
f or a 11
D
in
in blocks
?
?
and
as a single
Since the preservations for such requirements
only involve keeping elements out of
conflict v:ithin a given block.
whole block
t
P as a
si~gle
with our ability to recover
associated with
P
A,
there can be no
Moreover, considering the
requirement will not interfere
D
should the preservations
be unbounded.
Finally we will have
the determina:tion of' the size aDd number of these blocks
interwoveD with the construction in
suc~
a way that the
blocks progress through the list of reduction procedures
cofinally with the progression of preservations and injuries
through
a.
1.2
actual construction:
~he
Since
and
f
entire
explicitly only the
that s
par~.
:
It is of course understood
lIar steps are to be taken on behalf of
.,
bec;:tnnin c the constructlon ",[e let - f:o.
cr.-recursive pro(.iecticn.
W~
will be
~n
the least
x
for
with argument
in
De
an
~.Jill
For
eDC h of 1,'!hic h
e~ch
~e
y
fin0
y - A
reouiren--jent
associated with a
y - A
active reduction
y
A active reduction nrocedure
-
1
1'0
[f- eJd ex)
for "'lJhich
such computation and create a
_u
=
y -
D (x), we take the least
A
the elements required to be out of
of
*
there 18 no
If there is a
nrocedure.
a~
of
r;0"
Y
blocks
h~ve
will
~'!hich
x
('J.
Before
1;i1tr the const:ruction.
se~nont
initial
l_--..L)
B.
of our terrn1nolor:",1
T'::~--rest
si:'~'ulta~eousl~T
be defined
...
by the chosen
A
comDutatiQn.
If at any
ment
into
cedure
is a
x
A
E <
y
":le destr<?"y'
a*
A
is
x
the requIrement.
Y - A
require~ent
associated with
1.e.
we put an element of a
sta~e
E
active at
y -
reauctton nro--
A
a
sta~e
A require-
unless there
(as yet unde~3troyed) Hith arpument
such that
r-.
,..,-1
LIE:
J ..,.p.~(j
I
\. X
)
I,,;
has been enumerated in
D
since the
( )
r....t -,0
J
.X
_.-
reauire~ent
was created.
The idea is that as long as we seem to have
a computation
showin~
that
r-.;',
L-l.
-1
E:
In"
r n
-t
jj
\V'e need Day no
'71
( ..L
further attention to
€.
Now for the definition of the blocks
where
Y~
stage
T
= U{T<Y'
V
C at stage
(~8<y)
(a
P
5-requirenent is created at
6-re~uirement
an element of a,
is enumerated in
T}.
The idea is that (for some
A) ea'~h block
eventually reach a constant value which will reflect (via
injur~es
a bound on all
f)
c-requiremc;nts for
shOvT
D
that
caused by our having to
C < A.
:Jnce v;e have such a ()ound
alluded to in
1.1.
:'ioreover if
U Py = a*
A
:Ls the least ordi--
then every
P
handled and we will succeed in getting
n
Y<A
Finallv we take
sider the sets
y ed r-u :1~
Io(I o )
,4....
D,. lU t:
-
be the least ordinal
a
y-requirement.
A(B~
of
J.._'
t' l' n• ,0",'._
.
If
c (, rr)
.i
-~
y <
~
8
~
3
or
. fI-
2
so
;;~ore precisel~Y) ~'Te
j~"\.
(
;3 'I
J
J
•
3uch th2.t
8
A, B.
con-
requirements which would
j .. n" t 0
'.1
A
'Nill be so
y
and put it into
C(0)
as to preserve as l':luch as Dossible.
'0" pst
~.. 0
~...
can
N2
is bounded vic. the arguluent 2.bout recovering
nal such that
b '-~~
preser~e
we put
L~ P~ ,...v
.r ( U.r B )
u.;1
contains
c(o)
into
d;
require;:'"1ent priority ave!:' a
y
2 8.
structiorJL.
0 - B
r'equirener.t iff
our description of tho con-
~he
2.
Priority Arsument
Our prima.ry concern is to show that enough bloc I\:8
eventually settle down so that we can 8rgue that
[f-1sJA ~ D
s < G*.
for every
Lemma 2.1:
If by some
stabilized at values
N*
v..
~
<
.L ()
0
an d
sta~e
then
~ _1
pO
1 Y
U
.J
cr
S
V
have
respectively for
Yya
~.nd
Co
C < "'(
eventually stabilize
\'t~~
and
a
respectively).
U ?" < a* irnDlies that
o<y o
since no sequence unbounded in a can oroject
T';ote that
U Y8 < a
8<y
down to one bounded in
a*.
To establish the
a
clearly suffices to shov that
For
and
and
o<y
(of course at values less than
Proof:
We proceed inductively.
Yv
le~ma
it
eventually stabilizes.
a limit ordinal this is immediate from our assump-
y
We
tians and the definition of
=
consider
the~efore
v + 1
and show that there is a bound on
the stages at which
v-requirements are created and at
the case
y
which elements of such requirements are enumerated in
Let
stage
for
y
8 < \)
y
= U Y6
o<y
onvlard.
and look at the
co~struction
Everythlns connected vi th
has stopped
actln~
up by starre
f;
y.
we never have to worry about nreserving any
for
8 < v
after
sta~e
y.
mhus any
v - A
C.
from
--requlrements
In nart:tcu1ar-
8-requirement
reauirement
711
I
'
existi~g
at stage
strayed.
or created thereafter is never de-
v
cO~Dutatiorl
The
associated with such a require1'\0
r f - 1 E:"1 0
ment is, therefore, correct, i.e.
-
_1
h
r.p-l e
JA(
= L.l.
x ).
(x)
liTe must sho'l>J that there are not too many of then.
Con ~) ide 1" the set
is
v-inact~ve
subset of
D
at t'J'hich e3.ch
and so
any
of
W.
y.
This is a
no
v -- A
e::
in
liV
requirement
Hhj
can be created.
ment created after stage
r-Tcreover by the
W
a-finite subset
is an
After stage
T.
ch is associa ted "'I'li th any
2T1Y
\)
A
-
requtre-
gives the correct value of
The point is that the computation from
changed while the only change in
"that a nev·; element of
D
is never
A
that can occur is
rrJ.
•
~~1lS
is enuTr:era.ted.
,
\'lould put the reduction procedure involved into
definition contradicting our choice of
Finally it is
the only way
~ould
cl~ar
ho"rE:ver,
T/T
by
T.
from the construction that
a-infinitely nany
v - A
requirenents
be created would be to e~~ntually have ODe associated
with a
v-active requirsffient for each
to occur we could calculate
cide
1
v--inactive are contained
Poreover
T
E
Of course the stases
becomes
Since
such that
-v
this hull is bounded,- sa',l.·'y bv..
T
D.
p
v-inc?ct i ve after
becomes
forever.
H
in
a-·flnlte.
€
30
in
E:
~l-hull
P'.) < a*
•
of
*
it remains
y
in the
J.
on~e
above remarks
stage
at sone stace after
<
v
of
~f
K
D
=
0) be
D
x < a.
Were this
recursively as fellows:
at stage
T
and proceed until a
to de-
v - A
requirer'",()'+in
ha
\tIith arfSUment
been
fer'
creG~ted
assoc;iated vrith an
x
xsi{.
evel:~Y
SLot
(2ince sU.l2h
cJ
stage is recurs i ve, th'2rt:2' is one stage by 1:-rhie :'1 it has all
stages at ""I]rd.ch
the collect1on of
::-,eaulrer'ents are created.
\,\ - A
which elerents of these
ct-fin1te
requi 1-'erJents 1s an
re0uire~ents
Thus the contribution to
Sf.~t
from
are enumerated in
also stabillzGa.
va 1 u e s bel 0 vI
'l hus
i
and
ct
P8
= a*
y
(J
attain
and hence
y
can let
Ne
T
U
.
y - B
res p e c t i v ely.
a ;:
In vie ,<: of this lenm,::l
ordinal such that
\..'
A requirements
y -
argument now shows that the contributions from
requireme~ts
I'
Eecinnins c,t :3uch 2 stase the sane
eventuall;:'" 2tabilizes.
con s t cLn t
:i~oreoifer
is reGular there is a bound on the stages at
C
and SlDce
A
v -,
A
0
be the least
~
and be assured that
is
8<A
does in fact exist for each
a limit ordinal and that
~
< A.
tion
~e
s
are now in a nosition to nrove that our construc-
Of)V:tOllS
such ·'tha.t
.
r;(J
'v
AU B
ir:lnediate "J/hile
f ..-' B --(J.
<
'I'o check if'
ri
A f\ 13
less
before
p\s noted
SUCCeE'ct80.
1"\
1 ...--
Cl..
=
r'
v
r\
y:
1'"
1\
1\
" ask
and.
== (f;
if
C
is only slip;htly
ju,st find a stage
{lCf
41.
clearly renresents a reduction procedure
.8
==
l''"\ K == ¢
.
~or
from
are re
A
:\s
(J
This
C
A
and
7' 6
Bare
a-r.e. by construction we only have to Drove the
follo:'rinr; :
Lemma 2.2:
Proof:
ts not
Let
that
E:
<
v - A active
[ f --1 £ JA
=
aftex')
D,
v - A
2.
s'
v - A
cannot be
E:
=
x
~
such
which is
requireT;lent associated '.vi th
at any stage after
1\
H
Yv'
As
inactive at any stage
is regular there is a stage
(J
by
which
[f-lEJ~ (x) = [f-1s]A(x).
course also assume that
\
D(x)
n
J.l,
be the least ordinal <
v
[I[oreaver since
a ~ Yv
l"
By lemma 2.1 there is a least
P'J "
not the arc:ument of
a
e cur s i "'I e
vor the sake of definiteness
Assume not.
say
Ct -!")
r... l·o-1c~.A\,X'"
e- _
quirement with
r;'~e
D0(x)
crea:t ,e, a t
ar~U8ent
x
active reduction nrocedure
our choice of
x.
0
= D(x).
S
t ,ap'e
T •
',Ie
Since
a, a
"
v A
_,
associated with the
E.
may of
::'e-
v - A
This of C0urse contradicts
7?
3.
Some Corollaries about
a-de~rees
In order to derive sone interestlnp;
about
splittin~
a-degrees from the
theorem
consequenc~s
~e
need a simnle
lemma.
If
LemT1~~~:
a .-r • e.
set s the n
Proof:
de g
it
suffic~s
SInce
and
is otvj.ous.
is
Av
( B) •
ir~
To prove the other direction
A and
......
observation that
( A) V de g
B
AC
are
(and
a-recursive in
from
Be)
l\,
U B.
c;.isjoint this is ilnnediate from the
a~e
B
are disjoint regular
a-recursive
is
that Vie can get
•
B
3 ) = de g
B
Pi (I
to show that
A () B, i. €
A
and
(1\ U
That
deg (1-\) V der; (B)
A
::::
U
(/\
a-r.e. and regular.
"r
,C
:3)
n
(J
tl
and the fact tha.t
13
0
We can now nrove the usual corollaries of the
~plitting theore~.
C<?rolla!'y 3.2:
such that
degrees
Ci
ia
b.
d
a
is not
and
b
Let
c
a!:1d
a-recursive.
such
tha.-!';
C
=
d
(x.-r.e~
degrees
there are
a-r.e.
be
~hen
a \( b, d J
+a a
0
and
b
and
-
...-.
-
~......
b, 0 <
"'.,)
C1..
0
f t
i f
the n
=
D
s et ~)
a
,3. l1C.i
V
and the
i saor
b
theore~
recursive
is incosparablc
a
and
b
Du t
b .
~.
complete
Let
a
the
theore~.
ar:~
and
b
Qe~rees
b
=
1.
a
e ..L..
.!:: b.
0
a V
a-r.e.
c
de~ree
of the
D
theore~
be a
~
be
reEular
set of aegree
re~ular
be the degrees of the sets given by
If both
a
~nd
b a r e comDarable with
recursjve in it and so
,::1
and
d.
Let the
and
be the
by the 1 e 2m 2.
c
a-r.e. set and let
d.
c
one is
a-r.e. degree then there is an
Proof:
a v t' = c
and
ruaranteed
assures us
1ncomnarable with
then ',)cth
B
Let
are comp&r2.ble " (2.
b
c
C'i
be r2gular set of degree
and
j~
<
t;
ct.
in the theoren.
0
:"'-'::0
.1 ....._
a
0 <
C,
e:t
Let
Proof:
let
<
(1
C't
a V b --a,
< d
~a c --.- a contradic t ion.
0
but
d
4•
4.1
be
Stren
fl.
enin~
~'.-e ~/'!i11 shoH 1101',r
In this section
strengthenec. to
fro!'!
or
f~
r'i8.1~e
Si~ce
hyperrep~ular
()lJre
usu~lly
not
i~
a--calculable
a-calculable
~rom
of OJ.r theoren
and
invoked to
specifi_cally,
[s]A(x)
::'
our ]:1a1n theorern car.
a re
set
He vV'i 11 dot Ii ish v ad dIn 0'
:~ore
that
a set is
suffice to 8ake the
vations
a-calculability
for
~:e
~:Jill
0
o(?rrerular.
n a s r e w r e qui r e r1 e n t s the
~~ke
sets
preSer"18
hyperre~ular
cOr:1putatto:n~"3
Dr e
s e r-
[lOJ.
for
roughly on the lon~est i~itial se~Qen~ on ~hich
r
it is total.
L€
on this sefment
a~d
so its
I'.
l. ,.
ran~e
essentially
a-recursjve
wIll be bounded.
Thus
(i
d.
will be hyperregular by definition.
4.2
~e
The construction:
notin~
besin by
that we can place
an a pr:i.ori hound on the 1erlgt:-l of an InitIal
which we must concern ourselves.
that there is no need to nreserve
ger than
s
.~1oreover'
cofinality in
[e JA
on
y.
add:i.ttc,ns to
At
the least
1s
L (1
we can
at;out
Trivial manipulations show
f\
on any
[E]n
if there is a last
y
se~~P1ent
~et
by
seg~ent
10n-
.'X-cardinal vThose
with preserving
Keeninq this in mind we make the following
OtlP
construction:
we find)
(J
for each
and
such that there is no
x < s(y)
..:'\.
ctS oci t
E"
each
v' - A requireIf there is such
an
x
and
1s converrr,ent
requirement with argument
x
crea tea
1:Jf~
associated with
s.
v' - Ii
~he
qUirement consists of the nega.t1ve facts about
bound the creation of
by
Of course a
v' - A
A.
ments into the priority
A
a~d
not
confl~ct
(Since"
listin~
v -
A
by
and
c(o)
.'
H
into
The priority arguDent;
that the
givin~
v -
that
e: <
and
any
Pv '
of
[r-1s]A.(x)
of
s(y)
troyed.
v'
E
y <
'J.
·-require~-nents
or
B
Our main
stabilize.
< a*.
Let
v
X
....
are made into
< <:,('''\1')
......
1
We now determine
as before .
~oal
is again to show
Once that has been done
A
as follows:
be the least ordinal such
Once we are beyond stage
for
requirements do
A
we can argue for the hyperregularlty of
~onsider
s for
Finallv we insert these require-
this presents no Droblems.)
whether to put
4.3
r -~equi renent
the saDe priority as the annropriate
B)
requlrer'1ent s .
y
requirement is destroyed if any cf its
elements is put into
(for
to
y'-requirements and the enuneration
of element s contai red In
C
used
\'!e adJust the defini tton of P~
in this conputation.
re-
il'
~I
v
any- cOTTloutations
covering an initial segment
require~ents
which are never des-
is defined at all we even...
:':orcover
tually get such a corroutation
(A
1s still rs-gl:lar).
Thus
we can compute
on which it is defined by just going through the construction
2ince t is is an
~-recursive
proce:Jure
:ts bounded on this ser:rr:ent and
hyperre8'u
ing
19m~a
~e
2.1
~herefore
this
i~
LeTTlrna J.l.li:
contex~.
stabilized at values
Proof:
=v
y
+ 1
conclude our proof by e2tablish-
If bv some staF(e
then
D
'-
and,'
o"
0.1
Y
t efore,
,f\.s
~'la
""no",
pO'
...
requi:re;;lent s are bounded.
sta~es
creat
j
,. Y
respectivel~,'1
0
-
and
eventually stabjlize.
need only considel...
\'7e
h3. 7 e only to
T'Ie
at which
case
•..1 .• ,
="...,.)
(;of
ShO\-'1
v-
that there
v'-requirerents are
takes
("',L'
CO!'HjOnent
~T
have
and
and we know that the contributions for
is a bound on the
1s
of the otier
ca~e
If ther'e is no last
t)/.-cardinc.l or trh;
.I
last
bound..
a-ca~dinal
is sinGular we can
subset of
01')
D
;. \)
X
V,
{X
*
r)"ll
as sumpt ton.
and so enunerated in
houn~ej
~lhl1S
~.- fini
the set is
v
j.:3
r e ~; u J.::t r
'J
c;; fir s teo n -
x
iT
'v
and so
t
E:~.J
a-finite.
f)
£ .
Si~ilarlv
te
tine.
~:~et
-.7ith
e.
each o~ which is strictly
,
If tl'le last
c~-~r.
(the associated reouireMent,
the argument of the requiresen~)
s idep the
directly for this
\), -requIrements created corresnoncl to 2.n
rrhe
les s than
ar~ue
\.
all these
')
is
< y
":hich
a,-r.2.
recuire~ents aSSOCl-
for -'lhi ch
ar~uments
t he
';Ie
cre~~t e
could arsue as above, since
\' · -rc'Qu ire:Y'":ent s, ',Ie
y < a*.
Or the other hand
if there t'lere no such bOLtnd, i're could
stve ma:9
y
-+
t\rith the flrst
P
v
x
by sendinr:r;
\) f -requirement
over
p,. \)
<
vI
of
to th2
~:.Jj_th
Since this would exhibit
~.
y
(~.efj_ne?in
argument
as an
y
in
T
..1./
a
deed conclude as in the nrevious cases.
Of course there is
n()~I;
created after
x
a-finite union
Thus
•
y
it i'!ould
'He
ean in·-
0
no difficul t;r in
the corollaries of section 3 for
\~Jell.
?:3soclated
a-finite sets bounded below
contradict the regularity of
as
E
(x-recur·-
crt v:in9: 8.1.1
ct-calc.ula.biJit:'l cier:y·ees
recursion theory surely the most; powerful (and difficult)
is the i.nfinite injury r:lethod developed by Sacks [6J.
PIS
a prime example of this method we cite Sack's theoree that
the
r e,
degrees
C~!
;::Jr'
E.
"-""
0'-, Y'J
......
c:;
[1
C\
Il.-,.............J....
'I·
J""'. •
dee d:. bee n see n a s a t 0 11 ~ h s ton;2 f
method in
C'(. -,
be s t res -l J t t
the
t er
r e cur s i
0
the 0 r y [ 1 0,
0 (l
cis. t e has [) e e 11 ~) r i
SC0
a-r.e. de[rees are dense for
\; e
2
i<: etc 1:1 apr 00 f
adrnissitle
u.
0
f
detailed
~heorer
t :~ e
~t n fin i
t e
. Lt J,
[ 16,
~~.
a*
=
w.
uy' y
')l
L
J •
The
In this chap-
[1 eo rem
Eopefully tilL::; hezinninEs
~-recur3ion
has, in-
11 ? s t [} e 0 r e ~J [1 5 J t 11 a t
the d t~ n~) i t Y t
some insight into the gener
argum~"?nt3
0 Y'
~his
~'lill
for eve".C' y
also give
L1etllcd of infinite inj \J.ry
theory.
iscussion will aDD2ar
A full
elsew~ere
~roof
[13J.
111th more
."
1.
Gene.t>al ;-:) an
Let two re2ular
a-r.e. sets
s < 0. C
be
~iven.
~\Ie
sueh that
<
--"Ci.
A
( 0)
=:.
{~A
Seconaly.}
II (('
~)
.yo.......
1 \./
'de
C.
To
~nsure
that
A
has
1 the
to nake su.re thclt
c·
(;..
ne:~~ative rei1u.irer.-:ent~3 rC3cF'blin~"S
\;r5.11 have
cuara~tee
those of chapter III to
c -La
{ t.
that
Pine.Ily
we
r·
,.'
into certain areas of
~
lines of chanter III.
The idea is that, should
out to be
of
C <
--a
a-recursive in
into
A <
-a
ns long as it
B
a~
in.
sir:lple
a~pears
that
turn
B, we VJould tben have put all
~·Ia~~.
This would imply that
a contradiction and so
As in chapter III, the main overall problem
',...,
l:::>
+-vO
arrange tl:e requirements in a sufficiently short sequence.
Because
the complexity of the construction we are
forced to
L
2 ( E) - c: f ( ;~ ):> as
0
ur bas i cor d e r in g
0
r
~) rio r i
tie s .
this is a natural choice it is difficult to implement
it 1s gi v,en tJy a
L3
~' h 011 SL
sin~e
fune t jon \'Jhi Ie our cons t:ruc t ion nus t
dure for the priority
T+.l..\.,
converges only in the sense that the correct value
least one returned to unboundedly often.
•
lS
+-1--
vue
This Dhenomena
is typical of the irtfinite injury ffi?thod in ordinary recurs10n theory and it waB only to b2 exrected that it be
manifested in the priority listing i tsel f :Ln
a-~recu.rs iOIl
theory.
Of course, we also have the usual prohlec of infinite
injury arguments.
We must not allow positive requirements
to be thwarted by an unbounded succession of temporary
negative requirements.
This probleD 1s solved roughly as
in [12J by not allowins elements associated with positive
requirements into negative requirements.
Instead we put
in a collect ien of other elernent S \'Jhic:n are also being
kept out by related negative
this is vague.)
require~ents.
(Admittedly
The machj_nations necessary to
carr~:l
this
out THill cause a split -in the proof (tl10UZh not in the
construction) accordin[ to dtlether or not the
w
equals
or not. The idea is that we will have to be
able to close under the operations involved in form1ng
negative
in
w
require~ents.
many steps of a
all will be vJel.L.
If, on
This closure is done naturally
E
2
tl-;.e
process.
~hus
other hand,
if
j~ ~
c::::.
-
E2 - cr(G) > w
cf
(C~)
=:
we will Drove that only finitely many steps are needed
(.,),
essentially be
/
T
'J-!
relative to
D,
a
~.)
..
::r~
recursions.
by restrictjr.g
We handle
ourselves to recursive cof'inality - (B)
several parts of the construction.
cial
well.
its
2ble to iter2te s
not be
as
~crking
le~=a
2.1 which allows us to
many steps for
Eve~
this urastic
~ark belo~
.,,
p ~~, p
kX
that
{v II
=:.
u
of
r ('{
1J.~
,~
,
'.f)
for every
Cf.
a preDel" i.Iii tJ ,;:;1
lS
..
x <
e
lJ
but
=
a.
x<:8
b)
=
Y
the 1e2.3 t
8
;::uch ths. t thE:reI.8 a Dar·'t ial
function, k(x,Y), a-recursive in
B
such that
k
is
.i~
a proper initial segment of
U
k
x<0
x < y
but
= S.
x
-f[·r
g,LYJlJ
every,S < y, f[ g[ 0] ]
2.4:
for each
is bounded in
t.'3 for aporoximating
approxir.--i3. ti n§~
that gives us
a-recursive in
=
that
[efJB
f(Jex)
= [efJ~
B~
at stage
IJ B
=
ep
-"-
>
h,}
Ide
jegin by
a-recursive enumeration
As
a.
there are
f, [e
and for
p.
and
f, L!"
as usual via an
3(J
p
is unbounded in
e
f-l, and
f,
,,-1
r-
l
and
[ehJ
and
e
h
c - h.
hare
such
We can
ftherefore approximate them q.ui te easily Ly set tin(~
p(J
for
f-
1
and
(x)
h.
for ever:r
x
and acting similarly
(Note that "He al-r:;ays tal:::e tl'.e least
(J
[eJ~ (x)
available computation giving a value for
ide approxlnate
determine a value 4)
lemma 2.2 ( a) •
As
and He set
c~ cr
tS
divergent.
We add
a later stage
rnation about
(x')
-\.1
(J T
Ie
is
least
g
via the
k
to
of
a-recG.rsive in
y
such that
the proviso, however, that if at
"He dlscover tnat some incorrect infor-
reverts to the least
abou.t
B
for
y
~'Jh:l.ch
"vas used in the cOrlnutat ion for
The approximation to
f
and
rect on each initial se7ment of" a
As ror
are eventually cor-
h
B
and
respectively.
f-l, the approximation is·eventually correct O~
for each
can say 1.s that any
to be incorrect.
y
tneorrect inforuat5.on
such that
<5
<
rr.
propo~:;ed
"Finally for
gO(x)
=
y
For
x
don f -1-
,t.
~
value is eventually seen
,.
\
x < u, g,Xj
is the
for unboundedly Many
o.
3.
~Che
Eequ
We will try to give an intuitive picture of the
~etails
requirements and leave the formal
this we supress t}le
facl1itat~
.....
and Nrite simply
f
.L,
-1
approxi~ation
and
"g
T,',1.
e
all other
3.1
<,~.
0 : x)
.
+ln~o
~
A.
requirerr~ents
anu
D
1.1I.;_
v
,-1"
_nlS
AG(x)
find the least
Ath(y)
froD
1 n I I I I . 2 .)
priority
square
as a
S
y <
is enumerated
"
caKes
:or
p~sceuence
~n
over
e2.C;1
x < \'
l!Jf2
(~on-
2.Y'ra.~T.
such that W2 canrot cODDute
B
via some
p-active reduction procedure
i;J e
the n i:n i t i
G:. t
e po sit i v (.; r e qUi r e:-:'c en t S "7 i t 11
r
\j
that incorrect information about
Arh(z)
I
C h ( y) -
the least
fran
be;innjn~
r
1}
~.
(v)
\
'-
into the
o
th
Of course, if at any later stage we discover
of
some
nrocedures
-(i,
that try to copy
x
x
To
;::) <
'l\i.e pos5 tive requtremr::nts:
sider
Be~ore
l1 •
in earnest we insist that when any
B
to [18J.
A
C
,7
was used to compute
we then give up trying to copy
lh (z )
v
B
<:
1 nt
B
Vla SODS
0
c~Js (x ) ) er •
such t
t
'J e
c 2. n not co::::.:' l~ t e
n-active roduction
prcc~dure <
fg(x).
For esch
us
:in the c
r~a:(e
these 1nto
about
111:::e to
'poulll
allo~.;
s tee:. d
a~.:
nr~;;ativc
eJerlent s to be
soc i ate
priority
,;e
x
i,~c;
t
requ1r?:;'.snt::3 but th2 t
O~). t
of
b~?
A.
~, J
r[lhe
a,x,z
ma:Ln
l."'e s t J.>j_,~
that no element with an associated positive
ments lnto
tive
~)
.:.\
0,X)Z
'o""x,z
is later put into
R
0,X,Z
Crh(z)
B used to compute
destroy
for
~he
construction:
ority
x
priority
~hich
SI1G[}
e1e--
w
p
La , x , z •
'l',":--
~J
aj~~~z
If any ele'nent
or any information about
is found to be incorrect, we
> z.
We SiMply act as dictated by the
requirements described above.
element for
than puttinr; any
of high priority whicll are currently
an operation to get
su~h
3.3
i~::tther
rcq~ir2Gent
Indeed, ue basically close
keeping then out.
of
.
ti.or: is
we put in all uther elements of nega-
~1
req~irements
under
a su cc e2· S lon
n e ~ ,:-1. t i v e r f~ qui reI:; e n t
2,
with each
gets put into
':JOU 1 e).
At each stage we put in any
we have a positive requirement
wit~
pri-
unless it belongs to a negative requirement with
<
x.
4.
~tline
The Proof -- An
rrhe priorIty Argument 3nc3 Droof t}l.-lt OUy' constructi.O"il.
s u c e e e d s a. r e qui t e l 0 n g and
we have
j
e =' e n ~j. so 1:1 e.-I h ,q ton de t a t1 s
rrhus 11'1 this section ~':e merely gi.ve
neglected.
the sequence of ler1rnas
\Firi
'\fhlcrl the proof proceed2 ':Ii th
only a general idea as to hO\\f one
Lemma 4.1:
begins to copy
C
a.-recursively 5_n
stoDS
tryin~
One caD tell
into
a
prove~3
the lel11ma:3.
a-recurs:'_";lelv :tf one eve:"
glven part of'
=·1oreover,
1\.
Bone ce;.n tell whether one ever
to copy
C
into this Dart of
,1\
J..j.
•
late~:-'
Finallv
" .'
if one never does give this up the assoc:i.ated computation
relative to
is correct.
B
Lemna 4.2:
'rhis lemma is irnmedinte.
If a negative requirement
R
da,X,Z
permanent the associated computation relative to
is
corIOlect.
The proof depends on the well-foundedness (derived
from the priority ordering) of the procedure used to forn
from
Na,x,£:'
n"
Now for the heart of the procf one proceeds by a 51mul tar:.eOllS induction on
Lemrri2. 4.
1)
x < y
We can tell
to establish the follo'>tJins:
a-recursively in
B
if
a given negative requirement of priority
< X
Xl
lS
per-
manent.
2)
a-fini. tely many :oe!·I"~a.n.ent nega.-
There are only
x' < x.
tive requriements of priority
3)
x'
<
x
A permanent positive requirement
gets put into
negative requirement of
A unless it belongs to a permanent
~riority
T.here are only
4)
C1.
x"
<
x'.
--·fini ~ely many eler1snts at VJhich
we permanently try to copy
x' < x.
of nriority
C
in
A
with priority
a-fini tely many elements
I"Ioreover, there are onl y
for which we actually create permanent positive requirements
with priority
x' < x.
At successor
(~
staces,
the fact that we can theA
\.1
)
(from
~
\
\j)
and (LI.) plus
0-recursively check if a nega-
tive requirement is destroyed by a
quirement.
t
en
positive re-
tempo~ary
Of course, in every part we can tell iMmediately
if any incorrect information about
B)
was used.
b
The proofs of (2) and (4) are similar to 111.2.1.
failed we could sho"J that
C <
-a
would give first that
If (2)
B., w11ile a failure of
~ind
c
then
B.
<
-a
(J.~)
In
either case v,,:e contradict our t)asic as sumpticr: that
B < a.
('I
V
•
e 1 e r:l en t
p
l"
"(1 a
..:....
."..to.
~r i
J l"l
_~.
(3
II . J+"'0
1 1-'---.
0"· 0~
.
J
.f'"
._ Y
'../.. ; 0:-'"
:. _..
t
~>
~
..L ... '_
t h a p 0 sit i v ere qui r e ;~J: n t
tive requirement thereafter.
to which it
belon~s
r>
_
0 r'.l--t.........
(-1 l t-V
.J,
4 n,
_.....
'_ .i.Y"1
+v
'r '1.....-\..
~ tV
Y'
,....
J. 1 V
i () r: uti n to aLe.::: a-
Thus all the
ne~ative
are eventually destroyed.
ones
At linit levels we note that the whole procedure
was
a --r e cur s i ve 1 n
ments of
LJ,
~_J. X C C
were also ne
C
r) t t L1a tee r t
eJ.
in i t i a 13 e [>-
25. 11
t~e
It turns out th2t
ar:lount of
funct1on.
the definition of
proper initial seccent of
set is
a-fini t2 and one
r',-I.
C(:.Hl
c
Since
no',-
,::;0
th:"'ou
to this sta.se uniformly to get the requ:ireJ bound:::;,.
ide are nOH in
pOf:jition to
2.
tLat the set
shO~J
~
.c\
has all the desired .properties.
Lemma 4. 1+:
I,
of course, well founded and
t~le
sure the tree generated ty any
a.
under are
If
only problem is to make
a·-fini t~; set
In general the oper&tions
l:
2
E2 - efCa)
and so
=w
Sar; icaJ.ly
(ine must, however, sco.:ccl'l
the idea is to use
below
1', ~~ C.
is regu:tar and
then i.t
out
lS
touLdcd
th~t
one nust close
>
a 11 is
0J
tha~
\I e 11
.
finitely D2ny
steps suffice to close off, since in thjs case all priorities are intezers and each sten
in priority.
4.5:
----"-.-'-
Lenr:1a
i~surcs
a strict decrease
I,
Lenma
,..
<+.0:
In both of
t~le3e
ler;;FIas
the id.ea is like that of
fl'he main ne,;,-] ohstElcle for 4.5 is to 81101'[ that
111.2.2.
for certai.n
corre(~t
CO::1Dutatj_ons
tually gets an assoclatec. ne
permanent.
cedure for
of
again faced with a
~ust
analyze the pro-
negative requirements.
We are
cperation problem and again
closu~e
the proof splits ac.eordin.g to Hhether
or not.
one even-
p"
ive requirement Hhich is
In addition to 4.3 one
constructin~
from
::'"\c - ef(Ct)
=
w
Since it is easy to see that the relevant correct
computations of
,~
H
from
B
eventually produce permanent
positive requirements, there are no extra difficulties
in the proof of 4.6.
As we insured that
B <
the required properties for
-a.
A
3, we now have all
in
B
<
ex
A
<
,....
c(.
v •
Bibliography
1.
R. B. Jensen, The
r;
n
uni:formizatlon lemma, unpublished
mimeographed notes.
2.
G. Kreisel and G. E. Sacks, Metarecursive sets, Jour. Symb.
Log. 30 (1965), 318-338.
3.
M. Lerman, On suborderings ·of the a-recursively
en~~erable
a-degrees, Ann. Math. Log., to appear.
4.
J. MacIntyre, Contributions to metarecursion theory,
Ph~D.
Thesis, M.l.T. 1968.
5.
, Minimal a-recursion theo:retic degrees, ,Jour"
Symb. Log., to appear.
6.
G. E. Sacks, Degrees of Unsolvability, Ann. Math. Studies
No. 55 (Princeton, 1963).
7.
, Metarecursion theory, in: Sets, Models and
Recursion Theory, ed. J. N. Crossley (North-Holland, 1967).
8"
, Higher Recursion Theory. (Springer-Verlag, to
-~---
appear) ..
------,
Trans.
124 (1966), 1-23.
and S .. G. Simpson, The a-finite
10"
..l \..rm...
11.
~~~S
Posts problem, admissible ordinals and regularity,
injurJ~
method,
!,:ath" Log., tc appear.
J. R .. Shoenfield, A theorem on minimal degrees, Jour o
S:rmb.
Log., 31 (1966), 539-544~
12~
-------_.-..,,--"'-;,
Degrees
0:'
Unsolva,bility (North-Holland 1971) ~
~! ;"
13.
S. G. Simpson,
A~m1issible
ordinals and recuTsion theory,
Thesis, M.I.T., 1971.
14.
C. Spector, On degrees of recursive unso1vability, Arm. Math.
64, (1956), 581-592.
Additions for Chapter IV
15.
G. Driscoll, Meta-r.e. sets and their metadegrees, Jour.
Symb. Log. 33 ,(1968), 389-411.
16.
M. Lerman and G.. E. Sacks, Some minima1 pairs of a-recursively
enmnerable degrees, Annals of Ma"::h. Log., to appear.
17.
G. E. Sacks, The recursively enlrrnerable degrees are dense,
Annals of Math." 80 (1964), 300--314.
18.
R. A. Shore, The a-recu!sive1y enumerable a-degrees are dense,
to appear.
""'.....•'.
',i'>·.
j
Richa~('cl
A.
Shore
on August 18, 1946.
.,. '1,
.lJc
til'l
SChO()'"
'1 fl
••
..L....
~'ras
He was
i"(')0
U LL.";;',
lr',c.c.,
>':)
the fol:l.o :!1ng
~)eptember.
1965
fY'OI:1
l
le8 ve
OE
..... ".~
J..ll
T'ln
(.,. ~ .... 8,
¢BK.
., ,.-..
.J... ')
~Ilhich
r< i...c))
,J
On'
•
e~
of Technolo;y.
f~~,JD'
and entered Harvard
there cons tit uted his
~) tl.lci;.;·
he recei
3.
3eDte~ber
~athe2atics
teachirL~~
he
be~an fo~r
years of
at the Massachusetts
7nstit~te
as.sist2.ntsllip 2.nd a staff
?ar't~ti.on ::~el,~tjon~~j,
~
wa~
scholc~r-··
supported by
rJ~r8.tneeshlp
21 ct'2c1 to f:'111 r:v::rnr)ershi.p 1.:::
,,-.
J~..tn1cT>
cum 18.U.d .•-:: 1n
anci Square Eracket ?a:etitioD
t.
1961~.-
B.J.Ed. In June" 1966.
rec ~2 i ved an
National Science }i1 ou21rlati:)n
'-
Unive~3ity
He spent the academic yea.r
In his third and fourth years he
Cardlnals3.nd
the~~oston
Dur
beJ.el a fulJ.···time
ship.
hi s
The following
graduate work in
J
uatecl
~"la~3sacrlUf;etts
Har,rard at the Hebre·~.f Univers i t~! in
I..T ex,usa 1 er:>.} Israel.
setts, from
=Ln Eoston,
bOl'"'r1
'-"
and Ii'.Te?k
CC;~T)actn2sS
_elat5()ns p
z~X
and is
2.
r:1cnber 0J.-' the
a