AND
T H E F O U N D A T I O N S OF MATHEMATICS
Editors
A. HEYTING, Amsterdam A. MOSTOWSKI, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford
Advisory Editorial Board
Y. BAR-HILLEL, Jerusalem K. L. D E B 0 U V 8 R E, Sanfa Clara H. HERMES, Freiburg i/Br. J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Brisfol E. P. SPECKER, Zurich
N O R T H - H O L L A N D PUBLISHING C O M P A N Y AMSTERDAM
0
LONDON
MATHEMATICAL LOGIC AND FOUNDATIONS OF SET THEORY PROCEEDINGS OF A N INTERNATIONAL COLLOQUIUM HELD UNDER THE AUSPICES OF THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIES JERUSALEM, 11-14 NOVEMBER 1968
Edited by
Y E 0S
UA B A R -
LLEL
Professor of Logic and Philosophy of Science The Hebrew University of Jerusalem, Israel
1970
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM 0 LONDON
@ NORTH-HOLLAND PUBLISHING COMPANY - 1970 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.
Library of Congress Catalog Card Number 73-97195 I S B N 7204 2255 8
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD -LONDON
PRINTED I N ISRAEL
PREFACE
on on
1968.
by
by
on
Y.
11,
on
on
no
on
no
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA* BY
0.
Nz= (Tyro,rl) $1).
L
a,p, y, A , B , C , ..., E P(T)” M2
...,
M2
.,
x, y , z, ..
definable F ( A , , ...,An)
F ( A , , ...,An)
weakly-definable by
[6]
on
Nz. special automaton on on H
H c P(T)” P(T)” -
c P(T)”
n.
by by P(T)” F ( A , , ...,
by 1,
G ( A , , ...,
MZ) Fz(A
..., F , ( A , , ...) A , )
,,..., 69 1
0192,
Office
MICHAEL 0.
2
[4]),
by [3] 24 1. Notations and basic standard definitions
0
= pl,
n n=
1 = {0} , 2 = (0,
...,n - l }
x: n +
by xi.
(xo, ...,
xy
l(xy) = l(x) xo
.
n-termed sequence on
x
x ( i ) , 0 5 i < n, l(x) = n
x
length
+ l(y).
.
x
y = (yo, ...,ym- 1) ...,x , - ~ ,yo, ...,y,- l).
x = (xo, ...,x,-
l).
(xo)
x = xoxl
... x,-
by A . i < o,
0
1.
projection pi
pi(x) = xi x = injinite binary tree xE immediate successors
i
by
< n. on (0,
=
xE
root of
A
x.
by A.
11.
10
.
3z[y =
x
subtree
xE
0
up
00
0
1.
1
on
5y
by x x # y roof x
6 y (x
initial x
by
=
1y E T,x 5 .
=
path
7c
7c
~ O E X
EX, 7c
c
x ~ n ;
c
X , ~ E R ,
Y E ?L,
T,
7c
x
5y
y 5 x.
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
frontier
c T,
Tx F , is bigger than x EF, x y. n c T, T F,, < F,,, 1 , n < w . = { x x 5 y f o r some y E F } frontier
-=
I
E (0,
ra(x) = xu , x E T .
z c T,
T,
c(n n = 1. F , c T, F , T, F, (F, < y EF, ET c ( S n n) = w S = Un
structure
successor function ra:T + T by of two successor functions
”% = ( T , r o , r 1 > . N,
.
a,p, y,
L,
r o ,rl
,
ro
T; A , B,C, ... , L,
T; tEV
L,
t
second-order theory of two successor functions ( S 2 S ) Jv; =! ( F N,). [6] by
R
on
P(T)” F ( A , , ...,
DEFINITION 1. definable (S2S) (1)
r,; E;
x, y , z , ...,
...,
[
= {(A19 ***,An)
F(A1, .-.,An)}*
weakly-definable
2.
automata
X DEFINITION 2. (u,T) ( u T, , T,). by V&.
I
alphabet.
Z-(ualued)tree (u,T,)
u,
(u, T,) ,
V,,,
3
u : T, + Z.
(u, T,)
x E T, by
MICHAEL 0.
4
3. A table M ) set of states, M M : x Z + P(S x table of moues ( P ( A ) A). A special finite automaton special C-automaton) 'u = M , So, Fi) M ) So E initial states, F S designated states.
r : T, + S
4. A run ( S , M ) on y E T, , , 'u on on t +:A+B
by t). = { b I b E B , c(+-I(b)) 2 o}.
5.
2l= <S, MySo,F ) accepts ( u , T,)
n
t
accepting r on ( u , 7 ') T', n) nF # 0.
by 'u
=
( u , T,)
E M(r(y),
r ( x ) E So
T(2l)
I
T(2l) = { ( u , T,) x E T , ( u , T,) accepted by A
A
c V,
defined
a}.
3 , A = T(2l).
s.f.a. definable
[6].
T(3)=
23
'u T(23). 3.
[6],
23 by
$3.
~'(xy= )
A A c V, invariant X E T ,t E A yET, A. is 5 A 'u ( u , T )E T('1I)
t' = (u',T,)
t = (u, T ) by
V, T(%) ( u , T )E A .
V, is s.f . a . definable, then there exists an automaton {so}, s ~ E S and , T('1I)= A . This 'u m a y be chosen so that (s1,s2)~M(s,cr)implies s1 # so, and s2 # so.
1. If A
E
'u = ( S , M , S o , F ) such that So =
2. If A , B G V, are s.f.a. definable, then so are A u B and A n B.
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
Proof. A = T('u), B = T(B) = (S', M', do,F' ); S nS' = !J. = (SUS',
'u
= (S,M,so,F),
5
B
M U M ' , { s o , s ~ )F, U F ' ) .
T('u U B) = A U B .
3 x B = ( S x S' x {0,1,2}, ii7,(so,sb,0),F) ((s1,si,b), ( s , , s ; , ~ ) ) E ~@((s,s',a),c) (slys2) E M ( s , c), (si, s;) E M'(s', 0);b = 1 =0 s EF , =1 s'#F'; b = 2 =1 ~ ' E F 'b; = 0 = 2, =0 s$F. F = S x S' x (2). T(%xB)=AnB. DEFINITION 6. t = (u,T) Z, x po propo(x,y ) = x . projection p o ( t ) , by (Po% T )* projection po(A) A c V,, x,, , po(A) = {po(t) t E A } . C,-cylindrification B c V,, A c Vz,xz2 P O ( 4 =B.
I
THEOREM 3. I f A E VzIx,, i s a s.f.a. definable set, then po(A) c V,, is a definable set. If B E V,, is definable, so is its Z,-cylindriJcation A s V,,X,2. Proof.
'3 = ( S , M , so, F ) Z, x by 'ul = ( S , M 1 , s o , F ) ,
u M(s,
M , ( s , a,) =
(01,
c,)),
T(%) = A .
g1 E Z,
9
sE S .
UZE22
T('u,) = p o ( A ) . 3. A counterexample B V, - B Z = (0, B n c T, 1E A = V, - B n c T,, 14 n).
A
=
I
T(3)= A
'u
( u , T,)
I n).
B (u, 7 ')
A
'u = ( S , M , s , , F )
c V,
MICHAEL 0.
6
DEFINITION 7. t = (u,T) t , = (ul,Tx),X grafting t , on t yET uz(z) = u(z) x $ Ty, uz(yz) = u,(xz) Ty = { y z z E T } , T, .)
I
DEFINITION 8. = (u, T ) N(x) 5 n
E
T, (uz, T ) zE T.
t, = (un, T ) , n < w , N(x), x E T, u,(x) = u(x).
4. Let t = (0, T )E T(%) and let r E t ) be a n accepting and run If there exist nodes x < z < y such that r(x) = r(y) = s , s E u(z) = 1, then there exists a tree t ' $ A which is accepted b y %.
xu. C' = S x (4, T ) +(z) = (r(z),u(z)), z E T . t, = (4,T,) on (4, T ) y = xu t , = (41,T ) . r(x) = r(y), po41 on (plcjl,T). p04,(xu2) = t, on rl xu2 t2 = ( 4 z , T ) . p0& on ( P ~ ~ ~ , Tpo4z(xu3) ) = s. n < w, t, on tndl xu" t, = (4,,,7). = f = ($, T ) , t' = ( p l $ , T ) . p04,, on (p14,,,T), J = po$ on t ' . n c T, n) n # J ! t' E T(%[). xu" E n n <w ~ ( x u "= ) s sE n) n n no xu", Jln = r l n = #!J. n <w x u " ~ n x, u n f l $ n . n' T, J(xu"u) = r(xu) X U E ~ .
Proof.
y
=
I
#.PI.
=
z t o,,(t)
=
xw
= (u,T) T . Do(t) =
p,&xu"w) {X
IU(X) =
=
1, n < w ; by
t'$
A. on n ,
D,, l ( t ) = D,,(t) n { X 1 c(o,,(t) n T,) = w } .
5. Let 3 = be special {O,l}-automaton; t = ( u , T )E A E D,, ,(t). I f r E t ) is an accepting run such that c(r(T - {A})) n , then for some x < z < y , r(x) = r(y) = s, s E F , and r(z) = 1. Proof. B y on n . n, A < x En r(x) E F . {x(n) n c T } E = {y y ~ ( n )for , some n c T }, z E Dn(t) z $E x
I
1
~(n) c(D,(t) n T ) = w , x = x ( n ), z < y , r(y) =
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
s$r(T, - { z } ) z E D,(t), r T, by
S E r(T
I
c(r(T, - { z } ) ) S n - 1. ( S , M , r(z),F ) on ( 0 , T,),
7
- {A}),
6. For every 1S n there exists a tree t, E A such that A E D,- I(t,,).
Proof.
t,
=
by
(ul, T ) , v l ( A ) = 1, u,(x) t , on t , by t,- on t ,
=
0, A <x.
,
t,
t,
lkO, 1 S k < w.
7. T h e set A is not s. f.a.-definable. A = T(%) by 3 .
Proof. tn+,
'u = ( S , M , so, F )
c(S) = n . x< z
5,
4. by 3 ,
8. Not every set of valued trees definable by an automaton ( i n the sense of [ 6 ] ) is definable by a special automaton. The set A is an example f o r this. 4. A closure result
9. Let 'u = ( S , M , s , , F ) be a special %automaton, and let T E E . Define D to be the set of all C-trees t = ( v , T ) such that t E D i f and only if f o r every x E T , u(x) = z implies ( v , T,) E T('u). T h e set D is s. f.a. definable.
'u by 3.
t E D, u(x) = z,
x
( v , T,) by yE x
'u different S. by
'u 'u
s
y,
3,
by
'u
u ES* on S . u(i) # u ( j ) j < i , u(i)' = A of u, by C(u) = u(O)'u(l)' ...u(m - 1)'.
i <m
on
(0,T ) ,
u(i)' = u(i) C ( u ) , the contraction
= Z(u),
MICHAEL 0.RABIN
8
+,,: l(u) + l(C(u)) by
contracting u(i) = C ( u ) ( j ) . i < l(u); u ES*
=
E(u)
Z(C(u)). 4u(i)5 +,(i)
2l
9.
u = {u IUES*,
H
=
r$u
zEX
= (S",
C(U) = u ,
u(i) =
{414: r n - r k , R ( 4 ) = k , k S m < n } .
M")
implies i = l(u) - I } ; S" = (U x H ) U { d } ,
d
dump state. M" by = {(d,d)} OEX. (u,~)ES", l(u) = k , O E Z . M5(u,4),a) = { ( d , d ) } u ( k - 1 ) = so a # z, u(k- 1) = z. ((u1,4J, ( ~ ~ , 4 ~ ) ) ~ M " ( ( u , 4 ) , a ) w1,w2eSk (wl(i),w2(i))~M(u(i),a) i < k ; u j = C(wj)ai a j E { A , s o } ,j = 1 , 2 ; q5j = $,,, j = 1 , 2 .
(Ul(4l(i)>
9
u 2 ( 4 2 ( i ) ) ) E M(u(i), a>9
i < Ku) * up
'u
x
T.
O , l , ...,k - 1 ~ ( x )=
d,
M"((u,
(u, 4) E S", x
u(k) = so 4),a) = { ( d , d ) }
.
n
l(u) = k , u(O),u(l),. . . , u ( k - 1 ) . xE
= z,
23
23
d,
'u
v ( x ) = z,
'u d
so
x0
i, i < k , xl
'u
wl(i)
(w1(i),w2(i))~M(u(i),o). i l < i2 < ... < i,,-l wl(0), ..., wl(k- l ) , ij = wl(i) (u,,+~) 8 x0, I(ul) = p Z(ul) = p 1 ul(p) = ul(j)= wl(ij), < p . 4,:k+p ul(i) = wl(ij). ( U ~ , ~ ~ ) E xSl .
+
= wl(ij)).
41(i)=
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
9
23.
= {(A,@),(so,@)}
10. t = ( ~ , T ) V,, E rERn(23,t) r(A) E Sg r(x) # d , x E T. r(x) = (u , 4,), x E T . l(u,) = k , m
,,
10. Let k be activated at r,(x) = so = u"(k), rx(y) = u"(m) x < y (u, T,)) . u(x) = z , rx E n
.
x En ( m , x ) -+ ( m ,y )
T
m
x)
x
x . DeJine r,: T, ( k , x ) + ( m ,y ) .
+
S
by x
m stabilized at x x < y E n.
11. Let k be active at Z E Z , there exists a z 5 X E and ~ an m such that ( k , z ) + ( m , x ) and m is stabilized at x along n. I f m i s stabilized at X E along ~ n and m , m then m , is stabilized at x along n. For every path n c T for which r(n) # {(A,@)}(i.e. z ~ u ( n ) there ) exists a xo ~n and an m, such that m, is stabilized at x , and every m > m, is not stabilized at any X E ~ .
s
Proof.
9. 12. t E D i f and only i s there exists a run r E such that r(A)ES$, d # r ( T ) , and f o r every n c T i f m is stabilized at Z E along n , then c((y1z y e n , u'(m)EF)) = o.
s
r. v(x) = z
Proof. by
.
rx(x) = so. n ( k , x ) -+ ( m , z ) r,(y) = u'(m). tE t E D . For x 3,(u, T,))
10, m S k z 5y ~ n (v, T,) E T('lI). 7,
E
.
T,
m In(r,l n) v(x) = z
d # r(T), k < n r, E ( v , T,)), x < z E n, z n. #
@. (0,
T,) E T('U).
~
MICHAEL 0.RABIN
10
on t .
m x ET, p(m,x)m x; p(m, x ) = y k , ( k , y ) -,( m , x ) , no z < y o(p(wr, x)) = 'c xE m x. p(m, x ) by r ( y y 5 x } . on l ( x ) , X E T , rERn(23,t). by r(A) = (so,@) u(A) = r ; r(A) = (A,@) r(x) = (ux, 4x) x l(x) 5 h ux(m) = f , , ( m , x , ( ~ ) , m < Z($), x. Z(x) = h Z(u") = k. w,+ l(m) = fp(m,x,(xa), m < k, a = f,,(m,x, on (u, T,(,,x,), ux(m) = f p ( m , x ) ( ~ ) , (w,(m), w2(m))E M(u"(m),u(x)). y = x0 y = xl uy = ... ~~(k-1)) u ( y ) # r , 4y uy = ... w,(k-l))s, u(y) = z. ((uxo, po),(uxl, ~ ' ) ) E M " ( ( U ~4x), , u(x)). m 4"(m') = i ) p(i,y) = p(m,x). u'(i) = wl(m) = fp(m,x,(y)= fp(i,y,(y). r r:{xIl(x)5 h l} + S" r : T + S% origin m x , y <x
p(m, x ) = x
1
I
+
rERn(23,t) d$r(T) u'(m) = fp',,,,,,(y) y E T, m y. ncT m z E n, p(m,z) = x E n z 5 y , p ( m , z ) = p(m, y ) = x . uy(m) = f x ( y ) z 5 y. n) nF # pl, c ( { y [u Y ( m ) E F , y ~ n ) )= O . 9
I
s
13. Let R be the set of S"-ualued trees ( r , T ) such that d # r ( T ) , r(A) E Sf; for every path 3t[r E Rn(23, t ) A t E V,]; n c T , x E n and m which is stabizized at x, c ( { y x < y E n, u'(m) E F } ) = O. The set R is s.f.a. definable.
1
R, by 6 ( r , E R, nR , 6 n + 1 = {O,l,..., n } . @(n,s) = (O,O), S E S ~ ; W ( m , ( u , + ) )= (m 1,m + I), m < n, (u,~)ES", I(u) 5 rn 4(m) < = (n+l,&f,l,{n}>. h : T + n + 1, h(A) = on (r,T)ER,nR, ncT by 11, n$ z) X E ~ m x , c ( { y [ x < y ~ n u, ' ( m ) ~F } ) < u. R =R,nRbnT(6). Proof.
R,
+
I
Proof of Theorem 9.
12, t = (0, T ) E D (r,T)ER.
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA -
Z = S” x Z.
(a, T )
A
B 3.
13
=
11
pofi E Rn(23,(plfi, T ) ) ,
I
T ) (pea T )E R } D = pl(A n B ) .
by
5. Automata on finite trees by EcT
%
a
P(S x S ) ,
=
so E S .
An r(Ft(E)) = {f}. [l, 5 , 71
A finite (u, E ) u : E - Ft(E) + X. on ( S , M.s,,f) M:(S- {f}) x Z + on % = ( S , M,so,f ) accepts e = (u, E ) r on e r ( A ) = so e by % by TI(%). on
14. Let % = ( S , M , s o , f ) be a Z-automaton. DeJne a set B c V, b y : (u, T )E B if and only if there exists a sequence (E,),,, of and ( ~ , G , ) ET I ( % ) , n < o , where frontiers of T such that En < G, = { x l x 5 y f o r some ~ E E , } .T h e set B is s.f.a. dzfinable.
F)
fl = S x S ; by ((sl,si),(s2,s;))E (s1,s2)€ M ( s , a ) , (s;,s;)E M(s’,o) s‘ # f , s’ = f. T(23) = B . t = ( u , T )E T(23) i :T + S (E,),,, T i(E,) G F ,E, < En+ , n <w . G, = { x I x S y f o r some y E E n } , n < o. r’: G,+ + S r’(x) = po(i(x)) x E G,, r’(x) = pl(?(x)) x E G,+l- G,. r‘ on (u,G,+J IT,,+^) = { f } . (u, G,+ 1) E TI(%), n < o. (u, T )E B . (u,T)EB. (E,),,, (u,G,),,, r,: G, + S , n < o , on (u, G,) r,,(E,) = {f } . T, = { x Z(x) = n}. (n(i))i<, n(i)
Proof. (so,f); i@((s,s’),a)
So =
F=S
23
=
i@
x {f}.
I
I
MICHAEL 0.RABIN
12
A <X
rJ
1
=
xE
Jo(A) = (so,f') = S o .
E
r
=
I
m(k
- Gk(k). fk+,
I
+
1x
=
=r
=j ; (Jk)k
= (0,
< EJ
I
p l f k + l ( x ) = rJ(x)
{f}, k < 0. (u, T ) T(B).
=
J =
B
(m(i))i5k
E
T),
T(233).
6. Weak definability and s.f.a. A.
A
Z = (0, l } . X, ( A l , ..., An) z ( A ) = (u,, T ) T : P(T)" -,
T.
=
11.
R
G
A
uA(x) = (x,,(x), ...,x,,,(x)) , x
E
T.
2l rep-
P(T)".
resents R @) = T('u)n VZ",,A.
15. Let K , 5 P(T)" be the set of all A = (A1,...,A,,) such that c(Al) = 1; and let K , E P(T)" be the set of all A such that c ( A , ) < w. T h e sets K , and K , are representable by special automata.
Proof. n = 1. =
C
212 =
C"-'
by
( { ~ 0 , ~ 1 , ~ 2 } , ~ 1 , ~ 0 , { ~ 1 } ~ ,
Ml(soyl)= {(so,so),(s
=Z x
({so, sl, s2}, M z , so, 190)
Ml(S0,O)
=
= {(sl,sl)},
1,s l > > ,M,(s
3, =((~O,~l),(~~l,~O)},
M2(s0,
= {(s 1 J I > > ,
191)
=M,(s, 90) =M2(s,r
by
Ki
= {(s2,s2)}.
= =
M2(s0,
=
1) = {(SZ ,sz)>.
!Ii, i = 1,2.
12. A = (Xi,...Y X , , ~ , + ~ , . . . ~ ~ , + , , ~ ~ + , + ~ , x i € T 1 5 i 5 m , ui G m + 15 i 5 m + n , Ai c T m n 1S i S m n k. A is type ( m , n , k ) . Ai A by o(A) = (Ai)15ism+n+k A i = { x i } , 1 5 i 5 m yA i = a i , m 1s i 5 m n , k o(A) A.
+ +
+ +
+
+
16. Let K ( m , n , k ) = {o(A)l A is of type ( m , n , k ) } . T h e set K ( m , n , k ) i s representable b y a special automaton.
Proof.
15
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA
13
17. Let R c P(T)" and Q be, respectiuely, represented b y the special automata '% and 23. T h e following sets are representable b y special automata.
RuQ; RnQ; R , = {(A1,...,An-l)l ~ A ~ [ ( A , , . . . , A ~ - , ) E R ~ ) ; R , = { ( A l , ..., A,- 1) Va,[(A,, ..., A,- 1, 4 E R ] ) , here u, ranges over allfinite subsets of
I
T ( Ru Q )
Proof.
=
T ( R )u z ( Q ) , T ( R n Q ) = z ( R ) nz(Q),
2.
..., x,p, p o : C " - ' x C -+ C n - ' ,
p
x,)
(XI,
-+
( x , , ..., x,- ,)
.
R , = pR
C" = Cn-' x C
W4) = P
(W). T(Rl)
=
z(p(R))
=
by
R,
p(t(R)) = 3.
x:
(u,
C" = X"-' x
VZn,A);
x,
.
+ ( 0 ,l } (u x ( u x x ) ( x ) = u(x)x(x), x (u, E ) .
= C"-' x ( P ( S ) '% = ( S , M,so,F ) . q
A
E
VF
((poO)x q,
by ( S , M , A c V? by
E
A
S q: SEP,~(X), -+
A
EA
EA
A (~,G)EP y E Ft(G) y = x l ; 2) x : H + (0, , r : H -+ S on ((poV) x x,H ) r(x)Ep16(x) x€Ft(H).
9
all x E
x€Ft(H) x(Ft(H)) = r(A) = so EP
P H c y = x0
P
=
Tf(Q
(E
~~71). 14,
by
B c Vf
Ft(G,) < Ft(G,+ ,)
EP
po(A n
by
, n < o,
Vxn-,,A= z ( R 2 ) , 3.
by by
MICHAEL 0.RABIN
14
t = (u,T) t € p o ( A nB).
t = T ( ( A ~. . ,. , A , - , ) ) .
= fEA
poO = u ,
2"" x ( P ( S ) ci
l,ci)) = ( u x xz,T ) . ( 0 , T )E B , H c 1-2 x E ( T - H ) u F t ( H ) , x,(x) = 0 .
= (i?,T)
T
T ( ( A ~..., , A,-
P
n < o. r(A) = so r(x)EplO(x) (V,T)EA, x,(y) = 0 y e T, x€Ft(H), x€Ft(H). r(x) p,V(x) ( u x xz,T,) by ( S , M , r ( x ) , F ) r ' : T + S by r'(y) = r ( y ) x E F t ( H ) ; r,: T, + S y e H , r ' ( y ) = r,(y) YET,, xeFt(H). r,(x) = r ( x ) x E Ft(H), r' r' '% on ( u x x A , T ) . T((A,,...,A,,-~,U))E T(%) a. ( A , , ..., A,,- ' ) E R , . t = (0, T ) = z ( ( A , , ..., A,,- 1)), ( A , , ...,A,,- 1) E R , . q : T + { 0 , 1 } , q(T) = (0). XET s(x) = {sIsES, (u x ~,T,)ET((S,M,S,F)). q = xpl ( A ,,..., A , , - , , @ ) E R , s(x) # @ x E T. i= T) poi? = u p16(x) = s ( x ) x E T. i~ A n B t E po(A nB ) T(R,) = p o ( A n B) n Vzn~ E A s(x) 0. H T = H u { x u x e F t ( H ) , a = 0, l}. x: H (0, x(Ft(H)) = , ci c x,IH = x, xol(y)= 0 y#H. ( A , ,..., A , - l , c i ) ~ R , ( u x x,, T )E T(%). rE x xz,T ) ) rE x x , H ) ) , r(A) = so. ( u x x,, T,) by ( S , M , r ( x ) ,F ) , x E F t ( H ) . x,(y) = 0 y E T,, x E F t ( H ) , r ( x ) E s(x) . E = { x l(x) S n , n <w , EP . f EB. =
x x,,H))
I
--f
I
+
LEMMA18. Let and P E P(T)" be the set of n-tuples ( A l , ...,A,,) such that Ai = { x } f o r some X E Tand x w € A j . Let Q be similarly dejined with x w # A j . T h e relations P and Q are representable b y special automata. Proof. 3 n = 2, i = 1, = 2. w = E , ... & k . S = {so,sl, ...,sk,d}, w =A S = {so,d}). 6 = 0,1, a€Z2. M(so,(0,6)) = {(so,so)}, M(so,(l,6)) = E, = 0, M(so,(l 6 ) ) = = 1. 1 5 i < k, = E , = 0 , M(s,,o) = ci+' = 1 .
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA
15
M(Sp,(d,l))= {(so,so)}, hf(sk,(d,O))= M ( d , o ) = { ( d , d ) } . n T ( ( S ,M,so, {so})) n K(1,0,1)= z(P). 16 P by M(s,, G ) .
A
=
19. Zf F(x,a, A), where x = (x,, ...,x,), a = (a,+ ',...,a,+,), (A,+,+l,...,Am+n+k), S2S p
=
{(x,a,A)IX,
o ( P ) = R(F) ( s x
Proof. by Va[t,E
12)
by t , = t, ,
F a + t,
€a].
t,, t,
u = u, by Vx[x E u tf x E v ]
u
.
u
positive
v [-F
F(x,a,A))-,
A.
v -GI,
by -VuF
t Eu
F
= 3u
-
by
- -F,
t Eu ,
[ F V GI 3uF = Vu
t
= [- F N
F, F
A
- - [ F A G] = GI,
u
by
u
F. F by on xiw E A j , xiw E ai, w E T , or R(F) by 18, 16, 3. R(F,) R ( F , vF,) = R ( F , ) y R ( F , ) R ( F , A F,) = R ( F , ) n R(F,), R(F, V F , ) R ( F , A F,) by
F(x,a, A ) R(F)
(I)
(m,n,k).
F(F) F;
i
S rn + n .
ui
The idea of pushing negation signs next to the atomic formulas, was suggested
to me by M. Magidor.
MICHAEL 0.RABIW
16 ci c T R ( F ) E K ( m , n , k ) , so
i
(4)
(4)
5 m + n,
(3) m
+
i n
Ai
5 m y Ai Ai
,A i , ... , A m + , + & E R(F)
(A,,
R(3uiF)
no
R(Vu F )
by by
on
20. T h e partial order x 5 y on T , and the lexicographic order x y defined b y x y = x 5 y V3z[zO 5 x S y ] , are weakly definable.
<
<
Proof. Va[VuVu[uEa A [ u O = u V u l = u] x
x
5y.
x
+
u ~ a A] y E a
+X E ~ ] ,
5 y.
21. T h e relations ci c T, is a frontier of T,; frontiered tree; and ci = Ft(/?);are weakly definable.
/? c T is a Jinite
Proof. Vy[yea+x5y]AVyVz[yEaAzEaAySz+y=z]A V y 3 z [ x ~ y - t z ~ a r A [ y ~ z A z ~ y ] ] , ci
c T,
T,.
ci
22. A relation R E P(T)" is representable by a s.f.a. if and only if f o r some m there exists a weakly definable relation P E P(T)"+" such that
Proof.
P
R 19
z(R) = T(%)n Vx",h, % = ( S , M,so,F>. S, S = (0, m < o. r = (B,, ..., B,)EP(T)" r:T-+S r(x) = ( x ~ , ( x ...,zBm( ), x)), X E T . 1-1 P(T)" ST. r(x) = s , ES ,x T, r(x)EF G S , r ~ R n ( ' % , z ( A , ..., , A,)) A r(A) = so , r = (Bly..., B,,), N
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA
VTc[In(r
1
17
$1
by frontier of T, and V y [ y E a + r ( y ) E F ] ] G(B,, ..., B,)
S2S z ( A , , ...,A , ) € T ( % ) , S2S by
3a. (Al7...,A,,)~R, 3B1 ...3 B , [ H ( A l , ...,A , , B , , ...,B,)
A G(BI, . . . , B , ) ] .
7. Applications to decidability and definability [ 6 ] by
on [6], [6].
23. There exists a n efectiue procedure of deciding f o r a special &automaton % = ( S , M , s , , F ) whether T(%) # 0. If c(S) = n , then this procedure requires n4 computational steps.
M'(s,cr)
Proof. T ( ( S , M ' , so, F ) ) # 0 .
E c T. R(H) r:E
=
UmELM(s,o)),
{a}. T(%) # M'(s,a) = M ( s ) .
%
H ES, E # {A}
SES
-,S
0
r(A) = s R(H)
by
r(Ft(E)) E H . H , = 0,
i <w,
Hi+1 i <w, Hi= H i + 1 Hi= H i + k = R ( H ) , k <w. Hi E S, H , = H,+ = R ( H ) . Hi, Hi+1 n2 c(Hi) 6 n . H, = R(H), R(H) n3 F, = F , i w , Fi+,= Fi n R(Fi). F i + lE F i i w . Also, F i = Fi+, Fi = Fi+k k < w . F, = F , , , ; F, = G . G c(F) 6 n R(H), n4 T(%) # 0 so E R ( G ) . T ET(%) so E R ( G ) by 27 H,
E
,
-=
-=
MICHAEL 0.RABIN
18
s,ER(G) r: T +S, T ET(%). to = ( r o , E o ) ro E E , ) , ro(A) = so, ro(Ft(Eo)) sEG t , = (r3,Es) r , e Rn(%,E,), r,(A) = s, r,(Ft(E,)) G. to T A; ( r o ,T ) . x EFt(Eo), (ro, T ) x; (rl,T). El c T rl r,(Ft(E,)) E G. xeFt(EI), tr,(*) ( r l ,T ) x; T). (r,, T ) , i < w Ei = D(ri), i <w. E, # {A} sE G, Ft(Ei) < F Z ( E ~ + ~i )<, w . T = Ui
24. There exists a n efective procedure of deciding f o r every sentence G of the f o r m 3 A 1 ... 3A,F(Al, ..., A,) VAl ... VA,F(A,, ..., A,,), where F is a formula with quanti3cation over just individual of finite-set variables, whether M 2 k
G
Proof.
G M, k G
not Jv; k
G = 3A ... 3A,F(Al, ...,A,,), M2 k G, by
19 (0, {(Al, ..., A,,)1N2 k F ( A , , ...,A , , ) ] .
F
%
.V; k
T(%) # gl
by
23. [4]. A
[6]. 25. T h e weak second-order theory W T O of linearly ordered sets is decidable.
Proof.
G P = (A, 5 ) k
G E WTO P
[6]
c(A) 5
w.
B
cT
(B,
<)
19
20) A
P
P
21
G x
5y
by x
A. A ET by As
T
(A,<).
S2S by
G(A)
G E WTO 24. [6],
M2 t= G ( A ) .
upon [4]. [6].
THEOREM 26. The weak second-order theory WSU of a unary function is decidable.
Proof. F(x, y , C )
2.4
[6],
xE T C ET M2 t= F(x, y , C ) . ET x T C s T , F ( x ,y , C ) D ( f ) = {x IM2 t= F ( x , y , C ) } T. A T , C E T , M2 k Al(A, (A, A ) (A) E A . P = (B,g) A E T, C c T M2 I=A l ( A , C ) P N (A, A). F(x,y,C) Al(A,C) do
Al(A,C)
y
E
T
I
1
= VAVAG(A,C)
S2S,
G E WSU
G M2 t=
no
G.
8. Characterization of weakly definable relations R
R E P(T)” P(T)” - R
T(2l) n T(23) # @. % = ( S , M , so, F ) , 23 = ( S ‘ , M ’ , shy F’).
by
(u, 7’) = t E T(%)n T(B),
MICHAEL 0.RABIN
20
rE t) r‘ E t). E c T T) < Ft(E), < Ft(E), r(G) F r’(G’) E F ‘ . x E Ft(E) El c T,, T,) < Ft(E,), < r(G,) F , r’(Gi) F’ . on, x E Ft(E,). S x S’ H , = S x S’. Hi+, on (s,s’)€Hi+, (s,s’)€Hi e = (u,E), E # {A}, < Ft(E) < Ft(E), rE (a,e ) , r’ E e) (1) r(A) = s, r’(A) = s‘; (2) r(G) c F , r’(G’) F ’ ; (3) x E Ft(E) (r(x),r’(x))E Hi. H , 2 H , 2 .... Also, Hi = H i + 1 , H i = Hi+k k <w. c(S)c(S’) = m, H , = Hm+k k <w.
.
27. Proof. 7~ c
T,
T(%)n T(23)#
Id if and only
if (so so) E H , .
( u , T ) = t E T(%)n T(23). r(A) = so, r’(A) = sb
I x) n F #
rE
t)
I x) n F’ #
(EJis,,, Ei c c T, i <m, E , = {A} <m, T ) Gi,GI Ft(Ei) 5 Gi < F t ( E i + , ) , r(Gi) E F , Ft(Ei) 5 < Ft(E,+ 1), r‘(Gi) E F’ . xeFt(Ei), 5 m - 1 x) E = T, n,Ti+,. Ft(E) = T, n F t ( E , + , ) Gi n T, n T, E T,). r E r‘ E on ( u , E ) . r(Gi n T,) c F r’(Gi n T,) F ‘ . ( d x ) ,r ’ W E H , * 5 m-2, +1 5 by (r(y),r’(y))E H , y E Ft(Ei+ l). by (u, E ) , (r(x),r’(x))E H , . (r(x),r’(x))E H, x E F t ( E ) , i 5 m -k . A E Ft(E,) (so, s;) = (r(A),r’(A))E H , . (so, s6) E H , = H,+ 1, EH, e(s, s’) = ( u ( s , s’), E(s, E(s,s’) # {A}, r(s, s’) = r E e(s, s’)), r‘(s,s‘) = r‘ E < Ft(E(s,s’)), G’ < Ft(E(s,s’)) (1) r(A) = s, r’(A) = s’; (2) r(G) E F , r’(G‘) F ’ ; ( 3 ) x E Ft(E(s,s‘)) (4x1, r’(x))E H , t (u, T ) x E T ,
I
I
21
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA
t = (u,E)
y $xE ~ ( x z= ) u(z) z EE . t, e(so,sb) (!& T ) A . t , = (w,, T ) rl by r,(x) = r(s,si)(x) x~E(s~,s~), r; by r ; ( x ) = rf(soysA)(x),x ~ E ( s ~ , s A ) . x E Ft(E(s0, sb)) ( r l ( x ) ,r;(x)) ~ H r *n t, , x E Ft(E,), E(so,sb) = E l . e(r,(x), r;(x)) tz = (w,, T ) . w, E,. rl rl r2 E Rn(%,(w,, E,)) , r l E Rn(%,(w,, E,)) y E (E2- E l ) u F t ( E , ) x E Ft(El) y E xE(s, s’), rl(x) = s, r;(x) = y = x z , z EE(s, r,(y) = r(s, s’)(z), r;(y) = r’(s,sf)(z). ~EF~(E,) rl(y) = r,(y), r;(y) = r;(y). rz r; r, r;, Also, xeFt(E,) (r,(x),r;(x))EH,. by (ti=(wi ,T)),<,,, ((w (ri)i
(w, T )
w(y) = u(y)
9
.
I
I
COROLLARY 28. If there exists a %tree t =(u, T ) and a strictly increasing sequence (Ei)i6mof finite trees as in thefirst part of the previous proof, then T(%)n T(%) # 0.
Proof. T(%)n T(%) #
(s0,s;)€H,.
@. X n (= (0,
A = ( A , , ...,A,)
E P(T)”
T(A). t = (u, T ) by rE t) (Gi)i<m
r‘ E Rn(’u,(u, G))
by ‘u. a
t
of r(A) = so, r(Ft(G,)) F , i c o. Ec E G r’(A) = so , r’(Ft(G)) c F .
T
a
G = Gl
MICHAEL 0.RABIN
22
i,
r'
=
rlG.
E' G' T, r"(x) = r ' ( x ) , r"(Ft(G'))c F . j >i, r" = r G' . on.
1
KO = S X E
r' T, r"E G' = G j n T, E'
x€Ft(G)
x,
by
x T x Vz,A,
K,(s,x,t)
SES,
<w;
T, (u, T )
K i + , ( s , x , ( u , T ) )= VE3G3r[E finite subtree of T, + E c G is a finite subtree T, A r E G)) A r(x) = s A r(Ft(G)) E F A VY [Y F t ( G ) K i ( r ( y ) ,Y , ( u , T ) ) l l * i <w sES , Ki Ki c T x P(T)". M2. q : S + Xk +
by
T,
Kh,
Y 9
by u, p . (Al, ...,An) x 5y G T x E T): G x, r:G +S q , by k ul,...,uk.A r(x) = s K , ( r ( y ) ,y , t ) by A s [ r ( y ) = s + G, E,
(6), (0,T )
t,].
P(T)" and Q = P(T)" - R are representable b y 29. If R special automata, then R is weakly-definable. Conversely, if R is weakly definable, then R and Q are representable b y special automata. ( 2 )
Proof. 19 no R by 'u -- ( S , M , so, F ) , T(%)n T(23) = Id. 23 = ( S ' , M',sb, F' ); c(S)c(S') = m . t = (u, T(%) K,(so,A, t E T(%)
K i ( r ( x ) ,x , t ) K , ( s , x ,t ) .
by t T('?I), t E T(23). r' E E , = {A} Gb K,(so, A,t) on (u,Go) ro(A) = so ( 2 ) K,- , ( r 0 ( x ) , x ,t ) x
+
(2)
by
< w ,x E T , by K,(so, A, t ) K,(so, A, t ) t = (u, T ) t) on t . r'(Gb) E F ' . Go r o : Go -+ S = Ft(Go), E F E,
This theorem was suggested by A. Tarski.
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
23
Go,Gh < F t ( E , ) . G; Ft(E,) 5 G; r ' ( G i ) F' . (2) x E Go ' 7 n El El G, r 1: +S ro rl on ( u , G , ) G , = F t ( G , ) , (1) r l ( G l ) E F (2) K m - 2 ( r O ( ~ ) , ~ , t ) x EG,. up En,Ft(Em- ,) 5Gm-,, Caw,. rm-,: Gm-, + S rE t). ( E i ) i < m ,r r' 28. T ( 3 ) n T(B)# @, Km(so,A,t) t E T('U)
30. A formula H ( A l , ..., An) of S 2 S is equivalent in S2S with some formula H , ( A , , ...,A,,) which has no arbitrary set quantifiers, i f a n d only ifthere exist twoformulas G , = VBl ... VB,F,(A,, ..., A,,,B,,...B,,) and G 2 = 3B1 ... 3B,F2(A,, ...,A,,,B,, ...,B J , such that Fi has no arbitrary set quantifier and H is equivalent with G , in S2S f o r i = 1 , 2 . 22
Proof.
29.
by H. H ( A , , ..., A,,)
Is
S2S H
H , ( A , , ...,A,,) S2S?
§7
P
(A,<)
=
(A, 5 )
A c
S2S.
so
REFERENCES [l] J. E. DONER, Decidability of the weak second-order theory of two successors, Notices Amer. Math. SOC.,12 (1965), 819. [2] A. E H R E N P E U Decidability C~, of the theory of one function, Notices Amer. Math. Soc., 6 (1959), 268. Decidability results in non-cZassicaZ logics I , Technical Report No. 29, [3] D . M. GABBAY, Applied Logic Branch, Jerusalem, 1969, p. 42. [4] H. LAUCHLI,A decision procedure for the weak second order theory of linear order, Contributions to mathematical logic, K. Schiitte, editor, North-Holland, Amsterdam, 1968, 189-197. [5] M. 0. RABIN,Mathematical theory of automata, Proc. Sympos. Appl. Math., Vol. 19, Amer. Math. SOC.,Providence, R.I., 1968, 153-175. 161 , Decidability of second order theories and automata on infinite trees, Trans. Amer. Math. SOC.,141 (1969), 1-35. [7] J. W. THATCHER and J. B. WRIGHT,Generalized finite automata, Notices Amer. Math. SOC., 12 (1965), 820.
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM YIANNIS MOSCHOVAKISt University of California, Department of Mathematics, Los Angeles, California, U.S.A.
R x R,
A
5 A
realized
=
5
prewellordering $3). o(A) = supremum {C:
R = A
5
A}
A’s,
DC
AD
A:
If AD, then for each k 3 1 , o(A:) is a cardinal. Zf AD and DC,then for each k 2 1, o(A:) 2 Kt; also for each odd k 2 1 , o(Al) is a regular cardinal and each subset of o(A:) is “ll: in the codes”, in a reasonable sense of this expression. R xR
A:, §$1,2
. $3,
A =R x R
A
$4,
$5.
§$6,7 A;,
I
I The author is a Guggenheim fellow. Work on this paper was sponsored in part by an N.S.F.grant. 24
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
o
$1. Preliminaries. R = Ow numbers.
{0,1,2, ...}
=
real
3 = XI x
Xi w R; k, I, m , n , t , s, ...
x
xk
x, y , z, w
o
a,
y
x
R.
p,
y,
...
( x ,y )
i
i
+
x
X
y.
9J = Y , x
=
X, x
x Xi
x Yj,
Xx9J = x,x
... X X i X Y , X ... X Y ] .
1952 enumeration, iteration (S,")
cursion on X
9J
re-
3, 9J 1959.
As
( n o ,..., nk> = 2"0+'* 3"'+'. ....p;l<+l, pi
(n)i = supremum { j : p i f 1 divides n} , (a)i = Ata((i, t ) ) .
Z(0) = 1 , Z(t
+ 1)
= (L%(O),...,a(t)),
ii(t) = n ,
x = (x,,
..., xk)
X ~ E W
x~ER,
n(t) = ( n , ( t ) , . . . , a , ( t ) ) E o k .
A
: X --f o g(n,, ...,nk) , h(n,, ...,nk) f(x)
N
g(f(s)), s
N
pt[(Vt'
< t ) [h(f(t'))
:3 + o :X g : X x o --f w
N
f(x) x
A
25
+R
01 & h(Z(t)) > 01. on
26
YIANNIS N. MOSCHOVAKIS
f ( 4=
Mx,t);
D o m a i n ( f ) = {x: Vt[g(x,t) is defined]}. f :X + 'I) = x ... x & f l , ...,A on 2-l w R
fW
= (fl(X),
...,A(x>)* :
by
f :X + 9
p
g : R x X + 'I)
= X
g(P,x).
a
X
o
:X + 'I) f:X
p.
type(X) = 0
X
X
+
9
continuous
o ,type(X) = 1 type(X) = type('I)),
R . One n:X+'I)
on X x X
X
type('I)) = 0 , then a partial function f : X+ 9'J is continuous if and only if Domain(f) is open and f is continuous on its domain. If type@) = 1 , then a partial f:X + 'I) is continuous if and only if Domain(f) is a Gd set (a countable intersection of open sets) and f is continuous on its domain.
on w
w
(the analog of enumeration). For each X, 9 there i s a recursive partial function @: R x X + 9 such that each continuous partial f : X - 9 i s given b y
27
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
f(x)
=
W , X )
To
f o r some
@(E,X)
=
{&}(X)
N
{&}xqX),
ITERATIONTHEOREM. For each X there is a recursive function S x : such that f o r all 9,3 {El
(x, Y )
=
{E}XX993(X,y)
{SX(E,
=
+
(Y) 9
{SX(E, x)}9*3(y).
RECURSIONTHEOREM. For each X there is a recursive function F X : such that if E* = FX(&),then f o r every 3 { E } (E*,
x)
2
A,
(XI.
X,
A
A (1-1)
[X]A = { A c X: f o r some B E A and some recursive
f:X [X]A “ A A”
“ A EA”
A =f-’[B]}.
+
by A
E
X
X
X
=
A
on
&A = { A n B : A , B E A } (conjunction, &)
VA
=
i A
=
{ A U B : A , B E A } (disjunction, V) 1
- A : A E A } (negation
dual, 1)
XOA = { A :f o r some B E A , B s o x a E A o (3n)[(n, a ) B ] } (existential number quantification, 3 m )
-
X’A = { A : f o r some B E A , B 5 x a EA (3p) [(B, a ) E B ] } (existential real quantification, 3a) IIOA =
II’A
=
1 x 0i
A (universal number quantification, V m )
TA (universal real quanrwcation, V a ) .
28
YIANNIS N. MOSCHOVAKIS
EA = { A c R : A is open}
=
Xi+'
=
A;
=
172; ,
c'n: , f: R
A A =
--f
o
light-face (0) by recursive
R +w,
R : f o r some recursive partial
Zk = ( A
A e f ( u ) zO]
C' , 1, n
72;
f:R
B k > 0,
no.
+R
&, v ,
af-'[B]
C'
A second order number theory =,
analysis.
', +, * .
0
a,P, ... ,
n, m , ...,
a ( n ) , p(a(n)) 3a,Va.
on w
by on R no bound
by
k
=- 0 ,
A
R
3alVaZ
0(P,a,al,az,a3)
Ei 3a30(p0,
by
is a,
no
aZ, a 3 )
Po go
third order number theory F, G,H ,
...
by
Rw,
29
F(a), P(G(a)) 3 F , VF A A
.
Z:
R
by
a, F )
3F O(B, a, F ) do
ll:
As
=
Po
no bound
Po, Z:,A:
by
(k > 1)
72;
3 F VG O(Qo, a, F, 92.
X:.) Et,
ll:.
= Z:n
i
Structure properties. R
. X. A, R
A (1-1).
R xX
2.1. Universal sets. A
=
G,
[X]r
G
{G,: ~ E R ) ,
G,
a-section
G, [X]r GI.
(R-)unioersal
= (x:(~,x)EG}.
G
(R-)parametrized
c R x R.
by
X,
h(x) = A t ( Z ( t ) ) ;
h:X
--f
R +X E X,
R
h-' on =
A
E
~
=
=
f), H = ((a,x ) : (a, hx) E G }
(2-1)
[X]r.
ir,
nor, Z,,!, ll:
MOSCHOVAKIS
YIANNIS
30
CA
by
G = {(&,a): { & } ( a ) =
by
A
6ir. EL- II; #
k,
hierarchy theorem,
2.2 Separation and reduction. separation property A,BEr AnB= C E r n i r A E C, B nC = reduction property A, BE E A,B, B,A,nB, Reduction(T) 3 Separation (ir)A Reduction(T) 3 not Separation i
CA, II: on C
Cl II
k
>= 3 1959b
k 2 3.
Reduction (Ci)
A ER x R
2.3 Determinacy. I1 a(O),
...
p(O), a ( l ) ,
(a, p)
(a, p) $ A , I1
A,
A
on 11
strategy
I
p.
a * [,4] = a where a(n) = 11 [a]
A 11
r~
*T
winning
a , (a, [ a ]
=
p where p(n) = r@(n +
* z) $ A .
p, A
(a*
p) E A ; z
determined
I 1953,
I1
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
31
by 1964
R x R is determined.
Axiom of Determinacy, A D . Every subset AD
Countable axiom
choice f o r
sets of reals, Vn3a(n,cr) E A + 3aVn(n, (a),)
( A E o x R)
EA
AD R
by A D AD
for R
AD-
by L [ R ] = the class of sets constructible f r o m R AD if there exist measurable cardinals, then every II: set is determined. 2.4
A;
1968 dejinable f r o m a real is determined.
every subset of R x R ordinal no
by A Determinacy (A): every A E A is determined, A E A for A E R x R of Determinacy(A)
A E [R x RIA. A
YIANNIS N. MOSCHOVAKTS
32
2.4 Prewellorderings. A prewellordering of a set A (with field A )
x
4
4
5Y *
5 4(Y),
5
on
4 :A + 5
5,
5 on A x, y A ,
on
5
5
5
length
5).
canonical surjection
5
A
-
5, lack of infinite descending chains, Vn[xn+, 5 xnl
=>
3n[xn 6 xn+1] s
{x~}.”=~
5
A.
dependent choices -
X,
A EX
A ER x R,
Dependent Choices, DC. V U ~ P P) ( ~ ,A
=>
3aVn((U)n, (a), + 1) E A
AD,
AD. DC L[R])
DC,
R
.
AD
DC. by G .
Prewellordering - and 5 in and 5
There is a prewellordering 5 on G and relations
i r respectively, such that W E G => V z { z 5 w z 2 w e [ z ~ G & z5 w ] } .
(2-2) G’
yo. (El,
PI> 5 ‘ (UZY P 2 ) * 5 ’, 3 ’ .
5’ on
on
(UlY P I ) )
5 ( Y o J (U2Y P 2 ) )
Prewellordering
Y
DETERMINACY A N D PREWELLORDERLNGS OF THE CONTINUUM
33
&,
Prewellordering
e- Reduction
Prewellordering
, Prewellordering (ZJ),Prewellordering (II:) Prewellordering (ZA) by
in 1964 Prewellordering (Xi)
II:
Z;.
[o]Ai,
by Suzuki 1967
if i s parametrized, closed under recursive substitutions, &, v, Vcc and Prewellordering then Prewellordering (C'T)
.
1967
of
1969.
II:
-
do
if i s parametrized, closed under recursive substitution, &, v, 3m,Vm and Vcc and some relation in wellorders R with order type K,, then Prewellordering (C'T).
k 2 2 , Prewellordering (Xi). 1968
If
1968.
i s parametrized, closed under continuous substitution, &,
V,3m, Vrn and 3a, if
Prewellordering then Prewellordering DC Prewellordering (Xi)
(II;) X
odd k .
II
k
D C a n d Determinacy (rni r ) ,
k
Prewelfordering on
34
(II:))
(IV), on
X: (111)
L[R],
Prewellordering ( X I ) , do Prewellordering ( X i )
53. The basic lemmas. A &
R,
A
.
[
i
S
realized
R
[.
A
A [ R x RIA)
A
5 is realized in A & < [ is realized in A a [ [>0
&c
*5
+ 1 is
is realized in A ,
realized in A ,
is realized in A 5 5 is the length of some prewellordering R in A .
X.
on
0th) = supremum {[: 5 is realized
o(A),
A}.
A is
=
in this theory.
All results stated thus f a r in this paper a r e provable AD DC, o(A)
= supremum =
{[: there exists a surjection ~ : R - B [ )
supremum {[: there exists a n injection
by
+ ‘2);
35
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
X;
A (3-1)
... 3a,8,
3a,3a2
8
no
Et
X , by X. Ct
o.
on R
j(
by
m” ,
N
= m”
(3-1) subformulas R
occurrences
0 no the f o r m
.
only positive
&(x)
m. by X : ( x )
N
x N i
N
m
m ) o 3n[n # m &x(S)
N
n].
formula 0 there i s a Z: formula Y(a,P) 1. For each (with no occurrences ofx) such that
0 * 3a3P{Vn[x(a),
(3-2)
i.e. the universal closure
=
P(n)l 8z ‘y(a,P)}
9
(3-2) i s true in the standard interpretation.
0
Proof.
by up
x
no
&, v , 3 m , V m ,
m by
N
no
3 a 3 P { V n [ ~ ( ( a ) , )N P(n)] 8c
m
0 o Vt@*(t) ,
by by
= 6 8c P(0) = m }
by
@*(t>
* 3 w q V n [ x ( ( a ) , ) = P(n)l 85 Y * ( %P, t>>
Vt@*(t)
*
* ~ r ~ ~ { v n [ x ( ( r=) ,6(n)l > V t 3 ~ ~ P { V n C ( a= ) n (Y)I &k W P ( n ) = s ( ( t , n > > l
Y * ( %P,
o>> Y
.
YIANNIS N. MOSCHOVAKIS
36
by (up LEMMA2. Let G c R4 be universal f o r [R3]3c:, the ternary relations in x;,f o r each partial x: R + o put (4@ G(X)
(3-3)
* 3a3P{Vn[x((a),) 21 P(n>I & ( E , 8, a, P ) E G ) ;
then G ( x ) is universal f o r
X
S n: R 1:R
type 3E
=
1,
+X
5
-, w
S, R
S ~ ( L x&) n~ = 01
~(aN ) n e-
(3-4)
v [n(a)o S n(.>o
S
&
&
n(co1
S 71(a>,l
1 >.o(.[
& n = 11.
W S )
(3.5)
=
x;w,
x,
5
4
5 4(x)
x~Field(S)
on
5
4: F i e l d ( S ) -+ 5 5 3, =q <5 9,
+ 92
q.
by
X x ‘1) C o d ( f ; 4 ) = { ( x , ~ ) :x
f:5+32 q
5 x&yEf(4(X))}.
A choice subfunction o f f
g : ( + 92
<5, g(?)
f(?)7
f(?> #
* g(v) f 2. q < 5 ,f (q) #
65,
g(q)
f(q). Assume A D . Let S be a prewellordering LEMMA 3 with field a subset of some X and length 5 , let f : ( + b 2 be a function.
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
37
Then there exists a choice subjunction g o f f such that Cod(g; 5 ) is a E l ( 5 ) subset of 3E x '2). Proof. Pr on
[
5 5,
[,
fc
5-[, fc(1)
=
1
f(1)
= @ [S
Cod
Ei ( 5 ),
[ j q < t .
5
fc
5.
I
4 : Field ( 5 )+ 5
1 = [+ 1
I fc,
=
Pr
gc
gc
y o E ([)
g,
gc - { ( C Y @ ) }
=
"
{(CY
{YODI
fA
(x, Y ) E Cod(g,;
5 ) * (x, Y ) E C o d k c ; 6 ) v C4W
=
5
Y = Yo].
4(x0) = [ ,
xo E Field ( 5 ) $(x) = [
-=. x 6 x,&x,
j x,
1. [X x '2))IE:( 6 ) G E R x 3E x '2) a
Cod(g,; 6) Z; ( 5 ) 2, Zi ( 5 ) Z: ( 5 ) X x '2) As
G , = { ( x , Y ) :( ~ , ~ , Y ) E G } . CY
I1 wins e
i (3q)[g,
v ( 3 1 < A)@[
By
is a choice subfunction o f f , & G, = Cod(g,; 51
c 1)(3g,)(3gC)[g,is a choice subfunction off,
& g c is a choice subjunction of f c & 1 < & G, = Cod(g,; 5)& G, = Cod(gc; S)]. u
11 j.
code
g
G, = C o d ( g ; g ) ,
YIANNIS
38
MOSCHOVAKIS
AD,
Case 1. I has a winning strategy gvts,,
p
0.
q = q(p)
fvt,,,
GO*[,, = COd(gq(,);9
-
supremum {q(/?): p E R } < [ < A i, I1 0 by p = Cod(g,; I), g, f, p by 2. 2 supremum {q(fi): /? R } =
g, is
f, ( x , ~ ) ~ C o d (5 g ,)~* ; ~P[(~*[P],x,Y)EG],
Zt(6)
Cod(g,;x) is
Case 2. 11 has a winning strategy G 3 x $9 by (x, y ) E
o
w I w & 3z[z S w & i ( w 5 z ) & { E } ( z ) i s defined (z), x, Y ) GI 7
82 {E}(z)N
X
R.
Ae,w
= w & ~ ( c 1 ) ~= z & x(a)
R
E
{ E } ~ , ~ ( Z ) ,
on
Ei(S)
N
3ct[n(~),, on z E Field(5).
w w, 5 w , g:R x X R
c>
E,W,
A&,W = G g k w ) = { ( x ,Y ) : (g(E, w), x, Y ) E
by
4 z)
i(w
x
13
,
o
E E R ,W E X ,
T.
GI * by
G Zi(x)
&, v,
~CX,
on
w EX,
E*
{&*I (w) _N
on R x 3 ,
[g(E*, w ) ] * T
21
ME*, 4 1*7
5
g
as =
SUBLEMMA. For each w E F i e l d ( I ) there i s some choice subfunction g,(,) of frcw,such that
5
= I;(w) and some
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
39
and by Field ( S )
Proof
on 4 ( w ) .
4(z) < 4 ( w ) ,
E
A&*,,
u
=
{Cod(g,(,);3 :4 b ) < 4(w)l
C o d k , , ; 51,
=
q = supremum { [ ( z ) : 4(z)
[
< q, g,K)
u {&(,)(o:
4(z> < 4 < w > > .
=
f,,;
g,, q
4(z)
< +(w)}
-= 4 ( w )
Y
2 4(w), 5(z) > 4 ( z ) *
q =2, q <2,
by
A&*,, = GE(&*,W)- C o d k , , ; 5 ) z ‘[g(E*,W)]*T
‘Od(g[; 5 )
=
5 > q 2 $(w) [g(e*,w)] * z = { E * } ( w ) by
fr.
gc E*,
=
u
WE
Field(
S)&(w,(O
9
fA
gA
( x , y ) ~ C o d ( g L5; ) *
6 w &({E*}(w),x,Y)E
~ W [ W
Zt(5)
Cod(gA;5 )
5
;i
5
5’’ -,‘4’2
4 : F i e l d ( 5 ) + 5.
on
5
‘1).
Cod(f;5 ) =
{(XI, * * xn,
Y):~
1*,
xn E
P on Cod(P; 5 ) = {(x,,
.. .,x,):
Cod(A; 5 )
=
xl,
Field( 5 ) & Y ~ f ( 4 ( ~ 1 * * ) ,4 ( x 3 ) } *
5
A
5,
.. ., x, E Field( 5 ) & P ( 4 ( x l ) ,...,$(x,))},
{ x : x E F i e l d ( g )& ~ ( x ) E A } .
YIANNIS N. MOSCHOVAKIS
40
Iff:
5" -92,
g : 5'' -+ s2
choice subfunction
ql,
g,,
<
f
g(q1, * * * q n ) 5 f ( ~ l , * * . , ~ nf )
-. V n ) ;
( ~ 1*,
9
d V l ? . * * , V n ) f @.
-+
4. Assume A D . Let 5 be a prewellordering with jield a subset of some X and length 5 , let f : 5" -+ g2 be a function. T h e n there exists a choice subfunction g o f f such that Cod(g; 6 ) is a subset O f X X O .
so
Proof.
f on
5
92.
g
f f*(g)
" ( ~ 1 ,*
f
( ~ 1 * ,*
+
qm
{ a : f o r some choice subfunction g"
=
G, = C o d ( g " , S ) ) ,
Z;(S) (X>n x '1). g* Cod(g*; 5) E i ( 5 ) .
G
f* (xl,.
=
qn)
*
n
< 5,
. . , x n , x, y )
* 3a[(x, 4
3,
Cod(g*; S ) 8~(a, X I , . . .,x,,,Y>E
Ei(S)
A
A
=
C o d ( g ; S)
g
5 . Assume A D . Let S be a prewellordering with jield a subset of some X and length 5 , let P be an n-ary relation on 5 , A a subset of 5 . Then Cod(P; S ) , Cod(A; 6 ) are 4
Proof.
P on f(ql,..*,gn>
a,,,al is f (xl,
so by
5 = {ao)
f'(ql,**.,vn),
= ( ~ 1 )
lP(g1,***,qn),
4 Cod(f;S )
...,x , ) ~ C o d ( P ; s )0 ( x l ,...,x , , a , ) ~ C o d ( f ; 6 ) . 3
41
6 . Assume A D . Let 5 be a prewellordering withjield a subset of some X and length let x be the partial function associated with - via (3-4). Let be a class of sets containing all singletons, parametrized I and closed under continuous substitution (preimages), &, v, 3m, tlm and 3a and containing ( ( 6 , m ) : ~ ( 6 E ) m } . Suppose A = where each A,, is a
Proof. ‘2).
E
subset of some ‘2). Then A E ~ . x ‘2)
f(V)
by Cod(g;S )
Uq<&
= { a : G, =
5 -+
3
A,}
R2,
S). y
Cod(g;
A o 3x3a[(x, a) E C o d ( g ; 5 ) & (a, y )
AEr. 54.
How large is 0 ( ~ 2 ) . 0 ( ~ 2 )= (2”’)’
AD
2”’
= jirst aleph greater than 2 ” O .
2”’ 0(~2)
R
no
R A D , 0 ( ~ 2is)
by 1968.
nxl)
AD)
nxI AD
DC)
uk
n,,
Solovay games,
6,
uI
by by
by 3
Friedman
“Large ordinals
the axiom of determinateness”.
as
42
THEOREM 1. Assume A D . I f there is a surjection 4: R is a surjection 4*: R + { 2 . 5 < 0(~2)
Proof. on R by
4: R .+ 5 ,
+
a
5
A c
by
=
5B *
5 +(PI. 5 Cod(A;5 ) E;(s); +(a>
by E
Pr otherwise. Assume A D . I f 5 < 0 ( ~ 2 )then
COROLLARY 1.1.
Proof.
5
+ 5' A , if A
x(A) = order type
E
5
5' < 0 ( ~ 2 ) .
t+h : R -+ct;xe)2.
5 <0 ( ~ 2 )
=
#
5
if G, = C o d ( A ; 5 )for some A
A
5
then there
5
+*
+*(a) =
5,
by
x 5 is a wellordering,
0 otherwise,
x
is
o
$: R
+ 5' ,
COROLLARY 1.2. Assume A D . I f there is
5' < 0 ( ~ 2 ) . a surjection 4: R+,5
which
i s ordinal definable j r o m a real (in L [ R ] ) then there is a surjection
$: R + { 2 which is also ordinal dejinable f r o m a real (in L[R])sothat each subset of 5 is ordinal definable from a reaI (in L [ R ] ) . L [ R ] is
t+h
Proof L[R])
I
0(~2) nA 0(~2).
67.t 1967, 1969. 2 o(R2),
no 6 < x
x
2 3.
43
L
denumerable ordinal variables.
aA, A
'uAby
('&,A)
A<
K ,
A
A
-
L'
B
K,
(1,A )
<5(K,B )
B(x,, ...,x,)
cl, ..., 5,
A,
..., 5,)
(%,,A)
1
wl, ..., 5,). 'uA
B,
A =
!!IK.) reflects with respect to { A : (A, A n A) << ( K , A ) } K
and
{'?IA}
A
K ,
K.
{aA}
2. Assume AD. Let a denumerable L a n d a class of structures be given. There exists a regular cardinal K < 0 ( ~ 2 which ) reflects with respect to L and
{aA}.
7. Let x1,x2 be partial functions on R to w , assume x1 E x 2 , i.e. x1 is a subfunction of x 2 . If G(x) is the canonical universal set for X i ( x ) subsets of X dejined by (3-3) and (2-1). then G(xl) E G ( x 2 ) .
Proof.
R by
Xi(x)
G(x)
X's by
(3-3)
X:(x) ~ ( 6 N)
x,
,
Proof of the theorem.
us
on
YIANNIS
44
MOSCHOVAKIS
by by
F on
5 >w,
5 5
F
5
<5;
L,
w.
F
{aA}
L, K
5
no
K {%A};
4 : ~ + ~ 2 , codes 5.
4(5)
5
q <5, g5tP
for
4
K ,
* (3rl<5)(31:<5)[ccE4(?)&PE4(r)&? SK K . 5
T h ( 5 , A ) = { q < (: F($ = ( m , t l , ...,(,) of some formula '4xl, ..., x,) of L' and
551.
and m is the Godel number
.
b W1,...,
F
q
<5,q
B c 7,
(q,B)
n:w x R x R
<-= (
5
7
4
*
Th(?,B) E W 5 , A ) .
+R
(
5
=
0. Put
Case 2 . 5
=
5 +1.
Case 1.
Case 3.
4 ( ( ) = (n(l,a,a): Put
4(5) = {71(2,a,cc): cc~4(4')).
4 is a singular limit ordinal. Put
c
[ < 5 and some cojinal m a p f : + 5 , cc
+(() =
$(c) and
f o r some = C o d ( g * ; &),
where g* is a choice subfunction of the function f *(q) = 4 ( f ( q ) ) } .
G(St;) R x R, (3-5). As
G,( 5 [)
=
((7,s):
(p, y , 6 ) G( S ,)} .
5
Case 4. 5 is a regular limit ordinal. Put +(C) = {n(4,a,P):f o r some c 5 and some 5 < 5 there is no A 2 5 so that (A, A nA) << ( 5 , A ) and a 4(5) and q?(5<) = C o d ( W 5 ,4;St)}. A
45
by n(n, j )
4(5># r]
R.
El( St)
H ( &)
<5
on
4(5)
5
5
@,
G-
4(r])n4(5) = @. 5 4,
(<
$3.
5 5
3, 5
1, 2 3, 4
5
1, 2. Proof of
when
5 f a l l s under
hyp. on
3,
4(()
f,: [-+r], f 2 :( - 5 g:, g: f = 4 ( f l ( v ) ) ,f T ( v ) = 4 ( f 2 ( v ) ) . by Cod(g:; sc) v < (, f : ( v ) nf2(v)# @, hyp. f l ( v ) = f 2 ( v ) .
[< r] GP(5 = Cod(g:;
gr = g ; , C o d ( g 2 ;Sc), n 4 ( f 2 ( v ) )# @ r] =
r]
Proof of
5 i) =
by
5, r]
n(3,cc,P)~4(1]) n 4(5), hyp., r]
Case 3.
5,
r]<
when
5 c),
<5 .
5 f a l l s under
4 4 , a, p)
Case 4.
4(r])n 4(5)
<5,
hyp., ( < r]
4(()
LY
A E r], B
Hp(S r ] ) = C o d ( T h ( A , r ] ) ;ZV),
H/?(St) = C o d ( T h ( B , S ) ;St).
x,,,x,:
&, St
S V is
(3-4). by
of
x,,
&,
7, Cod(Th(r],A);5 J E C o d ( T h ( 5 , B ) ;St).
&,
SV Th(rl,A) A ) << (5, B ) dyadic second
c Th(5,B),
4(5).
(r],
order
language
5 ,=
2, = (A;'2, ("""2, 5 , ~ )
for
one
ordering
r]
5
YIANNIS
46
m, V1lA
MOSCHOVAKIS
=
5
=
each closed, unbounded A
=
is r e g u l a r } ,
c 5 intersects m,},
n,,
COROLARY 2.1. Let be the dyadic second order language f o r one ordering, the class of its standard models a s aboce. If K is regular and reflects f o r this then K E ~ J I , . Thus if we assume AD, there i s some K < 0 ( ~ 2 such ) that K E ~ X , .
{aA}
{aA}
{(q, 5): q, 5 ordinals}
Proof.
by (q, 5 )
G((? 5 ) )
=
(V,O.
5
5.
G
5
5
A,B
x
K ,
A * B = {(q,O): qEA}
VEB}.
( A* B ) nA)<%(K, A * B ) , B).
K
A nA ) ~ < ( K , A ) ,
u {(q,l):
(A,B nA)<<(K,
5<
{A,,},,
K
K ,
*
{An},<, = { ( v , C ) : V < ~ & ~ E A ~ } *
q <5
A
K ,
(21A,An E K
~
(A,A, n
~
~m,,,
&,A),
A
AEA. on
5
. K
(2) K Proof. (A,A n < < ( K , A ) . “the universe is regular” (3) K E llJz, + , then K 9X, + . Proof. K ,
(A,*{As},,
4<(~ A J,< < (A,A n < ( ( K , A ) ;
“ A is unbounded” A by
LEA,
K
>5
K
<< ( K , * {A,,},,<<),
m,
A,
LEA
K
A
81, n K
(A,( A * DI,)nA) <4 ( K , A * %I,). A nA) << ( K , A ) , EA .
(A,YJl, n
47
<<(K,!IJ~,).
K
~m,+
intersects !IJl,”
“every closed unbounded set
(211,!IJl,n !Ill,+, ,
(91K,Zm,), A
K
KE!Bic+2.
(4)
If
Proof. (A,B n
is limit,
C
KE!IJ~,+,.
KEXR,,
and
A <<(K,
>
K ,
<
B), B = *{!IJl,,r\lK},,<~*A.
(A,A nA) < < ( K , A ) , A E A . Also v] < 5 , (A,!IJl,, nA) <<(K,W,,n K ) , “every closed unbounded subset of K intersects !IJlb” (9lK,93,,n ~ ) , KE%JI, E 9X,,+l, (a1, 9?,,n I.), A E n,,9X,+, = ‘$I,. K !DIE,, K E!IJ~,+,. 55.
R
The effect of structure properties. A =
I‘
o(A) by
THEOREM 3 . Assume A D . Let contain all singletons and be closed under continuous substitution, &, v, 3m,V m and either 3ct or V a . Then o(A) is a cardinal. Moreover, i f is a prewellordeing in A with length is in A . then f o r each A c 5 , C o d ( A ,
s s)
Proof. by
i
TI‘,
5
Q(v],
5
Cod(P;
f:
+K
>0,
A
Field( 5 ) = R . P(V,
3ct, o(A)
=
on
* f(?) 6 f ( 5 ) *f(v]) < f ( 5 ) .
9
s ) , C o d ( Q ; 5)
Zt(s)
by
I’ ct
s’p
e- ( c t , b ) E C o d ( P ;5 )
R
IC ,
4 :R
3)
6,
s ‘ P * f(4W 5 f(4(P))-
48
MOSCHOVAKIS
YIANNIS
6' 1 ( a S ' PI
* f(4(B)))
* ( P , C o ~ c o d ( QS; ) , 5' K
,
5'
i
A
-t 1
A, 5,
5-A .
A $7
AD, Zz
L[R] AD,
3a, V a ;
o(Ai)
3m, Vm,
o(Ai) AD L[R]), odd k
o(Ai)
o(A:) dyadic second order language f o r two orderings
5 ,,S2,=,
o(Ai) k > 0. six
by
1 , ~ %.K
=
A ( 1 9 4
K
2, 2
(AxA)2
(AXK)2,(KXA)2,(KXK) Y
S A ,S K $3 ~(6) m
on 1
2, SA, S K4 ,
7
K
by P:, .. ,P,*,
by
P,?
Xi.
analytic P r , ...,P : . A by P:, ..., P: P , , ..., P , P:, ..., P: is analytic in P , , ..., P,(projective in P , , ..., P , R 5 1, S R, 4 : Field (S + 1, : Field (52) -+K A c1 x K , Cod(A;S
S 2)
= {(a,
p): a
Field( IC x
Cod(A;S Cod(B;S y o
A G 1, B
IC,
1,
,) & p
Field( S
1. jz) = C o d ( A ; 5 ,), 2)
=
Cod@; 5 J ,
2)
& (4(a),$(P))EA}
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
49
L E M M A8. Assume A D . To each formula 8(x,, ..., x,, y,, ..., y,) of the dyadic second order language f o r two orderings with the exhibited individual variables and set variables A,, A,, ... we can assign a formula B*(a,, ..., u,,,P1, ..., p,), analytic in the symbols 5 S 2*, A,*, A: ,... with the following property. I f S 2 , are prewellorderings of subsets of R with lengths I , K and canonical surjections 4 : F i e l d ( 5 ,)++I,$: F i e l d ( S , ) ++ K , i f B,, B,, ... are values f o r the set variables A,, A,, ... in the standard model and i f we interpret S : , S;,A:,A,*, ... b y Cod(B,; S , , g , ) , Cod(B,; 5 5,), ..., respectively, then f o r a,, ..., a,, E Field( 5 ,), p,, ...,p, Field( 5 ,) we have
,,
,,
O*(a,,
...,anrPI,
**-,Bm)
*
3 A . K
Proof by O(x,, ...,x,, y , , ...,y,) = 8 .
C 6 ( 4 ( a l ) ,..-,4tan),+(Pl), * . . ? $ ( P r n ) ) .
on 8*
8
y l = y, (x,, y,) EA ,
pl) E A,,
(a,,
I 3xO(x)
3a[a
S ,,5 ,, (a,p)
8
~ T f i &f12s;p1 , K
5 ,a & 8*(a)].
5
on
s ( a ‘ , p ’ ) o a , a ‘ ~ F i e l d ( S , )& P , P ’ ~ F i e l d ( s , ) & { [ a s ,a’&-(a’S ,a’)] v [ a S ,a’ &a‘ sla&p~ , p ’ ] ) 5
K-I,
f :, Ix
K
x:Field( S ) + K .I
I x
K
by
++ K . A
Field(
s)------x
++ K . 1
(4,$)(a, p) = (4(a),$(p)). I x
B C o d ( f [ B ] ;5 )
=
K ,
{(a,/3):a~Field(~,)&p~Field(S,)&(~(a),$(~))EB}
= C o d ( B ; 51, 5 2 ) .
50
YIANNIS N. MOSCHOVAKIS
5, R x R.
by
c1x
B
K
5 ) is
, C o d ( B ; 5 ,,5 ,) i s Z t ( 5 )
5 5 ,. by
(3-3).
Q(P)
(5-1)
0
(3B c 1 x
5 ) = Cod(B; 5 1, 5 ,)]
K)
5 ,,5 ,, (VY,Y’,696’) [(P, Y,
(5-2)
5 )&Y 5 17’ &Y‘ 5 1 Y &
O*(A*) 3AO(A)
O(A),
S26 =-
E
1x
A
IP[Q*(P) 8~O*(Gp( 5
Q*(P)
K,
9
Q(B).
1
K
1
K
5.
4. Assume A D . Let A be a class sets containing all singletons and closed under continuous substitution, &, 1,3 m , 3u. Then o(A) is a limit cardinal. Moreover, if 5 i s a prewellordering in A with length 5 , thenforeachAc(,Cod(A;s)isinA.
Proof.
A
o(A) A,
5
%A,
A+
A.
A , , A,, ... O(Al,A,)
1x 1
“ A , , A , are wellorderings iland there is some subset 1 x 1 which is a n isomorphism A , onto A , or onto an initial segment A,”.
5 *, A,*, 5)
O*(A:, A:)
by
G( 5 )
Z;(
5,
A . As
Q ( E ) 0 ( 3 A C il x 1)[G,(5)
5,
O(A,, A,) R x R. 8,
= Cod(A;5)]
A.
5 ’ P * Q(a) 8~Q(P) &
5 1, C,( 5
-
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
4‘
51
A+.
A,
5.
THEOREM 5. Assume A D . Let be parametrized, containing all singletons and closed under continuous substitution, &, V, 3 m , ’dm, 3u and V u . T h e n o(A) is a regular limit cardinal (inaccessible). Moreover, if 5 is a prewellordering in A with length 5 and A E 5 , then C o d ( A ; 6 ) is in A .
Proof.
K =
o(A),
K
5. K
6
1 f:1 + K is
> 0.
q,
R
A
G
R xR g(q) = {(Ply):
G, = R x R - G , and G , is a prewellordering
.
of R with length
g(q) # Pr q, g Cod(g*, A , Cod(g*, EA. (a, p, E Cod(g*; 5 ), G f( ~ ( L x ) ) , 4 :R + I 6 ; 4,: R + f (4(a)) g*
s)
(u,P,y,b)
by
3
s)
Zt(s),
R G,.
5 ’ (a’,P,y’.6’) * (u,B,Y)EcOd(G*; 6 ) & (a’, p ’ , y ’ )
4/?(6)6
E Cod(g*;
$0,
6)
(6’).
R4
5‘ $(u, P, 7, 6)) =
5’
by
K,
$/?(4 *
A. y , y‘, ...
x, x‘, ... $(x, y )
x6y, there exists a subset of the cross product of the two domains which is an order preserving bijection of { x ’ : x ’ 6 , x } onto { y r : y ‘ 6 z y } . B * ( a , p , j E 5;)
52
YIANNIS N. MOSCHOVAKIS
ST,5 ;
O ( x , y ) by
(a,p, y, S), (a’, p‘, y ’ ,
5’ ,
4p(@ 5
4p*(S’)
s;,s;)
* 0*(4S‘,
5; =
S2 Gi = R x R - Gyp; 5 I” (5;) R x R-
R xR-
ir, (a,p, y , S), (a’, p’, y’, 6’) 5’
8.
8
Gpt.
( R x R - Gyp)
5 ; (ST), .
4p(S)5 q5a.(S’) (a, p, y), (alp’, y ’ ) ECod(g*;5 ) .
5 ’,
A
K
both 3a Va,
6 . Assume A D . Let be parametrized, containing all singletons and closed under continuous substitution, &, v, 3m, Va, assume Prewellordering Then o(A) is regular. Moreover, if 5 is a prewellordering of the universal set with length K which establishes the prewellordering property, then K = o(A) and f o r each A c K , C o d ( A ; is in
s)
Proof. K
1< K ,
5 o(A),
4 : +K
4(xo) =
(5-3)
{ ( y , z ) :y
5
z & z j xg &
l(Xg
5
z)}
A
5’ rl/: Field( 5 ) + 1 , supremum{f(r]):r] < A} = K .
I
A
f:A + K o(A)
5
K
K
4(x> =
= {x: x E
by
3
(r])}
* Cod(g*;I ‘) C o d ( g * ; 5 ’) ir.
g*
Xi( 5 ‘).
5‘)& y
-s 3 a 3 x [ ( a , x ) ~ C o d ( g * ;
5
ir
5
5 x],
53
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
G
(2-2),
ir,
2.1 A
K,
-
Cod(A; 5) 5,
3a. 1< K , Cod(A;
sA)
sA
5 up
Xt(SA),
-
c 1,
by A E 1, Cod(A; S A )
R xR A EK,
a
A
ir
H
I1 wins
by (5-3).
1,
p,
I1
7ca $ G v [nu E G &
n:R+R x R
(31 >-+(a))[ H , = Cod(A (7 1;
4: G ++K o, n(o* [ ~ ] ) ~ G a n d s u p r e r n u r n { + ( n ( a * [ ~ ] ) ) : ~ ~ R } = ~ , p , q5(n(o*[PI)) < 1, I1 by p H , = Cod(A n &).
I1
G i by
r, t,
xECod(A;
s ) * XEH[,-l,,,r
Cod(A;x )
, 5 unio(A)
formly,
K =
AD. o(A)
by G ,
{a:G , = A }
AD. AE
= {Ga:ct E R }
codes
A.
(2-1)
[X]r
54
YIANNIS N. MOSCHOVAKIS
A E X,
A up
X.
&, v ,
uniformly closed continuous
f:R x R + R Gf(,,,)= { p : { E } (p) . f :R x R + R v
{ E } ~ ; ~
&, x,P,
.
=
G, n
3m
f:R + R
a, = [w x R ] T ,
{ p : 3m(m, p) H a } ,
V m , 3a, Vcr.
LEMMA 9. Let be parametrized b y containing all singletons and uniformly closed under continuous substitution, &, v , 3m, V m , Va, assume Prewellordering via a canonical surjection 4 : j) K . There i s a continuous function R x R + R such that i f A = R and i f f o r each p E A , { E } (p) then f ( E , cr) and (Vp A ) [ 4 ( { ~(p)) } <
< 4 u( E , 4 1 1 * G,
A
uniform version
Proof.
* 3Y[YEA $
&(P,P>
is
5 {E}(Y)I
S
il-
(2-2).
B
ir; g , :R x R + R
A
=
R - G,,
A =R by = fl(c, M)
5
B = R - Gg,(E,a),
- G,
p E A , { E } (p)
gl(E,cr)
B,
y E A , (gl(E,a), g l ( q a)) fl(e, cr)
{E}
S~(E,M)EG. l (fl(E, a) 5. E l (Y),
gl(E, a ) $ B , 1
a) .$
a>
s (4 (Y),
E
G,
y A,
4({E}
(7)) < 4(fl(&, a>>.
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
55
LEMMA10. Let be parametrized b y G , containing all singletons and uniformly closed under continuous substitution, &, v , 3 m , V m , V a , assume Prewellordering via a prewellordering S on G with length IC and canonical surjection 4 : G ++IC.There is a continuous function f z ( a ) such that if H is the universal set f o r [ R x R ] T associated with G via (2-1) and 5’ = H a = R x R - H , is a prewellordering with length then fz(.,P>EG and S +(fz(a,P>>.
A K,
Proof.
5’ = H a = R
gl(a,/3,y),
A,
x R
- H,
5’ y
y
R - Gg,(a,p,y) =
6 2 ’ 7 & -I(Y
S’S)}. E*
f,(g,(x,P,Y),SRxR(&*,4 P > > ,
(5-4)
SRxR
f, (5-5)
=
{E*kP?Y) { E * } (a,
{S
RXR
(E
*, % P ) } ( Y ) .
P,
g, x R - H, I):Field( 5 ’) + . y Field( ’),
S ’ = Ha = R on I)(y) $(y)
S
+({E*}
(a, P, y ) ) :
6
(5-4)
(5-5)
$(a) < $(YL < 4({&*} (a, P, Y))
by
{&*}(a, P, G $(S) < $ ( y ) ,
4({E*} S
9
(a, P, 6 ) ) < 4({E*}(rn,P,Y)), 4({&*}(a, P, Y)) *
6 I)@) <
by
5‘
g,:R x R + R R-
fz(% PI
= Ha = R x R - HB,
GgL(a,p)
=
{Y:Y S ’ Y }
= f i ( g Z ( 4 P), S R X (&*, R a, PI) *
THEOREM 7. Let be parametrized b y G , containing all singletons and uniformly closed under continuous substitution, &, v , 3 m , Vm, Va,
56
YIANNIS
assume Prewellordering T h e n K = o(A). Proof.
K
MOSCHOVAKIS
via a prewellordering 5 on G with length
K.
S o(A) 5.
56. Projective ordinals.
o(A) 5
K
k >0, 6: = o(A:),
8. Assume Determinacy ( A i m ) , DC. Let 5 be a prewellordering of of a universal lT:,+, set which has length K and establishes Prewell1 ordering (ll;,,,+,); then K = 62m+l. Assume A D . Each 6: is a cardinal. If 5 is a projective prewellordering with length 5 and A G 5, then C o d ( A ; 5 ) is a projective set, hence A is dejinable ( i n set theory) f r o m a real parameter. I f 5 < 6; and A = with each A , , E E ~ ,then A E E : . A set A is in El if and only if it is the union of Kl Borel sets. Assume A D , D C . Each 6im+l is a regular cardinal. If 5 is a prewellordering of a universal IIfm+, set which has length 6f,+l and establishes Prewellordering then f o r each A c 6i,,,+,, C o d ( A ; 5 ) is II;,+,. (For m = 0 this is due to Solouay.) For each m 2 1, 6;,,,-, < 6f, 5 6:,+,. For each m Z 1, (6:,< 6:,+, . For each k, K: 5 6:.
U,,<&,
7
Proof.
$2.
6
3,
El
K,
6
$2. $2
6 6:,,,-1
ll:m-l,
A:,
57
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
SZrn-'
Proof of (8.3.4).
on GZm-', Prewellordering (IIi,- 1) by (IV) Cod(A ; 5 " " - l ) II;,,,- 1 , A on 62,- 1 . l-Ii,,,-l,
1 nzm,
6, by
$2.
A
HZm-l a E WZrn-' o H:"'-'
= Cod(A; 5)f o r some wellordering A of 6iml
G2"'+'
Hi,,,.
+2m+1:G2m+1 + 6:m+1
prewell-
I:,
ordering (IIj,,,+ l).
Aim+
10,
f(a) a E W Z m - ' + f ( a ) E GZm+l
H:m-l = C o d ( A ; S Z m - ' ) A +'"'+'(f(a)>. 9 1 + A, S q, (6zm-l) S q .c S:,+l 6; = 6: = K1. AD DC 6: = K: for
tAS
1
6,,
=
.
1
=
K:,+,
tA, q<
k 2 1
m. AD odd k
8:
DC
k.
K,
K,
AD
DC
AD. 61m+
ti;,,,.
6&+l
0 .)
k 2 1,
$7. A:.
6," = o(A,"). 6:
Et,I , :
At.
1 &
Z: v
II:
. How-
3m Vm,3a, Va
F: R +w FJB) = W ( a , B)),
58
is
z: R x R + R
A E R x Ru
Va3F(cr,F ) E A
(7-1)
* 3FVa(a, F,) E A
Ei, nz
(7-1) do
3m, V m , la,Va. AD;
(7-1)
collection L[R], Collection. For each A
G
R x
Va3F(a, F ) E A
"0,
*
3FVa3/?(a,F,)
E
A.
LEMMA11. Let M be a class which is a model of Z F (without choice), which contains R and such that some surjection f : O N x R -+M is definable in M with parameters f r o m M ; then M satisfies Collection. I n particular L[R] and the class of sets hereditarily ordinal definable f r o m real numbers satisfy Collection. Proof.
V
=
M
Va3F(a, F ) E A , E Rm &
v(a) = infimum
G(% 79
=
A
( v ( 4 , Y)
if
cR E
x Ru.
4 1,
Y) E Ru,
f
= 0
F(&) = G(a,y,G) a, y
a,
/?,y , 6
n(/?,6) = E z(a,y) = 8 . ( ~ ( a )y ,) E Rw (a,f ( ~ ( a )7)), E A ; /? = n(a,y ) ,
by F,(@ = F(n(/?,
F,
= (~(a),
,
(a, F,)
E
=
f Ma), ?)(a 7
A.
THEOREM 9. Assume A D and Collection. Then each 6: i s a regular limit cardinal (inaccessible). Moreover, if 5 is a A; prewellordering with length 5 and A 5 5 , then Cod(A; 6 ) is A:, hence A is definable (in set theory) f r o m a real parameter.+ o(R2) o(R2)
do o(R2) 11.
o(R2) o(R2)
L[R]
59
5.
Proof
9,
6:
is
DC.
6:
THEOREM 10. Assume AC, DC, Collection. Let L be the dyadic second order language f o r an ordering,for each 1 let 2, be the standard model of L as in $4. Then thefirst regular K which rejects f o r this L, is less than 612.7
{aA}
Proof.
71: R
F
WF
x R
0
-,R
{(a,P): F(n(cl
DC
p))
=
F
O } is a prewellordering.
W F is
analytic
by
$F:
Rm. F WF, Field(5,)
SF -+ 1,
by F , L, 2
Q(F)
(7-2)
FEWF&& =
*F
R(F,
(7-3)
0
=>
{a,},
AF. $ : ~ + ~ 2
K ,
W F & VclVp [m E Field ( 5F)
[G(n@,P)>
=
0
R(F,
Q(F)
*P A; ,
by Z;
SK
do by
2 ~1
sKB*
3F3G[R(F,
& 3 y 3 6 [ F ( ~ ( y , 6 ) )= 0
& G(n(y,CO) = 0 0
G(n(6,B)) = 011
V F V G [ Q ( F ) & R(F, G) * 3y36[F(n(y,6)) = 0 & G(n(7,a)) = 0 & G(x(6,P)) = 0 ] ] .
Q ( F ) , R(F,G) At
6:.
by
AD)
60
MOSCHOVAKIS
YIANNIS
SUBLEMMA 10.1.
Put
P , ( n , m , F , G , ) 0 [ n is the Godel number of a formula O(S*,A*,uI, . . . , u r n )analytic in the binary symbol and the unary symbol A* and with m f r e e variables] & S *, A * , ( U ) .~. .,, (cl),) is true when we interpret 5* b y S F and A* b y { P : G ( p ) = O } ] .
s*
T h e n Pl(n,m,F,G,cc) is a At relation. by
Proof
(SF),
SUBLEMMA 10.2. Let H ( F ) be the canonical universal set f o r PU t P 2 ( F , f i ) 0 f o r some B c I F , H p ( F ) = Cod(B; SF). T h e n P,(F,P) i s analytic. Proof. H ( F )
P,(F, p)
F
0
vcc[u E Hp(F) => CL E Field ( S ,)] & V c t V y [ c c ~ H ~ ( F ) &~ a F &Y y
YEH~(F)].
SUBLEMMA 10.3. Put P 3 ( n , m , F fi,cc) B(x,,
0
[ n is the Godel number of a formula
...,x,, A ) of the dyadic second order language
f o r one ordering with m f r e e individual variables and one f r e e set variable] & F E W F & Vi[l & [‘u,, k O ( $ F ( ( ~ ) , ) ,
S i5m
( ~ ~ ) ~ s F i e( SZFd) ] & P 2 ( F , P )
...,~ j ~ ( ( x ) , ) , B ) ,when
B is such that
ffp(F) = C o d ( B ; S F ) ] .
T h e n P,(n, m, F , f i , a ) i s A;.
8.
10.1, 10.2
Proof SUBLEMMA 10.4. Put
P,(F,
p, y )
( 5F) & P,(F, 8) B G A, is such that H&F)
o F E W F & y E Field
& =
then Then P,(F,P, u ) is A;.
C o d ( B ; &),
($F(y)
7
$F(Y))
(lF,
B)l *
DETRMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
Proof.
61
P,(F,P,y)
VnVmVa{[Vi[l S i 5 m
=>
[(a)i S F y & l Y s F ( a ) i ] ] & P3(n,m,FY,flY,tI)] m, F , P, a>>
=>
7
sFv sF),
FY,PY
H,(F)
y
=
SF
Cod(B;
H,y(FY)
=
5FY) .
Cod(B n
SUBLEMMA 10.5. The condition Q ( F ) defined b y (7-2) is A:. Proof.
P,(F) o F
W F & ,IF is regular & VflVY[P,(F, P ) & y
3Y“Y
Field
F)
SFY’
“AF is regular” A:, n,,p,,a,,
P3(no,B,F , Po, ao)
(s
Q(F) o P , ( F ) & V y [ y E Field (&)
4: K -+ R2 5
P,(F)
At.
1P5(FY)].
by
on
5
A;, A: by F ) .
by
10 A:
there i s some
K ,
5 5 K < S:,
6:
such that
K
f o r each 5 < S:, reflects with respect to L, {aA).
mK.
K
9 A:,
n >2.
E:, Ill: L[R];
3m,Vm, la,Va Sl: = o(Ai)
(n 2 2, k 2 REFERENCES Separation principles in the hierarchies of classical and effective [l] J. W. ADDISON, descriptive set theory, Fund. Math., vol. 46 (1959a), pp. 123-135. [21 , Some consequences of the axiom of constrirctibility, Fund. Math. 46 (1959b), pp. 337-357.
62
141
161 171 [8]
191
Some consequences of the axiom of definable determinateness, 59 pp. D. Infinite games with perfect information, 28 pp. The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Ann. 3, 1940. Introduction to metamathematics, 1952. , Countable functionals, 1955. The axiom of determinateness and reduction principles in the analytical hierarchy, 74 pp. Hyperanalytic predicates, 129 pp.
--,
Abstract first order computability. 11,
138
pp.
On the axiom of determinateness, 53 pp. JR., Theory of recursive functions and effective computability,
1121
Y. 70
1967. complete classification of the A; functions, Bull.
pp.
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABLLITY PART I: A SURVEY C. E. M. YATES The University, Manchester, England
on
on
do
on
by
[9];
by
to
63
[lS],
C. E. M. YATES
64
no
O(l)
$12,
1
Q. 31.
s
O(l)
O(l);
[4], O(l!
Part I :
$1. $2. §3. $4. $5. $6. $7.
A survey D.
D. L)
( S @I)). D ( 5 a)
. D.
a.
Part 11: Minimal degrees. $8.
A $10. A $11. A $9.
O(2). O(').
Part 111: Finite Boolean algebras and degrees with no minimal predecessors.
$12. $13. $14. $15. A $16. A
$17. $18.
D D no OC1)
( sO ' 2 ) ) . ( s0'"). no
Part IV: Distributive lattices. D ( 5O'2)). D ( 50'").
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABILITY
$19. $20.
65
D.
5 O(I) V
D ( 5 0“)).
VI
$1. Introduction to D
by no book [ S ] on do fo,fl,
. a ’ ,
(0,
N
N
N.
[13] string S
0 no n+ n by a* 1
by
length by o(n)
+1
by
.
*z a*O a* @
z
by
=
by ‘ n ’ .
n
c =
a(n) f
n
alz,
a’)
X c
G
n
*z
z f !2.
a(n) #
N
n,
= X[n]
X
X[n] n 1.
+
S; This elegant approach is due essentially the Shoefield [13].
C . E. M. YATES
66
on S
on N ,
S
do
N.
X(n)
N. Y E.N ,
X
Y,
on n .
B,,
DEFINITION 1.1. S. F F
F is an S-map order-preserving F(o)
F(z)
B,
z
z.
B
F
X cN,
F*(X) Fo,FI;.., F:, F:,
R.H.S.
, X
DEFINITION 1.2.
recursive in Y , F
X,
N,
X = F*(Y). X jTY
X is
X Y
B
X
=
Y.
Y,
B
X. Y
X
sTY
ST
sT
Y
sTX ; 5
s
zT
D D.t
We use D( a), D( 2 a), D( < a) and D( >a) as abbreviations for the degrees of unsolvability s a , 2 a, < a and > a, respectively, for any (I 5 D.
INITIAL SEGMENTS OF THE DEGREES
UNSOLVABILITY
67
D b a
b
a.
6 S
D)
D D D
0,
1.u.b.
D a
a
0 by
by (As
g.1.b. 0.
do
D
aeD a < a('); a ( ' )
not
a)
jump
a
6 a(').
a
no
5 O('). n
a(") a
D no
D
!
on on D : no
on [ll].)
p.0.
D
A more direct proof was verbally communicated to the writer by Professor Solovay at the Colloquium.
C. E. M. YATES
68
on
5)
1.3. initial segment
5’)
p.0.
S’
E
x
p.0.
(D, 5)
5y
yE
xE
topped topless. x
E
closed
.
y 5
S
x, y e 1
5
5
D
D D D
5 ’)
(p, S ’ )
p.0.
1.4.
embeddable in
S),
p.0.
S),
3
5)
( p , S ‘)
bottomed,
-C
5’) Z-
segment of’ ( D , S ) ,
S ’ ) embeddable as an initial S),
5)
5’)
1).
1.4 by 5 ,
S’)
[lo, $31
S K1 K,
p.0.
S’) by 2 ” O
512
[lo]),
by by
S-isomorphism
1.5. 0 0, z
ES
T
.
cz
t)
F(a) E F(z) S.
tree G
subtree
69
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABILITY
F* (F*)-l
F* X
F*(X)
1.6.
splits a string (r f o r e Fe
2
e, splitting tree f o r e e,
T(z)
T(z*
z.
[13].
p
Let T be a recursive tree. I f in such that no two strings in is recursive.
split /?
is a substring of
f o r e , then
Let then
be a recursive splitting tree f o r e . If and has the same degree of unsolvability as Fe*(Y). Fe
There are 2N minimal degrees.
1.7 S} E
S; * i (4) no c
i
c
T,*o, T, , T, *
...
T, no
c*i(4)
=
*i
on n , T,,
3
T,, 3
.*. 2N 2N do
on
1.8. If ao,al,... , is a sequence of nonzero degrees then there is a continuum of mutually incomparable degrees, each of which is incomparable with every degree a,.
70
C. E. M. YATES
b
a.
a >0
1.9. If a a n y nonzero degree then there are 2" degrees b such that the g. 1. b. o f a and b exists and is 0 . by
$51 1.7 1.10. There are 2" degrees b such that b is not the join of two incomparable degrees. 1.7,
d< b
b
d
d no
c.
cb x
not
c
b = x(')
d = O(l)
c
5 d";
b d.) A
2
"
as p.0.
$2.
Finite initial segments of D .
D
(P,5')
D. (P,5') (L, 5 )
D ( P , 5') S ( L ,
Sacks' conjecture:
s)?
5)
p.0. p.0. 912,
(L, 5 ) ; D .
61
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABILITY
71
2.1 (Spector-Titgemeyer-Sacks-Shoenfield-Lachlan). Every Jinite distributive lattice is 2 D .
1956
D.
1962
by
is
D;
no
$111
on
by no no
53 §19. good
on
by
by
>D.
by 1
1
0 (a) Fig. 1 Some further generalisationshave more recently been obtained by S. K. Thomason (Sublattices and initial segments of the degrees of unsolvability, to appear.)
12
C . E. M. YATES
n
a , b, c b
5
b
c
c.
[5]
S
.
[5]
+
2.2 n = p k 1f o r some prime number p then the n-th Chinese Lantern is 2- D . n = pk + 1 p n on n = 3.
no
n = 7,
not 2. by
T h e Pentagon i s 2 D .
2.3
111
$13
2by
$14
by
( S O(I))-
5 O(l). IV.
do
( S O(’)), 53. Countable initial segments of D .
_I ’)
p.0. p.0.
5 ’) % D ( L , 5)
S ’) 2 (L, S ) ?
73
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABILITY
$2.
LACHLAN’S CONJECTURE:(L, S ) is a countable bottomed upper semi% D. lattice then (L,
s)
do
no
do
SUBSIDIARY CONJECTURE: If (L, 6 ) is a bottomed countable lattice then ( L , 5) is 5 D .
2 C
a0
Fig. 2
b , u b2 = c
.
g.1.b.
a,, a , , b, , b2, b, b2
,
2 D;
do
THEOREM 3.1 distributive lattice then (L, S )
If (L, 5) is a bottomed countable
: D.
C . E. M. YATES
74
by
3.2. If ( L ,
s ) is a
[3]
bottomed countable chain then (L,
s)
D.
is
3.2 3.3. If a is any countable ordinal (with the usual ordering) then a 5 D .
3.4. If ( B ,
s )is a countable Boolean algebra
then (B, 5 )
D. by [lS]. by D ( 5O ( ' ) ) .
D
3.5. There is a nonzero degree which has no minimal predecessor.
(1, - o)
Proof. 3. b0 i b, b2;
7
... ... 0. Fig. 3.
bo, b , , ... .
0 3.2 b,, b,, ... ,
D. no bi is rn
[lo, $12, Q. 31. by by
[6].
d {b:(3a)(u 5 b & U E d)}
INITIAL SEGMENTS O F THE DEGREES OF UNSOLVABILITY
75
no
on
can $1
A is:
a. < a 1 <
a. < a 1 < ...
COROLLARY 3.6. There is a n ascending sequence of degrees a , < a l < ... which has a n upper bound b with no minimal upper bound 5 b . Proof.
(0, -0)
4 b e
62
.
.. ..
. . ...
. .. ..
.
a2.
I I 0 . a1
Fig. 4.
b > 0 , > 6, >
3.2 bound s ’ b .
2 D.
0 < a < a , < ... . a , < u2 < ...
no
do by
b O(l).
3.1 [4].
by
EDU
A?<: This question was proposed by
Joan Moschovakis.
EDL
76
C.
YATES
S . EDU ( D , 2 ).
9, -
3.7. EDL is recursive in EDU, and hence EDU is not axiomat isable.
Proof.
9$.
(s i (EDL
t (s)
++
(EDL U { (s} i
by by by i (EDL
t @)
“ ( V x ) ( 3 y ) ( x < y)”
t--f
(3a)(D(> 0) nD(< a ) t;
D ( > 0) C D ( < a ) “ ( V x ) ( 3 y ) ( y < x)”
i (EDL
t-
(s)
++
i ‘3); i (s
EDU t ( 3 x ) @ ’ ( x ) ,
by
W(x) @’
x.
i (s
3.1
EDL
(s,
EDU. EDL
[2]
EDU
EDU
IV
D ( 5 0‘”). $2
D.
odd
$4.
Initial segments of D ( 5 O(’))
As
5 O(’);
no
INITIAL SEGMENTS OF THE DEGREES OF UNSOLVABILITY
by
< O(')
77
[12]
no
0.
[19]
< 0") 0
O(').)
$2, D(
O(l').
D ( 5 0'")
(As
by D ( < 0'") .)
; D ( S O(')). [lo, $41
p.0. 5 ,
p.0.
a D( S
0")).
5 SB
S)
p.0.
DEFINITION 4.1.
SB) by
S
B
SA
a.
sB [14]
5. (u,,, a,,, ')
nE
S
n$S
SA) S(A)
S(A) A(').
S a(')
~'i This was used in the work of Kripke, Martin, Sacks and later Jensen on the consistency a A' non-constructible set of integers.
C.
78
YATES
u4
u3
a2
b*
c2 a1
00
Fig. 5.
sa.
$3.
5 0":
C.2.
2 D(5
5 O(l)
C.3.
2 D( 5 0'") C.2 by
50'''
p.0.
no p.0.
D ( 5 0")) .
O(l)
5 O(4';
C.3 by 0(4)
on C.2 4.1. C.2'
C.2' p.0. C.3, C.2
C.3'
C.3
C.3'.
IV 4.2. Every bottomed distributive lattice of degree
2 D ( 5 O(1)).
5 0"' is 0")
L. Feiner has recently proved that there is a linear ordering of degree 5 Q's' which is not of degree 5 Q ( 4 ) . It easily follows from his result and a trivial generalisation of Theorem 4.2 that D ( 5Q ' 5 ) ) is not isomorphic to D ( 5Q"'). (If these structures are extended to take into account the jump operation then the corresponding weaker negative result is course trivial.)
79
by 0(4)
0")
5 0(4)
C.4.
5 D ( 5 O(')).
by 4.3. There
a nonzero degree
predecessor.
3.5, 4.1 (1,
3.1 3.5
-0)
EDU ( 5 0")) O('),
s
$3.
EDU
( s0"))
4.4. EDL is recursive in EDU ( 5 O ( ' ) ) , and hence EDU is not axiomatisable.
3.7,
5 O(') do
D ( 2 O('))
on $5.
Initial segments of D ( 5 a ) for arbitrary recursively enumerable a
by
$12,
1.1
[20].
do A n y bottomed distributive lattice of degree f o r every nonzero r.e. degree a .
D ( 5a ) a = O(').
a
s a is 2 D( 5 a)
80
$6. Uncountable initial segments of D
D
D M
0;
2'".
by 2'"
D;
by by
$31
3.
2'".
D
K,
D
K,.
D
K,
D?
K,
=>
D;
D
on 6.1. I f on
" K , 2 D".
of
no
D
K,
(P, 5 ) (P, 5 ) (P, 5 ) (P, 6 )
_-+
by
D
.
D.
K,
81
N,
(P, 2 ) (P, 6 ) S D . ( P , S) (P, 5 )
K,
D.
If ( P , S )
(P,
fs,
s)
D. U.3
U.l
do. K,
by
U.l
> 57.
by
a.
Some general problems of isomorphism and indiscernibility
on
4.2.
( 2a)
D ( 2 b)
( 2a )
D ( 2 b)
a
b?
conjecture)
5)
5) ( I , S)
5 ’) 5’) D
f
D . Unfortunately,
f’ ( I r ,5 ’) this conjeclure false. (I’, S ’ ) is O(’)
f‘ a
M. YATES
C.
82
(I, S )
D(
I’
a
’)
s 0“’)
> O(l).
a
1.u.b.
a
(I’,
D ( s 0“)) by
a,
not
a2
2 O(l)
$51). by f ’
D.
no
4.2. Q.l
by do
M = {X
c D : (3a)(Vb)(b2 a
-P
bEX)] X
X
D -X
M
M.
7.1 T h e Axiom of Projective Determinateness implies that there is a degree a such that D ( 2 a ) and D( 2 b) are indiscernible all b 2 a . T
Proof. by
@E
T
-
5?< (3a)(Vb)(b2 u @
+
$3)
D ( L b) k @).
5?< -
{ b : D ( z b) k @}
Addison-Moschovakis [l] actually formulates a more general axiom but it is the the axiom above which plays the crucial role.
83
{ b : D ( L b) k
A4
i (D}
M
{b:(V(D)((DeT * D ( 2 b ) I= a)} a
M.
9, -
D ( 2 b)
b 2a ,
for
w
T.
of
Q.2 by is
of
of and Y. Some consequences of the axiom of definable determinateness, 59 Undecidability of some topological theories, 38 Initial segments of Turing degrees, of unsolvability,
fur
19
Distributive initial segments of the degrees und 14
Some non-distributive lattices as initial segments of the degrees of unsolvability, 34 D. Category, measure and the degrees of unsolvability D. The axiom of determinateness and reduction principles in the 74 analytical hierarchy, Theory of recursive functions and eflective computability G. 416419.
minimal degree less than O"), Degrees of unsolvability,
67
no. 55
of
Measure-theoretic uniformity in recursion theory and set theory, Foundations of Mathematics On the degrees of unsolvability, of 69 theorem on minimal degrees, 31
On recursive well-orderings, On degrees of recursive unsolvability,
20
of
theorem on initial segments of degrees, Untersurhungeniiber die Struktur des Kleene-PostschenHalbverbandes der Grade der rekursiven Udvsbarkeit, zu 1962. Density and incomparability in the degrees less than O"), 31 Recursively enumerable degrees and the dcgrees less than O"), Set, Models and Recursion Theory Initial segments of the degrees of unsolvability, Minimal degrees,
SOME APPLICATIONS OF ALMOST DISJOINT SETS R. B. JENSEN
R. M. SOLOVAY*
AND
Seminar fur Logik, Bonn
University of California, Berkeley.
Z$
[7],
1. Introduction. (I).
If of
[9]).
At
A:.
O‘,
lTi
A:.
a,
A:
[3]
lli uses
of o $2.
93,
+ V = L.
M M. M[x] p [1 x J
* (1)
6 x,
=
6.
by sets, see (21.
(2)
84
85
SOME APPLICATIONS OF ALMOST DISJOINT SETS
6
M MCxI$4,
$5,
A:
by
$4.
2. Almost disjoint sets
2.1. 01
:
Let M be a countable transitive model of ZFC. We suppose that K f = K, , in M . Let A E K, , A E M ( 3 ) . Then there is a Cohen extension of M , M [ x ] , obtained b y adjoining the sei of integers x such that A is constructible from x and M and M [ x ] have the same cardinals.
[4]. x {Q,( ), u < K,}
2.2.
is
A.
x
do R(x, y ) : x n y
As y w , R( , y )
Ec, R( ,y )
2.3. R( ,z ) A
y
z
R( , y ) y
y
z
z
almost disjoint.
R( , z )
y nz
us.
There is a pairwise almost disjoint family, 9, of subsets of w of power (3) We frequently use set-theoretical concepts relative to a model without explicit mention. Here for example, A is, in M , a set of countable ordinals.
R. B. JENSEN AND R. M. SOLOVAY
86
{si}
Proof.
w
. j
S(f)
w,
{j:sj
S(j’) w
f , g S(f ) n S ( g ) , s j S(g)
f ( n ) # g(n).
w.
5 n.
if S(f )
S(,f) n S ( g )
% = { S ( j ) : f.w“)
w
2.4. 2.1.
Let be a countable transitive model of Let Y E be a subsei of P ( o ) ( ~ ) .Then there is a subset, x , of o such t h a t : 1 . M [ x ] is a model of with the same cardinals a s M . 2. I f y E Y , y n x is finite. 3. Suppose y E P ( ~ )and ~ ,y nx isJinite. Then y E y , U ... U y , u F where y i E Y f o r i = 1,.. ., n , and F is finite. 3 2.)
by 2.5. on
y
M , y nx
2.4 [6]. :
1.
on P
Y
p
>= p’
p
2.
p’.
on P
G [6].
P
Y. 1)
s
P,
(s,t), t.
p
t
w,
s
2 (s‘,i‘) snA =s’nA,
(s,t)
cs’; 2) t s t ’ ; 3) on 1) s E x
P,
p
An
P,, t)
x. 2 ) x nA
G
x
A E ~ .
=
M c
s nA , on P,.
=
{n13(s,f)EG
p A
=
nEs]
EM[G].
M[x] [6] (4)
10.2
P ( w ) is the power set of w , and o is the set
non-negative integers.
87
1. P , satisfies the XI-chain condition. p1 = ( s , t , )
p , = (s,?,) p, p, ( p , t , u t,).
Proof.
X
P,
M
P,,
X X
2.
Let A E Y . Then x n A is jinite. (s,t)~G
Proof. x n A =snA.
AE?.
3. Let A E M . I f x n A is jinite, then lhere is a jinile set s and a finile subset 1 of Y with A E s u
(Ut).
A nx
Proof.
“A n x
t)E
=
=
F”.
A n E A - (F u s u u t ) . t) “ n E X nA
(s, t ) 5 (s’, t ) , so .) 1
s‘ = { n } U s . x n A E F”,
2.4.
3
2.6.
2.1.
{f,,a < XI} x
w,
0)
y , = S( f,) w by 2.4. 2.4 and the fact that {y,,a < K,} A = {a y , nx 2.1
Y
=
{y,: a E A ) . 3)
1
3. Cardinal collapsing with reals 3.1.
4
6,
f:a+ 6,
a < 6,
regular
p < 6,
y+
5
(f) b . y.
weakly inaccessible.
is strongly inaccessible, A < 6, 6.
y
y+
6,
R. B. JENSEN AND R. M. SOLOVAY
88
6.A {f ( 4 ) : <
a
f :a + 6,
f
=
3,
f :6 +6
normal 6, weakly Mahlo
a. A
A
A
f :6 + 6
=f
(0,
5< 3.2.
$1, Let M be a countable transitive model of Let 6 be an uncountable regular cardinal in M . Then there is a real x such that: 1 ) M [ x J is a model of 2 ) K;["'= 3 ) i f 6 is either not weakly Mahlo or is strongly inaccessible, then every cardinal A of M , greater than 6 , remains a cardinal in M [ x ] . 1)
M,
3)
$1.
6
2)
,
6=
f:
+y
1.121. w
y
x
.
f
M[f] M
M[x] .
6
6
=
M[x],
y
K';""], 6 6
3.3.
M,
by
6 A. 1.33. There is a set A E 6 with the following properties: 1) M [ A ] 2 ) NyrA1= 6 ; 3) f o r a > 6 , a is a cardinal of M is a model of if and only if a is a cardinal of M [ A ] . f,: w 3.4.
a < 6,
by
A +
2)
a.
M.
6 M,
f: a
6,
< 6.
E
M,
L[E] G M , 6,a
M
by
3)
a
6 f(a) a
L[E] .
89
SOME APPLICATIONS OF ALMOST DISJOINT SETS
{
a,
5
o
R,
a
=
at
,
(2'3': ( i J ) e R } . M[A].
on
5,
by
5
f(5).
at
M[A].
6. N
=
(a,, a
<
L[E, (a,, a < A ) ] .
N N.
N. N,
f(A)
f(A) =
Ia <
<
a
L[E]
f(A)
L[E]. f ( A ) < Ny.
M.
N , sof(a)
a,
5 Ny. f ( A ) = ,:N
al N.
3.5.
$2, x
f :R
-+
1) f(4)= a,; 2) f ( a , )
R (a,
Ia<
3) x
= a,,,;
6).
y E o ,n E O . 1) f ( n , y ) : w - t w ; 2 ) f ( n , y ) ( O ) = n ; 3 ) f ( n , y ) ( k 0 R(x, = { n S ( f ( n , y ) ) nx $2,
I
f(n,y)
+ 1) = 1
k ~ y ,
There is a real y such that 1) M [ A , x ] is a model of ZFC with the same cardinals as M [ A ] ; 2 ) R ( x , 4 ) = a , ; 3 ) If 5 < 6 , R(x,at) = a,+1. 2.6. Lemma. The sequence ( a , (E, x> *
I 5 < 6) is constructible f r o m the pair
L[E,x].
Proof.
a, by 2 ) at+ = R(x, a , ) . 3 ) If
on
5 by
y
y = f(A).)
( E , (a,, a <
1) a.
(a,
a,
y
Ia <
=
I
R(x,O);
{a, a
<
R.
90
JENSEN AND R. M. SOLOVAY
6 < 6, 6 = KyCx1.
M[A, x] a, E M[x] ,
u
8,
M[x]. u
u
M[x].
M [ A , x]
u
M[x] .
M
6 by
3.7.
Let M be a countable transitive model of 6 a weakly Mahlo cardinal. Then there is a m a p f :6 + 6 such that 1. M [ f ] is a model of 2. 6 is weakly inaccessible in M [ f ] . 3. If 6 is strongly inaccessible in M , M and M [ f ] have the same card ina 1s. 4. T h e m a p f i s normal. 5 . If u < 6 ,then f ( u ) is singular in M .
3.2. 6
A E 6 by x
by
f
6 3.3 f (u)
6 = K, M
M[f], M[f,A]. a
3.7.
3.8.
< 6. f P
hEP
u
<6
6.
h h
P
2) 3)
h(P)
u
{ h(P): P < a>
h‘:a
3)
+ 1+6
1)
h 2).)
no
P by Let h E P . Let h‘(5) is dejined.
Proof.
u =
5 < 6 . Then u u
+1
h has an extension h‘ such that h by h(u) =
P < a}).
SOME APPLICATIONS OF ALMOST DISJOINT SETS
So
91
a = $’
a
P.
+1
1= h’: 5
+ 1 + 6 by h’(Y) = h(y)
h’(y)
I.
=
5 P,
y
+y
P < y 5 <. fi < y 5 5 , h’(y)
h’
f
on P .
=
f a
< 6,f ( a )
u
. 6,
6
M. M[f]
f
6
3.9. M[f].
1. Let a be a limit ordinal. Suppose that h , E P , that a h , , and that {hc, < }. is a strictly increasing chain of conditions. Then lies in P .
UC<&
<
h = U5
Proof.
5
6
y
< 6.
h(<)
<
5
h.
=
. = a
YI
*
h P.
y1 is
2. Let a < 6 . Let g : a - + M , g E M [ f ] . Then g E M . g
Proof.
g.
h
a
h’ 2 h ,
g:u + M . 1,
h” 2 h’
h”E G
g. g M.
g,
10.6 2 M[f].
M
hEG h.
6
6 M[f].
10.2 3.7
6
a
< 6 ,a
M[f].
is
[6].) 3.2
2
M[f].
56 6, no
P M[f]. 3.10.
[6].)
M
6
6
3.7
6 3.2
92
SOLOVAY
4. A non-constructible A\ real : consistency
4.1. $5,
$1.
$1.
+
Let M be a countable transitive model of V = L. Then there is a transitive model N of ZFC such that: predicate, S ( x ) , which will be described below, 1) For a certain we have N k ( 3 ! x c w ) S ( x ).
Let U E N be the unique subset of o such that N k S ( a ) . Then a $ M , and N = M [ a ] . 3) a is A:. (This is clear since a is a singleton b y 1) and
4.2.
N.
on
T,(x),
lli
ai,
1) M [ ( a i , i ~ c o ) ]1 T,(ai). 2) EO. af
i #j,
= ai
M[(a:, iEw)] k
i
a3 = 0 .
(3x)Tj(x).
(9.
(3x) Tj(x)
N a,
M [ ( a i , i w ) ] . For N = M [ { ( i , a i ) :i E a } ] .
by
[5].)
K.
4.3.
Ti. om
{fa,i:cr
< K,, < w }
T,(x)
M. (Vcr) (x n S( fa,j)
f+
j = i)
.
T’s do
&’s look fa,i’s
M. 4.4.
for (5)
{
.
by
93
SOME APPLICATIONS OF ALMOST DISJOINT SETS
M.
T
1) 2) 3)
v
=
L.
I
I
LA L,, I= T .
L.
V
T
M',
{x} .
t
x
= L,
x E M'.
M'
.
K, x No
4.5.
K, .
N, x KO.
(cI,,~,)
1) A A
M,, o.
o
fa
CI
T.
<5.
T
M,
<
{(cr,fa):cr
T M;
V =L
by P f:o + o .
T
P t
s
s.
t M,,
#
,
faEMc.
f(ac,nc)
K(x)
LEMMA.T h e predicate T,(x) is
94.3.
(in the iwo variables i and x).
T'
Proof. by
5,5 {faqi}
T',
T.T'
M'
fa,i
a
T
L,, k T ' , T'. T '. M'
T.
is
M, s,,~
f.
c5,
by
My
5,
on
CI
.
t L,, t E M".
t L,, M".) T
M"
on o by o R, on o , T"' II:
M. "(o; R,,) a.
94
R. B. JENSEN AND R . M. SOLOVAY
T,(x)
by
T‘,
ns(ffj’
)
e,
j = i).
T’, ( v a )(Sl(a>
S,
S,(a, x, i ) )
+
A;.
S,
4.6.
{(ai,i
< a)}.
by i
< o.
Qi
Qi
T,(ai).
ai
t)
o
s
t
a < X,}.
on Qi by
$2.2. P
Qi’s.
f
o i.
. f ( i ) Qi P by
i,
f(i)
=
P ($,$)
S
i,
S f2(i). LEMMA1. P satisjies the K,-chain condition. EP
Proof.
,
w by
n,(f)
n,(f)(n)
P
f(n). n,( f)
f) = 1
g
$2.2.)
G , on P . a i = (n
w
I
=
( s , t ) A n s)> . $2.
LEMMA2. Let N ,
=
M [ ( a i , i E w ) ] . Then f o r all i , N , k 7;:(ai). T,.
LEMMA3. Let N ’ be a model ZFC with M c N ’ c N , . Let x N ’ . Then N , k iff T,(x). (This result is a special case a general absoluteness result f o r statements due to Shoenjield [7].)
95
4.7.
$81).
T
P,,P, P , x P , by
( p , , p,)
5 (pi,pi)
5 pi
p,
p2
on
5 p;.
x P,
on PI
x
on on n , on P , x ... x P , . M [ G , , . .. ,
... x
x
i , Gi
on P i . x
P
on P”.
{ 1, ...,
7c
P”,
... x .
II
Gnfl, x ... x G,(,, = ,
9.11)
on P”.
7c
M [ G , ,..., G j - , , G j + , , ...,
Gj
by
4.8.
N,. N’ = M [ ( a f , i < w ) ] .
#j .
a; = a,
a; =
aj
4.
<w)
(a;,
a:.)
Rj
P
=
P
=
Qj x R j
Gj on R j .
so Gj
=
Proof. X E N’ 1 p
Nj, x
c o.
Njk
N’ 1 q(x).
$4.6,
z z
G
by M[Gj].
i Tj(x).
.
x.
p
E
Ti(z).
Gj
. M’
1. M’ i s Proof. f:w + 5 (6)
nR j . N’ Rj
N,
THEOREM.
(4,4).
L,,
p, z
(6).
M’ .
5 w work
5
L,, ,
5
5.
5 is
R. B. JENSEN AND R. M. SOLOVAY
96
L,,
f
9
4
=
M‘.
M’ .
{f ( n ) : n w }
w G M‘.
M’ M‘,
I/ = L
%,,
M. Rj
“n
“nE z”
T”
F,, %,,
%,,EL,,. 8,E M ’ .
$4.6, F,, is p z,
1
L. L,,
R j be generic over M . Then G‘ np,, is non-
2. empty.
Proof. X = { q E R j : q’ 5 q
q’
4.
M. G’nX # ~’EG‘, M’.
q 2 4’.
4
x
F
p’
~ E G ’ ,q ’ E P , , Tj(x)
N’,
n S(fc,j) = F
Gj
z n S(f{,’)= F .
fA,k
p‘
P’ 2 P . p’(n) = ( s , t )
S( fL,J
t
.
f,,’
n , s, t , n=k. P p‘, p‘ ER j .
3. Let g l , . . . , g m be the set of functions which appear in p ’ but do not lie in M . Then fC,’ is generic over M ’ [ g l ,. . . , g , ] (with respect to the set of conditions, P , of 94.5).
{ h l , . ..,h,,,+l}
Proof.
{ g l , ..,,g,, fC,’}
$4.5. M ’ [ h l ,..., h j - J
94.5
1
j
$4.7. N’
M ’ [ g l ,. . . , g , ] .
Y
=
{nEw: i p ‘
ngz}. z
Y 4.
YEN‘.
p’.
m + 1.
hi
SOME APPLICATIONS OF ALMOST DISJOINT SETS
97
2 Y
=
{ n : (3q' €9,) (q'
q' Itn € 7 ) .
p'
{ F n~,E
M'.
W }
M'. p'
L,, N'
N'.
p' q'
5. Y n S(f<,,)E F . p" 2 p'
Proof. n E S(f{,,)
p" I t n E t . t n S(fS,,) =F.
f<,, Y n S(f) c F . k 1) f<,k
k # j , 3)ftvk
sk
by
3)
N'.
T,(x).
x
n $F . p" 2 p' ,
p" G , p'
G,.
3.)
Y nS( f{,k)G F . n S(,f<,k)
by
n S( f<,k),
f
fA,,
so
fc,k
n
s,
p' ,
{fA,,} k
p'
N'.
f:o -+ o
s
N' ,
n$ F p" 2 p'
p" It n E t .
n E Y.
Y n S(f,,k) sF. 4.9.
N
by
4.1. As N,.
N,.
a,
N
M[a]. {s,} {s,} s , s'
s
a 2 j , 2j
b, = a z j + , 2 j , 2j
+1
2j
+1
+1
s'
so
E o, b, = a2, 2j a ,
a.
b,
a.
on j ,
2j 2j
+1
> 0.
a.
k <j ,
.
a.
by
I > 0. bk
{s,}
s,
sk
n,
a.
s,
s,.
1- 1. W e put
a
98
n j E b,;
+1
2j
a.
a.
1. M [ a ] k (3x)?(x) if and only i f j
a.
j
Proof.
a.
aj
k
b,,
b,
j/2.
n b,,
a.
j
s;n by
3
sj
a j M[a].
a.)
2j
54.6,
M [ a ] t Tj(Uj).
.
j $
MCa] c N j . i(3x)Tj(x),
aj
4.8
M[a] k
4.6.3 $M . a, $ M .
4.8
a,
k.
b, N',
a,
4.1.
2.
In M [ a ] , a is the unique solution of a II: predicate.
1
Proof. J' ct)
p)
j
n aj
a,
+
j 20 jEa,
c
=
{n Ij(.j,n)Ea},
4.5,
2 j , 2j 1 Tj(c). rx)
f ( j , n) a .
a.
p)
S.
bEM[a],
M[a] kS(b), b Ea. b =a.
b
=
a.
by
b
p
S, c1
S
1 b Ea ,
5. A A: recursive upper bound for L. 5.1.
Suppose that has countable transitive models. Then there is a set of integers, a , and a counlable transitive model of' N, such that 1) U E N . The following propositions hold in N : 2 ) Every set is constructible f r o m a .
99
SOME APPLICATIONS OF ALMOST DISJOINT SETS
For a certain Il: predicate, S , we have (Vx E w ) ( S ( x ) t , x = a ) . (Zt .follows f r o m this that a is A: .) I f y E w is constructible, then y is recursive in a . ( I t follows f r o m this that a is not constructible. For example, if a is constructible, is recursive j u m p , a’, and a‘ is known not to be recursive in a . ) on no
4.1.
4.1 5.1 4.1 5.2.
+ V = L. N,.
N , , Kf;
Kf;
gi,
Kf
Kf { q(x)}
gi
N 1.
ll:
(3x)T(x)
N,
Kf;
N,,
N
5.3. 1) All 2)
v
T, = L. w
M, V
T,
T.
M, L N 2k T .
L.)
=
K,
5 K ,.
5 K, ,
x
K, < 1 < K,
1
x.
T
$4
by
T. K , < I < K,.
T w
5.4.
Kf K, .
LA
LA w,
Kf;.
P,
f P by
LEMMA. W e assume V = L . Let 1 be an ordinal between K, and K, such that LA is a model of T . Then P E LA.If p E P I there i s an LA-generic .filter G on such chat P E G . Proof.
P
5
LN,c L A . LA,
P(w) ELA.
L,
R. B. JENSEN AND R. M. SOLOVAY
100
P
Kl .
P (Cf. [ 6 ,
f:KF
--+
Kk
=
U G
G , on P . 5.5.
{ p r , 5 < Nl}
$4.5.
S, x K,
P
{ M r , 5 < K2],
$4.5,
T
(a,P>
K,} .
K, x
K,
x Kl
M, ,
.
fu,a
pa.
M.) Kk --+ Kk.
g , , g, g,
.
K, < O R M ‘ < K,
T
1) M , 2) (a,p) Kt K; (Of 5.6.
g,
alnzost disjoint
I
{a gl(a)
= g,(a)}
< Nf .
h : Kf; + Kf; S ( h ) : Nf -+ N; c(
f Ia
< K, .
c(
I
.
f a
L,,
W>(.) = 5. Kf;
g
S(f)
Kf.
S(g)
T,.
5.6.
f
$5.2,
K: f: Nk --+ Kk. (Va) (f
So
i
S(fU,J
/? = i).
Pi,
M,
positive
f
7i(f)”.
A
Qi
qo
(qo,ql).
Kt. (fu,i;y)
jinite
qo
q,
a
<
q1
< Kf;.
SOME APPLICATIONS OF ALMOST DISJOINT SETS
q1 , A
(&;y) qdn) f q2
(A) Qi
s(fu,i)
(4,)
y
101
< A,
*
( q l , q 2 ) 5 (q;,qi)
q , E q;
c 4;.
LEMMA. Let M be our fixed countable transitive model of ZFC + V = L. Define Q i , within M y as above. Let G be an M-generic filter on Q i . Put g
= U(41: ( W ( ( q l Y q 2
) E GI>*
Then g : Kf;-+ Kf;.Moreover
5.7.
q g 1
m).
f~ Qo
f
Qo
f: K O
E K f. KO, f
Qo by g = vG,
Kf. on
,
Kt.
KO
g
Qi’s.
GI, Gi
on go = U G , . 5.6.
on Q i .
i > 0,
g i : K’;+ Kf
N , = M[(gi, i e w ) ] . 4.8.
4.8.)
LEMMA.For each. > 0 , we have N , C T,(gi). Let now j > 0 . Put gi = g i , f o r i # j . Let g i = 4 . Let N ’ = M [ ( g i , i < w ) ] . Then, N’ C (Vg) i q(g). 5.8.
M [ g o ] , Kf;
ai
2)
-
i
a, E w
ai~M[go,gi]:
go(r) < g,(s). gi(go(r)) = g d s ) . by M[ai] = M[g0,gi]. N,, a $4.9. N = M[a]. a,~M[a]. 5.6,
5‘7~ai
n€ai
= { j : g j E N } = {j:(3x)q(x)
=
01.
{ai>,
R. B. JENSEN AND R.
102
SOLOVAY
a
4.9.2, There is predicate S ( i , x ) , such that for i > 0 . 1) N k If N k S ( i , x ) , and i > O , then i E a . 2)
5.8.
as two place predicate,
5.7.
1) i 2) 3)
R = { ( r , s): 2'3' g : w s' h
=
S(i,x)
(o,R) a} .
{ ( g j r ) , g ( s ) ) : 5'7s T(a). S,
h
Nf. N;.
o
R
a}.
5.7
1)
Il:.
R
R.
R.
[l].)
L,
c1
1S Nf.
R. = c1
Nf,
A',
R' LA,,
<
L, <, I
6 B(R,
3).
on w ; R
R
(w;S)
(0;s)
T,
Nf
T 3)
ll: 4.5
2),
x
R.
ll:
x.
by (7).
6. Some open questions
6.1. As
5.1
by (7) Actually, we should introduce a theory T', and use the corresponding fact about codes for models of T'.
SOME APPLICATIONS OF ALMOST DISJOINT SETS
103
ll: S(a, b),
b
Ew, x, y, z
S(a, b ) .
E w
x
y,
x
y,
y
S(x,z),
z.
x,y
x
Kf‘”’
x
K,
L[x]
x.
1
ZFC +
ZFC + 3)
ZFC +
+ 2)
.
ci
0’
6.2.
L[O#].
L.
tl
ZFC + : ZFC
+
L” 6.3.
y.
A;
+
a,
ZFC.
M M[x]
M M.
x
ZFC M[x]. M[x].
GCH
GCH GCH
M[x]: I/ = L
M.
M,
K, M[x]
M
GCH
M[x]
1)
6.4. ZFC + V = L.
M
M[a],
N,” by
$6.3.
104
K,
Proof.
K,
A,
K2 A,
E M[A,]
.
K1,
A,
K,
A,
M[A,]
M[A,] A,
A,
EM[A,]
,
M[A,]
KO
M[A,]
W. Some consequences of the axiom of constructibility, 46 (1958), 337-357. The consistency of the axiom of choice and of the generalized continuumt21 hypothesis with the axioms of set theory, no. 3, 1951. Definable sets of minimal degree, pp. 122-128. Internal Cohen Extensions, 1966. Unramijed forcing, on , The problem of predicativity, on 132-139, 1961. model of set-theory in which every set of reals is Lebesgue measurable,
t11
, A nonconstructible A: set of integers, 50-75.
127 (1967),
ON LOCAL ARITHMETICAL FUNCTIONS AND THEIR APPLICATION FOR CONSTRUCTJNG TYPES OF PEANO’S ARITHMETIC HAIM GAIFMAN
$0. Introduction.
$1.
$1. $1
92 $1
[l].
93. by
by
of by
t,
“t(u)”,
u, 105
106
HAIM GAIFMAN
by
by
"<" "u1
< u2"
u1
" u , = u2"
u2. t(u)
unbounded
T
z
t
t
end-extension
t
23
'u,
'u
by
91
b,
'u
t,
23 23
'u 'u; t
23
23
(X 'u.
23
% A
3.
'u,
antiminimal
23
'u 'u. A
tl(u) depends on tl(u)
tz(u).
T
$(u)
~(T(u))
t , = z(tz).
pu4
by
b
z(u),
t2(u)
4 u,
23
'u,
minimal
t
4,
u
0 t
23,
23
r(b)
~(t)
ON LOCAL ARITHMETICAL FUNCTIONS
B
b T(b)
107
2,
t
~(t),
'u
T
'u 'u
by
by b by b . on t , t , on t , ,
'u on t,
on t , .
t,
t,
91
.
t t
1 '1
t,
'u,
t
is
t
'u, 'u
'u
up
'u
by
on t2(u)
t,(u)
t2(u)
by t,
t,(u)
t,,
T, u = T
by on
t,
1,
on t , .
t,
t,
t;
t,
t,
'u.
Bi, i
ti
t;
'u
= 1,2,
'u
by
23,
ti
B,,
'u,
do
t;
'u,
f,,
'u.
t(u)
T(U)
T(u),
=u
~ ( t #) t .
t
T.
X
=
( X , <) x, X
f
,
x X}
'u A,
23, x
,
wi X
'u ,
by A
X , UX ,
108
HAIM GAIFMAN
<.
X ,
23
'11,
up
by ( X , <)
23
'11,
x,f(x)
23
23
'11, A
by
uX ' ,
1 '1
X' c X ,
X on
23,
4(u)
+(u)
z < 0,
z
4. 2 "O
All
no
on
A
z1
I A 1"'
i (tl =
z2) ICI'"
C
All on
$1. $1. Local functions and their iterations.
i,
J,
k , I , rn, n , w, x, y , z
X, Y U , I/
u, 0 Q,
Q,(X) by
z2,
X , Q,,
W,
A,
109
ON LOCAL ARITHMETICAL FUNCTIONS
@O(X) = x , W + l ( X ) = CD(@”(X)).
Ax X ( x CD(X))
@
by by
X
x
U
U(u),
u
do
U”,
“u
n , CDn AxyX(x @’(X)) @
X , @ ( X )c X , 3,xy(x
@’\o)), o Axy(x E W(o))
AxyX(xW ~ (X)) on CD AxyX(x @’(X)) on
an,
AxX(x CD(X)), AxyX(x @ ( X )
on, w
(0,
o; X ~ 2 “ ‘ “, X ( i ) = ‘L X(i) = “i$X.” (X(O), ..., X ( x - 1)) , x > 0 ,
“2””,
DEFINITION. local X
EX" X rx x = 0. c w x 2”,
on w x o , x,
(“ “V”i” “3” ,
“y = X
X
m.
m”
y
R,
is R, ,
110
HAlM GAIFMAN
on X
Q(x, X ) , (X(O), ..., X ( m - 1 ) ) ,
bound
DEFINITION. A
@: 2”’+
local
$(u, U )
u
Ax X [ x @ ( X ) ]
on, P DEFINITION. A p,(u, u )
u
p 2 ( 4 v)
4
P . In x
U
U P-provably local u
AxXQ(x, X ) , X ,
4(u,U )
f-,
ti
Ri.
pi ( i =
3u, u’[u = U
r v’ A pl(u, u)].)
“P-provably local”, 1. (1) l f @ is local then A x y x ( x ~ @ ’ ( X ) )is arithmetical. THEOREM (11) I f $(u, U ) is provably local then, regarding 4 as asserting that x @ ( X ) , where u corresponds to x and U to X , the definition of @” can be formalized in P , in the sense that a formula ~ * ( u , vU, ) exists such that the recursive conditions f o r @“formulated with the help of 4 and 4* are theorems of P . (Here 4*(u, v, U ) i s to be regarded as asserting that X E @ ~ ( X with ) , ti corresponding to x , u to y and U to X . )
Proof. (I) z ‘‘z = (z(O), ..., z ( x x z i < x , z(i) z. z by “lh(z)” 0 0, z y (z(O), .. ,z ( x - 1 ) ) , x = Ih(z)). y, z1 = z2 i y , z1 z 2 , z2 z1. R, R, z x in m 5 lh(z) R,(x, z r m). z x m 5 /h(z) R 2 ( x ,z m ) . z x x
r
-+
* This concept can be generalized by requiring R, (i = 1, 2) to be analytic. set theory one gets an analogous definition by regarding X as a proper class, m, x, y as sets, X ‘ j m as X n m and R, as properties expressible in first-order set theory.
111
ON LOCAL ARITHMETICAL FUNCTIONS
0's
z x
z
z.
x
X
(1)
1's
no
(2)
m
X
rm
x.
n
x
n
x,
n
z,
n 5 Ih(z)
i,
i
z
i
.
z
Y(z)(i) = 1
i z
z1 + z 2
(3)
3
Y(z) Y(z)(i) = 0
n, z 1's
Y(zl) + Y ( z 2 ) ,
i Y(z) = 0.
z,
z2
Y(x r m ) $@WP m , n,
X
Y ( X p m)
m
n. Y;
(3)
(5), mi mi, i < n . Y ( X r m),
(2). n, m (3),
X
m
=
r mi
0,1, ...
@(XI * w
AzY(z)
w,
by
AznYfl(z) on k z1
k
z2,
z1
(6)
Y"(X
(3)
r m)
X
by
Y k ( X p m ) , m = 0,1, ..., X = Y n. Y p m $ Y " ( X P m*).
$z2
n
=>
Yk(z,) $'rk(z*).
m n. on k ; k Y(O),.. ., Y ( n ) ,... x Y o 3m[Yk(X r m ) ( x ) = 01. m Y(Y r m ) k m* Y k" ( X r m*)
Y
n.
112
I-IAIM GAIFMAN
@*(n, X)(O), ..., @*(n, X ) ( i ) , ... Y " ( X r m ) , m = 0,1, ... ,
@*(n, X )
x
@*(n,x ) o ~ W [ Y ~ r( m X ) ( x ) = 01.
@*(n, X ) @*(O,X) = X . (5) Y(@*(n,X ) m,) + Y ( @ * ( n , X ) 'i m 2 ) Y(@*(nX , ) m ) , m = 0,1, ... @(@*(+,X)). @ * ( n , X ) r m + Y " ( X r m*). Y ( @ * ( n , X ) r m ) , m = 0,1, ...,
@"(X).
n =0 m, 2 m2
m*
m = 0,1, ... .
@(@*(?I, X ) ) = @*(n
(9)
(11)
@"(X) = @ * ( n , X ) $(u, U )
p2,
m
Y"+'(Xr m),
+ 1, X )
@*(n,X) p i ( u , u ) (i =
4 , p,
(1)
P.
P.
Ixz[x Y(z)] I x y z [ x Y'(z)]. $*(u, u, U ) I x y X [ x @*(y, X ) ] . P
P for @" w x
wk x DEFINITION. (1 5 j 5 I ) , Ri wm x X l , ..., X ,
Q
wk x
x w x (2")'-'
Q (i
=
Ri(i = 1,2) x l , ..., xk
xj r m A ~ , ( 2 , 2 , Z , ? ) ] , 7 ~ ( 2 i j, -, 3 2 , m [ z = xj r m A R& 2,z, 211. '2 'X' ' x , , ..., xk' 'x,, ...,x , ' , ~ ( 2 , ; ) 0 3z,m[z
=
+
-+
' X , , ..., Xi.-,' 'X"' ' X j + , , ..., X l ' . Q ( x , , ..., x k ,X I , ...,X , ...,X , ) is Q
X,
+
'Xf7
113
ON LOCAL ARITHMETICAL FUNCTIONS
AZ[ x
..., X , ) is local in X j
..., @(x,, ..., x,,
U , , ...,
xi.
x,, ..., x,)]
4(u,, ..., u,,
provably local in U j .
2. I f @ ( X ) i s local in X then the iteration @ " ( X ) , as a function of n and X , is also local in X . (11) 4 ( u , U ) is provably local then the formula ~ * ( u , vU, ) of Theorem 1, which is supposed to define the iterate of the function defined by 4 , is provably local in U .
1
Proof. (1)
X E @ " ( X )e 3 z , m [ z
=
X p m & Y " ( z ) ( x ) = 01,
x & @"(X) o 32,111 [ z = X
@"(X)
r m & Y " ( z ) ( x ) = 11,
X .
P.
(11)
AxXY @(x,X , Y ) , 1.
( I ) Assume that @ ( x , X , Y ) is local in X . Consider w as a finite sequence of natural numbers and dejne @'")(X, Y ) by the folloiz~ingrecursion : @")(X, Y ) = X
(where 0
@("n(J'))(x, Y ) = @(y,
=
@'W)(X,
the empty sequence),
Y ) ,Y )
(where w n ( y ) is the sequence obtained f r o m w by concatenation with the sequence ( y ) , where ( y ) is of length 1 and its member is y ) . Under these assumptions, @'"'(X, Y ) , as a function of w, X and Y . is arithmetic and it is local in X . (11) l f Cp(u,v, U , V ) is provably local in U , then, regarding 4 as asserting that x @ ( y , x ,Y ) , the definition of @'"'(X, Y ) can be formalized in P , in the sense that a formula 4*(u, v , U , V ) exists such that the formula corresponding to the recursion schema is provable in P. T h e Jormula ~ * ( u , vU, , V ) is provably local in U. 1* 1. ''z x" "z, y Y x" Y
z
z, y
Y.
Y"(z)
1's Y ( w ) ( zY , ),
HAIM GAIFMAN
114
Y y z ,Y)
=
z,
Y)
=
Y ( y ,Y ( " ) ( Z , Y ) , Y ) .
y ( w n W )( z ,
@*(w,X , Y ) , @*(w,X , Y ) @*(w,X , Y )(0), @*(w,X , Y )( 1 ) , ... Y ( " ) ( X r m , Y ) , m = 0,1, ... . 1, @*(w,X Y )
@(")(X,Y )
(11) @
@(")(X,Y ) ; 3. (1) Let @ ( x , X ) be a function which is local in X , and let g be an arithmetic function such that, f o r every x and X , g ( x , X ) is a number. DeJine @l(X) b y the recursion:
@(X)
=
q +' ( X ) =
x, W ( n ,@:(x>), @WN*
T h e n AnX@,"(X) is arithmetic. (11) Let ~ ( u , vU, ) be provably local in U and let y(u, U ) be a term. T h e n , regarding 4 as defining the relation A x X [ x ~ @ ( y , X ) and 1 y as defining g , there exists a f o r m u l a ~ ; ( u , u U , ) such that the recursive schema above, when formalized using 4, y and 4:, is provable in P . (In this formalization 4 y ( u ,u, U ) should correspond to x E @ ~ ( X with ) u, v and U corresponding to x, n and X , respectively.) @, g,
4 4;.
@i(X). Proof. ( I ) by
y
1*,
AwX@("'(X)
a+O)(X)
=
x,
@(n'n(y))(X)= @(y,@(""(X)) h:
x
+
w by
h(O,X)
=
g(O,X),
h(n + 1 , X )
=
g(n
+ l,@(g("+i*x)) (X))
g ( k , X ) = ( g ( O , X ) , ..., g ( k - 1,X)).
7
LOCAL ARITHMETICAL FUNCTIONS
115
on
X
h @D'fi('".x))(X), n X , @;(X). @;(X) = @ ( h c " , X ) ) ( X ) , @:(X)
P,
h
by
h, @(w)(X),
@('(".')'(X). P.
by
3
g g ( x ,X )
g ( x ,X , , X , )
n 2 1,
@i(X), :
@;m @:-'(m@ i ( X ) )
q g " " ( X >= @(g(n,
n =0 X
7
by
.
7
@i(X)
g ( x , X , , ..., X , )
2k -1.
n :
q + ' ( x >=
@ ( S ( ( @ , o ( X ) , . . . , @ ~ ( X 7> >@i,"(X>>, )
(@:(X), ..., @ : ( X ) ) (i =
0,..., n ) n
mg(X)
Qg(X) ( i S n). {(n,i,j): i < n&jEXi}.)
(XO,...,Xn-,)
:
DEFINITION. Q Q ( x , , ..., x,, X , , ..., X , Y,, ...,
x-j)
x-j :
Y,, ...,
Q c wk x (2")', local in ( X l , ...,X j ) R i , Ric w x m i x = x l , . ..,xk X , , ..., X j ,
~ ( xX ,, Y ) 0 32, m [ z = 2 4
4
-
-
-
4
i Q(x,X,Y
a
)
+
-
+ + +
+
4
3z,m [z = X
- 4
-t
G & ~ , ( x z, , Y ) ] , +
r m&
+ + +
R 2 ( x , z ,Y ) ] .
HAIM GAIFMAN
116 +
+
‘3, ‘X’
‘Y’
‘x,,
+
-t
‘Y,, ..., q - j ’ ; ‘z’ ‘m’ ‘mi, ...,mi’, ‘ z = X I‘ rn’ (i = 1, . . . , j ) . + Q(G,X1,..., X j , Y ) Q*(G,X, ?) X, Q* Xl,...,Xj X. 4
-
..., xk’
‘ X , , ...,Xi’
‘ z , ,. . . , z j ’
+
‘zi = X i p m,’
( X , , ...,X i ) Q
U,,..., U j . @(xl,...,x k , X 1 ,. . . , X j , Y,, ..., q - j ) is in -+++ + + + l z x X Y [ z ~ @ ( x , XY,) ] X , , ...,X , .
( X I , ..., X i )
1, 2, 1* QD,(X,Y )
@,(X,Y )
3 (X,Y )
@;
@Z @;“(Xy
r>
@Z+
Y ) = @,(@XX,
V
Y
= @l(@XXY Y ) Y
n, X
@:(X, Y ) ( X , Y ).
Y
@W, r>>
Y
@W, r>)
Y
Y 3
by
92. Construction of minimal types.
$1 X,
f: w + w . X,
X’,
X’ X‘,
X,
X‘ x’s
X
X’ = {z:
.
w,
f(x)=w y x X , x Iy # f ( z ) . zo zo > y f(x) # f(zo) xE X E X y < z => # f(z)}, X’
P,
X
z z
x
< zo .
X’
P
ON LOCAL ARITHMETICAL FUNCTIONS
117
f:w x
w +w,
y,
...
w,
Lxf(0, x) w,
& f ( O , x ) = w,};
= {x: x
Axf(0,x)
,
,
+
f(n
x
= w,,,,
w,+,
w
if
w
Ixf(n F(n,
=
no
w
+ .
F
P f(x,y),
z(u,u),
~ X Uu,, U),
,
F(y,
P. ( ),:
( ),: x
x + (x),
w (x), = y
+ (x),
,
w
x
(x), = w (x),
>0
&f((x),,
o zE
@(x,
=
(x), - 1.
y< =n
+1
= @(x, w
+
(x), = w 1, = w} (x), = 0 ,
&f(n
+
(x),
x
.
@(x, x
g z
(x), = n &f(n, =
(x)~ (x), = w
no
y
y
+ 1, y
w (x), = 0.
g.
x,
= =
3,
F(n,X) = 3(I) I
118
HAIM GAIFMAN
F 0 P. f(n,X ) f ( n , y X ) =f ( n , F(n, X )
P
3(II)
f*(n, X ) xo xo = x F(n,X)
F(n, X ) =
x X
F(n,X)
x
X F(n,X), f*(n,f(n, X ) = x
x
=f ( n , X ) ,
-
f
F ,f
F(n,X),
f*. P.
f*
X X , F(n,X) F(O,X),
3
F(n
..., F ( n , X ) , ...
+ 1,X) by on.
F(1,X),
D,(X),
:
P
D,(X). D,(X), n , D,(X)
D,(X)
X
on by
f
n , F(n, X )
P F(n, X ) 3 F(n 1, X ) . f”(n,X) x E D,(X) x F(n,X ) . :
x
+
3
X
n = f ( n ,X )
f
F(n, X ) . P,
X
x>z,
D,(X)
(11)
.
F ( z , X ) & y < x}
F ( z , X ) & { y: y
x E D,(X) o
. x, f*(n,f(n,
A(u, U ) P:
x >f”(n,X), X)
=
x.
z(u, u) f ( u , U ) , z*(u, u, U ) , z #(u, U ) , D,(X)] f,f * f .
”
119
LOCAL ARITHMETICAL FUNCTIONS
[ V U ~> U uU(u)]
-+
[ V U ~> U uAr(u, U ) ] ,
Ar(u, U ) A u > z # ( u , U ) -+ [z(u,u) = ?(u, U ) v T * ( u , T ( u , u ) , U ) = U ] .
(11*)
2, P , 23,
I ~ ( a , ..., , a,)
a,,
ui
ai
A, z(uI, ..., u,)
B. =
..., a, E B , 1, ..., n ) .
b EB
... ,
a,,
1 ~ ( a , ..., , a,, b ) 1% ,
A u{b}
n
T
a,,
= 0,1,
23,
T
by
..., a, E A .
..., a, z
A
by
A
by b .
23
a A.
k Vu 3u > u+(u),
?+(u), ?(u, U ) , z*(u, u, U ) ,
b
Ar,+(u)
EA,
-+
~ ( ab,) = ?+(a) V T ; ( U , ~ ( U ,
1% E A ; 1%
A, tz(a) b' = z(a, b ) ?+(a) A . A b',
I
Vu30 > uAr,+(u).
23 k A r , + ( b ) ,
b > zz(a)
1
+(u)
23 C
(11*)
I Ib
T ( U , o)
Ar(u, U ) by
Ar,+(u) U(u), u T$(u,u) z$(u) T#(u, U ) . b
(12)
+(u)
a A,
b)) = b .
I
b' = ?+(a) A, by
b' b. by A u { b ' } .
1%
23 1 Ar,+(b),
u)
x> Ar,+(u) Vu3u > uA,,+(u)
'u.
23,
T:
23 CAr,+(b)
2l.
'u
1%.
by A U { b }
,
23,
I
b > t$(a)
I z$(a, b') I%=b. b
T(U,
+(u)
23, 113:
Vu3u > u+ ( u ) d(u)
z(u,u),
Ar,+(u) T(u,u), To, TI, ...)T", ...
.
HAIM GAIFMAN
120
40(u)
P !- Vu3u > U
~ ~ ( U ) .
n , P t Vu3u > uA,(u) P t Vu[A,+,(u) {A,,(u):n = 0, 1, ... } U { u > 6:(r P.
+ A,,(u)].
P P'
P
no
0
P' {A,(u): n = 0, 1, ...} . P. 'u 93 P' , by A
1,
P, P
'u ,
U{b}
$ 1 , so
P' , P. by
$1, {t'}'
E {Ai(u): n = 0, 1, ... ,[ E E} A$(u), ...,Af(u), ... Ao,A,, ..., A,, ... [ # t,~
z(u)
n
A: 3,
,
P:
by
9
1,
121
[l] R. Proc. Symp. on Foundations of Mathematics 1961 pp. 257-263.
Infinitistic Methods,
1959), Sets,
Models and Recussion rneory,
1965),
by J.
1967, pp. 122-155. up
[3] H. 1967,
on
connecLos
Set
Soc.,pp. IV.
1-IV. R 16.
DEFINABLE SETS OF MINIMAL DEGREE RONALD JENSEN Bonn
ZF
$1.
ZF,
by
ZF
4 L[x] ,
x E L[a]
x
by on A:.
A:
ZF
ZF
ZF
+ I/ II:
by
A:)”.
A: and on
L,
V
L.* w1
L; :
*
Cf. Souslin’s Hypothesis is incompatible with V = L, by
inJ. S.L.
122
Jensen; to appear
123
If o1 > m i , then there is a sel a c o of minimal degree of constructibility such that a is the unique solution of a II: predicate in L [ a ] . 52.
(I 1 I.
P= P ,5 P ‘P 5 a P P
I
I I
) Q’
on ZF A
forcing conditions
5
Q’ . P , Q P compatible A A c P dense P A. X P pre-dense P P P X . X c M-dejhable X P P-generic ‘P
P
, over P 2 P’ A G n A # Pr
4
PEG A.
n X # Pr
X .
on M.
ZF
X i( i < o)
P.
A
V i X ic A . M
ZF
X P,
M
+ CH ,
P
CCC
w, > m y ,
5 o1
.
ZF M[G,].
A nX
P
A X ,
cX
is P,,
product lemma M
G , x G2
M[G,] nM[G,]
=
P, x
..., P,
RONALD JENSEN
124
by by by
w,
#
I
u, U ' I u
u u' on,
'5'
E
on
w
a a
u, u'
uE
!ij
T
I
w a
{a
SE
=
P
T"
--+
P
M
.
E
G = Ga = M[G] = M[a]. P; G, M. X l = l Gai P a (T, s s c s r -, 2
1
a co
M, a
{ T EP I u
. a cw (ul, M M,
ZF a
n
nu n
n
n
T,
Is1
Is1 = n
us
S
II
S,
A U E Sn T
u c s.
n <w
Is1
= ls'l = n ,
+ V = L,
w
lemma,
M[a].
E
=
M
...,a,)
s # s ' + T,I IT,..
125
DEFINABLE SETS OF MINIMAL DEGREE
P
$3.
P
P=
L,
u
P,
UC",
P, (a < 0 , )
P, by
L.
on a
I
Po = ((2'")" u € 2 < " } PA =
U P,
2
vcl
L,
by
y = y.
P,EL, 'p(o>n L,+1
+ L,. P
P,
wx
P,, 2
P,,
P 5
u,uE~<"u , c v
.
on C =
-Df
6
L,
G@) S = S'"' -Df w x 2'"
S
u
L)
G")
h <w
P,.
n <w IuI = IuI = n A u + u .
-, s , , I s , , .
by
P,+~ = P,U
P=
{ S : ( n < w , UES,}.
u P, PI
L.
P
LEMMA1. I f A c P: is L,=-definable and dense and closed under extension in P:, then for k, m,, ..., m, < w there exists h 2 k s.t. if s1 ,..., s , E ~ ' " , Isi = h a n d si # si for i # j , then
1
126
RONALD
1
S'"',
2. If bi c w goes through S2,) ( i = 1, ..., n) and b l , ..., b, are distinct, then ( b l , ..., b , ) i s Pngeneric over L y 2 . 2 , no P a + l\ Pa Pa+ c Pp L, ? ;
p > tl
r
If X
3. pre-dense in
Proof.
+
Ya
then X is
c P," is LyT-dejnabIeand pre-dense in P:,
P a + l \ Pa Pa
I
{Sn4 n < w }
by
P E Pa+ \ Pa
\
Sn4.
P
X.
=
(S:, ,..., S:,),
P
8
=
X
REP:^
Pa)"
Q
.
VQEXRSQ}.
Pa. I u iI < k ( i = 1, ..., n ) . 1, ...,u, ~ 2 < , , I ui 1 = h , vi # u j ( i # j ) , X. u l , ...,u, ui3 ui Q = (S;'!, ..., S z )
8 h 2 k ( S , , , , , ...,Smnv,) ( i = 1, ..., n ) .
u1
4. If X c P," is Ly7-dejnable and pre-dense in P:, then X is pre-dense in P".
X
Proof. by
on
Pp
p,
tl
S /3 < w l .
3
Yp.
5. {SP'I n < w } is pre-dense in P. Pa+, \ Pa P. P,+ , 4.
Proof.
\ P,+
Pa
\ Pa by
6. Proof.
Pa
3.
P" satisjes CCC in
V = L.
X
P
c Pm
X M
L,,, E)
SF',
\
EM.
P, X
127
DEFINABLE SETS OF MINIMAL DEGREE 71:
p.
ci
ci
=
n(wl)
Y
E
M +-+ ( L , , E ) M
.
:
M
+
9
4.Y)
=
Y E M , CL,,+~(Y)
u P, n(P) u P,
P=
Y = YnL,.
P,
v
< wl,
V<W,
= Pa.
=
v
p6
p > ci
a.
.(X)
X n
=
c
I)
X 4,
P, n ( X ) = X n P: P. X X = X n P,!'
X n
L,.
'!J~(w) nL , , ,
Pa E v(w)n
y
ya
2
y, y,.
P,".
X
:
7. Let b,, ..., b, c o. Then ( b , , ..., b,) ..., b, are distinct and
i s P"-generic over
A c i V h ( b i goes through Sy) for
=
1 , ..., n . 5.
by
Proof.
A
( b , , ..., b,) r
X
n
c P,
\
A
X
ci
X
Pa
Pa,
X
by
2.
X Q.E.D.
7 8. ff a , b are P-generic over L and a # b , then ( a , b ) is PZ-generic over L.
128
RONALD JENSEN
be an inner ZF model s.t. L c M . Then the set lTf in M .
9.
1
A
{ a c w a is P-generic over L } is
II,
Proof. z = wf.
n
(~:’Icr
L,.
El
z, L ,
Al
Ha,. a
t ,
Au
< z V n (a
Sf’)
8, 10. Zf a is P-generic over L , then A = { a } in L [ a ] .
Proof. 8 a)
b# a
b # L[a] , by L[a] .
L. L;
b
11. Zf a is P-generic over L , then a is of minimal degree.
Proof.
a # L[x] .
x E L[a]
x x EL.
X
x
2;
P
do L[x]. c P,
nX EX,
P,
c,G L[b]
.
w1 > wf.
X.
X. X G, # X , b b # a
L[x] = L [ X I . E
L w1
b
=- of, L.
a E L [ X ]= L[x]. G, c X . 2
(a, x E L [ u ] nL [ b ] c L
x
DEFINABILITY IN AXIOMATIC SET THEORY 11* BY
Hebrew University of Jerusalem
0 1. Introduction and
ZF
statement of the results. Z U ZF
GCH
do
w
1 ([S, 11). If ZF is consistent, so is ZFC + GCH together with the following additional axioms: (1) There exists a nonconstructible real number. Every hereditarily-ordinal-definable set is constructible. There is real number a such that V = L [ a ] , every set is constructible f r o m a . (4) Every constructible cardinal is a true cardinal, or, in other words, if u < p are ordinals such that there is a mapping of u on p then there is also a mapping of u on p . [S]
1
N
ZF
+ GCH + (1) +
by GCH ZF ZFC: no
(5)
(4)
p.
(6)
p.
[S],
P(f)
f E w"
no
(5)
no
*
a
all [5].
[5]
[6]. [61, [ll]. by
N00014 69
0192. 129
AZRIEL LEVY
130
At, GCH
(4)
N, = (5)
up 2. If ZF is consistent, so is ZFC ( 3 ) and (7). In ZFC (1)
+ GCH
together with
. p. 1881)
II
P(f,g )
Q
Q(f,g) Q ( f ,g ) .
g
by Q .
by
R(f,g)
P(f)
R(f,g )
by
=
g
R
by
0 A P(g). R ( f ,g ) ; P(g),
Q ( f ,g ) g
(3)
by V = L[a], a.
P(f,g ) on ow “P(f,g), no g’ €corn L[a] P ( f ,g’) (3) At a.
by
by
Q ( f ,g )
g
P
a.
by up (3) by
(8) by 3.
(8)
If ZF i s consistent, so i s ZFC by
+ GCH together
with (8).
(3) (8),
DEFINABILITY IN AXIOMATIC SET THEORY I1
131
no
. ZFC
+
w1 0,"
o1=
<
01.
w1
<=q + = ol,
w:<
wi w
1
q, r
05
o
0,"
nE
R 3k3l (kRI A n = 2
c)
L[r]
w1
3')).
. ol,
o L[r].
r
05
*
wt
no
L L[r]
wf
w1 = wtCr3. wf"] = w , by
L[r]
L[r].
r, ZFC
+
go
ZFCl
ZFCl
(ll), ZFC by
p. 306]), ZF
on
ZF ZFC
ZFCI".
+
ZF.
ZFCl .
4. ZFCl (i.e., ZFC together with the existence of an inaccessible number) is consistent, then so is ZFC (10).
+
do definable
GC H,
KO < K, < ZNO; K, ?
THEOREM 5. a suitably defined ordinal the absolutely definable ordinals of Hajnal [4, p. 324]), if ZF is consistent and
AZRIEL LEVY
132
“A > 0 A A is not confinal with o” is consistent with ZF, then ZFC is consistent with = K,, together with and
Every real-ordinal-definable set of real numbers has cardinality
5 K, or K,
.
by
5 KO 2’0.
6. If A is a suitably defined ordinal, i$ Z F is consistent and “ A > 0 A A is not confinal with 0’’ is consistent with ZF, then ZFC is consistent with = K, together with 6 by by IK, -
K,
ZFC
do ZFC,
ot
w:+~
o
cot.
r
4,
do, a,# o:+ r K, . T
,,
q.
o,=
T
6
7 (SOLOVAY If ZFCl (i.e., ZFC with the existence of an inaccessible number) is consistent, and it i s also consistent with “A>O and A is not confinal with o”,where A is a suitably defined ordinal, then so is ZFC together with = K, and Every real-ordinal-definable set is of cardinality 5 KO or includes a perfect set (which is of cardinality 2”)).
[2,
S. IV, $103.
8.
8. If Z F is consistent then it is consistent with The continuum is a countable union of countable sets,
DEFINABILITY IN AXIOMATIC SET THEORY I1
(17) w1
of;(where 0," is the a-th injinite constructible cardinal) by [12,III, $31, (16) :
=
(18) w1 (19)
(18)
133
= wl.
a,,
(19) (17),
R(n,f) . ..)
(
no
(Vn EO) ~o")R(n,f), (Vn E w ) R(n, .
(20)
R(n,f),
by p. 188]),
(Vn E w)(3 E wO)R(n ,f) (Vn E o ) R ( n ,
...
ma,
{(k, 1 )
by (17) is an ordering of natural numbers"
"
(2k- 3') = 0}
w,
R(n,f) "f is a well-ordering
f.
L, a, .c w,L = wl, o n < o,
1 0.".
.. .)
(f o ,
I
-
((2" 3k,2" * 3') m < n or m = n andf"(2k-31)= 0 } a. u
0,"
2 at).
u
2
2". 3 k,
wk
n no
T,
+
y,
(L,
2 wt".
0."
2 0.:
by
order-type a n < w,
2
+ (T,E),
L,
y
M,
u
L
[2, 1111.) R(n,f),
k <w, L = o, = wl.
2 ZF, T = L,,
wl.
AZRIEL LEVY
134
2 of”,
“f is an ordering of natural numbers of order-type
(21) f a well-ordering of natural numbers, whose order-type we denote with a, and every countable transitive model M of $, which contains the ordinal a,has n + 1 injinite ordinals a, which are cardinals in the sense of M .
s
2 wf;.
a
$.
M $
M
y,
05
M,
no
6 < w k on wk,
+1
no
M wf;,wf, ..., wf; S a,
M, (21) 0 < m <w .
a < wk
L,
M
M, n
=
(21)
a < w1
a S/’ < w k .
p p on
g y
< p.
g ~ w f ;by , y x, [2,
n
g E p x y,
p <wi
13).
I Ip 1 04
p,
xcLB
16 5
6
y L, 11, $4,
Lmk. by (21), Lm; Lu;, a 2 w!;, is
$.
Luk
+1
y < p < wk,
5a
. [2, p. 731) $, by
(21),
u A $,
IJ
(21) (22) Vg, h E
O ~[f
A g E‘
M Ew A u A $ ( M ,E ’ ) A h
on
( M , E ’ ) A (Vk E M ) ( k k -, i 5 n, Zg‘k (22)
on
n
+1
l o , . . .,I ,
li
( M , E’))]. is
(22),
DEFINABILITY IN AXIOMATIC SET THEORY I1
II:,
+
135
2;.
+
“f
“g
II:.
(22) E’)
=xin II: .
At
2.
52. Proof
by
[ll]. [11].
ZF.
by ao. conditions.
Sp.
P 0.
Sp (23) n(p)
p, p ’
II
E
p’
n(p),
qE
S q).
G M[G]
=
ZFC.
L’ ZF,
N . L’
S,
S(x)
G x
x
E)
. [5]
C
L’
L“ C,
.
[2]
[5].
LEMMA1. Let Q be a sentence of L’ which does not contain G, and assume that (23) holds. Q is true in N , if and only if, 0 It Q .
[ll, xl, xl,
...,x, LEMMA 2. Let Q(x, Y
...,x , E N , 0
...,Y,)
3.51. kQ[x,,
..., x,]
(N,E).
be a formula of L’ which does not mention
136
AZRIEL LEVY
G with the only free variable x , and with no constants f o r members of M , except y l ,..., y,. Let B be the class of N , such that f o r all x , x E B t-) x E N A N k @ [ x ,y,, ..., y,] . I f holds then, there is a formula Y ( x ,y,, ..., y,, S p ) of set theory such that f o r the given y,, ...,y,, and f o r all x
sp,
xEBnM
t ,
X E M A M k Y [ x , y , , ...,y,,
I f , in addition, B E N , then also B n M
&].
M.
Proof ( [ I S , 621). B n M = { x I x E M A N k @ ( x , Y I ,..., Y,)}
I
= {x x E M
A 0 I t @(X, Y l , ..., Y.)} M
ZF
XEM 0 I F W X , Yl,..., Y,J
p -= cio ,
B nM
R(a) M . R(p)'M), BnM.
-
c R(P) nM
1)
..., y,,z)
Y ( x , y,, ~(X,Y,,...,Y,,s =
P )
B E R(P) R(p)'M)
BEN R(fi)'M),
M,M Y ( x ,y,, .. ., y,, 5 p ) ,
M by
3. Assuming (23), then f o r every x e N and EM, if x is hereditarily-ordinal-deJinablef r o m y in the sense of N then x E M .
by
Proof ( [ 5 , T ( y ,ci, x)
[S], 3ciT(y, ci, x )
y, ZF t
r(y,a, x ) A r ( y ,a, z ) -+ x
=z.
N-M y
N.
s EM .
c1
N k r[y,a,s]. N by
N s X E S
y , ci E M
by
L' , N,
t ,
ci E
s
N,
ci
< go.
~ ~ [ X E A ZT ( y , a , z ) ] .
2, s = s n M
E
N -M .
M,
ZFC + GCH
+
M
+
ZF
+V = L
137
DEFINABILITY IN AXIOMATIC SET THEORY I1
2 by
147-1481,
[ S , pp. 132-1331.
ZF
M w on
P [ l l , I, 1.12]), miM) G UG = F M
+ V = L.
P
M ). S p P . As w on miM’.
w1
.
M
[2, p. 1431, on
w
[ll, I 1.121, N N
V = L,
M
M, M. N.
N,
N V = L[G] FEN, N , G = { f ~ P l Ff} , ~ N = M[G] = M[F] . N ZF, N R on w w miM’, r R. F, G, M[r] F M[r] = M[G] = N , V = L[r] N, (3) N. (23) Sp n: o, PEP n*(p) {(n(n),p ( n ) ) n E p ) . n* (P, p , p ’ ~ P , n: w on . n:*(p) p‘ (23) (P, Sp). N,M by 3, N. TEN A A EA a T”. EA N; by (2), A N. M on miM). N, wiM) N, No N, A N. N.
1
rp).
$3. Proof of Theorem 3. by
ZF + V = L. [2, p. 1301
P z
w,
z = miM’.
AZRIEL LEVY
138
z = w\')
z,
P z x w
sp
F = U G. 2, F N = = M[F]. z =
dM) 1
=
9
N, V 21, N.
=
t x w
L[F]
F
c w1 x
K,+,
FEN N, N by [7,2.8 0 < c1 E N . GCH
41
w.
N
c z.
PA P A x w , G A = { p PA p G , FA = F ( A x o), F B.
p's
I
I
, FA = U G A. F B
I
A E
4. For every real number a N , there is a set A which is countable in and such that a M [ G A ] . Proof. L"
L:
d
71).
z
.
@,
+
Q,
nE d .
p pE
t,
a. p,
p. 134,
1.
{0,1}
p. 134,
K, . =
A
A
on
0 P,
z,
A, A = U,,,A,;
(p) E A,.
A
n < w, nEa
t-*
(3p~G,)(p nEd)
GA G nEa n
nEa
d. (4)
nE d is
N
by pE q p q Ep q . Q,, q Q , c A, x o c A x o, q E GA, a = { n o ( 3 p E G A ) ( p It. n 6))
1
.
5. Let A E z; every real number b which is ordinal-dejnable f r o m GA is constructible f r o m G A . Proof. P [ll, I 2.31,
.
=
PI-A, GA
by
PA x P I - A , Gr-A
A
b GA
,
DEFINABILITY IN AXIOMATIC SET THEORY I1
by
M y bEM[G,]. G,. “ g i s not by
M[G,] b
3
V= L
M,
139
3. constructiblefromf”. $1
S EN, S 4
AEM
GA
z
N G, G,
x
N,
.
d. d E My
d
a
S
N.
G,
r, A on o p E G,}; P(x) R(o) on a). r. gEoO S S G, 5 5 by g
My N h {{((h(a),n ) ,p ( ( a , n ) ) ) (a, n) E R(w) R(B) = U Y c B P ( R ( y ,) )
I
M[G,] = M [ r ]
S r
f
S(f,g ) ” .
g
r ,g
f
r,
g
G,. g by
S, g
A
I
.
r, $4. Proof of Theorem 4. M ZFCl V = L .
+
[ l l , I, $31. p on
R
-=
Rxo G
on R x o ZFC
+ GCH. PA
P
R [ll],
.
P P R p(a, n) a . 5 P, F = uG. F is a < R F(a, n) w N = M[G]
R by
[ l l , I, 53.4,
r
r < S2,t
P, P Ax Q, (R -
P P G on P 11 , EM[Gc]. N,
G A= G nPA,G*
= E
a . As
Q x o.
G nQ .
N by
G‘ N
r
G‘. N.
140
AZRIEL LEVY
N . As
a
$2,
bET Gr .
a,
5, b T $3.4,
Gr , N
21
r.
KO N.
T
$5. Proof of Theorem 5. M Z F +V = L P of z w, p. 134, $3. 21 N.
A >0
A
by
AtM' 41
p. 1301
z.
K,; by
N
p. 132,
6. T h e following holds in N : I f T is a set of real numbers which is real-ordinal-dejinable then there is a real number r, such that either T G L [ r ] , and then, TI 5 K1,since there are at most K1 reals constructible f r o m a single real, or else 7'1 = K,.
I
I
5
6.
of
Proof of Lemma 6 (beginning). d N. nEd nEw, A ={aI(3p~Q)(3k~w) A z 4 A support d . go on 6.
a
$3).
7. I f d is the name in C of a real number a of N , and A is the support of d , then a L[GA] holds in N . Proof. M
4 =
$3
a E M [CAI.
N
L
8. Let @(x) be a formula of L' with, no free variables other than x, and which does not contain the symbol G . Let A E z A E M ; by @(GA) we denote a sentence of L' which asserts, in a natural way, that GA satisjes @(x) . I f ' p P and p IF @(GA),then already p ( A x w) It @ ( G A ) .
I
I
p (A x
Proof.
IF @(GA),
=
I
p ( A x w)
P, c(
{((a,n), k)} E G {((a,n),
M [ G ] 1 (D(GA). q n <w k E {0,1} 3.21 q . q p ( A x w)
I
DEFINABILITY IN AXIOMATIC SET THEORY I1
G
P
I ( A x w)), M (p’)
141
(p) 2
=
I
q 2 p ( A x w). o p ‘ P o(p‘)
by
=
(u, n) E
[ll, = N .
2.11 o(G) o(q) = p , qEG P, p It @(G,), in M[o(G)] = N , @((o(G)),) N. on A x w, (o(G)), = G,,
M[o(G)] = M[G] p E o(G) . o(G) o(G) @(GA) p 4 @(G,) N,
LEMMA9. LetA,B 5 z , A , B ~ M , a n dl e t x ~ o , x E M I G , ] n M I C B ] ; XEM[G,,,] $3 Proof. M[GA]. u
x EM[G,]
C
u
y
by
M[z]
u
O1(y,z) M .) Q1(y, z)
u. (u
x
L’
Q1(y, z)
M.
L‘
G.
O1(y,z)
n
(24)
-= w , N
k O1(n,G,)
nEX. @,
XEM[G~] N k (Vn E w ) ( ( N k @,(n, G B ) ) f+ n E x) Q2(n, GB)). (G, L’
N k (Vn w)(@,(n,GA) m1(n, GA), p EG
f+
L’). (25)
P It ( v n w, [@l(n, GA)
(26)
x = {n E W (% E GAnB)(q u P It G,(n, GA))
-t+ @ 2 ( n , GB)]
*
I
(26), q € G A n BE G
N, n EX ,
I
n PEG q uP E G by (24), R E X . p’ G p ’ It D1(n,G,).
q’ = 4” ( A x w ) ; by
p’
p uq ” . 8
*
(26). @,(n, GA) is
p p‘ p ’ 2 P, p u It @,(n,GA). p u 4’ It @,(n, G,). p , p’
EG
142
AZRIEL LEVY
q p Uq
I
=
by pU ( B x 0); by
8
(25)
Q2(n.G B ) . =
q
ZF; p u It Q2)n,GB). It- Q2(n,GB) p u q It- Q,(n, GA).
(25) pu
Q 1(n, GA)
=
c p' G
q
n
GAnB.
(26). x
M[G,],
x
Proof of Lemma 6 (end).
(26)
MLG,]
.
T
N Y(x, y )
L',
T = { b c w l b ~ NA N ItY(a,b)). "(a,
L'). b T N.
A N by
r.
A do
L:,
B
d.
T
G
M , GA L[r] N
r,
6
6
6,
N 6 E w A h $ L[G,], p It
(27)
A 6 $ L[G,] A
ct
psi ( B )
ct
# /3 E
Y(d,
pa,
M t
ct,fi<~,
pi' x w
w
p
M
B ct
6c
<
A p,(y) = 0,
y
pBi ( B - A ) n p , - ' ( B - A )
A.
ct
=
< q
P
6,
,
=
I
{ ( P A Y ) , n> (Y, n>
(Y, n > (y, n> s.
n)
-
N
u
G P
(28)
(B)
G
L: .
u
7, F(a, s) L[(o,,,(G)),] G,;i
M.
n)
P. F(a,
=
(oa,s(G))B
N.
o,,,
by L[(ga,s(G))B]
=
M LIGp;l
(B)]
*
C
143
DEFINABILTTY IN AXIOMATIC SET THEORY I1
F(a, s>
l(B)]
*
z x o,
pLi
pa€G
=
pLi
,
=
.
= sa
=
1 p a ( y , n ) = 1 - p(pa(y), n)>
(Y, n>
Pa
*
F(a, s,) is
6
.
A,
z, z = KLM); by
z
by 7
sa
=
p.
KY,” oa,s
oa,Jpa) oa,,,(pa)= p .
by
p
P
oa.,
P 2.11.
=
pa€G
by oa,,(G)
by
=
~ E C T ~ , , ~As
oa,s,(pa)= p oa,JG),
=
,
p
E
oa,,=(G),
F(a,sa) =
L[GPi1 L[o,,,(G)),] = L[G,]. a, f l < T , a # fl F(a,sa)= F(P, ss). =
pa on by F(a,sa) 4 L[GA]. F(a, sa) # F(/?,ss) F(a, s,) E LIGPLl,,)]
9 F(a, sa) E L[Gp;i(B)np-j~ pa p,’(B) n p S l ( B )
.
=
.
AZRIEL LEVY
144
L(G,). (pa- ' ( B ) nps- ' ( B )
F(a, s), G p , l ( B ) n p B I ( BL) [G,].)
F(a,
# L(G,),
F(a, sa)= F(P, ss)
F(a,
a
. o!
F(a,
"(a,
F(a, s,))
.
N, S) =
(O,,,*(G))~
= a.
A
=
d
I
S ) = { n w ( 3 p G,) ( p It n E =
I
n)} =
w (3P E (~z,sz(G)).4) (P
=
n
@>
7
(a,,sw(G)), = GA.
by
q GA.
=q
c Axw
q E G,, on A ,
pa
by q G,, y A
(y, n ) E
=
=
= p,(y) = y.
by
s, n = q(y, Jq)) =
c
IT,.^=(^)) ( y ,
s,
nA
x w = 0, (y,
(9);
a, J q ) = q ,
YEA;
#
by
s, nA x w = 0 , #s,; p s,, $ s,,
s,,
pa
=
by
REFERENCES W. ADDISON,Some consequences of the axiom of constructibility. Fundamenta Mathernaticae 46 (1959), 337-357. COHEN,Set Theory and the Continuum Hypothesis. New York, 1966. [2] P. [3] S. FEFERMAN and A. LEVY,Independence results in set theory by Cohen's method I1 (abstract), Notices of the Amer. Math. SOC. 10 (1963). [4] A. HAJNAL,On a consistency theorem connected with the generalized continuum hypothesis. Acta Mathematica Academiae Scientiarum Hungaricae 12 (1961), 321-376.
I1
145
[ll]
Definability in axiomatic set theory I . 1964 1965, pp. 127-151. Independence results in set theory by Cohen’s method I , III, IV (abstracts). of 10 (1963), 592-593. A generalization Giidel’s notion constructibility, 25 (1960), 147-155. SCOTT, Ordinal Definability. 1967 on On the independence of the axiom constructibility. of 81 (1959), 536-540. Mathematical Logic. 1967. modelof set theory in which every set reals is Lebesgue measurable,
1121
Zur Axiomatik der Mengenlehre (Fundierungs- unda Auswahlaxiom).
[5]
[6] [7] [8] J. [9] [lo] J.
3 (1957),
173-210.
