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Projective Logics and Projective Boolean Algebras (*)
Non-Classical Logics, Model Theory and C o m p u t a b i l i t y , A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland P u b l i s h i n g Company, 1977
F'ROJECTIVE LOGICS AND PROJECTIVE BOOLEAN ALGEBRAS (*I by R I C A R D O hlORAIS
I, INTRODUCTION, Lusin and S i e r p i n s k i s t a r t e d , i n 1325, the development of the theory of p r o j e c t i v e sets ( c f . Lusin 1925 and Sierpinski 1925) b u t soon afterwards t h i s research p r a c t i c a l l y ended due t o the complexity of the problems and the lack of b e t t e r t o o l s t o work with. I t was not u n t i l t h e l a t e s i x t i e s t h a t some new r e s u l t s were ( c f . Fenstad 1971, Moschovakis 1973, and Kechris 1973) using t h e proposed, b u t s t i l l questionable, axiom of Addison and Moskovakis 1968). Recently A. Nerode thought of develop
obtained recently
" P r o j e c t i v e Determinancy"
a l o g i c LA t h a t would b e s u i t -
a b l e f o r t h e study o f t h e a n a l y t i c s e t s , o r p r o j e c t i v e s e t s of level L
WlW
(cf.
was t o Bore1 s e t s . This was done by h i s s t u d e n t P . J . Campbell,
1 , as and
f u r t h e r strengthened in E.Ellentuck 1975 u s i n g d i f f e r e n t methods. This paper follows a sugestion of Eilentuck of t r y i n g t h e g e n e r a l i z a t i o n of t h i s approach t o a l l l e v e l s of the p r o j e c t i v e hierarchy. [ * ) The a u t h o r i s i n d e b t e d t o P r o f e s s o r E l l e n t u c k f o r t o Rutgers U n i v e r s i t y (U.S.A.)
and
support.
20 1
h i s o r i e n t a t i o n , and
CAPES ( B r a z i l ) f o r
their financial
202
R I CARD0 MORAl S
11, PROJEC I V E LOGICS, 0, PRELIM NARI ES s
Let [hey]’ denote’the s e t of a l l n - t u p l e s (uZ, ...,w n ) of f i n i t e s e quences of natural numbers satisfying e ( ~ , )= ... = e ( ~ ~where ) , L(u)denotes the length of u . Throughout t h i s t ex t 6 and g will denote elements of %; n ; m, and k will and f o r The subsets
be natural numbers and u will stand both f o r elements of [neqJn”, elements of ney the s e t of f i n i t e sequences of natural numbers. most important tool developped in t h i s work i s a pair of families of of [hey]’, denoted respectively by FLLeen and F u l l ;
.
D E F I N I T I O N 1 . a) F E F L U , ( F 0 a Fd2l-n b e t ) i d and vney ii F c [neq]’ and F 402%6iu t h e 6aUvwing cvndition giuen by a 4.7XLng 0 6 quant.idiehn
w i t h n aetmnntionn: (3
I
d,)(v dz)*..(Q6,)(Q’k)((d,
I
k,**-,&,
k ) E F)
whme Q and 2’ ahe din-tinct yuanLL&iem. S , i m d ! d y we
b)
dedine
G E F u l l ; (G 0 a F L L e e - v ~ - n Z m n e t ) id and vney i d G c [ n e d n und
(v
61)(362 )-.*(Q’6,)(Qk)((d,
I
k 9 * * * r 6 n
I
k) E GI.
F and G will denote elements of F u l l and F u l l * respectively,and the word countable will always mean e i t h e r f i n i t e or denumerable. Finally, i f @ = {@iI i 8 I ) i s a countable s e t of formulas o f a formal I\ $ . or simply I \ ~ $ ~ language, the conjunction A @ will also be written i e 1 4. i f no confusion a r i s es .
1, THE
LANGUAGE L~~
We s t a r t with a f i r s t order logic L with countably many re la tion, funcbe the infinita rylogtion and constant symbols, and w l variables. Let L W1W
by adi c over L as defined in Keisler 1971, p . 6. We obtaion L from L Pn WIW joining two new operators P, and P; i n the following way: F o m a of L ar e those of L with the addition: i f @ i s amap from Pn W 1W [neq]” i n t o formulas of L then Pn(@) and P i ( @ ) are formulas of L Pn‘ Pn Extend the notion of n u t h 6 a o t i o n by defining:
203
PROJECTIVE LOGIC
i f f ( 1 F € F L L W M ) ( w w € F ) CR C$(w) [ h ]
[h]
(1) CR CP,($)
( 3 G € Full;)( W w € G) d ! = $ ( w )
iff
d bpi(@) I h ]
O c c m e ~ c ao f v a r i a b l e s and c o n s t a n t s i n P a ( @ ) and $(u) f o r
be those i n t h e c o l l e c t i o n o f a l l
w € [neq]'
[h!
P i ( @ ) are .
defined t o
D e f i n e d - w a e i d i t y and w a e i d i t y i n t h e usual way. The n o t i o n o f nub&mnula o f K e i s l e r 1971, p. 11 (2)
= {PM($)j
Sub(P,($))
i s extended by
uw S u b ( $ J ( v ) ) .
u
Uw S u b ( $ ( v ) ) *
S u b ( P ; ( @ ) ) = { P l ( @ ) }IJ
Extend t h e n o t i o n o f t,iouing Rhe c z e g d o a i n ~ i d e(cf. K e i s l e r 1971, p.11) as f o l l o w s : (3)
where
(P,J@)) 1
is
Pi(%)
(Pi($))1
is
P,(l@)
I
l$ i s t h e map t h a t takes w i n t o
Axivmn o f L
P"
l$(w).
w i l l c o n s i s t o f t h e n i n e axioms f o r p r o p o s i t i o n a l
as p r e s e n t e d i n B e l l and Slomson 1969, p . 36, p l u s t h e s i x axioms f o r as i n K e i s l e r 1971, p. 15.
logic
Lwlw
The Rda ad Irz@tence w i l l be (Rl)
Modus Ponens
(R2)
i f I-$
(R3)
i f f o r every
$ S @ t- J, +I$
(R4)
i f f o r every
F € FullM
(R5)
i f f o r every
G
P;($)
Vx$
t h e n + $+
+$
€
I-
if
x i s not free i n
t h e n I-$
AF$(w) + $
Full; I- J, + VG$(w)
+
$J
.
AQ.
I - P n ( $ ) + VJ.
then then
I- $ +
P,*($).
ble o n l y have t o w o r r y about i n t r o d u c i n g P,($) because t h e f a c t s a r e e a s i l y g o t t e n f r o m those about P,($) u s i n g t h e f o l l o w i n g
tance o f t h e axiom + $ 1
PROPOSITION 2 .
w
I-Pi(l$)
H
1 $
.
Now b e f o r e we proceed w i t h t h e s t u d i e s o f L
b.tatemk?VLt
abvut w
me Rnue meta
-
ins-
lP,($)
p o r t a n t p r o p e r t i e s o f t h e f a m i l i e s FuUM and
P R O P O S I T I O N 3.
about
l e t us c o l l e c t some
Pn FILee;
.
in-
Suppane t h a t dvti ewmy W E [ n e q ] ' , x(v) o a m e t a l& Q and Q' be bn in U e 6 i n i t i o n I . Then t h e ~vUviu&g
.
htatemenb.
204
RICARDO N O R A I S
a) ( 3 61)...(2dn)(2'k)A(d1
I
k,...,6n
b) ( V ~ , ) . . . ( 2 ' l i , , ) ( 2 k ) A ( d l I k , . . . , c) ( 3 F E F u U n ) ( W v E
F)A(w)
I
k)
dn 1 k )
id6
4 5 6 (3 F i d 6 (36
Ffin)(Vv
E Fuu;)(Wv
E F)A(v) 8
G)X(u)
( W G E Full;) ( 3 w E G ) A(") ( 3 G E FuU;)
d) ( W F E F u l l n ) ( 3 u E F ) A ( u )
E
(Ww 8 G)A(w)
I t i s important here t o note the complete symmetry between t h e s e two f a m i l i e s of sets. I t i s tnis symmetry t h a t w i l l make possible the majority of our proofs, besides e l e g a n t l y reducing a l l the work i n h a l f . we Now, in order t o b e t t e r understand the behavior of t h e s e f a m i l i e s , have:
PROPOSITION 4 .
16
IT
> 1,
and H c : s e q ] "
E
= { ( " p . . . , " , )I
(5 I
[(",),
, Let
"2,..*>"11)
E
HI.
Then
a) H E F u U n
.id6
(36) (Hd E
b) H E Full;
.id6
( v 6 ) (HS
Fa;-,)
E FLLeeYl-,)
F i n a l l y the next proposition shows t h a t the f a m i l i e s F u l l M and n.
a r e well mixed t o g e t h e r . The proof i s by induction on
2,THE CONSISTENCY PROPERTY
F&f;
I
Let C be a countable s e t of constant symbols not appearing i n L . Let c E C t o L , and from bl c o n s t r u c t t h e l o g i c M PM A banic tm i s e i t h e r a constant symbol of bl o r a term of the form P" function d(t t k )where R L,...,t k a r e b a s i c terms and 6 i s a k - a r y symbol o f L . but The d e f i n i t i o n t h a t follows was taken from Keisler 1971, , p . 11, clauses here, besides adapting i t t o the present s i t u a t i o n (namely adding C9 and C9) we a l s o modified, t o simplify the proofs, the notion o f b a s i c bl be t h e f i r s t order l o g i c obtained by adding each
.
,,...,
teroi, and clauses C10 and C11.
DEFINITION 6. n e s:
A CoiuDtency PhUpULty D a be* S batin6y.ing
doh
each
205
PROJECTIVE L O G I C
to
The d e f i n i t i o n o f Consistency P r o p e r t y i s t h i s l o n g because we want have:
THEOREM 7 . and
40 E S ,
PROOF:
(Model E x i s t e n c e Theorem). 16 S 0 a C a ~ n O t e n c y P h u p e h t g
then
Without
han a modet.
40
l o s s o f g e n e r a l i t y we way assume t h a t each subset o f an e l -
elllent o f S i s a g a i n i n S . To c o n s t r u c t t h e model s a t i s f y i n g w i t h t h e s m a l l e s t s e t Y o f f o r m u l a s o f E.4
(i)
no
(ii) Y
Let and
T =
f o r which:
i s c l o s e d under subformulas.
(iv) i f l@ E Y If
c E C
then
a b a s i c term and
1
@(t)E Y then I$(t') E Y
c = t E Y.
be t h e c o u n t a b l y i n f i n i t e s e t o f sentences o f Y ,
{to,tl, . . . I be t h e s e t o f b a s i c terms. S t a r t i n g w i t h
h0
construct
an i n c r e a s i n q sequence o f elements o f S as f o l l o w s . Suppose we have 4nl+l
4,,, :
.
$1 8 Y .
and R i s a b a s i c t e r m t h e n
X = { I $ o , I$,,...
and we b u i l d
start
Y
(iii) I f t i s a t e r m , t '
(v)
Pn
oo we
h,,,,
R I CARD0 MORA I S
206 (1)
i f A,, U
{@,,,Ig
(2)
i f sm U
{a,}
=
A;+,
;
i s V@ then f o r some $ E @,
(2.1)
$ ,,
(2.2)
$m i s 3 x $ then f o r some c E C, A;+,
(2.3)
@m
(2.4)
i s P,($)
IAF$(v)}
U
{$,,,I
U {AG@(v)}
f i n a l l y , s i n c e i n any case Am+,
Next d e f i n e
e sw
c = d
L e t [c]
u
= A;+,
sw
=
.
{c =
um sm
fml
&A+,
s+ ;,
=
E
S,
=
o;+~
=
sm u {$,,,I E S; t h e r e i s c E C such t h a t
8 S,
E S.
and d e f i n e an equivalence r e l a t i o n on C by: c+d
c E C and l e t A =
be the equivalence c l a s s o f
Now f o r each k - a r y r e l a t i o n symbol P,
6,
l
sm u {$m}U { $ ( c ) }ES,
G E F a ; ,
This i s the universe o f the model t h a t w i l l satisf.y symbol
@
E S,
i s any o t h e r formula,
$,
=
E S,
i s P i ( @ ) then f o r some
$,
(2.5)
U
u {$mlu I
= A,,,
F E FU.ee,,s;+,
then f o r some
u {@,,,I
= A,,
(3)
sm
we consider t h e f o l l o h i n g cases:
E S
= A,,
iff
let
S
o f L define a r e l a t i o n
so
I [c] i
.
o f L and each k - a r y
c E C}.
function
Rm on Ah and a f u n c t i o n F,, from
Ak
i n t o A by:
(a) (b)
,..., r e k ] ) E Rm Fm( [c,] ,. ..,[ch]) = [c,] ( [c,]
Note now t h a t i f is A@,
@ E 6,
then
iff
Pm(cl
,..., c),
iff
co =
b,,,(c,,
E
...,ck)
E
.
and
0 E Y
f o r each
(a)
@
(b)
@ i s any o t h e r formula, then
8 E @ ;
$ E Y.
Then use t h i s f a c t t o show t h a t the s t r u c t u r e
a satisfies
=
so
1
m 6 wl, {F,,,
I
m E w}, A >
.
Theorem 7 i s a n i c e t o o l t o use i n the p r o o f o f
THEOREM 8. (The Completeness Theorem f o r
L
)
Pn
16 @ 0 a s e n t e n c e
06
207
LOGIC
PROJECTIVE
To show t h a t e v e r y theorem i s v a l i d we Drove t h a t t h e r u l e s o f i n -
PROOF:
ference (R4) and (R5) p r e s e r v e v a l i d i t y . Rule (R4).
(a)
Suppose
3F E FuU,,
VF
E
F u l l R , 02 t=
U? b AF@(u) A 1J,
A F @ ( w ) + J, t h e n i t i s n o t
t h e case
.
i m p l i e s t h a t 3 F E FuRe,,
B u t by D e f i n i t i o n 1, Ce CP,(@) and t h e r e f o r e i m p 1 i e s n o t UL C P,($)
A 1$ o r e q u i v a l e n t l y ,
that
U? I = A F $ ( w ) ,
CL c P,($)+$.
Rule (R5).
(b)
d C J, * VG $ ( w ) .Then a!= 1ji o r Suppose VG E F u l l ; , ( 3 u E G ) OZ k = $ ( w ) a n d h e n c e b y P r o p o s i t i o n 3, CR
( W G 8 FLU;) 11) or
U? C J ,* P,($).
( I F E FullR)(W w E F) CE i=@(v), which i m p l i e s
Now we have t o show t h a t e v e r y v a l i d sentence i s a theorem. I n o r d e r t o do t h a t we l e t S be t h e s e t o f f i n i t e s e t s o f sentences n o f o n l y f i n i t e l y many
c E C
o c c u r i n n and n o t I-
MA.
M
YJn
such t h a t
We t h e n show S i s a Consistency P r o p e r t y and t h e r e s u l t f o l l o w s hence
lip,
t h e n @ i s n o t a theorem i n
cause i f @ i s n o t a theorem i n L
r7M
{ I $ } E S. By t h e Model E x i s t e n c e Theorem @:
beand
has a model and t h e r e -
fore @ i s not valid. We e x e m p l i f y t h e p r o o f t h a t S i s a Consistency P r o p e r t y be p r o v i n g (C8) and (C9). (C8)
Suppose P,($)
Full,
U {A,@(u)}
WF
(WF E F u l l , )
( I- A F @ ( w )
1An); t h e n by (R4)
since
E n,
P,(@)
P,(@)
P,*(@)
Suppose
I-An-,
( I-
S.
I- P,(@)
+
1A 0
and,
lAn, a contradiction.
E n b u t ( V G E FLU;)
I- l A ( n u { A , @ ( w ) } ) ( W G 6 FILL$)
I-
+
E
e
I- lA(n U { A F @ ( u ) } ) , and so
E n, we have
Since
(C9)
n b u t (WF E FuU,)(n
E
f o r every
(n U { A G @ ( u ) }g S ) ; t h e n again
G E Full;
and
so
An + V G l @ ( w ) ) , which i m p l i e s , by (R5),
P,(l@).
Therefore, by P r o p o s i t i o n 2, a contradiction.
I-
An + l P i ( @ ) o r , e q u i v a l e n t l y , I- l h n ,
206
R I CARDO MORA I C
There i s another p r o j e c t i v e l o g i c of i n t e r e s t t o us, namely: DEFINITION 9 . h ~ n bowm w,
The logic L
P
0 dedined t o be t h e u n i o n oh & L
o h i n othetr ~0oh.d~:
(a)
ln L
(b)
The &en
UA
n
Pn($) 0 a domda doh ewmy n .
P '
06
(R4)
(Vn E
(R5)
( i n E w ) id
id
w)
Pa
indmence (R4) and (R5) now head
( W F E FuUn) I- A F $ ( w )+ $
then
I-
(VG E FLU;)
then
I-$+ P,($).
I-$+ VG@(w)
i s complete s i n c e a l l L are. P Pa There i s one important theorem p a r t i c u l a r t o L
Pn($) + $,
Obviously L
THEOREM 1 0 . (R4')
In L
P
0 a h.u&
t h e @f%LCLing
= 1)
id
V6 I- Ak'$(d
(b) (doh. n > 1)
id
Vd I-
(a) ( d o h n
whehe
$d
(w2,.
..,wn)
= $( 6
:
06 in&?kence: k) --f
I .t(w,),
P
+
$
$
w2,.
then
then
I-
+
PI(@) + $ ;
pn($)+
n-
;
.., w n ) .
We conclude this s e c t i o n with the remark t h a t t h e downward Skolem-Tarski theorem holds f o r both L and L P Pn *
111,
$J
Lowenheim-
PROJECTIVEBOOLEAN ALGEBRAS,
1 I NTRODUCTI O N , I
In t h i s s e c t i o n we d e f i n e a new kind o f Boolean a l g e b r a s , c a l l e d n-proj e c t i v e Boolean a l g e b r a s , which a r e g e n e r a l i z a t i o n s of t h e S u s l i n algebras introduced by L . Rieger in 1955 ( c f . Rieger 1955). Our work, however, i s patterned a f t e r a recent paper by E . E l l e n t u c k (Ellentuck 197+) i n which he s t u d i e s the S - a l g e b r a s o f Rieger based on his previous paper on S u s l i n l o g i c (Ellentuck 1975). R i e g e r ' s idea with t h e S u s l i n algebras was t o provide a s t r u c t u r e i n which one could model nn1 a n a l y s i s .
1. BASIC
nt
a n a l y s i s . Our algebras a r e intended t o help model
RESULTS,
Let B be a Boolean algebra.
209
PROJECTIVE LOGIC
The joim and nieeA of B w i l l be denoted r e s p e c t i v e l y by Sup and 7ng. The iizditzite j o i n of t h e family {bi 1 i E I } i s denoted by Sup bi or simply by Sup bi
i
is7
i f i t c l e a r which s e t 7 i s .
I f Q i s a map from [neq!" i n t o 6 we s h a l l use the n o t a t i o n Pit($) f o r the following element of 73, provided i t e x i s t s : Pi*($)
=
and, s i m i l a r l y ,
SUP
F
Ini( $ ( w ) , wEF
where, a s u s u a l , F runs over The symbols P,(@) and and t h e previously defined confusion.
DEFINITION 1 2 . (W
- PBA
ijoh n h h t )
F u l l n and G over
FU.eek
.
P;($) w i l l be used both f o r t h e above suprema formulas o f Lpn, b u t t h i s should l e a d t o no
A u - B C J C J ~d~ gU e~ b~ t ~ ~ U M w - I 3 4 O j ! L d W t 600tea~d g c b h n id Lt 0 it - PBA dot C V U i ~ i E W.
Formula ( 4 ) i s a very powerful d i s t r i b u t i v e law and not a l l algebras closed under Pit and P i s a t i s f y i t . In f a c t , t h e r e a r e complete B o o l e a n algebras i n which ( 4 ) f a i l s . In our work, however, we need t h i s d i s t r i b u t i v i t y t o t i e t h i n g s up ( s e e f o r example condition ( 6 ) below), and we a r e thus forced t o introduce i t a s p a r t of t h e d e f i n i t i o n . To g e t an example i n which ( 4 ) f s i l s s e e Morais 1976. Another way t o see the importance of ( 4 ) i s t h e next proposition which presents t h r e e e q u i v a l e n t formulations of ( 4 ) .
PROPOSITION 13. 7 6 B 0 c( Bootenti d g e b h a i n t o B , tt,t -I$ be t h e tNnp deijined by (-
whehe - 0 t h e nytnbat eqUiWdent:
604
Q)( w )
=
-$ (w)
c a t i i p L ~ i e n t d L oi ~n ~ B
aid
Q 0 a riiap
. Theit tlze
6hotti
[Aeq]'
60ttCJdt7g
ah&
210
RICARDO
MORAIS
Now, using these equivalences, we can get several properties of projective Boolean Algebras, namely:
PROPOSITION 1 4 . A u - B o o L ~ ~dMg e b h a B 0 n-PBA i6 m d Only i6 d a s e d u n d e h the P i a p e h a t a h and (4) holds. PROPOSITION 1 5 .
16
M > 1
PROPOSITION 17. Evehg
whehe
w
PROPOSITION 1 8 . PROOF:
Now
M-
PBA, then B 0 ( n - 1) - PBA.
an example.
The cornple*e B a o l e a ~d g c b m 2
= {O
,I} 0 w - PBA.
Since 2 i s complete we have just t o show ( 4 ) holds i n 2 .
P,($)
=
0
i f f sup In6 - $ ( w ) . = G uEG
.
B 0
- PBA ~ a t i n 6 i e A :
iotoak-n add
iff
i f f (by Proposition 3 )
holds
M
aMd
8 0
Sup In6 $(u) = 0
F uEF ( 3 G E FuRe;)(Wu
1 iff Pi( - $ ) = 1
i f f (WF E FuRen)(3w E F)(@(u)=O) E
iff
G)($(u) = 0 ) i f f In6 Sup $(u) = O G uEG
- P i ( - $ ) = 0 and therefore (6)
211
PROJECTIVE L O G I C
The most i m p o r t a n t example o f an w - PBA however i s g i v e n by t h e
fol
-
lowing: The Lindenbawl dgebaa L
THEOREM 1 9 . w
- PBA.
PROOF:
Let
1
@
I
06
P
,the w - pfihujedue Logic
LP
denote t h e e q u i v a l e n c e c l a s s o f t h e f o r m u l a @ i n L
P
.
i n t o L and d e f i n e a map $J from [bey]" P by choosing f o r each W E [hey]' a r e p r e s e n t a t i v e f o r -
L e t @ be a map f r o m [hey]" i n t o f o r m u l a s of L
P
mula @(u) o f t h e e q u i v a l e n c e c l a s s
(8)
P,(@) = !pn(@)l and hence
T,
We t h e n show
(li(w).
,
F i r s t we have t o p r o v e t h a t t h e f o r m u l a choice o f t h e map
Pn o p e r a t o r .
i s c l o s e d under t h e
P
Pn(@) does n o t depend on t h e
@.
I t s enough t o show t h a t f o r any o t h e r map
'Ju E [ ~ e y ] " I- @ ( w )
+
$(u) t h e n
I-
JJ
:
if
Pyz(@)+ P , ( ~ J ) .
By (R4) t h i s f o l l o w s f r o m (9)
VF E F a n I-
AF@(u)
+
pn($)9
which i n t u r n f o l l o w s from, (WG E F u R e V : ) ( V F E
Fan)
(by R5), I-
AF@(w)
+
VG$(u).
B u t t h i s i s t r i v i a l s i n c e by P r o p o s i t i o n 5, g i v e n any F and G , F n G # @ . T h e r e f o r e (9) h o l d s . Now t o f i n i s h t h e proof of Theorem 19 we have t o show t h a t t h e d i s t r i b u t i v e law (4) holds i n L
P'
We s h a l l need,
(10) P i ( @ ) = I P p 4 which i s e q u i v a l e n t t o ,
S U P I AG @ ( u ) I = I P i ( $ J ) I G and so we have t o prove: (i) ( W G E F f i i )
I
I- h G @ ( V )+
Pi(@)
and
(ii) I f (WG E Fufl;)
I- h G $ J ( u + )
11 t h e n
I-
P i ( @ )+ $
.
212
RICARDO MORAIS
PROOF of (i):
From p r o p o s i t i o n 5 g e t
(WG E FU.eei) (WF E F d n )
+
AF l$(v)
+
VG l $ ( v )
now a p p l y (R4) and use P r o p o s i t i o n 2. S t a r t w i t h the hypothesis
PROOF of ( i i ): (WG € FULL;)
I-
1
$J
+ VG 1 $(v),
t h e n a p p l y (R5), and use P r o p o s i t i o n 2. F i n a l l y ( 8
,
(10) and P r o p o s i t i o n 2
give
Pn(@)
=
-
P,*(
- 0)
and t h e r e f o r e ( 6 ) h o l d s , which i s e q u i v a l e n t t o ( 4 NOTE: L
PM
E v i d e n t l y e x a c t l y t h e same p r o o f shows t h a t t h e Lindenbaum a l g e b r a of
(denoted L
PM
) i s n - PBA.
3, FREE n - PROJECTIVE BOOLEAN ALGEBRAS, DEFINITION 2 0 .
An nP
-
BooLean d g e b t a 0 a a - ho-
Izornornotpkm b-een
momohpkinm t h a t pk,hedmve~t h e Pn opehatoh. An W P - homomotpkinm 0 a u - honiomohpkintn
&at p u e h v e n Pn d o t evehy n
DEFINITION 2 1 .
L e t B be m
- genehaten B i d
(a)
G nP
(b)
G dheely
E w.
n - PBA and G
c
B. Tken:
B 0 t h e nm&ent
n - PBA containing G
.
nP- genmaten B i d G nP-genehaten B UJ~C! in a d d i t i o n given m y o t h m n - PBA B' and m y map h : G + B' t h e h e i b an nP - homomohpkidm H : E + B' w h i c h extend6 h
.
- net
06
gen -
An n - PBA 0 a dhee nP- d g e b h a i d contaia n P - n e t ad g e n ma to a . S . i m . 2 d y , dedine a 6hee ~ P - u Q e b h a .
a
dhee
(c)
S .in i. 2 dy dedine W P - neA 06 genehatom and 6hee
WP
ehato4,5.
DEFINITION 2 2 .
I f i s a common p r a c t i c e . i n any t e x t about " f r e e " s t r u c t u r e s t o
first
t a l k about i t s uniqueness and a f t e r w a r d s t o prove i t s e x i s t e n c e .
The
l o w i n g two p r o p o s i t i o n s a r e proven i n t h e same way i t i s u s u a l l y
done f o r
general Boolean a l g e b r a s . See f o r example Halmos 1963, p. 42.
fol-
213
PROJECTIVE L O G I C
PROPOSITION 23. 76 B 0 u dhee n P - d g e b h u , G t h e he,t 06 6hee nP-genefu7Xoh.S and h .the given map 6honi G into .the n - PBA B',then t h e nP - hamomahpkinm H : B + B' t h a t extendh h 0 unique. PROPOSITION 24. Any &oo 6hee n P - d g e b h a whohe .the hame catr&&y atre nP-0oma5pkic.
h d
0 6 genmatom
have
Now t o p r e s e n t an example o f a f r e e U P - a l g e b r a ( t h e e x i s t e n c e o f a f r e e n P - a l g e b r a i s proved s i m i l a r l y ) we proceed as f o l l o w s . F i r s t d e f i n e a phOpOh.iJ%onCdl o g i c LK f o r each c a r d i n a l
K
and t h e n
show t h a t t h e Lindenbaum a l g e b r a L~ o f LK i s a f r e e U P - a l g e b r a
we
with
K
generators. LK i s g o i n g t o have a s e t o f
{Pa j
c1
< Kl
K
variables
,
and t h e p r o p o s i t i o n a l c o n n e c t i v e s 1 and A o p e r a t o r s P,? and P;
. As
in L
PM '
we i n t r o d u c e
and l e t t h e s e t o f f o r m u l a s be t h e l e a s t s e t such t h a t
.
(a)
pa
(b)
if @
(c)
i f 0 i s a c o u n t a b l e s e t o f f o r m u l a s t h e n A @ i s a formula.
(a)
i s a f o r m u l a f o r each o r d i n a l
c1
<
K
i s a f o r m u l a t h e n so i s l @
i f @ i s a map from
[heq]' i n t o f o r m u l a s t h e n P,(@) and Pi(@) a r e
formulas ( f o r every
n E w).
Define "riaving t h e negation h i d e " f o r formulas o f LK as we d i d
LPn
the
for
with the addition:
For axioms t a k e t h e n i n e axioms o f p r o p o s i t i o n a l l o g i c as i n B e l l and I- @l*l@ and I- A @ + @, where @ i s a c o u n t a b l e
Slomson 1969, p. 36, p l u s s e t o f f o r m u l a s and @ 8
@.
For r u l e s o f i n f e r e n c e t a k e those o f L p n w i t h t h e e x c e p t i o n o f (R2). A r e a l i z a t i o n o f LK i s a map 2 =
6
f r o m t h e s e t o f v a r i a b l e s i n t o t h e w-PBA
{ o , 1 1 , which i s i n d u c t i v e l y extended t o a l l f o r m u l a s as f o l l o w s :
(4 d(l@)
=
- 6(@),
(b)
d ( A 0 ) = In6 d ( @ ) ,
(c)
6 ( P n ( @ ) ) = P n ( 6 ( @ ) ) and
$80
b(P;(@))
= P;(d(@))(
214
RICARDO M O R A I S
6(@) i s
where
the map defined by
6($)(u)
=
6 ( @ ( u ) ) for u
€ [neq]"
. 6.
We say t h a t a formula @ i s valid i f d ( @ ) = 1 in a l l realizations Now, before we prove t h a t LK i s an UP-algebra on K generators, need: 8 be an w P - d g e b h a and
PROPOSITION 2 5 . L e t ablu
06
LK & t o
B . EXtend
6
.to a l l d
o
6
we
any map dhom .the u a h i
m by~ trdu (a) X h h o u q h
-
(c).
Then
imfiu
I-dl
A($)
= 1.
In pa)Lticdah, by PhopohLi5on 18, eue-hy theohem oh LK 0 v a l i d . F i r s t note t h a t because of properties (a) and ( b )
PROOF:
(11)
d ( @ + $)
= 1
i f and only i f
d satisfies:
d(@) 5 6 ( $ ) .
I t i s routine t o show t h a t the axioms are mapped into 1 , b u t we check, as an example, t h a t the axiom $1 -l@ i s mapped i n t o 1 f o r the case @ i s Pn($). By (11) we have t o show, B(PYl(VJ)1 ) = 6 ( 1 P n ( $ ) )
.
But 6(Pn($)1 1 = P;(
-6($))
6 ( P,*(l$))
= -Pn(6($)) =
= p;(6(1$)
-
=
6(Pn($)) = 6 ( 1 P n ( $ ) ) *
where the fourth equality follows from ( 6 ) . Similarly, using ( 1 1 ) i t i s easy t o prove t h a t the rules of inference preserve the property of being mapped into 1 . As an example we check f o r (R4). Suppose W F € FU.een,
6 ( h F @ ( u *) $ ) = 1
and
we
have
to
show
d ( P n ( @ ) * IrJ) = 1 Sy (11) and property ( b ) we have ( V F E F a Y l ) In6 Therefore
UEF
b(@(U))
5
A($)
*
d(@(u)) 5 d($). F u€F B u t by definition t h i s i s P n ( 6 ( $ ) ) 5 6 ( $ ) , and hence 6 ( P n ( @ ) )5 ~ ( J J ) T h u s by ( 1 1 ) , 6 ( P n ( @ ) * $1 = 1 . We therefore conclude t h a t every theorem of LK i s mapped into 1. SUP ,In6
.
PROJECTIVE L O G I C
215
We a r e now i n o o s i t i o n t o show THEOREM 2 6 .
LK 0 a
64ee wP - d g e b t u an exac.tQ
K
genmcLtau.
F i r s t i t i s c l e a r t h a t t h e same p r o o f used t o show t h a t
PROOF:
L
w-PBA (Theorem 19) can be r e p e a t e d h e r e t o show LK i s w-PBA. Next l e t G = { ! p a l la gebra B t o g e t h e r w i t h a map
h : G + B.
6
=
~ ( u J , )
6
and l e t t h e r e be g i v e n an a r b i t r a r y wP-al-
K)
Now u s i n g h d e f i n e a nap
and e x t e n d
f r o m t h e v a r i a b l e s o f LK i n t o B by
h ( / p a1
ly
i n d u c t i v e l y t o a l l f o r m u l a s o f LK
.
By P r o p o s i t i o n 25 and (11) i t i s easy t o show t h a t every equivalence class
was
P
I @ 1 , and
is c o n s t a n t
so t h e f o l l o w i n g i s a w e l l
in
defined
map
f r o m LK i n t o 8 :
H(i@ This
I1
d(@).
=
H i s t h e d e s i r e d U P - homomorphism e x t e n d i n g h , and hence i t o n l y
remains t o show t h a t t h e c a r d i n a l i t y o f G i s given
a ,B <
K
with a # 6
i s n o t a theorem and hence
4, A
,
K
.
But t h i s i s e a s y ,
P r o p o s i t i o n 25 can h e l p t o show t h a t pa
I pa I
#
for H
I pB 1 .
REPRESENTATION THEOREM FOR FREE nP-BOOLEAN ALGEBRAS
pB
I
We s t a r t t h i s s e c t i o n w i t h a completeness theorem f o r L K . T h i s i s done t h e same way we d i d f o r L
Pn
and so we o m i t t h e p r o o f , a l t h o u g h we p o i n t o u t
t h e b a s i c p o i n t s . F i r s t we d e f i n e :
216
RICARDO M O R A I S
ththetle
1 5 S 0 a K - CoMnOtency P h O p M y and oo E S t h e n a h e ~ z c L t i o n 6 0 6 LK doh rukich d ( $ ) = 1 doh & @ E no
PROOF:
T h i s p r o o f i s p a t t e r n e d a f t e r t h e one f o r t h e Model E x i s t e n c e The-
PROPOSITION 2 8 .
.
orem (Theorem 7 ) . We s t a r t no and c o n s t r u c t a sequence (A,) o f S w i t h t h e d e s i r e d c l o s u r e p r o p e r t i e s . Then l e t map f r o m t h e v a r i a b l e s o f LK
d(PJ
d
Then e x t e n d quence (A,)
no =
i n t o 2 by
iff
= 1
o f elements o f
u nm
m
and d e f i n e a
Pa e
i n d u c t i v e l y t o a l l f o r m u l a s and because o f t h e way t h e
se-
was c o n s t r u c t e d we have
A($)
= 1
+ E nu .
for all
F i n a l l y , we have :
PROPOSITION 2 9 .
16 $ 0 not a
6(@) 0.
theatem 06 LK then doh
bOMe
tluLizaLLon,
J u s t l i k e we d i d f o r 1 we show t h a t t h e s e t o f a l l f i n i t e s e t s Pfl o f f o r m u l a s o f LK f o r which n o t I-1 h b i s a K - Consistency Property.Then
PROOF:
use P r o p o s i t i o n 28 t o g e t t h e r e s u l t .
An n P - d i d d 0 6 A& 0 a 0 - 6 i & l 06 A & 16 a 0- d i d d 06 b& 0 cloned undm Pn w e c a t t it an w P - 6ieLd 0 6 ~ t . t b .
DEFINITION 30.
cloned u n -
dm t h e opetlatoh Pn.
doh
n 8 o
Notice
evmy
t h a t we d i d n o t m e n t i o n any d i s t r i b u t i v e l a w here. T h i s however
i s no s u r p r i s e because we have:
PROPOSITION 31. Eumy nP THEOREM 3 2 .
- @Ld 06
n&
0 n - PBA.
(heSpecFOX each cahd+u?l K t h m e 0 an nP - 6.ietd 06 A & 06 b d ) that 0 n P - g e n m d e d [ ~ P - g e n e h a t e d ) by K 06
L L v d y U P - d.ieXd ia%
dements.
PROOF:
Let
X = ZK be t h e s e t o f maps f r o m
K
into
2 =
l o , 11 and d e f i n e
PROJECTIVE L O G I C
a <
f o r each
Next, l e t let
BKn
taining
217
K
Q
9, =
r6 e
= {g,
I
(respectively
ZK
I 6(.)
=
11
a<
K}
BK)
be t h e s m a l l e s t n P - a l g e b r a ( U P - a l g e b r a ) con-
which i s a s u b s e t o f t h e power s e t o f X,and
2. BKM and BK a r e
Since t h e power s e t o f X i s a complete f i e l d o f s e t s , well defined.
Q i s K t a k e a # B and choose 6 ( a ) # tj(0). Hence i f , say, d ( a ) = 1 t h e n 6 E g,
F i n a l l y , t o show t h a t t h e c a r d i n a l i t y o f any map but
6B
6 E
2K
gB,
f o r which and t h e r e f o r e
g,
'go.
Now copying what we d i d f o r LK we c o n s t r u c t a p r o p o s i t i o n a l l o g i c LKn f o r each n E w i n such a way t h a t t h e i r c o r r e s p o n d i n g Lindenbaum algebras
LKn a r e f r e e n P - a l g e b r a s .
Our r e p r e s e n t a t i o n theorem f o r
M P - a l g e b r a s i s an immediate consequence
o f the next very importdnt proposition.
PROPOSITION 3 3 . ~ ~ P - i A o m o t ~ p kt ioc BK L~ 0 iA MP - iAomohipkic t o BKn. PROOF:
; and 6 o t ~e v w y
We p r o v e o n l y t h a t L~ i s WP - isomorphic t o BK
The w P - i s o m o r p h i s m H : L~
--f
n E w, LKn
.
BK we a r e l o o k i n g f o r i s d e f i n e d i n d u c -
t i v e l y by: (a)
For every o r d i n a l
(b)
H ( I I@ 1 ) = H (
(c)
H(!AQl) =
(a
I
fl
@
c1 < K , H ( I I ) ' , where A'
p a / ) = 9., denotes t h e complement o f A .
H(l@:).
@ E@
H(IPI1(@)l) =
uF v EnF
H(I@(U)I).
T h i s d e f i n i t i o n makes H an U P - homomorphism, and we have t o show i t i s o n e - t o - o n e and o n t o . of
To show H i s o n e - t o - o n e we d e f i n e f o r each LK by
d'(P,) ( o f course e x t e n d i n g
6'
=
6
E ZK a r e a l i z a t i o n
6'
5(.)
i n d u c t i v e l y t o a l l formulas).
Next, by i n d u c t i o n on t h e c o m p l e x i t y o f @,we show
H(l@l) = { 6 € ZK1f(@)=1).
218
R I C A R 0 0 MORA I S.
F i n a l l y , we have t o prove t h a t i f H( t h e 1 o f LK But i f
.
H( 101 ) =
2K t h e n f o r e v e r y
e v e r y r e a l i z a t i o n o f L, t i o n 29, and hence
]@I
satisfies
101) i s t h e
6
€ 2K
1 of
, 6' (0) =
@ T. h e r e f o r e
.
BK
then
1 $1
1 which means
is that
0 i s a theorem b y Proposi-
i s t h e 1 o f LK L a s t l y , s i n c e t h e image o f L , under H i s an U P - a l g e b r a which c o n t a i n s
2 , the U P - s e t o f generators o f
BK
,
Now g i v e n any U P - a l g e b r a 8 , l e t
we have t h a t H i s onto. K
be t h e c a r d i n a l i t y o f t h e s e t 8 .
Since LK i s a f r e e U P - a l g e b r a we can g e t an wP- homomorphism f r o m L, o n t o
B. Therefore the previons p r o p o s i t i o n gives: THEOREM 3 4 . (a)
(The R e p r e s e n t a t i o n Theorem f o r P r o j e c t i v e A l g e b r a s ) .
Any nP - dyebha h an nP - homomohpkic h a y e
n&. (b)
Any P - dgebha 0 an
UP - kotnomohphic
huge
06 an nP - 5 i e L d
06
an UP - 6 i e L d
o6
06
beh.
IV, CONCLUSION, Our r e p r e s e n t a t i o n theorem f o r f r e e p r o j e c t i v e Boolean a l g e b r a s p r o v i d e d US
w i t h a " b r i d g e " f r o m l o g i c t o s e t t h e o r y , b u t so f a r n o t h i n g was s p e c i
f i c a l l y shown so as t o g i v e a r e l a t i o n s h i p between t h e p r o j e c t i v e f i e l d
-
of
s e t s and t h e p r o j e c t i v e s e t s o f L u s i n and S i e r p i n s k i . Our t e r m i n o l o g y t h e r e f o r e l a c k s some j u s t i f i c a t i o n , which i s however g i v e n b y t h e f o l l o w i n g
and
l a s t theorem: THEOREM 3 5 .
Foh n > 0 ,
1
.i~ an n - phojeotiwe 6 i e l d
06 b&,
whehe
A,
6Zand6 doh "boLd6ace A". PROOF:
( f o r a d e t a i l e d p r o o f p l e a s e see M o r a i s 1976).
We w i l l show t h a t
i s c l o s e d under t h e Pn,
b u t t h i s i s not enough,
however, t o p r o v e t h a t . i t i s n - p r o j e c t i v e because t h e d e f i n i t i o n n - p r o j e c t i v e algebra s t a r t s w i t h a o - a l g e b r a .
B u t i t i s easy t o see
t h e same argument used below can be r e p e a t e d t o show t h a t L I ~ + ~i s
o f an that closed
under t h e P1 o p e r a t o r , and t h i s i n t u r n i s a g e n e r a l i z a t i o n o f c o u n t a b l e
219
PROJECTIVE L O G I C
unions and i n t e r s e c t i o n s ( c f . Kuratowski and Mostowski 1968, p. 341). L e t now @ be any map f r o m
x
E
1
i n t o Qn+l,
[AQQ;'
1
P,($) can be g i v e n b o t h by a Jn+l and a By P r o p o s i t i o n 3
x
P,($)
E
JA+l
a n d we
show
that
predicate.
has two e q u i v a l e n t f o r m u l a t i o n s , namely:
(a)
( 3 F E F U . e e n ) ( W w E F)(x E $ ( w ) )
(b)
(WG E FU.eei)(3v E G ) ( x E @ ( w ) ) .
and
We a r e g o i n g t o use (a) ( r e s p e c t i v e l y (b)) t o show t h a t g i v e n by a
+
F i r s t , since w onto
i s countable, there i s a
[bey]"
F and
s t i t u t e the sets
1- 1 r e c u r s i v e map
by means o f X
{O
G by t h e i r r e s p e c t i v e
tw
E [bey]"
, 1 1 , and as we d i d f o r
I g(w)
= 1) E
The e x p r e s s i o n range
(Wm
.
I n addition,
= 0
{ w 8 [neq]'
ik
.
)'i+lp r e d i c a t e .
@(k) E
[(tlange g
=
{O, 11
.
g(n1) = 1).
otl
I g(w)
= 1) E
1 4n+1 c $ + ~f o r
T h e r e f o r e i f we w r i t e W1 and
sub-
11 i s w r i t t e n Fulln
( ~ 6 1 ) ( ~ 5 * ) " ' ( Q 6 , ) ~ Q ' ~ ) ( 9 ( 6 1 1 m* ,* .
F i n a l l y since
we
we t h i n k t h e domain o f g as w .
@
F a n and g ( k ) = 1) + x E @ ( k ) ]
g = {O,
E w)(g(m)
The e x p r e s s i o n
which i s
X from
c h a r a c t e r i s t i c functions
T h e r e f o r e (a) i s e q u i v a l e n t t o ( 3 9 E "u) (Wk E w )
and
is
P,*($)
E
and so we can t h i n k t h a t t h e domain o f $ i s w . L e t _v be t h e
[bey:"
i n t e g e r a s s o c i a t e d w i t h w C [beq]" g : [hey;"+
x
1 ( r e s p e c t i v e l y iln+l)p r e d i c a t e .
1
i s equivalent t o
A,i)4
= 1)
a l l k we can make
x
E $(k)
a
31 f o r q u a n t i f i c a t i o n o v e r r e a l s and WO
and 30 f o r q u a n t i f i c a t i o n o v e r numbers, t h e statement ( a ) now reads:
31 WO [ ( W O A,31
... "
Q1 Q'O) j L 3 1
- Kuratowski
we s i m p l i f y t h e above t o 31 3 1 VO [ ( A
...
v n+1
) +
v
Q'l,QO]
.
n+1
n Then u s i n g t h e T a r s k i
...
a l g o r i t h m s ( c f . Rogers 1967,
1
p.
307)
R I CARD0 NORA I S
220
which i s a
zi+l predicate.
Now using (b), since
G €
FuRel
is
we, s t a r t w i t h
[
W130
... 2'1 20) +
(WOA W 1
u n
and end up w i t h a $+1
predicate.
JLi and s i n c e @ ( k )
V1
E
1 4n+l
1
... 21 2 ' 0 1
n+ 1
came
I t i s c l e a r by now t h a t one o f t h e most i n t e r e s t i n g notions t h a t
up along t h i s work was t h a t o f
Full;
and i t s counterpart FLLeen
symmetry between these two classes o f
. The
,
c ;n+l
generalizing Ellentuck's F u l l s e t s , sets
n o t o n l y helped c u t t i n g a l l our proofs i n h a l f b u t also, and more s i g n i f i c a n t l y , w i t h o u t t h i s symmetry most o f our p r o o f s
-
c o u l d n o t have
come
through, s p e c i a l l y our l a s t theorem i n which the simultaneous use o f
FuUn
FuRe;
and
was fundamental.
For these reasons we foresee an i n c r e a s i n g use o f these n o t i o n s i n
the
f u t u r e s t u d i e s o f p r o j e c t i v e sets.
To conclude t h i s work, among several i n t e r e s t i n g q u e s t i o n s f o r which a l l t h i s machinery i s applicable, we s e l e c t e d two t h a t we are p a r t i c u l a r l y i n t e r e s t e d i n i n v e s t i g a t i n g , namely: (1) (2)
L or L i f any? Pn P ' If. M. i s 3 universe o f s e t s and B i s an n - p r o j e c t i v e Boolean a l gebra, what can be accomplished i n s i d e t h e Boolean valued m o d e l
What k i n d o f i n t e r p o l a t i o n theorem holds i n
MB ?
REFERENCES Addison, J. A. and Y . Moskovakis 1968,
Some comequenca Nat. Acad. Sci.,
06
t h t axiom o d de6&abLe d e L m i n a t e n a s ,
B e l l , J. L. and A. B. Slomson 1969,
Proc.
Vol. 59, 708- 712.
Models and Ultraprqducts, North
- Holland,
Amsterdam.
Ellentuck, E. 1975,
The ~owzdatiom06 S w f i n Logic, The Journal o f symbolic Logic, v o l . 40, 567-575
PROJECTIVE
197+,
22 1
LOGIC
Fhee SwL& d g e b m , S u b m i t t e d t o Czech. Wath. J o u r n a l .
Fenstad, F. 1971,
The a x i a m
0 6 deLetuninateflcbb,
Proceedings o f t h e Second Scandinavian
L o g i c Symposium, Ed., J . E. Fenstad, N o r t h - H o l l a n d , Amsterdam,41-61. Halmos, P. 1963,
Lectures on Boolean Algebras, Van Nostrand K e i n h o l d Company, London.
Kechris, A. 1973,
i'leubwre and categohy i n eddeotiwe denchipLLwe
bet
theahy, Annals
of
Math. L o g i c , V o l . 5, 3 3 7 - 384. K e i s l e r , H. J. 1971,
Model Theory for Infinitary Logic, N o r t h - H o l l a n d , Amsterdam.
Kuratowski, K. and A. Mostowski 1968,
Set Theory, N o r t h - H o l l a n d , Amsterdam.
L u s i n , N. 1925,
S w l Lcb emembLcn p o j e o t i w e h de M. H e i d Lebcngue, C. R. Acad.
Sci.
de P a r i s . Morais, R. 1976,
Projective Logic, Ph. D. T h e s i s , Eutgers U n i v e r s i t y , U.S.A.
Moschovakis, 1970,
Y.
DeLmtirzaflcy und
pmLJ&Otld&Mgb
ad t h e co~dizuum,in Mathematical
Logic and Foundations of Set Theory, E d . Y . B a r - H i l l e l , H o l l a n d , Amsterdm, 24 - 62.
North-
Rieger, L . 1955,
Cancmning Suofin d g e b m (S - k e g e b m ) and t l z e i n (Russian), Czech. I l a t h . J o u r n a l , Vol. 5. 99
htphtbtntathion
- 142.
Rogers, H. 1967,
Theory of Recursive Functions and Effective Computability, MacGraw H i 11. Sierpinski, W. 1925, S w l une &abbe d'emembLcb, Fund. Math. Vol. 7, 2 3 7 - 2 4 3 .
-
l n s t i t u t o de Matematica U n i v e r s i d a d e F e d e r a l do R i o de J a n e i r o R i o de J a n e i r o , RJ.,
Brazil
F'ROJECTIVE LOGICS AND PROJECTIVE BOOLEAN ALGEBRAS (*I by R I C A R D O hlORAIS
I, INTRODUCTION, Lusin and S i e r p i n s k i s t a r t e d , i n 1325, the development of the theory of p r o j e c t i v e sets ( c f . Lusin 1925 and Sierpinski 1925) b u t soon afterwards t h i s research p r a c t i c a l l y ended due t o the complexity of the problems and the lack of b e t t e r t o o l s t o work with. I t was not u n t i l t h e l a t e s i x t i e s t h a t some new r e s u l t s were ( c f . Fenstad 1971, Moschovakis 1973, and Kechris 1973) using t h e proposed, b u t s t i l l questionable, axiom of Addison and Moskovakis 1968). Recently A. Nerode thought of develop
obtained recently
" P r o j e c t i v e Determinancy"
a l o g i c LA t h a t would b e s u i t -
a b l e f o r t h e study o f t h e a n a l y t i c s e t s , o r p r o j e c t i v e s e t s of level L
WlW
(cf.
was t o Bore1 s e t s . This was done by h i s s t u d e n t P . J . Campbell,
1 , as and
f u r t h e r strengthened in E.Ellentuck 1975 u s i n g d i f f e r e n t methods. This paper follows a sugestion of Eilentuck of t r y i n g t h e g e n e r a l i z a t i o n of t h i s approach t o a l l l e v e l s of the p r o j e c t i v e hierarchy. [ * ) The a u t h o r i s i n d e b t e d t o P r o f e s s o r E l l e n t u c k f o r t o Rutgers U n i v e r s i t y (U.S.A.)
and
support.
20 1
h i s o r i e n t a t i o n , and
CAPES ( B r a z i l ) f o r
their financial
202
R I CARD0 MORAl S
11, PROJEC I V E LOGICS, 0, PRELIM NARI ES s
Let [hey]’ denote’the s e t of a l l n - t u p l e s (uZ, ...,w n ) of f i n i t e s e quences of natural numbers satisfying e ( ~ , )= ... = e ( ~ ~where ) , L(u)denotes the length of u . Throughout t h i s t ex t 6 and g will denote elements of %; n ; m, and k will and f o r The subsets
be natural numbers and u will stand both f o r elements of [neqJn”, elements of ney the s e t of f i n i t e sequences of natural numbers. most important tool developped in t h i s work i s a pair of families of of [hey]’, denoted respectively by FLLeen and F u l l ;
.
D E F I N I T I O N 1 . a) F E F L U , ( F 0 a Fd2l-n b e t ) i d and vney ii F c [neq]’ and F 402%6iu t h e 6aUvwing cvndition giuen by a 4.7XLng 0 6 quant.idiehn
w i t h n aetmnntionn: (3
I
d,)(v dz)*..(Q6,)(Q’k)((d,
I
k,**-,&,
k ) E F)
whme Q and 2’ ahe din-tinct yuanLL&iem. S , i m d ! d y we
b)
dedine
G E F u l l ; (G 0 a F L L e e - v ~ - n Z m n e t ) id and vney i d G c [ n e d n und
(v
61)(362 )-.*(Q’6,)(Qk)((d,
I
k 9 * * * r 6 n
I
k) E GI.
F and G will denote elements of F u l l and F u l l * respectively,and the word countable will always mean e i t h e r f i n i t e or denumerable. Finally, i f @ = {@iI i 8 I ) i s a countable s e t of formulas o f a formal I\ $ . or simply I \ ~ $ ~ language, the conjunction A @ will also be written i e 1 4. i f no confusion a r i s es .
1, THE
LANGUAGE L~~
We s t a r t with a f i r s t order logic L with countably many re la tion, funcbe the infinita rylogtion and constant symbols, and w l variables. Let L W1W
by adi c over L as defined in Keisler 1971, p . 6. We obtaion L from L Pn WIW joining two new operators P, and P; i n the following way: F o m a of L ar e those of L with the addition: i f @ i s amap from Pn W 1W [neq]” i n t o formulas of L then Pn(@) and P i ( @ ) are formulas of L Pn‘ Pn Extend the notion of n u t h 6 a o t i o n by defining:
203
PROJECTIVE LOGIC
i f f ( 1 F € F L L W M ) ( w w € F ) CR C$(w) [ h ]
[h]
(1) CR CP,($)
( 3 G € Full;)( W w € G) d ! = $ ( w )
iff
d bpi(@) I h ]
O c c m e ~ c ao f v a r i a b l e s and c o n s t a n t s i n P a ( @ ) and $(u) f o r
be those i n t h e c o l l e c t i o n o f a l l
w € [neq]'
[h!
P i ( @ ) are .
defined t o
D e f i n e d - w a e i d i t y and w a e i d i t y i n t h e usual way. The n o t i o n o f nub&mnula o f K e i s l e r 1971, p. 11 (2)
= {PM($)j
Sub(P,($))
i s extended by
uw S u b ( $ J ( v ) ) .
u
Uw S u b ( $ ( v ) ) *
S u b ( P ; ( @ ) ) = { P l ( @ ) }IJ
Extend t h e n o t i o n o f t,iouing Rhe c z e g d o a i n ~ i d e(cf. K e i s l e r 1971, p.11) as f o l l o w s : (3)
where
(P,J@)) 1
is
Pi(%)
(Pi($))1
is
P,(l@)
I
l$ i s t h e map t h a t takes w i n t o
Axivmn o f L
P"
l$(w).
w i l l c o n s i s t o f t h e n i n e axioms f o r p r o p o s i t i o n a l
as p r e s e n t e d i n B e l l and Slomson 1969, p . 36, p l u s t h e s i x axioms f o r as i n K e i s l e r 1971, p. 15.
logic
Lwlw
The Rda ad Irz@tence w i l l be (Rl)
Modus Ponens
(R2)
i f I-$
(R3)
i f f o r every
$ S @ t- J, +I$
(R4)
i f f o r every
F € FullM
(R5)
i f f o r every
G
P;($)
Vx$
t h e n + $+
+$
€
I-
if
x i s not free i n
t h e n I-$
AF$(w) + $
Full; I- J, + VG$(w)
+
$J
.
AQ.
I - P n ( $ ) + VJ.
then then
I- $ +
P,*($).
ble o n l y have t o w o r r y about i n t r o d u c i n g P,($) because t h e f a c t s a r e e a s i l y g o t t e n f r o m those about P,($) u s i n g t h e f o l l o w i n g
tance o f t h e axiom + $ 1
PROPOSITION 2 .
w
I-Pi(l$)
H
1 $
.
Now b e f o r e we proceed w i t h t h e s t u d i e s o f L
b.tatemk?VLt
abvut w
me Rnue meta
-
ins-
lP,($)
p o r t a n t p r o p e r t i e s o f t h e f a m i l i e s FuUM and
P R O P O S I T I O N 3.
about
l e t us c o l l e c t some
Pn FILee;
.
in-
Suppane t h a t dvti ewmy W E [ n e q ] ' , x(v) o a m e t a l& Q and Q' be bn in U e 6 i n i t i o n I . Then t h e ~vUviu&g
.
htatemenb.
204
RICARDO N O R A I S
a) ( 3 61)...(2dn)(2'k)A(d1
I
k,...,6n
b) ( V ~ , ) . . . ( 2 ' l i , , ) ( 2 k ) A ( d l I k , . . . , c) ( 3 F E F u U n ) ( W v E
F)A(w)
I
k)
dn 1 k )
id6
4 5 6 (3 F i d 6 (36
Ffin)(Vv
E Fuu;)(Wv
E F)A(v) 8
G)X(u)
( W G E Full;) ( 3 w E G ) A(") ( 3 G E FuU;)
d) ( W F E F u l l n ) ( 3 u E F ) A ( u )
E
(Ww 8 G)A(w)
I t i s important here t o note the complete symmetry between t h e s e two f a m i l i e s of sets. I t i s tnis symmetry t h a t w i l l make possible the majority of our proofs, besides e l e g a n t l y reducing a l l the work i n h a l f . we Now, in order t o b e t t e r understand the behavior of t h e s e f a m i l i e s , have:
PROPOSITION 4 .
16
IT
> 1,
and H c : s e q ] "
E
= { ( " p . . . , " , )I
(5 I
[(",),
, Let
"2,..*>"11)
E
HI.
Then
a) H E F u U n
.id6
(36) (Hd E
b) H E Full;
.id6
( v 6 ) (HS
Fa;-,)
E FLLeeYl-,)
F i n a l l y the next proposition shows t h a t the f a m i l i e s F u l l M and n.
a r e well mixed t o g e t h e r . The proof i s by induction on
2,THE CONSISTENCY PROPERTY
F&f;
I
Let C be a countable s e t of constant symbols not appearing i n L . Let c E C t o L , and from bl c o n s t r u c t t h e l o g i c M PM A banic tm i s e i t h e r a constant symbol of bl o r a term of the form P" function d(t t k )where R L,...,t k a r e b a s i c terms and 6 i s a k - a r y symbol o f L . but The d e f i n i t i o n t h a t follows was taken from Keisler 1971, , p . 11, clauses here, besides adapting i t t o the present s i t u a t i o n (namely adding C9 and C9) we a l s o modified, t o simplify the proofs, the notion o f b a s i c bl be t h e f i r s t order l o g i c obtained by adding each
.
,,...,
teroi, and clauses C10 and C11.
DEFINITION 6. n e s:
A CoiuDtency PhUpULty D a be* S batin6y.ing
doh
each
205
PROJECTIVE L O G I C
to
The d e f i n i t i o n o f Consistency P r o p e r t y i s t h i s l o n g because we want have:
THEOREM 7 . and
40 E S ,
PROOF:
(Model E x i s t e n c e Theorem). 16 S 0 a C a ~ n O t e n c y P h u p e h t g
then
Without
han a modet.
40
l o s s o f g e n e r a l i t y we way assume t h a t each subset o f an e l -
elllent o f S i s a g a i n i n S . To c o n s t r u c t t h e model s a t i s f y i n g w i t h t h e s m a l l e s t s e t Y o f f o r m u l a s o f E.4
(i)
no
(ii) Y
Let and
T =
f o r which:
i s c l o s e d under subformulas.
(iv) i f l@ E Y If
c E C
then
a b a s i c term and
1
@(t)E Y then I$(t') E Y
c = t E Y.
be t h e c o u n t a b l y i n f i n i t e s e t o f sentences o f Y ,
{to,tl, . . . I be t h e s e t o f b a s i c terms. S t a r t i n g w i t h
h0
construct
an i n c r e a s i n q sequence o f elements o f S as f o l l o w s . Suppose we have 4nl+l
4,,, :
.
$1 8 Y .
and R i s a b a s i c t e r m t h e n
X = { I $ o , I$,,...
and we b u i l d
start
Y
(iii) I f t i s a t e r m , t '
(v)
Pn
oo we
h,,,,
R I CARD0 MORA I S
206 (1)
i f A,, U
{@,,,Ig
(2)
i f sm U
{a,}
=
A;+,
;
i s V@ then f o r some $ E @,
(2.1)
$ ,,
(2.2)
$m i s 3 x $ then f o r some c E C, A;+,
(2.3)
@m
(2.4)
i s P,($)
IAF$(v)}
U
{$,,,I
U {AG@(v)}
f i n a l l y , s i n c e i n any case Am+,
Next d e f i n e
e sw
c = d
L e t [c]
u
= A;+,
sw
=
.
{c =
um sm
fml
&A+,
s+ ;,
=
E
S,
=
o;+~
=
sm u {$,,,I E S; t h e r e i s c E C such t h a t
8 S,
E S.
and d e f i n e an equivalence r e l a t i o n on C by: c+d
c E C and l e t A =
be the equivalence c l a s s o f
Now f o r each k - a r y r e l a t i o n symbol P,
6,
l
sm u {$m}U { $ ( c ) }ES,
G E F a ; ,
This i s the universe o f the model t h a t w i l l satisf.y symbol
@
E S,
i s any o t h e r formula,
$,
=
E S,
i s P i ( @ ) then f o r some
$,
(2.5)
U
u {$mlu I
= A,,,
F E FU.ee,,s;+,
then f o r some
u {@,,,I
= A,,
(3)
sm
we consider t h e f o l l o h i n g cases:
E S
= A,,
iff
let
S
o f L define a r e l a t i o n
so
I [c] i
.
o f L and each k - a r y
c E C}.
function
Rm on Ah and a f u n c t i o n F,, from
Ak
i n t o A by:
(a) (b)
,..., r e k ] ) E Rm Fm( [c,] ,. ..,[ch]) = [c,] ( [c,]
Note now t h a t i f is A@,
@ E 6,
then
iff
Pm(cl
,..., c),
iff
co =
b,,,(c,,
E
...,ck)
E
.
and
0 E Y
f o r each
(a)
@
(b)
@ i s any o t h e r formula, then
8 E @ ;
$ E Y.
Then use t h i s f a c t t o show t h a t the s t r u c t u r e
a satisfies
=
so
1
m 6 wl, {F,,,
I
m E w}, A >
.
Theorem 7 i s a n i c e t o o l t o use i n the p r o o f o f
THEOREM 8. (The Completeness Theorem f o r
L
)
Pn
16 @ 0 a s e n t e n c e
06
207
LOGIC
PROJECTIVE
To show t h a t e v e r y theorem i s v a l i d we Drove t h a t t h e r u l e s o f i n -
PROOF:
ference (R4) and (R5) p r e s e r v e v a l i d i t y . Rule (R4).
(a)
Suppose
3F E FuU,,
VF
E
F u l l R , 02 t=
U? b AF@(u) A 1J,
A F @ ( w ) + J, t h e n i t i s n o t
t h e case
.
i m p l i e s t h a t 3 F E FuRe,,
B u t by D e f i n i t i o n 1, Ce CP,(@) and t h e r e f o r e i m p 1 i e s n o t UL C P,($)
A 1$ o r e q u i v a l e n t l y ,
that
U? I = A F $ ( w ) ,
CL c P,($)+$.
Rule (R5).
(b)
d C J, * VG $ ( w ) .Then a!= 1ji o r Suppose VG E F u l l ; , ( 3 u E G ) OZ k = $ ( w ) a n d h e n c e b y P r o p o s i t i o n 3, CR
( W G 8 FLU;) 11) or
U? C J ,* P,($).
( I F E FullR)(W w E F) CE i=@(v), which i m p l i e s
Now we have t o show t h a t e v e r y v a l i d sentence i s a theorem. I n o r d e r t o do t h a t we l e t S be t h e s e t o f f i n i t e s e t s o f sentences n o f o n l y f i n i t e l y many
c E C
o c c u r i n n and n o t I-
MA.
M
YJn
such t h a t
We t h e n show S i s a Consistency P r o p e r t y and t h e r e s u l t f o l l o w s hence
lip,
t h e n @ i s n o t a theorem i n
cause i f @ i s n o t a theorem i n L
r7M
{ I $ } E S. By t h e Model E x i s t e n c e Theorem @:
beand
has a model and t h e r e -
fore @ i s not valid. We e x e m p l i f y t h e p r o o f t h a t S i s a Consistency P r o p e r t y be p r o v i n g (C8) and (C9). (C8)
Suppose P,($)
Full,
U {A,@(u)}
WF
(WF E F u l l , )
( I- A F @ ( w )
1An); t h e n by (R4)
since
E n,
P,(@)
P,(@)
P,*(@)
Suppose
I-An-,
( I-
S.
I- P,(@)
+
1A 0
and,
lAn, a contradiction.
E n b u t ( V G E FLU;)
I- l A ( n u { A , @ ( w ) } ) ( W G 6 FILL$)
I-
+
E
e
I- lA(n U { A F @ ( u ) } ) , and so
E n, we have
Since
(C9)
n b u t (WF E FuU,)(n
E
f o r every
(n U { A G @ ( u ) }g S ) ; t h e n again
G E Full;
and
so
An + V G l @ ( w ) ) , which i m p l i e s , by (R5),
P,(l@).
Therefore, by P r o p o s i t i o n 2, a contradiction.
I-
An + l P i ( @ ) o r , e q u i v a l e n t l y , I- l h n ,
206
R I CARDO MORA I C
There i s another p r o j e c t i v e l o g i c of i n t e r e s t t o us, namely: DEFINITION 9 . h ~ n bowm w,
The logic L
P
0 dedined t o be t h e u n i o n oh & L
o h i n othetr ~0oh.d~:
(a)
ln L
(b)
The &en
UA
n
Pn($) 0 a domda doh ewmy n .
P '
06
(R4)
(Vn E
(R5)
( i n E w ) id
id
w)
Pa
indmence (R4) and (R5) now head
( W F E FuUn) I- A F $ ( w )+ $
then
I-
(VG E FLU;)
then
I-$+ P,($).
I-$+ VG@(w)
i s complete s i n c e a l l L are. P Pa There i s one important theorem p a r t i c u l a r t o L
Pn($) + $,
Obviously L
THEOREM 1 0 . (R4')
In L
P
0 a h.u&
t h e @f%LCLing
= 1)
id
V6 I- Ak'$(d
(b) (doh. n > 1)
id
Vd I-
(a) ( d o h n
whehe
$d
(w2,.
..,wn)
= $( 6
:
06 in&?kence: k) --f
I .t(w,),
P
+
$
$
w2,.
then
then
I-
+
PI(@) + $ ;
pn($)+
n-
;
.., w n ) .
We conclude this s e c t i o n with the remark t h a t t h e downward Skolem-Tarski theorem holds f o r both L and L P Pn *
111,
$J
Lowenheim-
PROJECTIVEBOOLEAN ALGEBRAS,
1 I NTRODUCTI O N , I
In t h i s s e c t i o n we d e f i n e a new kind o f Boolean a l g e b r a s , c a l l e d n-proj e c t i v e Boolean a l g e b r a s , which a r e g e n e r a l i z a t i o n s of t h e S u s l i n algebras introduced by L . Rieger in 1955 ( c f . Rieger 1955). Our work, however, i s patterned a f t e r a recent paper by E . E l l e n t u c k (Ellentuck 197+) i n which he s t u d i e s the S - a l g e b r a s o f Rieger based on his previous paper on S u s l i n l o g i c (Ellentuck 1975). R i e g e r ' s idea with t h e S u s l i n algebras was t o provide a s t r u c t u r e i n which one could model nn1 a n a l y s i s .
1. BASIC
nt
a n a l y s i s . Our algebras a r e intended t o help model
RESULTS,
Let B be a Boolean algebra.
209
PROJECTIVE LOGIC
The joim and nieeA of B w i l l be denoted r e s p e c t i v e l y by Sup and 7ng. The iizditzite j o i n of t h e family {bi 1 i E I } i s denoted by Sup bi or simply by Sup bi
i
is7
i f i t c l e a r which s e t 7 i s .
I f Q i s a map from [neq!" i n t o 6 we s h a l l use the n o t a t i o n Pit($) f o r the following element of 73, provided i t e x i s t s : Pi*($)
=
and, s i m i l a r l y ,
SUP
F
Ini( $ ( w ) , wEF
where, a s u s u a l , F runs over The symbols P,(@) and and t h e previously defined confusion.
DEFINITION 1 2 . (W
- PBA
ijoh n h h t )
F u l l n and G over
FU.eek
.
P;($) w i l l be used both f o r t h e above suprema formulas o f Lpn, b u t t h i s should l e a d t o no
A u - B C J C J ~d~ gU e~ b~ t ~ ~ U M w - I 3 4 O j ! L d W t 600tea~d g c b h n id Lt 0 it - PBA dot C V U i ~ i E W.
Formula ( 4 ) i s a very powerful d i s t r i b u t i v e law and not a l l algebras closed under Pit and P i s a t i s f y i t . In f a c t , t h e r e a r e complete B o o l e a n algebras i n which ( 4 ) f a i l s . In our work, however, we need t h i s d i s t r i b u t i v i t y t o t i e t h i n g s up ( s e e f o r example condition ( 6 ) below), and we a r e thus forced t o introduce i t a s p a r t of t h e d e f i n i t i o n . To g e t an example i n which ( 4 ) f s i l s s e e Morais 1976. Another way t o see the importance of ( 4 ) i s t h e next proposition which presents t h r e e e q u i v a l e n t formulations of ( 4 ) .
PROPOSITION 13. 7 6 B 0 c( Bootenti d g e b h a i n t o B , tt,t -I$ be t h e tNnp deijined by (-
whehe - 0 t h e nytnbat eqUiWdent:
604
Q)( w )
=
-$ (w)
c a t i i p L ~ i e n t d L oi ~n ~ B
aid
Q 0 a riiap
. Theit tlze
6hotti
[Aeq]'
60ttCJdt7g
ah&
210
RICARDO
MORAIS
Now, using these equivalences, we can get several properties of projective Boolean Algebras, namely:
PROPOSITION 1 4 . A u - B o o L ~ ~dMg e b h a B 0 n-PBA i6 m d Only i6 d a s e d u n d e h the P i a p e h a t a h and (4) holds. PROPOSITION 1 5 .
16
M > 1
PROPOSITION 17. Evehg
whehe
w
PROPOSITION 1 8 . PROOF:
Now
M-
PBA, then B 0 ( n - 1) - PBA.
an example.
The cornple*e B a o l e a ~d g c b m 2
= {O
,I} 0 w - PBA.
Since 2 i s complete we have just t o show ( 4 ) holds i n 2 .
P,($)
=
0
i f f sup In6 - $ ( w ) . = G uEG
.
B 0
- PBA ~ a t i n 6 i e A :
iotoak-n add
iff
i f f (by Proposition 3 )
holds
M
aMd
8 0
Sup In6 $(u) = 0
F uEF ( 3 G E FuRe;)(Wu
1 iff Pi( - $ ) = 1
i f f (WF E FuRen)(3w E F)(@(u)=O) E
iff
G)($(u) = 0 ) i f f In6 Sup $(u) = O G uEG
- P i ( - $ ) = 0 and therefore (6)
211
PROJECTIVE L O G I C
The most i m p o r t a n t example o f an w - PBA however i s g i v e n by t h e
fol
-
lowing: The Lindenbawl dgebaa L
THEOREM 1 9 . w
- PBA.
PROOF:
Let
1
@
I
06
P
,the w - pfihujedue Logic
LP
denote t h e e q u i v a l e n c e c l a s s o f t h e f o r m u l a @ i n L
P
.
i n t o L and d e f i n e a map $J from [bey]" P by choosing f o r each W E [hey]' a r e p r e s e n t a t i v e f o r -
L e t @ be a map f r o m [hey]" i n t o f o r m u l a s of L
P
mula @(u) o f t h e e q u i v a l e n c e c l a s s
(8)
P,(@) = !pn(@)l and hence
T,
We t h e n show
(li(w).
,
F i r s t we have t o p r o v e t h a t t h e f o r m u l a choice o f t h e map
Pn o p e r a t o r .
i s c l o s e d under t h e
P
Pn(@) does n o t depend on t h e
@.
I t s enough t o show t h a t f o r any o t h e r map
'Ju E [ ~ e y ] " I- @ ( w )
+
$(u) t h e n
I-
JJ
:
if
Pyz(@)+ P , ( ~ J ) .
By (R4) t h i s f o l l o w s f r o m (9)
VF E F a n I-
AF@(u)
+
pn($)9
which i n t u r n f o l l o w s from, (WG E F u R e V : ) ( V F E
Fan)
(by R5), I-
AF@(w)
+
VG$(u).
B u t t h i s i s t r i v i a l s i n c e by P r o p o s i t i o n 5, g i v e n any F and G , F n G # @ . T h e r e f o r e (9) h o l d s . Now t o f i n i s h t h e proof of Theorem 19 we have t o show t h a t t h e d i s t r i b u t i v e law (4) holds i n L
P'
We s h a l l need,
(10) P i ( @ ) = I P p 4 which i s e q u i v a l e n t t o ,
S U P I AG @ ( u ) I = I P i ( $ J ) I G and so we have t o prove: (i) ( W G E F f i i )
I
I- h G @ ( V )+
Pi(@)
and
(ii) I f (WG E Fufl;)
I- h G $ J ( u + )
11 t h e n
I-
P i ( @ )+ $
.
212
RICARDO MORAIS
PROOF of (i):
From p r o p o s i t i o n 5 g e t
(WG E FU.eei) (WF E F d n )
+
AF l$(v)
+
VG l $ ( v )
now a p p l y (R4) and use P r o p o s i t i o n 2. S t a r t w i t h the hypothesis
PROOF of ( i i ): (WG € FULL;)
I-
1
$J
+ VG 1 $(v),
t h e n a p p l y (R5), and use P r o p o s i t i o n 2. F i n a l l y ( 8
,
(10) and P r o p o s i t i o n 2
give
Pn(@)
=
-
P,*(
- 0)
and t h e r e f o r e ( 6 ) h o l d s , which i s e q u i v a l e n t t o ( 4 NOTE: L
PM
E v i d e n t l y e x a c t l y t h e same p r o o f shows t h a t t h e Lindenbaum a l g e b r a of
(denoted L
PM
) i s n - PBA.
3, FREE n - PROJECTIVE BOOLEAN ALGEBRAS, DEFINITION 2 0 .
An nP
-
BooLean d g e b t a 0 a a - ho-
Izornornotpkm b-een
momohpkinm t h a t pk,hedmve~t h e Pn opehatoh. An W P - homomotpkinm 0 a u - honiomohpkintn
&at p u e h v e n Pn d o t evehy n
DEFINITION 2 1 .
L e t B be m
- genehaten B i d
(a)
G nP
(b)
G dheely
E w.
n - PBA and G
c
B. Tken:
B 0 t h e nm&ent
n - PBA containing G
.
nP- genmaten B i d G nP-genehaten B UJ~C! in a d d i t i o n given m y o t h m n - PBA B' and m y map h : G + B' t h e h e i b an nP - homomohpkidm H : E + B' w h i c h extend6 h
.
- net
06
gen -
An n - PBA 0 a dhee nP- d g e b h a i d contaia n P - n e t ad g e n ma to a . S . i m . 2 d y , dedine a 6hee ~ P - u Q e b h a .
a
dhee
(c)
S .in i. 2 dy dedine W P - neA 06 genehatom and 6hee
WP
ehato4,5.
DEFINITION 2 2 .
I f i s a common p r a c t i c e . i n any t e x t about " f r e e " s t r u c t u r e s t o
first
t a l k about i t s uniqueness and a f t e r w a r d s t o prove i t s e x i s t e n c e .
The
l o w i n g two p r o p o s i t i o n s a r e proven i n t h e same way i t i s u s u a l l y
done f o r
general Boolean a l g e b r a s . See f o r example Halmos 1963, p. 42.
fol-
213
PROJECTIVE L O G I C
PROPOSITION 23. 76 B 0 u dhee n P - d g e b h u , G t h e he,t 06 6hee nP-genefu7Xoh.S and h .the given map 6honi G into .the n - PBA B',then t h e nP - hamomahpkinm H : B + B' t h a t extendh h 0 unique. PROPOSITION 24. Any &oo 6hee n P - d g e b h a whohe .the hame catr&&y atre nP-0oma5pkic.
h d
0 6 genmatom
have
Now t o p r e s e n t an example o f a f r e e U P - a l g e b r a ( t h e e x i s t e n c e o f a f r e e n P - a l g e b r a i s proved s i m i l a r l y ) we proceed as f o l l o w s . F i r s t d e f i n e a phOpOh.iJ%onCdl o g i c LK f o r each c a r d i n a l
K
and t h e n
show t h a t t h e Lindenbaum a l g e b r a L~ o f LK i s a f r e e U P - a l g e b r a
we
with
K
generators. LK i s g o i n g t o have a s e t o f
{Pa j
c1
< Kl
K
variables
,
and t h e p r o p o s i t i o n a l c o n n e c t i v e s 1 and A o p e r a t o r s P,? and P;
. As
in L
PM '
we i n t r o d u c e
and l e t t h e s e t o f f o r m u l a s be t h e l e a s t s e t such t h a t
.
(a)
pa
(b)
if @
(c)
i f 0 i s a c o u n t a b l e s e t o f f o r m u l a s t h e n A @ i s a formula.
(a)
i s a f o r m u l a f o r each o r d i n a l
c1
<
K
i s a f o r m u l a t h e n so i s l @
i f @ i s a map from
[heq]' i n t o f o r m u l a s t h e n P,(@) and Pi(@) a r e
formulas ( f o r every
n E w).
Define "riaving t h e negation h i d e " f o r formulas o f LK as we d i d
LPn
the
for
with the addition:
For axioms t a k e t h e n i n e axioms o f p r o p o s i t i o n a l l o g i c as i n B e l l and I- @l*l@ and I- A @ + @, where @ i s a c o u n t a b l e
Slomson 1969, p. 36, p l u s s e t o f f o r m u l a s and @ 8
@.
For r u l e s o f i n f e r e n c e t a k e those o f L p n w i t h t h e e x c e p t i o n o f (R2). A r e a l i z a t i o n o f LK i s a map 2 =
6
f r o m t h e s e t o f v a r i a b l e s i n t o t h e w-PBA
{ o , 1 1 , which i s i n d u c t i v e l y extended t o a l l f o r m u l a s as f o l l o w s :
(4 d(l@)
=
- 6(@),
(b)
d ( A 0 ) = In6 d ( @ ) ,
(c)
6 ( P n ( @ ) ) = P n ( 6 ( @ ) ) and
$80
b(P;(@))
= P;(d(@))(
214
RICARDO M O R A I S
6(@) i s
where
the map defined by
6($)(u)
=
6 ( @ ( u ) ) for u
€ [neq]"
. 6.
We say t h a t a formula @ i s valid i f d ( @ ) = 1 in a l l realizations Now, before we prove t h a t LK i s an UP-algebra on K generators, need: 8 be an w P - d g e b h a and
PROPOSITION 2 5 . L e t ablu
06
LK & t o
B . EXtend
6
.to a l l d
o
6
we
any map dhom .the u a h i
m by~ trdu (a) X h h o u q h
-
(c).
Then
imfiu
I-dl
A($)
= 1.
In pa)Lticdah, by PhopohLi5on 18, eue-hy theohem oh LK 0 v a l i d . F i r s t note t h a t because of properties (a) and ( b )
PROOF:
(11)
d ( @ + $)
= 1
i f and only i f
d satisfies:
d(@) 5 6 ( $ ) .
I t i s routine t o show t h a t the axioms are mapped into 1 , b u t we check, as an example, t h a t the axiom $1 -l@ i s mapped i n t o 1 f o r the case @ i s Pn($). By (11) we have t o show, B(PYl(VJ)1 ) = 6 ( 1 P n ( $ ) )
.
But 6(Pn($)1 1 = P;(
-6($))
6 ( P,*(l$))
= -Pn(6($)) =
= p;(6(1$)
-
=
6(Pn($)) = 6 ( 1 P n ( $ ) ) *
where the fourth equality follows from ( 6 ) . Similarly, using ( 1 1 ) i t i s easy t o prove t h a t the rules of inference preserve the property of being mapped into 1 . As an example we check f o r (R4). Suppose W F € FU.een,
6 ( h F @ ( u *) $ ) = 1
and
we
have
to
show
d ( P n ( @ ) * IrJ) = 1 Sy (11) and property ( b ) we have ( V F E F a Y l ) In6 Therefore
UEF
b(@(U))
5
A($)
*
d(@(u)) 5 d($). F u€F B u t by definition t h i s i s P n ( 6 ( $ ) ) 5 6 ( $ ) , and hence 6 ( P n ( @ ) )5 ~ ( J J ) T h u s by ( 1 1 ) , 6 ( P n ( @ ) * $1 = 1 . We therefore conclude t h a t every theorem of LK i s mapped into 1. SUP ,In6
.
PROJECTIVE L O G I C
215
We a r e now i n o o s i t i o n t o show THEOREM 2 6 .
LK 0 a
64ee wP - d g e b t u an exac.tQ
K
genmcLtau.
F i r s t i t i s c l e a r t h a t t h e same p r o o f used t o show t h a t
PROOF:
L
w-PBA (Theorem 19) can be r e p e a t e d h e r e t o show LK i s w-PBA. Next l e t G = { ! p a l la gebra B t o g e t h e r w i t h a map
h : G + B.
6
=
~ ( u J , )
6
and l e t t h e r e be g i v e n an a r b i t r a r y wP-al-
K)
Now u s i n g h d e f i n e a nap
and e x t e n d
f r o m t h e v a r i a b l e s o f LK i n t o B by
h ( / p a1
ly
i n d u c t i v e l y t o a l l f o r m u l a s o f LK
.
By P r o p o s i t i o n 25 and (11) i t i s easy t o show t h a t every equivalence class
was
P
I @ 1 , and
is c o n s t a n t
so t h e f o l l o w i n g i s a w e l l
in
defined
map
f r o m LK i n t o 8 :
H(i@ This
I1
d(@).
=
H i s t h e d e s i r e d U P - homomorphism e x t e n d i n g h , and hence i t o n l y
remains t o show t h a t t h e c a r d i n a l i t y o f G i s given
a ,B <
K
with a # 6
i s n o t a theorem and hence
4, A
,
K
.
But t h i s i s e a s y ,
P r o p o s i t i o n 25 can h e l p t o show t h a t pa
I pa I
#
for H
I pB 1 .
REPRESENTATION THEOREM FOR FREE nP-BOOLEAN ALGEBRAS
pB
I
We s t a r t t h i s s e c t i o n w i t h a completeness theorem f o r L K . T h i s i s done t h e same way we d i d f o r L
Pn
and so we o m i t t h e p r o o f , a l t h o u g h we p o i n t o u t
t h e b a s i c p o i n t s . F i r s t we d e f i n e :
216
RICARDO M O R A I S
ththetle
1 5 S 0 a K - CoMnOtency P h O p M y and oo E S t h e n a h e ~ z c L t i o n 6 0 6 LK doh rukich d ( $ ) = 1 doh & @ E no
PROOF:
T h i s p r o o f i s p a t t e r n e d a f t e r t h e one f o r t h e Model E x i s t e n c e The-
PROPOSITION 2 8 .
.
orem (Theorem 7 ) . We s t a r t no and c o n s t r u c t a sequence (A,) o f S w i t h t h e d e s i r e d c l o s u r e p r o p e r t i e s . Then l e t map f r o m t h e v a r i a b l e s o f LK
d(PJ
d
Then e x t e n d quence (A,)
no =
i n t o 2 by
iff
= 1
o f elements o f
u nm
m
and d e f i n e a
Pa e
i n d u c t i v e l y t o a l l f o r m u l a s and because o f t h e way t h e
se-
was c o n s t r u c t e d we have
A($)
= 1
+ E nu .
for all
F i n a l l y , we have :
PROPOSITION 2 9 .
16 $ 0 not a
6(@) 0.
theatem 06 LK then doh
bOMe
tluLizaLLon,
J u s t l i k e we d i d f o r 1 we show t h a t t h e s e t o f a l l f i n i t e s e t s Pfl o f f o r m u l a s o f LK f o r which n o t I-1 h b i s a K - Consistency Property.Then
PROOF:
use P r o p o s i t i o n 28 t o g e t t h e r e s u l t .
An n P - d i d d 0 6 A& 0 a 0 - 6 i & l 06 A & 16 a 0- d i d d 06 b& 0 cloned undm Pn w e c a t t it an w P - 6ieLd 0 6 ~ t . t b .
DEFINITION 30.
cloned u n -
dm t h e opetlatoh Pn.
doh
n 8 o
Notice
evmy
t h a t we d i d n o t m e n t i o n any d i s t r i b u t i v e l a w here. T h i s however
i s no s u r p r i s e because we have:
PROPOSITION 31. Eumy nP THEOREM 3 2 .
- @Ld 06
n&
0 n - PBA.
(heSpecFOX each cahd+u?l K t h m e 0 an nP - 6.ietd 06 A & 06 b d ) that 0 n P - g e n m d e d [ ~ P - g e n e h a t e d ) by K 06
L L v d y U P - d.ieXd ia%
dements.
PROOF:
Let
X = ZK be t h e s e t o f maps f r o m
K
into
2 =
l o , 11 and d e f i n e
PROJECTIVE L O G I C
a <
f o r each
Next, l e t let
BKn
taining
217
K
Q
9, =
r6 e
= {g,
I
(respectively
ZK
I 6(.)
=
11
a<
K}
BK)
be t h e s m a l l e s t n P - a l g e b r a ( U P - a l g e b r a ) con-
which i s a s u b s e t o f t h e power s e t o f X,and
2. BKM and BK a r e
Since t h e power s e t o f X i s a complete f i e l d o f s e t s , well defined.
Q i s K t a k e a # B and choose 6 ( a ) # tj(0). Hence i f , say, d ( a ) = 1 t h e n 6 E g,
F i n a l l y , t o show t h a t t h e c a r d i n a l i t y o f any map but
6B
6 E
2K
gB,
f o r which and t h e r e f o r e
g,
'go.
Now copying what we d i d f o r LK we c o n s t r u c t a p r o p o s i t i o n a l l o g i c LKn f o r each n E w i n such a way t h a t t h e i r c o r r e s p o n d i n g Lindenbaum algebras
LKn a r e f r e e n P - a l g e b r a s .
Our r e p r e s e n t a t i o n theorem f o r
M P - a l g e b r a s i s an immediate consequence
o f the next very importdnt proposition.
PROPOSITION 3 3 . ~ ~ P - i A o m o t ~ p kt ioc BK L~ 0 iA MP - iAomohipkic t o BKn. PROOF:
; and 6 o t ~e v w y
We p r o v e o n l y t h a t L~ i s WP - isomorphic t o BK
The w P - i s o m o r p h i s m H : L~
--f
n E w, LKn
.
BK we a r e l o o k i n g f o r i s d e f i n e d i n d u c -
t i v e l y by: (a)
For every o r d i n a l
(b)
H ( I I@ 1 ) = H (
(c)
H(!AQl) =
(a
I
fl
@
c1 < K , H ( I I ) ' , where A'
p a / ) = 9., denotes t h e complement o f A .
H(l@:).
@ E@
H(IPI1(@)l) =
uF v EnF
H(I@(U)I).
T h i s d e f i n i t i o n makes H an U P - homomorphism, and we have t o show i t i s o n e - t o - o n e and o n t o . of
To show H i s o n e - t o - o n e we d e f i n e f o r each LK by
d'(P,) ( o f course e x t e n d i n g
6'
=
6
E ZK a r e a l i z a t i o n
6'
5(.)
i n d u c t i v e l y t o a l l formulas).
Next, by i n d u c t i o n on t h e c o m p l e x i t y o f @,we show
H(l@l) = { 6 € ZK1f(@)=1).
218
R I C A R 0 0 MORA I S.
F i n a l l y , we have t o prove t h a t i f H( t h e 1 o f LK But i f
.
H( 101 ) =
2K t h e n f o r e v e r y
e v e r y r e a l i z a t i o n o f L, t i o n 29, and hence
]@I
satisfies
101) i s t h e
6
€ 2K
1 of
, 6' (0) =
@ T. h e r e f o r e
.
BK
then
1 $1
1 which means
is that
0 i s a theorem b y Proposi-
i s t h e 1 o f LK L a s t l y , s i n c e t h e image o f L , under H i s an U P - a l g e b r a which c o n t a i n s
2 , the U P - s e t o f generators o f
BK
,
Now g i v e n any U P - a l g e b r a 8 , l e t
we have t h a t H i s onto. K
be t h e c a r d i n a l i t y o f t h e s e t 8 .
Since LK i s a f r e e U P - a l g e b r a we can g e t an wP- homomorphism f r o m L, o n t o
B. Therefore the previons p r o p o s i t i o n gives: THEOREM 3 4 . (a)
(The R e p r e s e n t a t i o n Theorem f o r P r o j e c t i v e A l g e b r a s ) .
Any nP - dyebha h an nP - homomohpkic h a y e
n&. (b)
Any P - dgebha 0 an
UP - kotnomohphic
huge
06 an nP - 5 i e L d
06
an UP - 6 i e L d
o6
06
beh.
IV, CONCLUSION, Our r e p r e s e n t a t i o n theorem f o r f r e e p r o j e c t i v e Boolean a l g e b r a s p r o v i d e d US
w i t h a " b r i d g e " f r o m l o g i c t o s e t t h e o r y , b u t so f a r n o t h i n g was s p e c i
f i c a l l y shown so as t o g i v e a r e l a t i o n s h i p between t h e p r o j e c t i v e f i e l d
-
of
s e t s and t h e p r o j e c t i v e s e t s o f L u s i n and S i e r p i n s k i . Our t e r m i n o l o g y t h e r e f o r e l a c k s some j u s t i f i c a t i o n , which i s however g i v e n b y t h e f o l l o w i n g
and
l a s t theorem: THEOREM 3 5 .
Foh n > 0 ,
1
.i~ an n - phojeotiwe 6 i e l d
06 b&,
whehe
A,
6Zand6 doh "boLd6ace A". PROOF:
( f o r a d e t a i l e d p r o o f p l e a s e see M o r a i s 1976).
We w i l l show t h a t
i s c l o s e d under t h e Pn,
b u t t h i s i s not enough,
however, t o p r o v e t h a t . i t i s n - p r o j e c t i v e because t h e d e f i n i t i o n n - p r o j e c t i v e algebra s t a r t s w i t h a o - a l g e b r a .
B u t i t i s easy t o see
t h e same argument used below can be r e p e a t e d t o show t h a t L I ~ + ~i s
o f an that closed
under t h e P1 o p e r a t o r , and t h i s i n t u r n i s a g e n e r a l i z a t i o n o f c o u n t a b l e
219
PROJECTIVE L O G I C
unions and i n t e r s e c t i o n s ( c f . Kuratowski and Mostowski 1968, p. 341). L e t now @ be any map f r o m
x
E
1
i n t o Qn+l,
[AQQ;'
1
P,($) can be g i v e n b o t h by a Jn+l and a By P r o p o s i t i o n 3
x
P,($)
E
JA+l
a n d we
show
that
predicate.
has two e q u i v a l e n t f o r m u l a t i o n s , namely:
(a)
( 3 F E F U . e e n ) ( W w E F)(x E $ ( w ) )
(b)
(WG E FU.eei)(3v E G ) ( x E @ ( w ) ) .
and
We a r e g o i n g t o use (a) ( r e s p e c t i v e l y (b)) t o show t h a t g i v e n by a
+
F i r s t , since w onto
i s countable, there i s a
[bey]"
F and
s t i t u t e the sets
1- 1 r e c u r s i v e map
by means o f X
{O
G by t h e i r r e s p e c t i v e
tw
E [bey]"
, 1 1 , and as we d i d f o r
I g(w)
= 1) E
The e x p r e s s i o n range
(Wm
.
I n addition,
= 0
{ w 8 [neq]'
ik
.
)'i+lp r e d i c a t e .
@(k) E
[(tlange g
=
{O, 11
.
g(n1) = 1).
otl
I g(w)
= 1) E
1 4n+1 c $ + ~f o r
T h e r e f o r e i f we w r i t e W1 and
sub-
11 i s w r i t t e n Fulln
( ~ 6 1 ) ( ~ 5 * ) " ' ( Q 6 , ) ~ Q ' ~ ) ( 9 ( 6 1 1 m* ,* .
F i n a l l y since
we
we t h i n k t h e domain o f g as w .
@
F a n and g ( k ) = 1) + x E @ ( k ) ]
g = {O,
E w)(g(m)
The e x p r e s s i o n
which i s
X from
c h a r a c t e r i s t i c functions
T h e r e f o r e (a) i s e q u i v a l e n t t o ( 3 9 E "u) (Wk E w )
and
is
P,*($)
E
and so we can t h i n k t h a t t h e domain o f $ i s w . L e t _v be t h e
[bey:"
i n t e g e r a s s o c i a t e d w i t h w C [beq]" g : [hey;"+
x
1 ( r e s p e c t i v e l y iln+l)p r e d i c a t e .
1
i s equivalent t o
A,i)4
= 1)
a l l k we can make
x
E $(k)
a
31 f o r q u a n t i f i c a t i o n o v e r r e a l s and WO
and 30 f o r q u a n t i f i c a t i o n o v e r numbers, t h e statement ( a ) now reads:
31 WO [ ( W O A,31
... "
Q1 Q'O) j L 3 1
- Kuratowski
we s i m p l i f y t h e above t o 31 3 1 VO [ ( A
...
v n+1
) +
v
Q'l,QO]
.
n+1
n Then u s i n g t h e T a r s k i
...
a l g o r i t h m s ( c f . Rogers 1967,
1
p.
307)
R I CARD0 NORA I S
220
which i s a
zi+l predicate.
Now using (b), since
G €
FuRel
is
we, s t a r t w i t h
[
W130
... 2'1 20) +
(WOA W 1
u n
and end up w i t h a $+1
predicate.
JLi and s i n c e @ ( k )
V1
E
1 4n+l
1
... 21 2 ' 0 1
n+ 1
came
I t i s c l e a r by now t h a t one o f t h e most i n t e r e s t i n g notions t h a t
up along t h i s work was t h a t o f
Full;
and i t s counterpart FLLeen
symmetry between these two classes o f
. The
,
c ;n+l
generalizing Ellentuck's F u l l s e t s , sets
n o t o n l y helped c u t t i n g a l l our proofs i n h a l f b u t also, and more s i g n i f i c a n t l y , w i t h o u t t h i s symmetry most o f our p r o o f s
-
c o u l d n o t have
come
through, s p e c i a l l y our l a s t theorem i n which the simultaneous use o f
FuUn
FuRe;
and
was fundamental.
For these reasons we foresee an i n c r e a s i n g use o f these n o t i o n s i n
the
f u t u r e s t u d i e s o f p r o j e c t i v e sets.
To conclude t h i s work, among several i n t e r e s t i n g q u e s t i o n s f o r which a l l t h i s machinery i s applicable, we s e l e c t e d two t h a t we are p a r t i c u l a r l y i n t e r e s t e d i n i n v e s t i g a t i n g , namely: (1) (2)
L or L i f any? Pn P ' If. M. i s 3 universe o f s e t s and B i s an n - p r o j e c t i v e Boolean a l gebra, what can be accomplished i n s i d e t h e Boolean valued m o d e l
What k i n d o f i n t e r p o l a t i o n theorem holds i n
MB ?
REFERENCES Addison, J. A. and Y . Moskovakis 1968,
Some comequenca Nat. Acad. Sci.,
06
t h t axiom o d de6&abLe d e L m i n a t e n a s ,
B e l l , J. L. and A. B. Slomson 1969,
Proc.
Vol. 59, 708- 712.
Models and Ultraprqducts, North
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Amsterdam.
Ellentuck, E. 1975,
The ~owzdatiom06 S w f i n Logic, The Journal o f symbolic Logic, v o l . 40, 567-575
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197+,
22 1
LOGIC
Fhee SwL& d g e b m , S u b m i t t e d t o Czech. Wath. J o u r n a l .
Fenstad, F. 1971,
The a x i a m
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theahy, Annals
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Set Theory, N o r t h - H o l l a n d , Amsterdam.
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Rieger, L . 1955,
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Theory of Recursive Functions and Effective Computability, MacGraw H i 11. Sierpinski, W. 1925, S w l une &abbe d'emembLcb, Fund. Math. Vol. 7, 2 3 7 - 2 4 3 .
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