Back-and-Forth Systems for Arbitrary Quantifiers

I

a. )}.

and its semantics are defined similarly, allowing

infinitary conjunctions. The quantifier rank of a formula of this language is the ordinal defined inductively by: qr (q, )

=

0,

if

q,

is a tomi c

qr (--, q,) = qr (q,) qr (1\ e) = sup {qr ( q, ) 1 q, E

(J }

qr (3 x
(0.. j xl"'"

where Let

~

then

x11. (4)1'''''

l' ... , nk ) is the type of 0.. j'

(n

be an ordinal, or the symbol

L:

w (Q.j')

'E] j

q, 11.)) = m1 x qr (q, -i ) + max n , , -i -<-

considered greater than all the ordinals,

~

consists of the formulas of rank less than ~. Two quantifier '"

~ -elementarily equivalent, ta , Qj)jE] == (~; tr.j)jE]' i f they satisfy the same sentences of this language. The proof of the following lemmas is similar to that of Lemma 3.2.1 in Caicedo 1978. structures are

LEMMA 1.1

I6.the numbetr. 06 tr.e1.a.ti.on, 6u.nc.:Uon,

COIl.6.tant,

a.nd

Qu.a.n~-iMetr.

.6yrnboL6 M MnUe, ~hen L~w(Q.j) jE] -l.6 eQu.-lvalent to Lww(Q.j) jE]" Motr.eovetr., 6otr. each MnUe nand k thetr.e -l.6 a MnUe numbetr. 06 non-eQu.-lva.fen.t 6O!tmula.6 06 Qu.a.nU6.(etr. tr.a.nk equ.a.f.to n w.Uh ~ mO!.>~ R va.tr.-<:a.bfu.

LEMMA 1.2 L ~w (Q.j)j E ]

G-iven

ta.,

Qj)jE]' thetr.e-l.6 ~mO.6t

a.6et 06

6otr.mu.fa.6 06

wh-lch Me 1l0n-eQu.-lva.fent -in th-l.6 .6.ttr.uc.tu.tr.e.

A

A Und.6.ttr.om-Mo.6tow.6U Qu.a.nU6.(etr. 06 type (n l' ... , n k) is a function 0.. which A n m n assigns to each set A a quantifier interpretation Q.(A)c fI(A 1) x .•. x or (A r), with the proper-ty that if 6: A -+ B is a biject1on, then for all (S 1'"'' Sk)'

(Sl,,,,,Sk)EQ.(A)

iff

(6'(Sl), ... ,6'(Sk))EQ.(B).

Itisreadily

seen that

this corresponds to the definition of a quantifier as a c'ass K of structures of the same type, closed under isomorphism (Lindstrom 1966),. by defining

Q(A) = {(Sl"'"

Sk)

I

(A, Sl"'"

A Lindstrom-Mostowski family of quantifiers {Qj L+

of

Sk)

E

I jE]}

K}. defines

an

extension

L0 JW in the sense of Lindstrom 1969 and Barwise 1974 by taking the sen-

86

XAVIER CAICEDO

tences of

Loow(Q.j )jEJ

as the .6tjn.ta.x., and defining for an ordinary structure

if.

a When the quantifier

(a; Qj(A))jEJ 1= 1>.

iff

interpreting the symbol

Qj

the language writing

1>

for

Loow(lJ../)jEJ

L+.

Qj is understood, we will abuse

The existential quantifier is the Lindstrom-Mostowski quantifier

3(A)

defined by the function well know.

=

{S

C

-

A

Is*-

0}.

of type (1 )

The following quantifiers are

CMd£na.t qua.ntiMe.M, type (1), for each ordinal a: Q.a(A) = {S:=.Allsl ;;;. wa}. for each ordinal a and finite 11., the quantifier of Q.~ (A) = {S :=.An 13 I ~A such that In ~S and III;;;. w } . a

Mag~do4-Maiitz qua.nti6~e.M,

type

(11.):

Chang qua.ntiMeJt, type (1) : Q.(A) = {S ~ A I HaJz.t.i.g qua.ntiMeJt, type (1,1): H(A) HenlUn qua.ntiMeJt, type (4) : Hen(A)

lsi = IA\}. {(S, T) I S ~A, T ~A, lsi = ITI}. 4 {S ~ A I 3 6, 9 : A -+ A such that 6Xg ~S}

Note that the logics obtained from these quantifiers do not incl ude those where the meaning of quantifier is not determined by the domain of the structure, like Sgro's topological logic (Sgro 1977). However, if we consider only logics for classical structures with a finite number of relations, functions, and constants, then any logic is a sublogic of some Lww(Q.j)jEJ' Even more, if the logic L is closed under substitution of relation symbols for formulas then L is equivalent to some Lww (Q.j ) j E ] '

§

2 BACK-AND-FORTH SYSTEMS,

Through this section (a, q) and (~, 4) will be quantifier structure where q 4 are interpretations for a quantifier symbol of type (11. " " , n ). A and k 1 B are assumed to be disjoint. Sequences in An u s'' (11. E w) wi 11 be denoted

and

a,

a, a', a', T, T ' , h, h'; the value of catenation is denoted by juxtaposition.

11.

will be clear from the context.

Con-

DEFINITION 2.1.- A baek-and-6olLth be;tween (a; q) and (~; 4) consists of a linearly ordered set p = (P, <), called the set ofpa4amete4.6, and a family

{E~

I pEP}

of equivalence relations in

An U

s"

for each

nEw, subject

properties (i) to (iv) below. Before stating the properties, we introduce some convenient notation. will denote

(a, a')

E

E~; the value of

11.

will be clear from the context.

a

to p

a' For

~

each k, It and a E A k , we can restrict the equivalence relation Ek+n to elements of A n in the following way: R, iff a R, a ii' E An). For

a a'

k = 0 and a = 0 we get lence relation.

EP 11.

restrict~d

to

a

An,

a' (a,

Obviously,

R,a

is an equiva-

BACK-AND-FORTH SYSTEMS ..., p

..

ra Ta = Ix

n

A Ia

E

... P

X

'V

a

is the corresponding equivalence class of in ing the language, [X]~ = u {[al~ I EX}.

a

87

.. a a}

An.

If X ~ An we write, abus-

B.

Analogous notation and observations are valid with respect to sequences in PROPERTI ES.

0

(z)

p 'V

0

(Extension property). Let p', then for all sequences a in (-U:)

pf., =

exist functions

n': .{.

n.

A .{.

n.

-+

= max A and

.6

B .{., 1 <;,i

{n T

1 , · •• , n I<} and p < PI <; in B such that a R, T

... < there

< 1<, such that:

• p ~ n· (A) a a 'V T (i(a) for all a E A .{., 1 « t. < L n,i p p (B) If X,[~A ,l<,i
implies

([ n'l (X 1) If , ... , [nrl< (X I<)]~)E ".

A and B. p (,iv) (Isomorphism property). If (a ... , an) 'V (b ... , b then the assign1, 1, n) ment a,it-'>b,i isapartial isomorphism from 01- to&. As (-U:), interchanging the roles of

(.i..u:)

DEFINITION 2.2.(01-; Cij)jEJ

each

jEJ.

(p, {E~I pEP, nEW})

isa

:to ($-;Jr.j}jEJ ifitisonefrom (OI-;Cij) The existence of such a relation is denoted by

baek-and-6oJvth

to

(:6-';"j)

(a;Cij)jEJ~

61tOm

for

()&..;\)jEJ

REMAR K. We assume that any back-and-forth sa ti sfi es the extensi on property for the existential quantifier. It is enough to postulate in (,i,i) , for each p' p a 'V T and p < p' the existence of a function 6: A-+B such that o a. 'V T 6 (a). Property (B) holds automatically. DEFINITION 2.3.- {E I nE w} is a bael<-a.nd-noJt:th wUhou:t pMamUeJt1> 61tOm n ioi, Cij)jEJ to ($-; "j )jE] if En is an equivalence relation in An U Bn for each n , and properties (,i) to (,iv) of Def. 2.1 hold, dropping the parameter conditions.

§

Such relation is denoted by (or; Cij) jE]

'V

(~; "j) jE]

.

3 CHARACTERIZATION OF ELEMENTARY EQUIVALENCE.

Let or and!tJ., be classical structures. C = {Cij I jE J} and D = {\ I jE J} are interpretations in 01- and~, respectively, of the quantifier symbols

XAVIER CAICEDO

88

{Q.[ jE]}. j

THEOREM 3.1. -

(ot; C) ~ (~; D) then (ot; C)

I6

Suppose (ot;C) ~ (~; D). Bn define:

E

a

{J

(ot, a; C)

iff

"val

iff

(:e-,1';D)

(ot, a; C)

iff

and a, a'EA Yl

For every {J
PROOF.

r ,1"

[o ,« ) "V (%--; D).

{J + 1

(Ol, a' , C ).

-

{J+l

==

(~,1";D).

{J+l

(:e-,

-

T

;

We show that this gives a back-and-forth with parameters (e ,«},

(J "V

D) • It is clear

that

is an equivalence relation and properties (i) and (iv) of Def. 2.1 hold. Since Given a E An, dE Ak, {J
eLi) and (-UA:) are symmetric, it is enough to show (u).

let

t~ a

= {
and

(Ol,a;C) 1=


a

By Lemma 1.2, the conjunction /\ t may be considered a formula of Since the logic is closed under negations:

Loow (2..; )j E f (1)

Now we are ready to check the extens i on property. symbo 1 of type

) and

Yl

(Yll"'"

.

k

and {J < {J 1 < ... < {J -6 = {J'. (Ol, a; C)

1=

::I

-6

= m~x .{.

.{.

{J + n or;;;{J + -6 or;;;{J'. i

Since a "V

T

(Ol,a;C)

and so /\ t.( a

(g"

b),

rank or;;; {J

T; D)

and define

we have

a

{J' + 1

x..{. /\;t.a ( x.). .{.

::I

1=

6. (a)

.{. (Ol, a,

£;

=

b.

be a 1J'

Suppose that a"V

T ,

qua n t i f i e r with

T E

BYl

E A .{., (oz, a; C) 1= /\ t a. ( d ), and so last formula has qr = qr (/\ t.) + Yl • a .{. then:

For each

(x .{..).{J' This a

X. /\ t .

Yl.. .{. Yl.

Qf'

Let

Since

{J + 1 C) -

(~,T;D)

Choose t:»

a

bE

(2)

n.

B.{.

such that (~,T ; C) 1=

is a complete set of

formulas {J

(i6--, T, 6..{. (d); D ) and so a a"V

T

of

6 (d),

the first condition of the extension property.

n.

Now let x..{. c- A.{., i •V

aEX.

.{.

/\

c ; (Xl. a

=

1, ... , k.

By (1) : [ X .]{J

.{. a

For each i

let T;(x) ~

be the formula

89

BACK-AND-FORTH SYSTEMS Similarly, in ~ :

13

[ 6'.-L (X.) I = {x E B -L r

n.

-L

I

(lr,

T ;

D)

T;

F

~

(x )} .

Moreover, the formul a ~ xl , ..• , xk (T 1 ( xl) , .•• , Tk ( xfz )) has qr = 13 + .6 .;; 13 I Therefore, by (2) and the definition of quantifier, ([X J13 , ••• , [XI..]13 ) E q. 1 a '< a j implies succesively:

(01., a; C)

F

QjX 1 " " , xfz(T1(x1),

,TIt(XIt)) ,

(s-,

1=

Qj ~1 , .•• , Xfz (T1( xl)'

, Tit (x It)) ,

T; D)

([6'1

(x1)]~

, .•. , [6'fz

•

(Xfz)l~) E\. •

REMARK 3.2.It is clear from thedemonstration of the last theorem that the functions 61 " " , 6 in the extension property may be chosen homogeneously, k so they work for all quantifiers of type (n l' ... , n ) . Even more, it is possifz ble to choose for each p', a, and T, in advance, a family {6 n: An .... B n In E w} which works for all quantifiers.

16

PROOF.

(Ol; C)

Assume

a'VT

'V

(ex ,<)

(JIy; D).

'V

ex (01.; C) == (;G-; D).

(~; D) then

We prove by induction

qr(¢).;; 13 < ex, then for all

¢ ( !j) , that if

plexity of the formula

13

(ex ,<)

(al; C)

THEOREM 3.3.-

on the

com-

a

T

and

,

implies: (al; C)

(~;D) 1= ¢(r).

iff

¢(a)

1=

The result follows from property (~) taking follows from the isomorphism property (~v). let ¢ = Qj xl"'" suppose that (1) holds for ¢1'"'' ¢k' let p + .6 .;; 13. Define for each ~: conjunctions is trivial.

X.

-L

= {a

Y. = {b -L

< P + 1 < '"

Since

p

tions

6.: A -L

n.

-L ....

n.

ft. E A -L

E

n.

B

-L

(1)

a = T = 0. For atomic formulas, The inductive step for negations

xfz

p

(¢1"'"

= m~x

¢fz)

qr(¢~),

= m1x

qr';; 13 n~,

and then

-L

(01., a; C)

1= ¢. (Ci ) }

(:6-,

1= ¢. (Ii)}

T; D)

be of .6

(1) and

-L

-L

< p +.6 .;; 13 , then by the extension properties there are func-

B -L,

n.

g.: B -L

n.

-L ....

• p

A

•

aa'Vr6·(a)

• P

-L

•

Tb'Vag~(b)

If ( [Xl] ~ , ... , [X k J ~ ) E qj

then

-L

such that for all for all

n. Ci E A -L n.

bE

B

•

-L •

(I) (I I)

([6~(X1)]~,... ,[6~~Yk)]~)EItj" (III)

90

XAVIER CAICEDO

Xl. = [XJ.]~'

CLAIM 1.

Yl. = [Yl.j~.

If aal~aa· where (Ol,u;C) 1= tP.(a),then(~, T;D)l=tP.(fi.(a)) .{. .{. .{. because of (I), the fact that qr (tP) '" p , and the induction hypothesis. By transitivity: a at ~T fi. (a), and so we have again ia , a, C) = tP. (at). This shows .{. .{. [X..{. jPa eX.. The other direction is trivial, and the case of Y..(. is similar . - .{.

6'..{.

CLAIM 2.

(X.) c Y., .{. -.{.

g'. (Y.) -c x.{..• .{..{.

This follows from (I), (II), and the induction hypothesis for [fii (X;.l]~ = Yl.'

CLAIM 3.

tPl.'

[gi (YJ.)]~ = Xl.'

The inclusion from left to right follows from Claims 1 and 2. then

.P • P • T b'Vag.(b)'VTfi.g.(b) and so .{. .{. .{.

.P. b'Vfi.g.(b). T .{. .{.

Suppose

6E

Yl.

•

Butg.(b)Egt.(Y.)cX . .{..{. .{. .{.

and we have bE[fi'.(X.)]p . .{. .{. T By (III), (IV), and Claims 1 and 2, we have that (Ol, U; C) 1= Q.j ;1""';11. (tPl""'tPlz) iff (Xl'oo.,XIz)Eq. iff (Y1,oo.,YIz)E!Z.. iff ($-,T;D) 1= • •

Q.j "1"'"

j

"Iz (tP 1' ... , tP Iz ).

COROLLARY 3.4.00

(b)

(Ol; C ) =:

(:& ;

j

•

a ~ ,e) (Ol;C) =: ($--; D) J.fifi (Ol; C) 'V (~; D). (a ,e) l.fifi (01-; C) 'V (~; D ) fiM aU. MeUnaL6 a.

(a) D)

COROLLARY 3.5.Ifi the. fiamiliu C, D and the. !>.iJni1.aJU...ty typu ofi Oland lJ, Me. MnJ..te., the.n the. fioUowJ.ng coneUUoM Me. e.quJ.va.Ce.n.t: (or; C ) =: ( J,; D) (e.quJ.va.Ce.n.t fio!Z. MnUMy fioJunulM) , (w,e) (Ol; c) 'V (X-; D ) ,

(01- ; c)

(n ,e ) 'V

(Z-; D)

fiM

PROOF. (J.J.) => (ill) and (ill) Lemma 1.1 and Theorem 3.1. •

=>

aee

nEw.

(l.) ar.e trivial; (J.)

There are more convenient characterizations of Corollary 3.4 (b). THEOREM 3.6. (l.)

The. fioUowing Me. e.quJ.va.Ce.n.t:

(01-; C ) ~ ( ~; D ).

00_

=>

(J.J.) follows

from

elementary equivalency than

91

BACK-AND-FORTH SYSTEMS

(u) (Ol; C)

~

(-Lil) (Ol; C)

'V

PROOF.

(.i)

=>

( ;g,; D) (Xr; D) (-Lil).

wheJte P 1.1> non-well. ondened, (bac.k-and-6oJtth wUhout Pa.JLamet:eM).

The same as the proof of Theorem 3.1, but

simplifying

the definition of back-and-forth to: a 'V 0' iff (Ol, a; C) ~ (Ol, 0'; D), etc. Instead of t~ one defines t .. = {4> (y) I (Ol. 0; C ) F 4> (a)}, By Lemma 1. 2, a. a. fI t .. is a sentence of L (Qj') j' E ] ' The rest of the. proof is simpler because a. oow we do not need parameter conditions. Choose any non-well ordered set P and define 0 ~ t iff a » t , Let PI> P2 > ,., be an infinite descending sequence of P , One shows, as in the proof of Theorem 3.3, by induction on the complexity of 4> (y) , that for all n: => (U).

(-Lil)

(U)

=>

(i),

o

Pn 'V

implies:

r

(Ol; C ) F 4>(a)

iff

(~; D ) F 4>(r),

To use the extension property in the inductive step for one chooses

§

4

P n + .:\

< P n +.:\ _ I < .. , < P n ' •

Qj

xl.· ... xm(4) l' ,, . , 4>m )

INTERPOLATION AND MONADIC QUANTIFIERS,

A quantifier symbol of type n(x ,This includes the cardinal quantifiers Qa: "there are at least wa eln ments", as well as Chang's and Hartig's quantifier, but notthe Magidor - Malitz quantifiers. Throughout this section L M will denote the logic L oow (Q.). E ] , j j where Qj runs through all the possible Lindstrom-Mostowski monadic quantifiers.

»·

LEMMA 4. I . -

a =L

oow

16 Ol and :t; Me .:\:tJtuc..tu!l.u 06 poweJt at

(Q):tr impUu

I

a

=L;,t-

M

mO.6:t

wI' then

•

PROOF. Let 2n = {o I 0 : n ... 2 }, for seA 1et sa = Sand sl = A - S • For each nEw and F: 2n ... {K 'K cardinal, K" wI} define the quantifier:

~(A)={(SI,.. "S)',;/OE2nl n

l~~)I=F(O)}, Since a monadic structure n L« n ,(. (A, SI".'. Sn) is completely determined by the above set of cardinals, there corresponds to it a unique F such that (Tl'".,T n) E Q F (8) i.ff (8, T1,.", Tn) "" (A, SI" ." S). Let 3!K x 4>(x) mean that the truth set of, .4>(x) has exaclty K n elements (K';; wI)' Clearly, for structures of power at most wI' Ol=L (Q);6implies

a

=L

(Q

3,K)

Ir.

and so

oow l' , K definable from the former quantifiers.

oow 1 (Q),(",' because the QF' s are oow F F If ~O is 4> and 4>1 is I 4> :

a

=L

92

XAVIER CAICEDO

() () ) QF"l"'" ( '1>1"1"'1>" rt

rt

rt

=>

f\ I>E 2 rt

1 (,,)1> (i I 1 . [3.' F(I> ) x i
(en;

Q (A))F 'V (ho; QF (B))F' To conclude the proof, we show F that the given back-and-forth has the extension property with respect to each monadic quantifier, and apply Theorem 3.3.

By Therorem 3.6:

~r

Let

a

<~

~

'V T ,

I,

and let

6:

If

A -+ B be the function given

QF's

by the extension property for all the

homogeneously

(see Remark 3.2).

M=([X11~"",[Xh]~)EQ;(A),choose F

such that

MEQF(A),

thenM' =

([ 6' (Xl) J~ , ... , [6' (Xh) l~

(B, M')

and so

) E QF (B) by the extension property. Hence, (A,M) "" M' E Q.(B) by definition of Lindstrom-Mostowski quantifier . • j

The proof of the following lemma is analogous.

Qa

Let

runs over all the cardinal quantifiers.

en

L ww(Q), where

==

k

or == jJ.L M In Friedman 1973, Friedman gives a sentence '1>

LEMMA 4.2. -

L C

L = Loow(Qa)aEOr,where C

hnpUu,

of

Lw(v(Qa) (respectively,

Q is Chang's quantifier) containing a relation symbol

P, such

(a,

P) 1= ep.

that for every Ol

there is at most an interpretation

P for which

However, K = {en I (en, p) 1= ¢ for some P} is not elementary in L This is shown C' giving a pair of structures en EK,:f:.- f/- K which are elementarily equivalent in L It is observed that this is enough to show the failure of Beth's definability

C'

theorem for any logic between

Lww(Q",) (respectively

Lww(QJ) and

Lemma 4.2, we have: THEOREM 4.3.Beth (Lww(Q), LM)

Beth (Lww(Q",),L

6iLU6.

M)

6iLU6

60JLarty

a>O.

L C'

After

MAD,

For example, Beth's theorem does not hold in logic with the Hartig quantifier since it is between L ww (Q) and LW The first part of the last theorem is not true for a = 0

because

satisfying interpolation; (see Barwise 1974). ic

L

is a sublogic of w 1w hence, Beth's theorem.

L extending M

Lww(QO)

and

The same is true of l:!.(Lww(~))

The next theorem imposes a restriction in such logics. A log-

L+ has 4elat£vizat£ortO if for every monadic predicate V and sentence ¢ of L+ ¢ V such that ta , V) 1= ep V iff en,.. V 1= '1>. Here, V is

there is a sentence

an interpretation of

V and

m~ V

is the restriction of the universe and

rela-

tions of en to the set V. Almost all interesting logics have this property. An exception is logic with Chang's quantifier. The Lowenheim numbe4 of a logic L+ is the smallest cardinal of power at most

K.

K

L+ has a model it has one L+ satisfies the downward Lowenheim-Skolem

such that if a sentence of

It may not exist.

theorem if its Ulwenheim number is

co ,

BACK-AND-FORTH SYSTEMS

THEOREM 4.4. -

Le;t

L+

93

be a tog.£e between

L ww and L hav-i.ng Ile.eauv-i.M L+ has Lowenheim numbeJr. «, then 601l

16

za:tion and -baU66y-i.ng -i.ntVtpota:tion.

mI .;; "1m

any -bentenee ¢ hav-i.ng -i.nMnUe modetJ> Sup {] 1= q,} = x , 16 not have Lowenheim numbeJt such. !>entenc.e haJ> modetJ> 06 MbWuvt.Uy iMge -i..ty.

PROOF.

L+ doe!> c..etJLd,[nat-

q, with infinite models of size X, X < K

Suppose that there is

,

but not of any size between x+ and K (included). By definition of U5wenheim number, there is a sentence l/J with all its models of power greater than X. Let e be a sentence whose models are the equivalence relations. The classes K 1 (respectively K ) of models of e having as many equivalence classes as a model of 2 q, (respectively, a model of l/J) are PC classes of L+, defined by the projection of the sentences: 8 /\

"6

is a function onto V" /\ 'ifx 'r1 Y (x E Y +-+ 6(x) = 6(Y)) 1\

q, V (resp.

1/1 V).

Since these sentences do not have common model s of power 1ess or equal than", K 1 are disjoint PC classes of L+. However, they are inseparable in L+.If 2 has X has equivalence classes of power X', and (A', E ') E K has (A,E) E K 2 1 X' equivalence classes of power X', an easy back-and-forth argument shows tha t and K

(A, E) =L (A', E'), and so (A, E) =L (A', E'). C M terpolation fails in L+. • COROLLARY 4.5. -

Le;t L+ be a tog-le between

tiv-lzaUon and -baU66y-i.ng -i.nteJtpotaUan, then Sk.atem theOllem.

From this we conclude that in-

L ww (12. ) and

L hav-lng IletaM L+ -6aU6Mu the dOwnwalld Lowenheim-

0

PROOF. There is a sentence in Lww(Q.O) ' and therefore in L+, which has models of power w but not larger. By Theorem 4.4 it must have Lowenheim number w .•

§

5

COFILTER QUANTIFIERS, A SIMPLER BACK-AND-FORTH.

n Let 12. be a quantifier symbol of type (n), q ~ 9(A ) is a eoMUeJr. -lnteJr.pllUaUon 06 Q. if it satisfies Monoton-i.U.ty: SEq and S ~ s' imply S' E q, and V-f.,f,Wbu.tiv-i..ty: SuS' E q implies SEq or S' E q. Obviously, q is a

q

cofilter interpretation iff the "dual" interpretation = {S I A - S €I- q } is a filter over A. In terms of the language, (m; q) must satisfy the schemata:

\!X(¢4l/J) 4(Q.xq,4Q.xl/J),and

q

tifier is w-eompte;te if u {Sn I nE w} E q implies

QX(¢V1/I)4(Qx¢VQxl/J).

is an

w- complete filter. for some n. •

Sn E q

Acofilter

quan-

Equivalently, if

Qa is w-co'mplete if its cofinality Ql' Magidor-Mal itz quantifiers are not

The cardinal quantifiers are cofilter, is greater than

w,

as is the case of

cofilter, however the quantifier

n H

o'

where

H~ xl'"

xn ¢(x 1"'"

eto x

is equivalent to the cofilter

quantifier

means "there is an infinite set n)

1

such

XAVIER CAICEDO

94

tha~

a 1, ..•• an

for all distinct


E

r

there is a permutation

'/1

for whi ch

holds".

The back-and-forth characterization of elementary equivalence may be simp1 ified in the case of logics with cofi1ter quantifiers:

DEF I NITION 5.1. A -6.{mp!e bac.k-and-60Jt.th 6/tom (al; q) to (~, Jr.) is defined as in Def. 2.1 (respectively Def. 2.3), except parts (A) and (B) of the extension property which are changed now to:

Ci-t)

aE An

For all

bE Bn such that:

there is

• P (A+) aa'VTb.

(B+) [ajP E q a

implies

[lij~ EJr..

(M:i+) The analogue in the other direction.

In the following theorem, the subscript S(Fin) indicates that there

tot,

p1e back-and-forth from classes of each

Ek

forth where each

q)

is finite.

Ek

(:&-; Jr.) where the number of

to

Similarly,

S(w)

is a sim-

e q uiva 1 ence

indicates a simple back -and-

has at most countab1y many equivalence classes.

THEOREM 5.1. - Le:t C and D be MnUe 6amL.UeJ.> 06 c.G6ille.!L quan..ti..Me.!L btte.!Lp/te:tatiOft6 and a.Mume the numbe.!L 06 /te!atiaft6 in Mc.h -6tJr.uc.tu!Le. is MMte, then: (a)

1'1

(aL; C) == (i:-; D)

r 6 in

ad~a1'1

the quan.ti Me.M

1'1

(b)

(al; C ) ==

(e.)

t

6

(z, ;

(oz; C)

(oZ; c)

in6

~

(

.t

D)

S(w)

i66

Me

(1'1,<)

'V (JY;D). S(Fin) w - c.amp.i'.e:te.:

(aL; C)

(aL; C)

(h-, D) with

P

(1'1,<) 'V

S (w)

(

lr ; D ) •

nan we!./'. OfLde.!Led, then

:c- , D ) .

PROOF. (a) Since a back-and-forth is a simple back-and-forth, from left to right follows from Theorem 3.1 and Lemma 1.1, plus the observation that the equivE~

qr at most k. For the q and Jr. are cofilter interpretations, then properties (~+ -M:i+) imply the original extension property. Given a ~T choose for each a· E An some 6 (a) = b such that (A+) and (B+) hold. Assume as in ~­ BofDef. 2.1 that XCA n and [Xjk E q• Since [Xjk =u{[ajklaEX}, and a k a the number of equivalence classes of the relation 'V must be finite, the union is alence classes of

result definable by formulas of

other direction one shows that if

of a finite number of classes. By the di stri butivity of q, [aj ~ E q fo r some dEX. By (B+), [6(a)jk EJr.. Bymonotonicityof n , [6'(X)jk EJr.. T

T

(b) and (e) are similar; the converse of (e) fails because the number of finable subsets may be too large. •

de-

95

BACK-AND-FORTH SYSTEMS

§

6 APPLICATIONS TO Lw w (Q1) .

Let al;. be a structure of type T;.' then [Oz.0" •. , a n] is the structure that has the disjoint union u {A;. 1;. .;; I'd as universe and whose relations are the corresponding copies of the relations of each al;.' A relation R between tures is PC in L+ if there is a sentence in L+ such that Oln)

R(Olj"'"

iff [0l1"'"

Oln'x;.]

struc-

1= ¢.

Let t : T ~ T' be an ;'nt~pnetat£on between si~ilarity types (languages) as t denotes the restri cin Barwise 1974. If oi is aT' -structure, then a tion (projection) of al to a T-structure, via t. In the case of a himple interpretation, we write a I' T.

r

LEMMA 6.1. Let C be a Mme 6amil.y 06 Und6tnam-MOhtoW.61U. quant-iMenh such. that L w w(Q1' C) hM neiativ-izaUon.6. Let t-i: T~ T.i. be .i.ntenpltetat£On.6 wh~e T .i..6 Mme. Then the 60llowing Iteiation between htJtuctunU al, $-, a.nd p

and

u

PC

.i.n the g.i.ven log-ic:

(al I'" t

1

; Qj' C)

p

'V

s(w)

(i&- I' t 2 ; Qj' C ).

PROOF. It is a routine exercise to write down sentences stating that some P {E~ I pEP, nEw} form a simple back-and-forth from a r t j to k I' t • 2

The only second order statement in the original definition, the extension property, has become first order. Q and the quantifiers of C are needed to state the ex1 tension property. Qj is also needed to put a countable bound on the number of equivalence classes. The infinite family of relations En(p, x j , ... '''n' Y1'''' ,Y n ) expressing the E~ 's may be reduced to a single one by a suitable encoding of the finite sequences, see Caicedo 1978 for details. Relativization is needed to express the meaning of each quantifier in o; I't j and :t:-l't 2 instead of [OZ,~,P, ... ] .

•

Let B(U, V, P, <) be the sentence given by the lemma where U, V, P, and
e-

In the next lemma, a class K is PC if there exists a cou.ntable set of sentences T and an interpretation :t such that K = {Ol t t I a 1= T}. L+ satisfies LS(w if every sentence (theory) having a model has one of power at most "'1' 1) LEMMA 6.2. L+

ex:tencUng

Mntence 06

L

WW

Lww(Qj)' then

(~; Qj ) w£:th

th~e

P non-weU Md~ed.

theo~e.6 Ol: and PROOF.

Let K j and K be PC &M.6e.6 06 It countably compact log.i.c 2 (Q 1) and hav.i.ng neiativ.i.zaUo n.6• 16 they aILe .i.n.6epaJtable by It

Let

:t;

may

t.: .{.

T

aILe

a

E K j , I(;- E

In CMe

+

L

K~huch:thcLt

.6ati.6Me.6

be chosen. 06 pow~ at mO.6t "'1'

~

T.

.{.

and K..{.

= {Ol ~t...(. I oi

LS(w j )

1= T .}, with ..(.

(Ol;

Qj )

P

'V

S( "')

60n countable

T...(.C L+.

By

countable compactness, we may assume the number of symbols in T and T.i. to be fi(lite

96

XAV IER CAICEDO

(see Flum 1975). Suppose K and K 2 are inseparable in L ww(Q.1)' Since there 1 are finitely many non equivalent sentences of rank at most n- in th-is logic, there

~

exist alnE K 1,

n E K 2 such that

~Lww(Q.1) .i:-n ·

Otherwise, we could sep(n,<) arate the classes by a sentence of rank n. By Theorem 5.1 (oZn;Q.1) 'V (-6- ; Q.1) ' n al n

Sew)

In this way, each finite part of the following theory of 8(U, V,

Here,

U T = {1>U 11>

of the full set.

P,<)U{T~, T~}U{P(cn)' E

n.

Making

L+ has a model:

c n + 1
By compactness, there is a model

a

= al' r t

1

and

J,

= Xo'

rYe= ra',

;t', P , ... ]

~ .t 2 , we have the result. -

It is the The next theorem is analogous to Lindstrom's theorem for L w w' best possible result in the sense that it is possible to have proper countabl y compact extensions of L ww(Q.1) by non-monadic quantifiers satisfying LS(w for 1), example logic with the Magidor-Malitz quantifiers Q.~.

THEOREM 6.3. - Le..t L+ be. a .togic betwe.e.n ltel.a.tiviza.tioYilJ, In L+ .6a-tW 6.{.v.. compac.tYle..6.6 and then

L+

L ww (12 ) and L having M 1 LS (w 1) n0lt cowu:a.b.£.e. the.oJti.e..6,

~ L ww(Q1)' +

1> E L - L ww (12 ) , then the cl asses K 1 = Mod (1) ) , 1 K = Mod ('(jl) are inseparable in lww(Q'l)' By Lemma 5.3 and Theorem 3.6, there 2 exists OZ F (jl and i&- F I (jl such that al ~ L /tY , a contradiction. M

PROOF.

Suppose

The next theorem shows that interpolation fails strongly in Theorem 4.3 for a = 1.

THEOREM 6.4.I

n they

Le..t

(jl

have. an inte.Jtpo.tant in

and

!J;

LM.

It implies

be.6entencv..on

L the.y have. One. in M,

L w w(Ql).6uch.tha.t(jlI=!J;. L ww (0. ) , 1

PROOF. Let r be the common language of 1> and !J;; if they do not have an interpolant in L w w ( Q1 ) ' then the PC classes K = {OZ r- T I or 1= 1>} and 1 K = {tr t T \;e, F 1/1} are inseparable. By Lemma 5.3, there are structures OZ I=(jl 2 and Z, F 1/1 such that (J1.. t r ~ L :t, ~ T. Therefore, 1> and !J; do not have interpolants in

L W

•

M

L ww (12 1 ) , In T ha.6 a (uncountab.tel model., it ha..6 a (u,ncouYl.ta.b.£.e) model. .6aU~nying at mo.6.t coun.tab.£.y n0lt each nEw. many n-.typv.. in L W THEOREM 6.5. -

PROOF.

Take

Le..t

K

1

T be. a couYl.ta.b.te. the.oJty in

= K 2 = Mod(T)

in Lemma 5.3.

Theyare

obviously insepa-

97

BACK-AND-FORTH SYSTEMS

rable; then we have of power at most

a-6-;'.

(a, Q1) If

wI'

t

i~wJ,

(JtJ-; Q1) with P non-well ordered, in

AIt , find

bEBIt such that

ia , a)=\

a

and f(;..

;-6;

then

(Q )(i6, b)=L (Q)(Ol, a'); see Theorem oow 1 oow 1 3.6. By Lemma 4.1, ta , =L (m, Therefore, it and satisfy the same M type in L Since the number of equivalence classes of - is countable, the same

Hence,

a)

a').

M.

is true of the number of types.

a'

•

In L w w we have that a theory with infinite models has an uncountable model satisfying at mostcountably many types OVe!L ea.eh eoun;ta.b£.e ~u.b~e.t. Thi sis no't true here as shown by the counterexample:

Q x P(x), --, Q x R(x), 1 1 If x If y(P(x) /\

p(y) /\

"<

-w

ct

UltcaJt OILdeJt" ,

x =1= Y -+ 3 z(R(z) /\ x

< z /\ z < y».

Any model satisfies uncountably many types over the interpretation of R. The last three theorems hold for any countable compact logic having relativizations and satisfying Linds.-Most. cofilter quantifiers.

LS(w

where

L w w(Q1' C ) C is a finite family of

1), In Theorems 5.5 and 5.6 one must change

LM for

L w w (Q1' C) all monadic quantifiers. However, we do not know of any concrete example. On the other hand, one can show analogues of these theorems for Stationary logic L(aa)·, see Caicedo 1977 b. So, any countable theory of L ~ct) has a model satisfying at most countablY many types in the infinitary logic L(ctct)M' L(ctct) is maximal in L(ctct)M' with respect to compactness and LS(w ) , and any pair of sentences of L(ctct) with an interpolant in the result obtained by adding to

1

L(act)M have one in

§

L(aa).

7 AN EXTENSION OF Lww(QO
PRESERVED BY PRODUCTS.

In Badger 1977, L.Badger shows that elementary equivalence of structures is not

Lww(o.o
Q8

For ea(h It define an It- variable quantifier symbol R It where R It xl' ... , x lt

98

XAVIER CAICEDO

The same quantifier is obtained if one asks the set (I, <) to be well ordered or just of type (w,<). By Ramsey's Theorem this is a cofilter quantifier. Let

Lww(R
where

runs over all permutations.

n

The extension is proper because well

is definable there by the sentence I R

2xy(y
L w w(Q.o
order

definable in

Usi ng our back - and - forth

for cofilter quantifiers we show: THEOREM 7.1.-

pIWduc.M

06

L ww (ReJtved by Mn.-Ue

~:tJw.c.:twz.e.6.

PROOF.

Let

x.-

IJl and

IX

(respectively

.."

and

I

IV

)

be

L

ww

(R
mentary equivalent. Since every sentence has finitely many relation and quantifier symbols, there is no loss of general ity in assuming that the similarity type is I m finite and restricting our considerations to C = {R , ... , R }. By Theorem 5.1: w (en; C) 'V ( it:-; c ), and the same is true for en 1 and x,. I Let S(F.tn) k. B I = (w, <, {E. I n, k. E w} ), ,t = 1,2, be the corresponding back-and-forths.

.<.,n

~

x

x'

For each pair of sequences = (xl'"'' xn), = (X'I'"'' x'n) define ((x 1 x'I) , ... , (x n x'n Define E ~ in (A x A,)1t U (B x B,)n by:

».

x+ x'~ y + y' We claim that the system forth from each

IJl x OZ'

E~ .

iff

B+ = (w,<,

to!6 x ~

I

,

xEk.I ,n y

{E~ liz,

It

and

E w})

X'Ek. 2 ,n

x + x'=

y'

is a simple

bacf<-and-

with finitely many equivalence classes for

The fact that

Ek. is an equivalence relation, and properties (,t) and (,tv) of n Definition 2.1 follow easily because they are expressed by universal Horn sentences, that are preserved by cartesian products, and we defined the new relations as the "product relations" are ordinarily defined in the cartesian product. EIz has finitley many equivalence class~ because for any the equivalence clas~ of this sequence with respect to En has the form: .. .. k .. .. ... .. 1<. .. ... k. [x+x'] = {u + U' IUE I x l and u' E l x"] l . Iz Therefore, if E~ has M classes and E~ has N classes, E has MN

x+ y

classes. IJ

=

X+

x'

pose that

It remains ton show the extension

E (A x AI)n and

k + Jt < Iz'.

T

=

Y+

proper't~

yIE(BxB,)n

(u+ -Lu. +) for ea:h R": Let

be such that

IJ~T, and sup-

By definition: ~

x

Iz' 'V

~

y

and

x'

Iz' 'V

y'

(l)

BACK-AND-FORTH SYSTEMS

Let

b=

a + a'

E (Ax A' )"l.

b'

(6 1"", bJt) and

By the extension property in B and 1 b~) E B,Jt such that:

B 2'

there

exist

= (bi, ... , ..

.. tjb

k.

of.

xa

of.

'V

and

[Gi11ERJt(A)

[a' l~, From (2):

99

~

E

k

x'a'

'V

(2)

[bl~ERJt(B)

implies

(A') implies

....

tj'b'

[b'l~,

E

R

Jt

(B').

(3)

.. .. k ..... (0. a + a' ) ~ (r , b + b').

It remains to show that (4) below implies (5):

s=

t

a-

.l'lk E RJt(AxA') o

(4)

[b+ b'lk E RJt(BxB')

(5)

r

Assuming (4), we get a linear order (1, <) of type tha t for all (u 1 wI) < ... < (u Jt W Jt) ; n 1,

(x,

U

l' ...•

(x', W 1'"

.•

U

Jt)

W Jt)

k 'V

k 'V

(w,<) with

1 ~AxA'. such

('x, Gi)

(x', ci')

If J and J' are the projections of 1 in A and A' respectively. one of them, say J. must be infinite. Choose a function 9 : J ...J' such that (u,e(u))EI for )). all u. E J, and define a linear order in J by: "i < u iff (u 2 1,g(u1)J«u2,g(u2 It must have type (w,<). Moreover u < < uJt in J implies (u )) 1 l,g(u 1Jt , uJt) E [iil~. < ... < (u Jt, g(u,,!)) in 1, and so (u 1 ' Hence, [aJ~ E R (A) • k Jt and [b 1. E R (B) by (3). Let (L. <) of type (w,<) with L c B be such that tj u1 < ...
(u W) < ... < (uJtw) in 1. Hence, «u W ) ••• (uJtW)) E S and so (x',w •••• ,w) 1 1 k • k • 'V (x', a~) 'V (tj', b'I"'" b~). By the isomorphism property of back-andforth. this impl ies b\ = ••• = b~ = b'. Define the infinite set 1+ = t x {b'} c Bx B', and order it by: (u b') < (u b') iff u < u • Then 2 1 2 1

ai, ... ,

(u b')<

<

«u 1 b' )

(uJtb'))

1

CASE 2.

Both

Then we also have

(u

Jt b') implies (u 1 ... UJt)E E

[6 +6'1:.

[bl~.

and so

This shows (5).

J, J' are infinite.

[b'l~, tj

E

RJt(B'), by (3).

Let (L', <) be a linear

order

XAVIER CAICEDO

100

of type (w,<) for which ui < ... < u~ implies (ui, ... ,U~)E [b']~' Let 6: L..... L' be a one to one order preserving function and define 1+ = {(u, 6(u)) I uE L} with the order (ul' 6(u 1)) < (u 2' 6(u 2)) iff "i < u2 iff 6(u 1) < 6(u 2). If (ul' 6(u 1 ) ) < ... < (uJt' 6(u Jt)) <6(u1),···, 6(u Jt))

E [

and shows (5) again.

then

(u 1" ' " uJt)

b' I~, which proves

E

~

[b

k

Ilj

and

«ul' 6(u 1)),··· (u Jt' 6(u Jt))

E

[b+b' I~

With this we finish the proof that

(oz. Now apply Theorem 5.1.

oz. ';

x

C)

w 'V

S (F,cn)

(~x ~ '; C )

•

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J. Barwise 1974

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J. Barwise, M. Kaufmann, and M. Makkai 1977

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A. Ehrenfeucht 1961

An appl-tc.a.ti.on 06 game.!.l .to .the eompR.etene.!.l6 pJtobR.em 60Jt 60Jtmal,(zed .theoJUe.!.l,

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Lee t .

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BACK-AND-FORTH SYSTEMS

101

J. Keisler

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1969

0

f Ma t h•

C. Karp 1965

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M. Kaufmann 1978

Some

results in Stationary Logic, Ph. D. dissertation, University of

Wisconsin.

A. Krawczyk and M. Krynicki 1976

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P. Lindstrllm okdeJr. pkedic.a.te togic with geneJr.a.Uzed quanti6ieM, Theoria, vol. 32, pp. 186- 195.

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F~t

1969

On exten6ion6 06 Etementaky Logic, Theoria, vol. 35, pp. 1 - 11.

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Quantifying over countable sets: Positive v s , Stationary Logic.

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J.A. Makowski, S. Shelah and J. Stavi 1976

tJ.-

togiCA and geneJr.a.Uzed qua.ntiMeJr., Ann, of Math. Logic, vol. 10, pp , 155 192.

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J. Sgro 1977

ComptetenC6~

theok~

60k topotogic.a..e.

mod~,

Ann. of Math. Logic, vol. 11,

XAVIER CAICEDO

102 pp. 173 - 193. S. Vinner 1972

A geneJUtUzation 06 EhJLen6w.cM'
Departamento de Matematica Universidad de Los Andes Apartado Aereo 4976 Bogota D. E. Colombia