Contributions to recursion theory on higher types (or, a proof of Harrington's conjecture),

(or, a proof of Harrington's conjecture) by

Leo A. Harrington

B.s.,

University of San Francisco

(1968)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF

DOCTOR OF PHILOSOPHY at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY September,

1973

' '

Signature redacted

Signature of Author •••••••••••••••••••••••••••••••••••••••••• Department of Mathematics, August 13, 1973

Signature redacted

"

Certified by •• .,... •• , •••••••• ~··•••••••••••••••••••••••••••••••

· .. 'A"ll • ·~. "· L-- . · / I Thesis Supervisor

Signature redacted

··

Acce.pted hY···•···~··•••••••••••••••••"••••••••••••••••••

·«,r,.,·, · ··· -Chairman, Departmental Committee on Graduate Students

ln;l'Ch.

1978.

Contributions to Recursion Theory on Higher Types

1. (or, a proof of Harrington's conjecture)

by

Leo A. Harrington

Submitted to the Department of Mathematics on

in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

ABSTRACT

We study recursion in higher type functionals, as formulated by Kteene, with particular emphasis on

functionals of type at least

3.

In section O a hierarchal definition of Kleefie's

recursion theory is presented. A set theoretic equivalent to this definition is mentioned.

In section 1 we restrict our attention to a particular functional, 3E, the equality predicate for type 2 functionals. We show that a natural question concerning 3E is independent from the ordinary axioms of set theory. As a corollary of our methods we obtain Theorem. There are models of ZFC in which:

w2

is large,_!.here is a

rr

1 _{2 well ordering of a set}

-of reals -of length •2, and there is a

~~

well ordering of w2. A similar theorem holds for the theory ZFC + ~there is a measurable cardinal". These results answer a question of Mathias.

In section 2 we study a formulation of Grilliot's

selection theorem in terms of stability. We investigate the exact extent to which stability holds. Along the way we prove Theorem. Let F be a type n+2 functional, n > 1. For any r.e. in F collection of type n objects, if this-collection 0ontains a member r.e. in F, then it contains a member recursi•e in F. Two applications of this result are given. The first is a priortty argument; the second

is the Plus Two Theorem: For m > 1 and for F a functional of type at least m+2, there is a-functional G of type

mention that throughout §2 we assume that the appropriate

equality predicate is recursive in each functional.)

In section 3 we examine. for each n > 1. a specific n+3

-functional, the type n+3 superjump.

s.

We note

that the type n+3 objects r.e. in n+ 3

s

are precisely the n~+l type n+3 objects, while the type n+3 objects recursive ih n+ 3

s

tall short of A~+l. We formulate a new recursion theory oJ1+3

s

and show that this new recursion theory

comes naturally from the old one. This new formulation in no way changes the recursive objects, but fewer

objects are r.e. under the new definition. This new collection of r.e. objects is shown to behave quite nicely. in particular it is closed under quantification over type n-1.

Thesis Supervisor: Gerald Sacks Title: Professor of Mathematics

Acknowledgments

I would like to express my gratilude

To Professor Hartley Rogers, Jr. who initiated my exposure to eecursion theory.

To Professor Eugene Kleinberg for presenting advanced concepts in set theory in a comprehensible yet enjoyable fashion.

To my present and former fellow students, with

special mention going to Frederick Abramson. The&r

pleasant and distracting companionship has added greatly to my stay at M.I.T.

To David MacQueen who, among other things, single-handedly (somehow) generated in me an interest in higher type recursion theory.

To

s. c.

Kleene, Y. N. Moschovakis, T. J. Grilliot, and all the others whose names are listed in the

bibliography of this dissertation.

To Alexander Kechris for many interesting and fruitful conversations to which, I must admit, his contribution usually outweighed my own.

And finally to Professor Gerald Sacks. Not only is he responsible for most of my formal and informal

education in mathematical logic, but also (although this may be the same thing) he is culpable for most of my mathematical prejudices, especially my firm belief in

the

power

or

absoluteness arguments, and the correctness

Abstract Acknowledgments

o.

Introduction Contents 1. An Independence Question 2. A Stability Phenomenon I. A Priority Argument II. The Plus Two Theorem

3.

The Superjump Bibliography Biography 2.

4.

1.

17.

31.

43.

53.

65.

93.

96.

§0. Introduction

This section will be devoted to presenting the basic

definitions of recursion theory in higher types,

men-tioning the main theorems of the subject, and establishing the notation to be used in later sections.

The finite type levels are defined in the usual way: Type (0) •~•the integers

Type (n+l) • Type (n)~ • functions from Type {n) to~.

A member of Type (n) will be called a type n functional. It will be convenient on occasion to identify members off

Type (n+l) with subsets of Type {n) or with functions from Type (n) to Type (n) or with various other entities. We will then say that these various entities are 'coded' by certain members of Type (n+l). It will be left to the reader to

choose

his favorite method of encoding. An important example of this 1s the ability to code a finite sequence of functionals as one functional. This ability is generally known as closure under pairing. We will let < > denote a ubiquitous pairing function.

For each integer n, a type n+l functional of considerable importance is the equality predicate for type n functionals, n+lE, defined by

0 if X = Y

1

if

X

~

Y

for

X,Y

£ Type

(n).

In [9] Kleene provided the type levels with

a

recursion theory. Our main concern will be studying recursion in type n+l functionals, F, together with n+lE. So, as is typical of this subject, we will not adopt Kleene's original definition, but will present our own equivalent formulation along lines similar to those in [19] ahd [22]:

Fix an integer n ~

o,

and fix a type n+2 functional F. Let I • Type (n). Members of I will be referred to as individuals.

Given XCI, let {e}; denote the eth type

n+l functional which is primitive recursive in X, n+2E. [This notion of primitive recursion has many equivalent

definitions. Perhaps the quickest and clearest is the

following: Identify I with some transitive set

(say I : R(w+n) • the collection or sets of rank< w+n),_ and consider the structure <I, E, X>. Via some natural

g6del numbering of formulas, let

le}:

be the eth

function from

I

to

w

which is first order definable over <I, t , X>.] Let

w; •

{a £ I

I

{e}:(a) = 0}.

We will now define a hierarchy which essentially embodies recursion in F, n+2E.

Definition 0.1

We

will

now

define, a subset,

~,

of

I•

J

a function

_I

l

•

.

e,F + Ordinals;

and for each ordinal a in the range of

_I

IF; a subset, ~, of I.

This will be done by simultaneously, for each

F

a t I,

inductively defining

a

set of integers, e:a, and

a function,

I I! :

~

+ Ordinals.

~. I

IF,

and H! will then at the same time be defined by:

r:/

= {<m,a> / m £ ~ a£

I}

a' HF = {<e,a,O>

I

a E

w

0 } LJ e F <H ,a>.

{<e,a,x+l> / F({e}P O _{) = x}, and}

Hi=

{<b,c> /

b

E ~ ,

lb IF<

A,

C £ HlblF}.

(For the sake of convenience, all superscripts will from now on be omitted whenever possible.)

(1) 1 E &a, Illa=

o.

(ii) x E ~a=> 2X Ee-a

(iii) given m,e £

w,

if m

Ee-a,

lmla = a, and if

<H ,a>

We cr C (Y, then 3m • 5e £ e-a, l3m •Sela= the first limit ordinal greater than a and greater than

lbl

for each

<H ,a>

b £

w

C1

e •

A definition of recursion in be extzacted from this hierarchy.

Definition 0.2 For e £

w,

a£ I, let

{ }e

'}'

\A )

...

.~ x mean

{e}F(a)

~

means {e}F(a)

=

X for some

x;

{e}F(a) t otherwise.

A partial function, • : Type (n+2) +

w,

is recursive in F,if for some e and for all G £ Type (n+2),

A subset of Type (n+2) is recursively enumerable in F if it is the domain of a partial function

recursive in F.

It can be shown fairly straightforwardly (as in [22])

that these definitions agree with Kleene's.

A rather immediate consequence of clause (iii) of definition 0.1 is that sets r.e. in F are closed under universal quantification over I. Not so obvious are the following:

{a) closed under existential number quantificatian {Gandy

[3)).

(b) not closed under existential quantification over I if

n

_-

> l {Moschovakis

[15]).

(c) closed under existential quantification over Type (n-1) (Grilliot [4]; see also MacQueen [12]).

There

are a

few important

ordinals

associated

wf.th F.

Definition

o.4

~•sup {!ml!/ m £~}•the sup of the o~der types of prewell-orderings of I which are

recursive

in

F. For

i

<

n, ~

_{1 • sup}

{lmlF / a e

Type (1),

- a

mt~}• sup {K~F,a> / a t Type (1)}; Ai= the sup of the prewellorderings of Type (i) which

are

recursive in <F,a> for some a e Type (1) • the order type or

{lml! /

a & Type (i), m £ ~ } . Notice that

~

will be denoted by KF. An ordinal a will be called recursive in

F

1ft

Ha

is recursive in

F

iff there

is a

prewellordering

or

I

or

length u

which is recursive in F. An ordinal a will be called constructive in F if for some

Wbenever the type n+2 functional F is clearly under-stood we will let, for a£ I and for i ~ n,

denote K<F,a> and _i Aa denote

i

l<F,a>.

i , also an

ordinal a will be called recursive in a (constructive

in a) if a is recursive (constructive) in <F,a>.

We will now present a more set theoretic formulation of recursion in F.

Definition

0.5

For an ordinal a, define L₀ (F) by :

definable with parameters over

LA(F) = Uo<AL₆(F). Let Ma(F) denote the structure <Lc/F), e:, Ft(Type (n+l) () L₀(F))>. Let ~ be the first order language appropriate to the structures

The La(F)'s of course simply constitute the

usual constructible hierarchy relativized to F and I. It should be clear that this hierarchy is very similar.

a < KF F

to the one defined in 0.1. In fact for , H₀ is first order definable over M₀ (F), and M₀ (F) can

be coded by a subset of I primitive recursive in

H:.

All of this can

be

done uniformly. By combining this fact with Theorem 0.3 (a) and by examining

definition 0.2, one obtains:

o.6

There is

a

1-1 recurstye correspondence, e <-> .... 'f'e' between integers e and t₁ sentelltes

t

e in

'.:t. ,

such that for all G c Type (n+2), {~}<F,G>(o)~~ iff

MK

<F,G>(<F,G>)~te•

0

For a c

I, M

₀ (<F,a>) is essentially identical with

Ma(F).

Thus, fixing F,

0.6

yields:

0.1

A subset, (J_ , of I is r.e. in F iff there

-is a _{I l} formula t(x) in ~ such that for all

a c

I,

a &

&

iff M.a(F)

J=

,Ca).

Ko

So

to study the subsets of

I

which

are

r.e. in F, which we will do in §2, it is Just necessary to have the function at-+~, a£ I. The fact that

sets

r.e.

in F are closed under universal quantification over

I

manifests itself

~Y

the following boundedness property:

o.8

For all a e: I and for all

r

₁ formulas

cf, ( X, y ) of ~ , if (? b E I MK <a, b > ( F)

I=

<f, (a, b ) 0

then for some

As was pointed out in [12], the function al-+ Ka

0 can be characterized by this property.

Namely, for any function g : I -+ Ordinals, if

g satisfies

o.s,

(that is if

.

.

for all a e: I and for all Il formulas ¢(x,y) of

;£ ,

if

~be: I Mg(<a,b>)(F)

I=

<j>(a,b), then there is

a < g(a) such that b'b e; I M₀(F)

I=

ct,(a,b)), then g(a) > Ka for all a e: I.

- 0

The Ka's

0 are essentially definable from the

hierarchy. For define a new _Il

6'a

EI M (F)

I=

ct,(a). Ka 0 each Il formula <f,(x) of

~

*

formula <P ( x) such that iff M~ (F) /= <I>

*

(a)

0

iff

(To get ct,,

*

find a formula e(x)

of -;;t:.. such that for all a t

m

and for all a M₀ (F)

I=

8(a) iff a = lmla for some m e: ~ ,

*

and let <t, (x)

=

3cr(M₀ (F)

I=

<j>(x) /\ 8(x)).)

we

L(F)

So far we have assumed that I= Type (n) where n 1s any integer >

o.

The case where n • 0 is

-fairly well understood, for if n • O then KF •

r1 •

~

for all a c I • w, and thus

0 0

MKF(F) is admissible. So we shall restrict our attention to the ease where n > 1.

Part (b) of Theorem 0.3 actually states that MKF(F) is not admissible. In fact there is a _l:l formula cf){x) of;;( such that for all

a e: I MKF(F)

F

•<a)

but M a (F) ~ +(a). Matters

Ko

are

different though if we look at _~-1 instead of KF. Part (c) of ~heorem 0.3 asserts:

0.9 For any _{t 1 formula} a & I, if M a ( F)

/=

+

(a)

Kn-1

cp(x) of

:Z,

and for any then M a (F)

F ,Ca).

Ko

§1. An Independence Question

As mentioned in

o.4,

there are various ordinals

associated with recursion in a type n + 2 functional F. To single out one, let us look at KF. A natural

question to ask here is what are the possible values of the cardinality of

,., +

the fact that ...i.n-l

?.

Very littl!:_is known beyond = Type (n-1)=+ <

7

< 2n = Type (n)=.

Translating this for the case F =

3

E, we get

5E

~l ~ K ~ w2. This is in fact hhe best result possible using ZFC, the ordinary axioms of set theory. In this section we will show that any reasonable pair of values for

5E

K and is consistent with ZFC. To be precise:

Theorem 1.1 If ~ and 8 are two cardinals of L

such that

«t

~ ~

::_

e,

(cof~)L > w, (cof

e)

1 > w, then there is a cardinal preserving generic extension,

N, of L such that

=N

(Jl2 w = 0. (Notice that if one shrinks from tampering with the cardinals of L, then 1.1 covers all possible values

for

~

K and

As a corollary of the method used to prove 1.1, we demonstrate that some results of Martin on the lengths of projective prewellorderings of the continuum are best possible.

Since this section will be entirely devoted to forcing arguments, we assume that the reader is familiar with

this technique.

[6]

would be

a

valuable reference. The following variant of Solovay~ almost disjoint set forcing

(7]

will be the main tool in the proof of 1.1.

Let M

~

ZFC + <~ l •

~

t •

Working inside M:

For h & 002 and i £

w,

let h(i) • the g8del number

of

<h(O), ••• ,h(i-l}>. Let S(h) • {h(i)/1 £

w},

and

let S • Uh&

w

_{2 S(h).} A member of s is called a sequence number.

Let Q. {q/q. <q ,ql>' q

0 0 a finite set of

integers, q

_{1 a}

finite subset

of w2);

for q, q' in

Q define q < qt

..:.Q by: q 1

S

q 1 ' , and

ti

h £ q 1

q ~ q'

~ is a partial ordering of Q.

< j -..ctA

Given A C. w2, let ~A = {q

be ~Q restricted to

At/A.

£ Q/q_{1 CA},} and let < .

.JA,

< 1 > is Solovay' s

-,..r;A

almost disjoint set forcing •

..dA

has the countable antichain condition (abbreviated c.c.c.).

Stepping outside of M for the moment:

A subset, a, of w will be called

4A

generic over M if G • {q £ .,4A/q₀ Ca and

S(h) (l a C q_{0 }} is ~ A generic over M

in the usual sense. Notice that M[a] = M[G]. The main property of Solovay forcing is that

A = {h £ (002)M/S(h)

/1

a is finite}. Thus A can be

defined from a and (w2)M. By appropriately iterating Solovay's forcing, it is possible to produce an

a

Cw such that A is definable from a alone. This we now do.

Returning to M:

Let

...

,

be the constructible

Define

0

A by: p £ @A iff

(1) p is a partial function from

<'<

₁ + 1 to

Q. The domain of p is finite, and

<Y

₁ £ domain of p.

(ii) For all a <

,'?

_{1 , if} Cl £ dom p then p(a) _£

-<fA.

(111) p (

<'< )

C (w2) L

l 1 - ' and if hwa+i e p(<\\>1 then a e domain of p, and 1 _{e p(a) 0 •}

Members of &A can be identified with total functions from

<'<

₁ to Q by setting, for

at

dom p, p(a) #

<,,+>.

by

Let !.A be the partial ordering on <?A defined

p < .p'

-A iff p(a)

!.Q

p' (a).

Notice that < &A,!,A> has the countable antichain condition. (If the reader has failed to notice this, the following should make it fairly clear.)

Leaving M again:

over M if there is G C

~

such that G is tJA generic over M in the usual sense and such that a= U{p(~1 )0/p £ G}. Given such an a and G, let,

for a<

~\?

_{1 , a0} • U{p(a)₀/p £ G} and let

Then 1s •a<~ ..JA generic over M, and a is ,dB

1

generic over M. Clearly G £ M[<aa>a<('\>₁

,aJ.

(In

fact (9A is nothing more than the partial ordering or M whose generic objects embody the process of taking a generic object, G', for •a<<'<.i,,JA and then taking a generic object for where

"

Ba,= {hwa+il

3p

£

a•

i £ p(a)_{0 } .} Since

.,,JA

and

.J

BG' have c.c.c., it follows that

o)A

also has c.c.c.; see

[6]

or

[27].)

For all a <

<V

1, i £ a₀

iff S(h(A)a+i) /) a is finite. Thus

<a0> a<

<"V

£ M[a], and thus M[a] = M[G] and also

l

{a₀/a <

N

_{1 } is}

~~

c

= {h £ (w2 )M[aJ / fia

in M[a]. Now, let

C is _-2

nl

in M[a]. Claim A s:

c.

Proof: Since each _aa is

~

generic over M, ACC.

Suppose h £ C

"'A.

Since h £ M[a] • M(G], h

is denoted by a term,

_!,

in

M.

For each

n cs,

let Dn be a maximal incompatible subset of

{p £ &A/plt-"n t S(t)"}. Since 6'A has c.c.c., each

Dn

is countable. Thus {a <

<'<

1

1

3n cs

3p t

Dn

p(a) ~ <+,+>} is countable. Pick _{ao < <~1} not in this set. Thus

b'n

c

s

b'p

t

Dn

p(a}

• <+,+>.

Since h £

c,

there is a finite subset, F, of w such that S(h) (1 a

ao ~ F. Pick Po£ G such that p0_H-"S(t)

n

_{a CF". Since}

ao

-h

t

A,

ht

po(eo)l. Thus taere is n E: S(h) such that

t/

I 0

'

_{Tl t S(h)}

n

t

F and h t p (a_{0 )1} n

t

S(h). Since

*

•

_{£ Sit)",}

there is p £ 6 stf/lh that P

H-"n

and thus there is pl t _G

n

_Dn. Define p2 by: for a < ~

- 1

So define p3 £

~

_by: p3(a) = p2(a) if a~ a_{0 ,} p3ta) =

0 <po(ao)o

V

{n}, po(ao)l>. Since n

t

S(h)

for all

h

£ po(ao)l, p3 _{'?_AP•} 2 But then, since

~

£ p

3

(a_{0 ) 0 ,} p

3

l+-"n £ aa "· Since p

3

!AP

0 , 0 p3H--"S(t) /) a CF". ao Since 3 1 3 p '?_AP, P

H--"n

£ S(t)".

Thus p3H-

"n

£ S ( t)

/7

a₀

~

F". But

n

was chosen so 0 that n

t

F. Contradiction_ To summarize Lemma 1.2 If

a

is is _-2

n

1 in M(a]. Now ~b prove 1.1:

l?A generic over M, then A

of L

Let M be a C!,!:dinal preserving generic extention

such that (

00

2)M =

&.

Let A£ M be a well

ordering of a set of reals of length <~. Let G be (/A generic over M. Let M' • M[G]. Since (?A has c.c.c., and since

0A

- M' \.\

(w2) •

<~,

and thus

adjoins only a single real,

- M '

jE:v:'

K ~ ~ • Let (y' E M be the

partial ordering designed to adjoin 8 many Cohen reals (see [6]). Let G' be

(J'

generic over M'. Let

=N

N • M'[G']. (w2)

=

8 since ~' is designed to make this happen.

Using the fact that K E 3 = (K 3E L(w2) ) , a standard

symmetry argument associated with

6'

yields that

3E N 3E M' ~ N

( K ) < ( K ) • Thus ( K ) ~

<:.'Y •

N • M[G' 6> G], and

o•

ED G is

_<S'

$

(YA

generic

over M. Thus G is _~A generic over M[G']. Thus

by lemma 1.2 A 1s

nl

-2 in

M[G'](Gll • N.

Thus

- N

,;:

_~

_{'V.

K

Corollary

1.3

There are models of ZFC in which

· ~ is large.

There is a

n

1 well ordering of a set of reals of -2

length 00_2.

There is a A1 well ordering of 00_2.

-3

Proof: Let M be a model of ZFC

+

<~i

=

<\>

₁ in

-which 00_{2 • K a regular cardinal > ~~}

1 •

Inside M, pick

r:

00_{2 <-> K. Let ~f be the}

cb-

= <LK(f), E~ f>. Let A C w2 be a universal

r

₁

set for ~ . Thus A is over

oZf-

and any other

over

d-

is easily

subset of 00₂

reducible to

which is

f

₁

A. Let (SIA be as in Lemma 1.2. Notice

that members of fS>A can be identified with reals.

will be identified with this set of reals from now on.

A is

r

₁ over

d-;

by examining the definition of

(YA, one can see that this implies that over

o&-.

Let G be

(YA

generic over M, and let a be

the o='A generic subset of w associated with G. We

claim that M[a] has the desired properties. In view

of 1.2, the only problem is to show that in M[a] there

is a A1 well ordering of (w2)M[aJ.

-3

Since

~

has c.c.c., (w 2 )M[a] = U · (w2 )L(a,b).

be:(w2)w .

For each b in (w2

/1,

let t(b) be the term

in M which denotes: 0 if b = <b_{0 ,b 1 5 and bo is}

not a well ordering of

w;

and which denotes the

th L(a,b 1 )

lb₀

I

member of (w2) otherwise.

of the form t(b) for some b £ ("'~)M.

(,

Let

!,

be the well ordering of (00_{2)M[a] defined}

by: d < d' iff 3b 3c[b., c £ ("'2)M;-6- b<~fc 4

ti

b ' < rb ( t

Cb' )

r) t ( b))

i

b' c'

<

re

(t(c'),,. t{c)) -f, t(b)

=

d ~ t(c) • d']. Claim < is 61 in M[a].

-

-3

A is

nl

-2 in M[a].

Thus

since A is tl universal

over

d-,

any subset of {(&)2)M which is

_~

over

ob-I l

-3

in M[a]. In particular, (w2)M.,

!r•

o'A and <A

is

are in M[a]. Thus to show

!

is

!~

(and hence

.!1>

we need only show that the predicate

R(b)

=

tfb• <rt>(t(b') ~ t(b)) is

t

1 in M[a]. -3

Consider the predicate T(b,p) E p e

~-+

b £ ("'2)M

"ti

pH-" b'b ' < rb

<

t

c

b •

> "'

t

c

b

» " .

Since R(b)

=

3p £ OT(b ,P),

and since G 1s easily seen to be tl

-3

in

M[a],

it will be enough to show that T(b,p) is Ill

-3 in M[a], and thus i t is enough to show:

First notice that since ()'A has c.c.c., every maximal incompatible subset of (YA is a member of LK(f). Let D be the collection of all maximal

incompatible subsets of

CPA.

Since

~

is 1:_{1 over}

c:d-,

D is A2 over

st-.

Next notice that the forcing relation can be defined by existentially quantifying over D. Thus:

For p E

~

let DP. {CED/ b'q E C(q ~AP or q is

incomparable with p)}. Then

g

a function with domain C and range a set of terms, such that t/q E C _{q SAp} => qJfcf> ( g ( q) ) •

pJ+-l,'xq, ( X) iff

ti

terms t

:3

C E _DP such that

t,q E C _{q ~AP} - > qfffq, ( t) •

plf<f> V 1/J if

3

C £ _DP such that

&q

£

e

q _!AP •> qF<J, or qff.11, •

And so on.

Using this definition of H- (and using the fact that K is a regular cardinal and thus.

~

is a _1:2

admissible structure) we get that for formulas $ which are limited or ranked from the point of view of

-:£-,

the relation pffi, is t_{1 over <LK(f),} E,

r,

D>, and thus 1:₂ over

d-.

The formula b'b' <rb(t(b') ; t(b))

±s limited as far as

;6-

is concerned, and thus T(b,p) is t₂ over

txf-.

Martin [13] has shown that any well-founded relation on reals has length <

<'?

_{2 •} Corollary

1.3

shows

that there is no reasonable extention of this result to

higher levels of the projective hierarchy unless one

uses a stronger hypothesis than ZFC. Martin has extended his result by assuming ZFC + MC (there is a measurable cardinal) from which he has shown that any tl

-3

well-founded relation on reals has length <

"'('<

_{3 •} This result,

too, has no reasonable extention:

Corollar1 1. 4 THere are models of ZFC + MC (assuming of course that ZFC + MC has any model at all) in which:

w2 is large. There is a

-length • 2.

nl

There is a 61 well ordering of w2.

-4

1.4 can be proven in exactly the same way as 1.3. It is just necessary to find a substitute for (w2)L, which is readily provided by (w2)L(µ) _{(See [24] for}

an exposition on L(µ).) Thus if M is a model of ZFC + MC +

<'<'

₁ = <'°<~(µ), and if A e: M, AC

_-

(w2)M

,

then if we let ho,h1,•••,ho,··· 0 <

,v

1 be the listing of

1 ~A'

r

₃

well ordering, and if we define

v

in exactly the same way as \YA was defined, only replacing the

shows: If a is

(r'

A generic over M, then A is

rr

1 _{in M[a]. The proof of 1.3 can now be easily}

-3

adopted to prove

1.a.

[We have recently discovered that the lightface versions of 1.3 and of 1.4 actually hold. That is, it it possible to find models of ZFC in which: w2 is large, there is a lightface Il~ well ordering of a set

of reals of length w2 , and there is a lightface ~l 3 well ordering of w2. This can be done by employing

the technique of

[7]

to arrange matters so that the parameter in the

a nl ₂ singleton.]

nl

§2. A Stability Phenomenon

Let n > 1 and let F c Type (n+2).

As was mentioned 1n 0.9, Theorem 0.3 (c), which is known in the literature as Grilliot's selection theorem, implies that for any

a£

I and any t₁ formula

,Cx)

in ; / , if Mic~-1 (F)

F

4>(a) then MKa(F) /= c>(a).

0

Students or a•recursion theory should be familiar with this sort of phenomenon (see for example (21)) which is usually referred to as stability (if one looks up from ~) or as reflection {if one looks down from K~-1).

In (20) Sacks used this result in a rather remarkable way - to perform a forcing construction in the setting of higher type recursion theory - which suggests that stability phenomenon of this sort can be quite useful. In view of this we now investigate how far this stability result can be extended.

Definition 2.1 For a e I and a < ~ call a an

As mentioned above, Ka is a-reflecting.

n-1 As

mentioned in §0, Theorem

0.3

(b) implies that KF is not a-reflecting.

Let J = Type (n-1). Members of J will be called subindividuals. An ordinal o will be called subrecursive (subconstructive) in F if for some'

i E J o is recursive (constructive) in <F,i>. If the type n+2 functional F is clearly understood

then for a EI an ordinal a will be called subrecursive (subconstructive) in a if o is subrecursive (subcon-structive) in <F,a>.

For a

EI,

let Pa = {i E J/i £ ~F,a>}.

n-1 P n-1 a

is just a complete r.e. in <F,a> subset of J. P a

n-1 and HKa are recursively equivalent modulo <F,a>,

n-1

and so

Theorem 2.2 Let a

HKa , epitomizes the structure n-1 be in I, and let a = a <P ₁,a> K n-n-1 Then a is a-reflecting. •

Proof

Let

+(x) be a

r

₁ formula in

.Z,

and assume

Ma(F)I== cp(a).

Let

w •

{b £ I/b codes a well ordering of J}. For be

w

let lb( be the order type of the well ordering coded by b. w is recursive in F.

A:.1

is the order type of the ordinals subconstructive in

a.

For b E

w

if

lbl

!

_{1~-l then there is 1}

t

J

such that in the well ordering coded by b the initial segment determined by 1 has order type exactly _). a _1·

n-Since K~-l is the sup

or

the ordinals subconstructive in a, this implies that K~a,b,i> > _{K:_1 , and thus}

So given b E w, either

(l)

lbf

!

i!.

_{1 , in which ease}

K~~ib>

!

a

and thus M~a,b>(F)

F

«Ha), and thus by 0.9

n-1 (2)

Jbl

< ~~-l in which case <a,b> 1-MK (F)

r +(a).,

or 0

3B

< K<a,b> such

n-l

that

a

is subcontructive in a, and

lbl

c the order

type of the ordinals < B which are subconstructive

Notice that by 0.9, (2) has the form of a predicate r.e. in <F,a>, i.e., there is a

r

₁ formula

w(x,y)

in

ot1

such that for b £ w, MK <a,b>(F)

F

tj,(a,b)

0

·&, we have b'b e: w MK <a,b> (F)

F

cf>(a) v tP(a,b).

0

By

o.8

there is a< Ka such that

0

tlb

£ w

There is such that >). a

_n-

1'

2.2

may

be restated in the following more recursion theoretic way:

Corollary

2.3

If

63

is an r.e. in <F,a> collection of subsets of J, and if

(D

contains a subset of J which is r.e. in <F,a>, then

d}

contains a subset of J which is recursive in <F,a>.

Proof: Subs•ts of J may be identified with members

of I. CY> 1s r.e. in <F,a>, so there is a t₁ formula

,Cx,y)

in

o!

such that for all b CJ b e:

63

iff MK <a,bi(F)

F

+(a,b).

{J;

contains a subset of J whlf:ch

0

in

~

such that b s {i _{£ J/MKa} (F)

J=:

1J,(a,i)} £ @. n-1

For each ordinal C1 let b₀ = {1 t J/M₀(F)

/=

w(a,1)}.

If a<

Ka

then _bO'

0 1s recursive in <F,a>. If

<J > a _then b • b.

Kn-1 Cl

b £ ~ . Therefore MK <a,b>(F) /:=' t(a,b).

Let

a

0 a <P ₁,a>

6e as in 2.2, i.e., a • K _n-1 n- _• Since b is

r.e.

in <F,a> b is recursive in _{<P 1 , F, a>, and thus} a

n-K <a,b> <

0 Cl. Thus there is a< a, a> K 1, such that

_n-

a

M₀(F)

F

;(a,b)

and

thus Ma(F)

F

,ca,b_{0 ).} By 2.2 there

is i < K~ such that _{M0 (F)}

f=

,Ca,bt5), and thus _{b 0} £ ~

and b0 is recursive in <F,a>.

The histroy

or

2.31 and of theorems similar to

it which

were

found in different settings, should be mentioned here. This type of theorem was first proven

for recursion in higher type functionals where, considering its similarity with the version of the Grilliot selection theorem stated in 0.9, it came as no surprise. It was then independently discovered by Martin and Solovay for

Il~n+l sets of reals assuming projective determinacy [14]. Kechris then proved it in a setting so broad as to include all other cases [8].

The limit of a-reflecting ordinals is also a-reflecting. Let Ka

r be the last a-reflecting ordinal. So for each a £ I there is a _{Il formula 8a(x) in}

L

such that MKa(F)

JI

ea ( a) but MK a+l (F)

I=

a

(a).

r r a

We will now show that

a

uniformity actually exists here. Theorem 2.4 _{There is a formula (not I 1 ) e(x) in}

d

such that for all a£ I MKa(F)

I=

8(a) but for all

r

Proof: To begin, notice that the ppoof of 2.2 can be adopted to prove

&emma 2.5 (Kechris) For a£ I, if

(J;>

is a non-empty subset of I which is co-r.e. in <F,a>, then there is b e:

(/;J

such that

Proof: If not,

pd:m1':

a _{I 1 formula <f>(x) of}

&(

such that MKa+1 (F)

/=

+(a)

r

b £

(J;

and MKa(F);'- 4>(a). For all

0

and thus MK<a,b>(F)

F

(f>(a)

and thus MK<a,b>(F)

F

~(a). 0

Since

(j;

is co-r.e. in <F,a> El formula tfJ(x,y) of

L

such that iff MK<a,b> (F) f:= ljl(a,b)). Thus

C7b

£

0

there is a

b'

b e: I(b I

MK<a,b>(F)

/=

f(a,b)v+(a), and therefore tfb e: I 0

t

63

MKa(F)

I=

lf,(a,b)v4>(a). But

Q?

is non-empty and so 0

3

b e: I such that MKa(F)

;£

lf,(a,b).

0

Thus MKa(F)

I=

~(a). 0

Contradiction.

Theorem 0.3 (b) will be the key fact used to

establish 2.4, but we will need an actual proof of 0.3 (b). Theorem 2.6 (Moschovakis) For any subset,

a,

of I

which is

r.e.

in F there is a relation ROf,y) on which is r.e. in F such that \:la £ I(a

t

a

iff

:/b t I R(a,b)).

Proof: I~ is enough to just consider the case where

a•

sf

hbe universal r.e. in

F

subset of

I. It

would be best if the reader reviewed definition 0.1 before continuing here.

Define an r.e. in F predicate, R'(x,y), on I

by: R'(b,c)

-(i) if b = <2m a'>

,

for some some m £ w, then c = <m,a•>.

a' E I and for

(ii) if b = <3m:5e,a'> for some a' EI and some m, e £ w, and if c # <m,a'>, then <m,a'~ £ (5-,

<H ,a'> lmla, = a, and C £

W

a _•

e

(iii) if

fia'

£ I b'm, e £

w

(b # <2m,a'> and

b "/, <3m.5e,a'>) then C = 0.

(iv) t/a' £ I (b ; <l,a'>).

Remark: The important properties of R' are

(l) if R'{b,c) and if b £&,then c £&'and

(2) if b

t

&' then

R'{b,c)).

3c £ I { c

t

&- and

Let R be defined by: R(a,b)

=

(viewing b as

<b_{0 ,b 1 , ••• ,b1 , •••} >₁ew a function from w to I) b₀ = a and

'd

i£w _{(R' {b 1 ,bi+l)) •}

To show that R has the desired property: If a£ &- and if R(a,b) then by (l} of the above remark

fal •

lb

0

I

> fb1

!

> ••• which is impossible.

rr

a

I

6-, then by (2} of the remark it is possible to such that

We

will now prove

2.4.

b • a and

0

Retaining the notation used in the proof of

2.6,

since R' is r.e. in F there is a _1:1 formula +(x,y) of ~ such that

&b,c

£ I (R(b,c)

MK <b JC> ( F)

F • (

b , C ) iff

MJ<! (

F )

/=

cp ( b , c )) •

0

also

a

l:Il: formula ,p(x) of

~

such that

'd

a £ I (a £

e-

iff MKa(F)

I=

lf,(a)). 0

iff

There

Let 8(x) be the following formula of

'cZ.

is

8(x) 1

b'm

£ fO [lj,(<m,x>) v

3

b E I(b • <b₀;•1b_{1 , •••} > and

b₀ = <m,z> and _{tii&w +(b 1 ,b 1}+_{1 ))].}

For any ordinal a and for any a£ I, if

Ma(F)

F

e(a) then b'm&w (m £ &a iff Ma(F)F lJl(<m,a>)},

Claim For any a£ I

To show MKa(F)F 8(a) it is sufficient, given r

m £ w such that <m,a>

t

lo/, to find b = <b_{0 ,b 1 , ••• >} £ I so that R(<m,a>,b) and l::/1£w (Kr <bi,bi+l> ~ K~).

We will construct such a b as follows. b = <m,a>.

0

Given bi

t

e--

such that bi < a _{find bi+l} ' &--Kr

_-

Kr,

such that _{R' (bi,bi+l) and Kr} <bi,bi+l> < Ka A

- r·

difficulty will arise in finding such a bi+l only

when bi= <3 m' •5 e ,a'>, for some a' e I m', e £ w, and when <m',a'> £ & (i.e., case (ii) of the definition of R'). Let

a•

Im' la,.

{J)

=

{c e

I/ct. & and

<H ,a'>

c £Wea } is then co-r.e. in <F,bi>' and thus by <b1,b1+1> lemma 2. 5 there is bi+l e

6!>

such that Kr

bi

< Kr < Ka _{r •}

The use of Theorem 2.6, (which really asserts that MKF(F) is not admissible), in the proof of 2.4 should be no surprise. The type of stability phenomenon we

have been studying, especially in connection with the uniformity established by 2.4, is intimately related

with admissibility. In particular, 2.4 implies Theorem 2.6. This in fact holds in general: Let g be a function

from

I

to Ordinals which satisfies

o.8.

So, as mentioned 1n the discussion following O. 8, I;;/ a £ I g(a);:

K:.

Let K • Uaeig(a). Suppose Theorem 2.4 holds for g, that is, suppose there 1s a formula

•<x)

in ~ such that b'a e I, if a is the first

ordinal such that Ma(F)

F

4>(a) • then g(a) ~ a < K and a is a-reflecting to g(a) i.e. for all

t1 ;(x) in

cZ

Ma(F) /=;(a) •> Mg (a)(F)J= w(a). Then

MK(F) is not admissible since the map a,-.. 1.10 [M₀ (F)f::: lf,(a)J

is

t

₁ over MK(F) and dominates g. But also Theorem 2.6 holds for g in that for any _{t 1 formula ;(x) in~,} {a e I/Mg(a)(F)

¥

tll(a)} is _{t 1 over} MK(F).

Theorem 2.4 has sevezal nice consequences. Corollart

2.7

Let _{K~ • Kr.}

(1)

b'a

t I K < K8

r - r

(2)

b'a e

I K _r < Ka _{r implies HK} is recursive r

(3) _{~iven a I:l} formula cf,(x) if

b'

b e: I MK ( F)

F

4J ( b ) r then there is a < Ko 0 such that

b'b

e: I M_{0 (} F )

/=

cf> ( b ) •

(4) Given

a

non empty co-r.e. in F subset,

_{63 '}

of I there is a non empty subset of

(B

which is primitive recursive in _HK (so _HK can •select'

r r

from non empty co-r.e. sets).

(5) (Kechris) An r.e. in F subset,

a,

of I

has a non empty recursive in F subset iff

3

a e: {l_ ( K~ ~ Kr).

Proof: Let a(x) be the formula mentioned in 2.4. (1) If Ka< _{Kr then MK ( F)}

_/=

:l

_{X e:}

r a (

_{X) and}

r _r

thus for some a< K0

0 and some b e: I M0 (F)F8(b), and

thus a > Kb

- r which is absurd.

(2)

if

K _r < _Kr, a then

3

a < K~ ( Mc/ F)

F

8 ( 0) ) ,

and thus _Kr = the 1st a such that M_{0 (} F)

/=

8 ( 0) is a recursive ordinal in

a.

(3) Follows immediately from (1).

(4) Given

cf7,

{a e I/K:

!

Kr and a £

d3 }

is

primitive recursive in

HK

(i.e. first order definable r

over MK (F)), and by 2.5 this set is non empty. r

(5) By 2.5 every non empty recursive {and hence co·r.e.) in F subset

or

I contains an

a

such that

If there is a c:-

Cl

such that Ka < K pick a

... o - r•

t

₁ formula +(x) of ;;( such that {;/b e I (b £

a

iff MKb(F) /:= c>(b)). 0 for some a < K0 o' Then MK

<

F >

I=

J

x c I ,

c

x

>.

r Thus a recursive in F, {b £ I/M₀ (F)~ t(b)}

is non empty and recursive in F.

We

will conclude this section by presenting two applications or Theorems 2.2 and 2.3.

I. We would like to show that 2.2 allows us to perform certain types of priority arguments for certain types of sets which are r.e. in F. In fact 2.2 will allow us to reduce such constructions to the setting of

a-recursion theory (see [21]) where the technique of priority arguments is fairly well understood (at least for the simplest type of such constructions, those that can be done by the 'a-finite injury method'). We will prove what is one of many possible formulations for recursion on higher types of a well known theorem

from ordinary w-recursion theory, the Friedberg-Muchnik solution to Post's problem, which asserts that there are two r.e. sets neither one of which is recursive in the other. There have already been such formulations [12], but the result proven here will be slightly stronger than these.

A little historical discussion would seem to be appropriate to justify our approach.

The first generalization of ordinary recursion theory which attracted any attention seems to have been the study of

n

1 and A1 sets of integers. (This

1 1

actually is part of the realm of recursion in higher type functionals, for, as Kleene noticed in [10], the

1 1

rr

_{1 ,} or Al* sets of integers are precisely the sets of integers r.e., or recursive, in 2 E; the relation on reals a,b: a is

Ai

in b, is just the relation:

~ is recursive in This result of Kleene's seems to justify the point of view that

n

1 sets

l

correspond to r.e. sets,

Ai

corresponds to recursive, and in corresponds to recursive in. Given this point of view, Spector (26] has shown that the

Friedberg-Muchnik theorem fails for

nl

l sets, that

1s, for any two

nl

1 sets of integers one 1s

the other. (Specter's argument immediately generalizes to higher type recursion theory: for any two subsets of I which are r.e. in F, one is recursive in the other together with F.) Kreiselhas argued somewhat strongly that Spector's result does not constitute a negative solution to Post's Problem.

In

[11] he and Sacks tried to justify this opinion. They extended the recursion theory induced

by

the

Ri

sets of integers to a recursion theory on sets

or

recursive ordinals,

2E

(i.e. ordinals <

w

_{1 ), called metal!eeurs1on theory.} They

then showed that there are two meta-r.e.

subsets of

2E

w

₁ such that neither is meta-recursive in the other. It is easy to see that this metarecursion theory can be defined in terms of recursion in 2E. Thus a subset, A,

of is meta-r.e. iff the type 2 object

2

{HaE/a £ A} is r.e. in 2E. We will now lift this

definition to an arbitrary type n + 2 functional F, (where n is still > 1.

Let S be the set of ordinals subconstructive

in F, (see definition 2.1). So S = {lml~/j £ J, m £ ~ } .

For ACS, let

A=

{H!/a £ A}. So A is a subset

of Type (n+l) and therefore may be viewed as a type n+2 object. Notice that S is recursive in F. Definition 2,8 Given A,B C S, A will be called F-r.e. if

A+

is r.e. in F. A is F-calculable in if for some C1 £

s

if is recursive in <F J

B,

~>.

Theorem 2.9 There are F-r.e. subsets of S such that neither is F-calculable in the other.

Proof:

Aa

mentioned earlier we will reduce the proof of 2.9 to the setting or a-recursion theory.

Let a= A~-l = the order type of

s.

Let

B

~ : a<-> S be the unique order preserving isomorphism between a and

s.

Define T C. w x a x a by

T = {<e,S,a>/e is the g8del number of a

t

₁ formula

cp( x) in .;;( ,

a

< a, and M-ro (F)

i=

q,( Tf3)}. Consider the structure <L (T),£,T>.

a

Claim l: AC. a is _{t 1} over

01

iff -rA is F-r.e. This is fairly easy to see: If ACS is F-r.e. then there is a _El formula cp ( x) in ;;t' with gBdel number e such that for a < _a, '[Q' £ A iff

M

Kn .. l (F)/= f{TO) iff

3Y

£

s

(y > 'tO and

?-\

(F)

/=

4>( TO')) iff <e,a,,. -1 y> E T. !on¥effseihy; given

a £

s

S(} a is first order definable over M₀ (F}

uniformly,and thus, letting

o •

t ·1a, the structure

<L 0(T), &, T/1 ~ x 6 x

o>

is first order definable

over M₀ (F) uniformly. Thus a t₁ definition

over

<rZ.

induces one over MKF (F). (Recall

n-1

Kn-l

=

sup S). Notice that this last fact implies that

m

is admissible, since any

_f1

over

<r1

map from an initial segment of al say from a < a, onto

an unbounded subset of a would induce a map, from to an unbounded subset of K

n-1' which is El over

MK (F) with some subindividual Rs parameter. 0.9 implies

n-1

that there is no such map.

This claim will allow us to use the recursion theory which

C"Z

induces on a, which by [25] will allow us to straightBorwardly apply the a-finite injury method.

For A, B £ « , l e t A< B mean A is

1

₁

-cri

definable

over

the structure <(t'l,B>. Then the uniform solution of Post's Problem (see [23), Chapter II) for

01

gives us _{A ,A1 C} a both

0 - t1 over Q1. such that

i • O,l. Thus TA₀ ,rA₁ arehboth F-r.e ••

If the reducibility "A

1

₀₁B," A,B ~ a ag:t,eed with the reducibility "tA is F-calculable in TB" then we

would now have proven 2.9. Unfortunately this 1s not the case. We shall present some properties which, if

possessed by B ~ a, imply that these two reducibilities are the same.

Definition 2.10 A subset, A, of a is 0'2-(hyperregular and regular) if the structure <(JZ,A> is admissible. A subset, B, of S is F-subgeneric if for all

F

<B,F,H

>

K _o a < K 1·

Claim 2: Given B C a, if B is <fl -hyperregular and regular, and if TB is F-subgeneric, then for all Ac a A ~<nB iff TA is F-calculable in -rB.

This too is fairly easy to see: F-calculable in TB is always at least as strong as

~& B.

If TA is F-calculable in TB, then for any a£

s,

the relations a£ TA, a t TA correspond to t₁ (with some member of S as parameter) assertions about the structure MB(TB,F,H_{0 )} where

<TB,F,H > _(J

B = K₀ • Thus since TB is F-~ubgemeric,

TA _{is h1 (with some member of S as parameter) over} MK (tif,F). So we just need to show that the fact that

n-1

B is

02

-hyperregular and regular implies that this induces a A,_{1 definition of A over <0'1,B>, which, by the proof}

of claim 1, it will if {<e,B,a>/B < o

1

a, e is the g8del number of a t₁ formula cf>(x) in

o:t,

and

MT₀fTB,F)~ +(TB)} _{is h1 over <a?,B>. So, given} a< a, since < 01,B> is admissible

anac

L₀ (T). Thus for

some

o

< a,

o

> <J , TB /) 'TO is first order definable

over MTo (F)' say by a formula with gBdel number eo. Thus TB /) TO £ _MK (F), and thus MTO'('iB,F) £ _MK (F).

..

So, given f3 < cr and given a formula cf> ( x) in

l;( ,

i f f

.:J

o

£ S

3

e _O e w :} X, Y E MK ( F) [ X 1 s

n-1

first order definable over _{M0 (F) by} the formula with gBdel number e_{O ,} and X = TB /} TO", and Y • MTO (X,F), and

YI=, (

Tf3)]. Using the predicate, T, this corresponds to a t₁ formula over <a?,B>. So we are done.

Notice that if B ca is

r

₁ over

(57.

then:

TB F-subgeneric implies that B is (J2 -hyperregular and regular.

Let B c: a be

r

_{1 over}

d'Z.

We would now like to find some property which would imply that TB is

F-subgeneric, and which would also be suitable for inclusion as a sequence of requirements in a priortty argument.

Since B is

r

₁ over (J2 we should think of B as

being enumerated in a-many steps: there is a total function f a~ at

r

₁ over

(Jl,

such that B • range off. Por

a< a, let B0 _{= range of f on rJ. We may of course}

assume that Bcr c a. Let

d'crz

be the first order language appropriate to the structure <d1,B>. The property we have in mind is:

2.11 _{For all r1 formulas} cp ( X) in _{L ~} and for all a < a,

3

0 < a,

o

~ a [if

3

y < a, y > o(<L (T),£,Tlv,BY>/=<J>(a)) _y then

3

y < a, y > o(BY = B

n

y and < LY ( T ) , £ , T f y , BY>

F

cf, ( a ) ) ] •

It is easy to work property 2.11 into a priority argument. Thus if we want a

r

_{1 over}

01

subset, B, of a which satisfies 2.11 together perhaps with some other properties, then given a priority argument which establishes these other properties, we can perturb this priority argument by adding requirements for eabh a< a and each

r

₁ formula (j>(x) in ;(

₀₁

which attempt, at stages of the priority construction, to ensure that BY= B

/7

y where y is the 1st ordinal > a such that

requirements guarantee that the resulting set B

satisfies 2.11. By applying the above discussion to the priority argument

ahat

establishes the uniform solution of Post's problem for (t2 , we obtain

r

_{1 over}

(Y2

such that Ai ~ <fl. A1_1

that both A_{0 ,A1 satisfy 2.11.}

A ,A1 C a, both

0

So all that remains to be proven is that property 2.11 implies that TB is F-subgeneric:

Given B

r

_{1 over}

fJl

such that B satisfies 2.11, let C = tB. C is F-r.e •• So there is a

r

₁

formula ~(x) in ;(_ such that for B < Kn-l'

MK ( F ) f:= 1J, ( B ).

n-1

B £ C iff For a < let

c0 • {B < o/M₀ (F)F iJ,(B)}. By the proof of claim 1,

~(x) can be chosen so that for all cr < n

Let S' = {S < KF/letting 6 = sup(S

n

S), o+S

=

S}.

Thus 6 £ S' iff sup(S

n

a) is small in comparison with

s.

Using the proof of claim 2, the fact that B satisfies 2.11 can be translated as follows: for any

a£ S and any

r

₁ formula cf,(x) i n d , there is

d £

s, o

> a, such that: if

3Y

£ S

Ii

S'(y

>

o

and

MY(cY ,F) J= cf,(a)) then: ( ]Y £ s (i S' (y .::, o and

-cY • c ~ y and MY(cY,F)F ~(a)), and thus

MK (

C,

F)

f=

4> ( a ) •

n-1

Now to show that C is F-subgeneric, given a£ S <U,F,H >

<C,F,H > ._/

lml₀ a < Kn-l" There is a

1

₁ formula <P(x) in o(.

such that for any ordinal

a>

o, 8 ES', <C,F,H >

iff S > lml₀

°.

Since m is in <C,F,H >

e-

a there is an ordinal

a

such that

0

M6 (C,F)/=cp(a)., and we might as well choose 8 so that

S > Kn-l and

a

ES', i.e. K + 0 ₌ 0 Thus

c

=

c

8

1 p P•

n-and so Ms(c8 ,F) Fcf>(o). By 2.2 we may choose such a f3 so that

a<

Kr, and thus there are arbitrarily large

Y' < K _n-l such that Y' E S 1 and _{My , ( C}

?

_{., F )} /=- _{cf> (} a ) •

For each such y' let y be the 1st member of S

~

y'.

Then y ES' and

cY'

=

cY.

Thus there are arbitrarily large members of S

11

S', y, such that

14-y(Cy ,F)

/=

cj>(cr). Thus the translation of 2.11 implies

lit?! ,F ,H >

that MK (C,F)/=cf>(o), i.e., lml₀ a < _{K 1 •}

n-1

n-II. The Plus Two Theorem.

Definition 2.12 For a functional F of finite type, and for an integer m, define the m + 1 section of

F by: m + 1 - Se(F) = {X ~ Type (m)/X is recursive in F}; define the m + 1 envelope of F by:

m

+ l - Env (F) • {X

f

Type (m)/X is r.e. in F}.

In [20] Sacks showed that for any integer m > 1 and for any functional F of type at least m + 1, there is a GE Type (m + 1) such that m - Sc(F) =

m - Sc(O). (Here we are using the convention which to an extent is implicit in our definitions 0.1 and 0.2, namely that for F c Type (n + 2) recursion in F means recursion in F together with n+ 2E. Thus F is to be identified with <F,n+2E>, and so all functionals are presumed to be normal, as in [16].)

For obvious reasons Sacks christened this result 'the plus one theorem•. This theorem essentially states that there is no type inherent in the m+l ~ sections. As Moschovakis suspected and proved, there is a sense in which the type of a functional is obtainable from its envelope: for G c Type (m + 2), m+l ~ Env (G) ~

m+l - Env (F} for any F of type> m+2T In fact: Theorem 2.13 (Moschovakis

[16)).

Given integers m,n,O ~ m < n, and given GE Type (m + 2),

then m+l - Env (G) ~m+l - Sc(F).

Proof: For each G' £ Type (m + 2) let

HG'= HG' = {Ab,c>/b,c c Type (m), b £ g..G', lblG' = o,

KG' and

r.e.

c £ H~'}. HG is universal among subsets of Type (m) in G. So by assumption HG is r.e. in F. We

must prove that HG is recursive in F.

Notice that for any G' £ Type (m + 2), if

HG' CHG then HG'= HG. _{This follows from the}

inductive nature of definition 0.1.

Let

C

= {HG' /G' £ Type (m

+

2)}.

L:'

is recursive

in F. Let

C/.

= {HG} = {X C Type (m)/X CHG and X £(:>}.

Since HG is r.e. in F the relation X CHG, _{is also}

r.e. in J:11_{, and} thus

a

is r.e. in F. Since

Type (m)

f

Type (n - 1) = J,

a

is an r.e. in F

collection of subsets of J, and

a

contains HG , a

subset of J which is r.e. in F.

tains a subset of J recursive in recursive in F 9

Thus by 2. 3 C{_

con-G

F, and so H is

Thus for all G £ Type (2) there is no F of

type > 2 such that 1 - Eny (F) = 1 - Env (G).