In [3] and [41 Skolem functions are used for proving the existence of constructive models [1]. In this paper we present a fairly general sufficient condition of ...
SKOLEM
FUNCTIONS
Yu.
L.
AND
CONSTRUCTIVE
MOb
ELS UDC 517.11+518.5
Ershov
In [3] and [41 Skolem functions a r e u s e d f o r p r o v i n g the e x i s t e n c e of c o n s t r u c t i v e m o d e l s [1]. In this p a p e r we p r e s e n t a f a i r l y g e n e r a l sufficient condition of e x i s t e n c e of c o n s t r u c t i v e m o d e l s f o r a t h e o r y des c r i b e d by a x i o m s in r e d u c e d f o r m , i.e., the condition of f i n i t e n e s s of o b s t r u c t i o n s f o r the c o r r e s p o n d i n g Skolem t h e o r y . This condition i m p l i e s a l s o the r e s u l t s of [3] and [4]. o , ..., 4 ~'~.... ,Oe > be a finite s i g n a t u r e (with e q u a l i t y o r without it) that does not c o n 1. Let thin functional s y m b o l s , and let 7" be a (consistent) t h e o r y of s i g n a t u r e ~ . L e t P b e a s e t of p r o p o s i t i o n s in r e d u c e d f o r m that c o n s t i t u t e a s y s t e m of a x i o m s f o r T . P ~ a r e E r b r a n o v f o r m s of p r o p o s i t i o n s b e longing to P , the P*" b e i n g u n i v e r s a l p r o p o s i t i o n s in the s i g n a t u r e
o" ~- ~ < : -¢" , o~, J~"~"7" , " • ",~ ' where .f~ are
s y m b o l s f o r Skolem functions. We shall c o n s i d e r the s i g n a t u r e ~ * . o~ U~cr~Q¢,...~ and n u m e r a t e all the t e r m s of t h e f o r m ~ ' . ( ~ • ,~, , } - ~ ¢ . in such a way that f o r ~ ' , = = , ~ . ( ~ ,~z )wehave ~:~,. JZ
"~7 "'"
"¢'¢'z."
,
"'"
,¢ -, z
,8z ,""
,ez,,~
$ ~ , ~ ~ . By r e g a r d i n g the t e r m s ~. as c o n s t a n t s , we shall define the s i g n a t u r e s c~" m n = ~ u < c~','"~°~-r> ~ / , ~ , . . . , ~ . ¢ ) and wrote ~ =O:~,nt . Let us note that ~ - ~ o = ¢r . If / ~ m a finite m o d e l of s i g n a t u r e c~ n , n ~ zrz, we shall denote by _~A(222 ) a f o r m u l a of s i g n a t u r e 0 ~', defined as follows. Suppose t h a t th~ m o d e l ~ c o n t a i n s ~ e l e m e n t s and that ~ : [ ~ , . . . , ,~- ;'.]--'ly/~I is a m a p p i n g of the s e t (~,y,..., ~ - ~ } into the b a s i c s e t of the m o d e l 2d2 • An e l e m e n t a r y f o r m u l a o r n e g a t i o n of an e l e m e n t a r y f o r m u l a ~T of s i g n a t u r e o "~ will be c a l l e d an a ~ 2 - f o r m u l a if the following conditions a r e s a t i s f i e d : ""~"
.
.
.
t,
,
1) All the f r e e v a r i a b l e s of the f o r m u l a ~ a r e contained in the set { ~ e o , . . . , :Z'~.¢ J . 2) If the t e r m ~ has an o c c u r r e n c e in /2/, it will have the f o r m :vt- ' ~',z ~ , o r 9~]. (~z'~,...,'vz~/' and there exists a term constant
z~ d,
d < zz • ~ =.~/. (~2,%, . .., ~ Z ~ j )
~-*i' ' i n t h e m o d e l ~
and ..e~ (~zsz.,), ~-. "~ zz., w h i c h is the value of the
c o i n c i d e s with the e l e m e n t p~ ('-r_.~ J.
3) In ~ the f o r m u l a d)Z will be t r u e if the v a l u e s of ~',. a r e the e l e m e n t s ~ ( ' ~ ' ) , and the v a l u e s of the t e r m s . ~ j d ' z z . , . . . , . ~ z . / ) a r e the v a l u e s o f t h e " c o n s t a n t s ~ 2 L = f j ( ~ 6 , . . . , O v , ~ j ) . w h e r e fld c o r r e s p o n d s to the t e r m ~] {-z~'"',-vzaj]' as in condition 2. If
A{~)
is a c o n j u n c t i o n of all ~ - f o r m u l a s ( t h e r e a r e evidently finitely m a n y s u c h f o r m u l a s ) , then ~. ~ ,t ( ~ )
.,-Tz~{ ~ z ) will be a p r o p o s i t i o n _~:r,, ...
Remark.
If the s i g n a t u r e Or' does not contain the equality sign, then _Y z~ { ~
) will likewise be with-
out it. A m o d e l /r22 of s i g n a t u r e ~ m , n , ,'Z_ (not c o n t a i n i n g functional s y m b o l s ) is said to be "V-finite if the u n i v e r s a l t h e o r y of any e x t e n s i o n T% T (of s a m e s i g n a t u r e ) is finitely a x i o m a t i z a b l e (by u n i v e r s a l p r o p o s i t i o n s ) . The following a s s e r t i o n is a r e f o r m u l a t i o n of T h e o r e m 2.3 of M a l ' t s e v [2]. P R O P O S I T I O N 1. A t h e o r y 7" is V-finite if and only if the u n i v e r s a l t h e o r y T is finitely a x i o m a t i z able and t h e r e does not e x i s t an infinite s e q u e n c e { ~'nz. I ~'~ ~ ' ) of finite ~ - m o d e l s that a r e s u b m o d e l s of m o d e l s of the t h e o r y T s u c h t h a t none of t h e s e m o d e l s is i s o m o r p h i c a l l y e m b e d d a b l e in a n o t h e r m o d e l . A t h e o r y 7" is s a i d to be s t r o n g l y V-finite if f o r any finite s e t [C~,,, ..., ~ ] of c o n s t a n t s y m b o l s the t h e o r y 7~ defined by the t h e o r y 2" in a l a n g u a g e of s i g n a t u r e o"'-.~u is V-finite. R e m a r k . It e a s i l y follows f r o m the definition that if a t h e o r y T is (strongly) e x t e n s i o n T_~ T of s a m e s i g n a t u r e the t h e o r y T ~ will be (strongly) M-finite. T H E O R E M 2. If a t h e o r y 7" is s t r o n g l y V-finite, then ~ t e m of a x i o m s ~v f o r a 7" c o n s i s t i n g of r e d u c e d p r o p o s i t i o n s . Proof.
V-finite, then f o r any
will have finite o b s t r u c t i o n s f o r any s y s -
L e t r, z 6 ~ be a n a t u r a l n u m b e r , with o~,,, = o ' U < C Z o , . . . , czz,_¢ > U < z', . . . . .
~,,~_, :Z, and let
T~ be a t h e o r y of s i g n a t u r e O'~,, defined b y the t h e o r y 7". Then ~,~ will be Y-finite. It follows f r o m the finite a x i o m a t i z a b i l i t y of the u n i v e r s a l t h e o r y 2,v that t h e r e e x i s t s a finite s e q u e n c e a~to,...,,r&~ of finite m o d e l s of s i g n a t u r e ~-m s u c h that a finite m o d e l ~z of s i g n a t u r e cr,~ can be i s o m o r p h i c a l l y e m b e d d e d in a
371
model of the t h e o r y ~'~ if and only if none of the models ~ z ' , £ ~ can be i s o m o r p h i c a l l y embedded in ~r~ . Let Am be the set of all finite models ~ of signature ~,w with a basic set I ~ l ~ - ~ such that can be embedded in a model of the theory ~ . whereas O~ is the set of all models belonging to A~ that are obstructions. On the set A,,~ let us define a p r e o r d e r ~ as follows: ~ ~ ~ " * ~ " can be i s o m o r phically embedded in ~ " . ~r~ ~ ~ , r , ~ ~ "~ & -~ ['~-~ "~ ~ ~ "/. Since the elements of ~,~ a r e finite models of finite signature ~'~. it easily follows that there does not exist an infinite sequence ~ , , ~ , , . . . , ~ , of elements of ~ such that ~ o > ~ > . . > ~ , > ~ r ~ , >. In other words, < ~,~, .~ satisfies a descending chain condition. Next, the V-finiteness of the t h e o r y ~,~ and P r o p o s i t i o n 1 show that t h e r e does not exist an infinite sequence ~ o , ,r~... of elements of A ,~ such that -1['~'~. ~ ~-~,.) and -~[~d ~ ~ . ) for z ' ~ : Hence has the finite b a s i s property. Let < ~ , ~ > be the set of all natural n u m b e r s with a natural o r d e r . It is evident that < ~ , ~ >
is an
o r d e r e d set with the finite basis property. Let X ~ < ~ , ~ > ~ b e the C a r t e s i a n d e g r e e of . Then X, will also be a set with the finite basis p r o p e r t y [5, 6]. Let .~-~ ~ ~ ~ o ] be a set consisting of O and of o r d e r e d collections of natural n u m b e r s with a partial o r d e r ~ defined as follows: O and < ~_,,...,~> a r e i n c o m m e n s u r a b l e and for
< ~ o .... , % , _ , > ,
Then ~
~'~:
~ ) will be a partially o r d e r e d set with the finite b a s i s p r o p e r t y .
Let ~
-. ~ ~
~ , ~ .... , ~_~ > be a finite model of the t h e o r y ~ .
s t r u c t a tree ~('~,/ over ~ 372
