Zeilschr. 1. math. Log2 und Orundlagen d . Math. Bd. 24, S. 467-419 (1978)

A FORCING APPROACH TO STRICT-17: REFLECTION AND STRICT-17: = 2:

by W. RICHARD STARK in Austin, Texas (U.S.A.)

1. Introduction A forcing approach to strict-17; reflection and strict-17; = 2: is of value to infinit-

ary logic. The results of this paper provide a link between the generalized recursion theory of admissible sets and the forcing methods of classical set theory which will be used (in [15] and [16]) to investigate compactness and completeness for the Ian- guages LA = A n

The trip from infinitary logic to the forcing methods of classical set theory has two parts. First, is the well known correspondence between the logic of LA4 and the recursion theory of 8, developed by JON BAILWISE in 131. For example: Lal is 2:-compact iff 'II k ))strict-l?i reflection((, and LA4 can be given a complete axiomntization when 'II k ))strict-17; = 2 : ~ The second step, which is the main item in this paper, uses an algorithm which reduces these recursion theoretic properties of 'II to questions of generic filter existence for certain forcing properties. Thus the methods of forcing in classical set theory are brought to bear on the model theory of the languages LA. Compactness, completeness and omitting-types results for LA, which are independent of ZFC, are easily developed (in [15] and [IS]) from the results of this paper.

Within any admissible set 'ZL, our algorithm constructs a forcing property F, for each strict-l7; sentence 0. If Fe is empty then it is absolutely impossible for 91 to reflect 0 (i.e. there is no model of ZF, containing 21, in which 8 reIlects 0). If F , is not empty then IZI reflects 0 in any model of set theory which contains a generic filter for F,. The uniformity of the algorithm makes H F ~ + O ( a l7: relation on 0, thus allowing the possibility of 'II k ,)strict-Z7: = 2:(( to be determined in various models of set theory.

Following the basic definitions, an informal presentation of the problem is given in which, for the sake of motivation, the algorithm is developed for one very simple case lacking the recursion theoretic complications of the general case. In the main section, the full description of the algorithm is given and its behavior is described in a sequence of lemmas which culminate in the main theorems. The final section, develops the basic consequence of the algorithm for theories in La( and describes it useful gener- alization of the procedure.

The investigation of the behavior of strict-17: sentences and formulas can be traced back to KREISEL'S recogriitiori that the strict-11; sets of integers are the same as the r.e. sets. KREISEL uscd the Brouwer-Konig Infinity Lemma, which is, in a 90me, generalized in this paper. BARWISE [2] used the compactness and completeness of the nfinitary languages LA, to prove that (strict-17; reflection) and (strict-l?: = 2:) itre jatisfied by countable admissible sets 8, At the 1972 Oslo Symposium on Generalized

where A is a large admissible set.

30*

468 W. RICHARD STARK

Recursion Theory, GANDY posed the problem ( # 13 of [12]) of finding a direct proof (one not using infinitary languages) of BARWISE'S results.

The solution to this problem of GANDY'S is given in the first half of the author's Doctoral Dissertation, written a t the University of Wisconsin under JON BARWISE, and summarily presented here. The compactness, completeness and omitting-types results which follow from the solution to GANDY'S problem make up the second half of the dissertation.

In connection with his graduate work a t the University of Wisconsin the author would like to thank Professors S. KLEENE, J. KEISLER, K. KUNEN and R. C. BUCK. Very special thanks go to JON BARWISE, and above all to JUDY INGRAO STARK.

2. Definitions and Notation A is an admissible set if ( A , E) is a transitive model for KP = (foundation, exten-

sionality, union, pair. A,-separation, A,-collection}. % is used for ( A , E) when A is admissible or for ( A , E. R,, . . .. R,,,) when A is R,, . . .. R,,,-admissible, 01 = ord A .

L(%) is the first order language with E, = and symbols for all of the relations in (11 and constants c, for each a E A . The intended interpretation of each c, is a. L+(%) is L(%) with class variables &.

The notation V y (and Ay) is used to stand for the B-finite disjunction (and con-

junction) of the sentences y in c. where c is a set of sentences in LA. For any sentence y, y' is the sentence formed from y by moving all 1 ' s all the way in to the atomic sentences. y7 is, of course, equivalent to i y . For a E A , P4(a) = A A P(a) . The basic diagram of % (in L,) is denoted dia(%), and dia(%) n e is denoted dia'(%). The strong diagram, v%, is the finitary theory dia(%) along with the infinitary (Vx(x E c , -+ V (x = c b ) ) [ a E L4].

L A = A ALmwo.

r C

bea

rr,' formulas are formulas formed by closing L+(%) under : finite conjunction A and disjunction V, all first order quantifiers, and universal second order quantifica- tion VS where 8 is a second order variable. stn: formulas are formulas built up from the existential formulas of L+(%) by: finite A and V, Vv E c,, 3v E err and VS. When interpreted in an admissible set % we have the following normal forms: for every n: formula Y(z) there is a A: formula y(S,, . . . , S,, , TC, y. z ) in L+(cll) such that for every a in A

8 b Y [ a ] iff % k VS, . . . A',, 3x Vyy(S,, . . . , S,, , R', y. a )

and for every s tn ; formula @(y) there is a A: formula 8(S,. . . ., S,,, x, y) in L+(%) such that for every a in d

% k @[a] iff cll I= VS, . . . S,, 3xO(S,, . . . , Sit, .r, a ) .

0 is stZ;' (or Z:) if 0' is s tn ; (or D;). We will usually be working with stZ; formulas or sentences, which we will assume are in the normal form 38 Vx8(S, x). s tn ; reflec- tion, in its stL',' form, is

Va(3S t J ~ 8 ) ~ -+ VS 3x8.

A FORCING APPROACH TO STRICT-^; REFLECTION AND STRICT-IT: = z;" 469

(21 k 3 s V . d if there is an L? _E A such that ((21, 8) 1 VxO(S, x). Whenever (3s Vx0(S, 6 , x))= is used it is assumed that a is an extensional set containing the parameters of 8 . In this case, its meaning is the same as 3 s a V.7 E a0(S, 6, x), since 0" = 0. This analysis i q assumed to be carried out in a model &' of ZFC. In this case, A E .A' and S ranges over the subsets of A in A?.

Sotation similar to ( F , 5 ) will be used for forcing properties in which the forcing conditions-denoted p , p , q, etc. -will be finite sets of L+((21) sentences. Actually, the conditions will be elements of A so the sentences will come from the part of L,, which codeq L+(%), rather than from L+((21) which, strictly speaking, is outside of ,4. In these definitions and in all that follows, p is used to denote extension of p and 5 denotes extensions of 17. Given ( F , s ) , the forcing relation It is defined in terms of the atomic sentences in the conditions of F by:

1) for y atomic, p It y iff [ y E p ] ;

2 ) p It i y iff Vp E F [not It y ] ; 3) p It V y iff 3y E c [ p IF y ] ;

4) p It 3xy(.r) if for some constant c, [ p IF y(c) ] . <

Weak forcing It"' is defined by: p IP' y iff Vp 35[?, It y ] . A subset D of ( F , s ) is open if V p Vq E F [ p E D & p q -+ q E D] . D is dense in ( F , s ) if V p 3q[p s q & q E D]. If T is a collection of subsets D of ( F , E) then the notation "0 It''' 3'' is used to mean "the sets of 2 are dense and open in ( F , s)". A subset G of ( F , 5) is %-generic if VD E '2[D A G =+ 01.

3. An Informal Presentation of the Problem and the Solution The first facet of the problem is that of reducing the second order sentence 3S Vx0,

or its relativizations, to first order sentences. An immediately apparent approach is forcing. since p 1 Vxe can be interpreted as 3S Vx0. if an appropriate generic filter exiqts. This approach ultimately works, but not without difficulty.

To see the difficulty and its resolution, let Vx0(S, x) be satisfied by a model ('ill. *?) constructed from a filter through ( F , s ) . Can the forcing relation on ( F , E) detect the truth of 3s Vx0-specifically, is there a condition p wch that p It Vx0(S, x)? This is what n e need, but unfortunately it may not happen. The failure of forcing to detect the existence of the filters we are after, even though they are in ( F , s ) . occurs when the collection of all desired filters is nowhere dense. In such a case every condition p has an extension p not contained in any of the desired filters, so not p IF VxO(S, b, x) and therefore 0 It iVx0(X, 6, x). This problem of detecting scarce filters in ( F . s ) is like the problem of detecting scarce to,-paths in a tree ( T , <) of height coo. To deter- mine whether or not (T, <) contains an (0,-path we construct the maximal <-initial subtree in which there are no terminal nodes. This subtree is the result of repeated removal of nodes which are not extendable.

Let 2 = [D,l I n E wo}, where D,, is the set of all nodes in T of height 2 7 1 . An ttjo-path through ( T , <) will correspond to a linearly ordered subset of T which is 2-generic. The maximal subtree described above is the maximal subtree in which the

47 0 W. RICHARD STARK

sets of 5D are all dense. I ts construction will use the relation 111 which is defined on <-initial subtrees T, of T. For p E T,,

p l l t T Y D, iff 315 E T,[p E D,,], p Ill T y 5D iff VD E D[p I l t T Y D ] .

Now the repeated removal of the nonextendable nodes is defined by: To = T, T,,, = { p E T, I p IltT, 5D) and T1 = n T, when A is a limit ordinal. For some ordi-

nal LX of cardinality 5 //TI/ we will have T , = T,+l , T, is the maximal subtree that we desire. T has an w,-path iff T, contains a %-generic path iff (in ZFC) T, + 0.

If we have a forcing property ( F ( Q ) , s ) and a collection 5b of open subsets of F(O), then the %-trimming, 111 $, defines a descending chain of 5 -initial substructures

Y < l

(F(O), E) 2 (F(1), s ) 2 . . . 3 (F(.]l) , s ) 2 . . . 2 ( F ( K ) , s ) where F(y + 1) = { p E F ( y ) 1 p \ltF(') %} and F(L) = n F(y ) . The only thing can be

said about the closure ordinal K is, IRI j IF(0)l. The importance of ( F ( K ) , s) is in the fact that it is the maximal s -initial subset of (F(O), 5 ) in which the sets of 9 are dense. Using IFw 9 to express density for elements of %, we can sum up the role of IltFp') 5) by:

y < a

I l t F ( Y ) 9 converges to IP 9, in < F ( K ) , Q, as y + K . Additional structure can be introduced into the argument a t three levels: 1) extra

assumptions about the model A of ZF in which it all lives, like did is countable or A 1 ZFC or A b %P @ MA; 2 ) gross (non-recursive) structural assumptions about (F(O), s ) and 9 such as (F(O), s ) has c.a.c. or cf((52l) = w,, etc.; and 3) recursive properties of (F(O), s ) and 3. Such additional structure can result in new g.f.e. argu- ments for ( F ( K ) , s ) or in finer estimates for the closure ordinals. Ultimately, here and in [15], arguments will be given using assumptions on each of these levels. How- ever, the next step in the development of 111 is to incorporate recursive structure to get rapid convergence.

4. The Algori1,hm Th0 trimming operator 111 will now be adapted to the problem of the existence of

expansions ('ill, s") of A satisfying a given sentence (such as VxO(X, 2)) or theory. In this case we will start with a collection F(0) of finite sets of L+('ill)-sentences in LA, and D will be a collection of sets D, defined for sentences pl so that if G is a $-generic filter then U G is a J-completion of the sentence or theory. The notation p 111 D, which was natural in the general case will be replaced by p 111 pl when all of the sets D, have syntatic origin.

Assume that ( X , r) is a &-initial substructure of (F(O), s ) , then p Ilt" y (read " p allows ly in X " ) is defined for p E X and y E': LA by

1) pl is atomic, p ]Itx pl iff ( p u {pl} E X ) ,

2 ) p IltX l p l iff (P u 3) P Ilt*y,v pl iff 3q-J E C ( P LJ {CP} E X ) ,

EX),

C

4) P Ilts A pl iff V p E C ( P u {v} E X I , C

A FORCING APPROACH TO STRICT-^^ REFLECTION AND STRICT-ZI: = 2; 47 1

5 ) P IlPY 3xp(4 iff 3CdP u {Q(Cn)} E 4, 6) P Ilt" Vq4x) iff Vc,(p u {p(cn)} E X ) ,

and bounded quantifiers are treated as defined symbols. For a theory T, p Ilt" T will stand for Vy E p u T [ p llts y]. Notice that we may have both p y and p Illx i y .

Recall the Henkin construction of a model for a theory T which uses a sort of down- ward completion of S of T. This set X has T at the top of it and a model As a t the bottom of i t and the intermediate sentences show As k T . A 1-completion S of a theory T is T and a set of sentences generated from T by HENKIN'S constant construction. Specifically, X is 1-complete if 1) for every atomic sentence y formed by substituting a constant used in S into an atomic formula used on X, we have y E S or ( i y ) E S ; 2) if (c = c') E S and (c' = crr) E X then (c = c") E S and (c' = c) E X; 3) if ( i p ) E S then (pl) E X; 4) if (A p) E S then p E S for every p E c; 5 ) if (V p) E S then there

is a p E c such that p E S ; 6) if (3xp(x)) E X then for some constant c (p(c)) E S ; 7) if (Vrq~(z)) E S then (p(c)) E X for every c mentioned in X.

LA, lT is a small subset of LA which contains every 1-comple- tion of T. Formally, Z T will be defined to be the smallest subset of LA such that 1) T E I T , 2) IT is closed under subsentences and sentences formed by substituting constants use in I T into formulas used in lT, 3) if (Ixy(z)) E I T then (y(c")) E Zy,, 4) if y E ZT then (y') E I T . In the solution to GANDY'S problem 1, will be used for 1, where T, is dia(r21) u {VxO(S, x)}. Since all of the existential quantifiers in this theory are bounded to elements of A , we may replace 3) by 3') if (3x E c,y(x)) E 1, then y(cb) E 1, for each b E A .

A filterG in (PT(LX), g ) is lT-generic if for every W E ~ T : D,nG + 0 and { { y } , ( i y } } A

n G $: 0. G is &-generic if G is l,,-generic. If G is lT-generic then U G is a 1-comple- tion of T. If G is &-generic then a model (a, s") of VxO(8, x) can be constructed from the basic sentences of UG. Unless stated otherwise, "G is generic" will mean "G is &-generic ".

C C

For each theory T

For an arbitrary theory T 5 LA,

FT(O) = {p l T I p is finite, vy E P[(y') # PI} 2

FT(1/ + 1) = (2, E FT(1/) I P IltFp'(Y) T} 3

FT(1) = F T ( y ) , when 1 if a limit ordinal. Y < a

F,(y) will be used for FT&) and lIty will be used for IltFe(Y). 1. Conservat ion lemma. If 3S VxO(8, x) then 0 E F,(a).

Proof. Whenever (21 k 3XVxB(X, x), there is a filter G in (F,(O), s ) such that UG is a 1-completion of dia(a) u {Vx6(X, x)}. Specifically, G is defined from a model (a, s") of VzO(S, 2) by G = {p E F,(O) I (a, 8) k p}. Assume G s F,(y). Let p be any condition in G, then

V"a, s") k p u { O ( f J , cA}l

V4P {WJ, Ca)} E Fe(Y)l

and so Vu[p u {O(S, ca)} E GI by assumption,

172 1%'. BlCHARD STARK

and finally p llty VxB(S, x). A similar argument begins with

\JY E (P u dia(W) [(a, 8) P u {Y>I and deduces V y E ( p u dia('21)) [ p ]ItY y ] . From these two facts thus p E F,(y -+ l ) , and so G g F,(y + 1).

By induction, G s F,(y) for all y . [7

One of the ways in which ]It and IF differ when defined over the same structure is: for p & ji, p It y implies 15 IF y , while p 111 ly implies p 111 y. A consequence of this property of Ilk is that each ( F ( y ) , 5) is an initial substructure of ( F , (0) , s ) .

2 . I n i t i a l s u b s t r u c t u r e lemma. For. each p , P , sentence p of I, and ordinal y :

a) ( F ( y ) , s ) is an initial substructure of (F(O), s ) , b) if l\ty 'p then p !ItY pl.

Proof . The lemma is obvious for y = 0. Assume (IH) that the lemma is true for all ordinals <y .

a) 15 E F ( y ) * V6 < y [ j i 11t' Vx8 and 9 !It' y for y E dia('21) u p ]

V6 < y [ p I l l6 Vd3 and p llt' y for y E &a(%) u p ] (by (IH))

* p E F t y ) .

P lltY p =.[F u {p} E W l =+- [P u b> E F W l (by a))

b) qi is atomic:

[P IltY PI >

p is V y :

P Ilt,' V Y 3 [3Y E cr15 LJ { Y } E Q)l l * [3Y E C [ P u { Y } E Q)ll (by a))

=+- p lltY vy.

L

L

For cp is Ay, V.r E a y ( x ) , 3x E a y ( x ) or i y the argument is the same as for V y . 0

The fact that the closure ordinal is 5 01 is a consequence (by Z: reflection) of F,(O) being no worse than n,". The proof of this property of F and 111 is an immediate

C C

and consequence of the following well-known lemma kiy GANDY.

3. Gandp ' s i nduc t ion lemma.

a ) If a(E. x) is a Z," formula with E positive a id the clmin EO 5 . . . 5 EY E - EYfl c - . . . i s defirzed from a Sf class EO by EY+l = {x I R' E EO v o(E', x)} and Ea = U EY i iAen 1

b) If z( F , s) is a I7: formula with F positive aiad the descending chain Fa 2 F' 2 =? - . . . z - F' 2 - FY+l z - . . . i s defined from a I?," class Fa by F Y + I = {x 1 (x G FO) A

A T C ( F , x)} and Fa =

in a limit ordinal, then the fixed point E K is a 2: (class. Y < A

- F' then the fired poitzt FK i s a I?," class.

y < A

A FORCIXG APPROACH TO STRICT-n; REFLECTlON AND STRICT-fl; = z: 473

In its most complete form GANDY'S lemma also specifies that the closure ordinal K is 5 x = ord(d). Furthermore, the I7: lemma which follows is really just a corollary of this lemma.

4. n," lemma.

a ) p E F,(y) is 17; in all three variables. b ) p E Fe(z) is IT: in p and 6.

c) p ]It7 y d) p Ilk;' V.u6 is n," i i b p and y .

Proof. The idea is to define a chain FG 2 . . . 2 Fa 2 Fd+l 2 . . . of approxima- tions to F = U (J (F,(S) x ( 6 ) x {e}) with fixed point FK = F, and then use (b) of

lemma 3 to deduce that F is a n: class.

is L!," in p , y and y E 1 0 ,

0 6 < U

With the preliminaries out of the way, we can now attack the heart of the problem

% k V43S VxO(S, x)Y -+ [O E F ~ ( ( x ) ]

This may be thought of as meaning that if a condition p is removed from F,(y) then there is an %-finite set e which contains the grounds for removing p . The proof that a set e contains the grounds for removing p from F,(y) depends on showing that p 6 F:(y + l ) , where Fg is a relativization of F, to e .

In the following, e will always be assumed to be a rudimentary set containing coo and 6. The importance of rudimentary sets is in the fact that they contain enough elenientq so that when F and IllF are relativized to e there will not be any mechanical mdlfunctions.

The theory dia(8) u (VxO(S, r)} is relativized to e by intersection, thus the trimming in e nil1 remove conditions which couldn't be part of a J-completion of diae(%) w { V d } . The appropriate relativization of 1, is 1; = e i\ 1,. For a 5 -initial substructure (Fg(y). s ) of (Fg(O), s ) , the relativization of the trimming formula [ p l l t Y Tele is defined by

Vc, E e [ p w {S(S, cn) } E F:(y)] 8~ Vy E (diae(8) w p) [p IIt''e"(Y) yl where

and

In the rlightly more general case of relativizing the machinery for any Z: theory T, T' would be the theory in e n LA defined by the normally relativized definition of T . In the following lemmas it is assumed that the notation also include the case e = A , in this case Fe@) = F(y) .

It is intuitively clear that the truth of pllj-' y depends on the existence in X of certain extensions of p which contain formulas needed immediately in the J-comple- tion of dia(%) w ( y ) . The formalization of this idea will be given by functions C' and D whose values C ( p , y) and D ( p , y ) are sets containing either p or extensions of p . Using the formalization of p IIt' y made possible by C' and D the appropriateness of relativizing the trimming to rudimentary sets will be established. The main lemma involving these functions is given before their definition for motivation.

Fg(0) = { p 1; 1 E FO(0)) = e A Fe(0)

F 8 y + 1) = {P E F;(y) I [P IIt" TOY}.

474 W. RICHARD STARK

5. C, D- lemma. Assume e is a rudimentary subset of A , then C and D are 0; func- tions mapping F,(O) x Ze into P,(F,(O)) in such a way that

1~ E F3y . + 1)1 i f f [P litY Tele i f f vy E Z"eC(P> y ) m y ) 4 0 & D(p, y ) F&J) =k 01-

The neighborhood functions corresponding to the trimming IfTe are defined by

{ p w { y } } ,

{ p w (p'}},

if y is a basic sentence in dia(%)

if y is 7 9 and y E p

r

{P u (y(ca))), if y is Y t C J and (Vzy(4) E P u { V X w J , 4) { p } , otherwise.

c(p, W ) = I { p u { y } ) , if y is a component of a conjunction / \ y in p I I I { p } , otherwise.

Notice that the fourth case covers bounded universal quantifiers and instances of V X B ( 8 , 2).

{ P ~ ( r p } I g , E C } > if y is vp and ( V Y ) E P e r

D(P, Y ) = { p u { ( c b E ca) & p(cb)} I b E a } , if y is 3% E cap(z) and y E p

The idea behind D is: if y is a disjunction or a bounded existential sentence in p then D ( p , y ) is the set of all possible extensions of p which contain the first step toward a 4-completion of y. Notice that rudimentary sets are closed under C and D.

6 . Rela t iv i za t ion lemma.

a ) [P 4 F e ( ~ ) l iff 3e E ALP E e & b) Va[3S VxB(S, x)la implies [0 E Fe(a)].

The proof of lemma 6 uses lemmas 4 and 5 and 2: reflection. The proofs of lemmas 5 and 6 are left to the reader (or may be found in [14]).

The last objective is to show that F,(a) += 0 implies 0 It VzB(S, x) in (Fe(a) , g) . This will follow from the fact that \Ity converges (in the sense described in lemma 8) to It"' in (Fe(a), 5) as y approaches 01. This may also be viewed as meaning that the trimming completes its job in a steps-which is the other half of GANDY'S lemma.

4 3'2y)l,

7. It" lemma. I f p E F(a) and p E p then p It" 97.

Proof . The result is immediate when p is basic. Let p = Vy . Assume (IH) the lemma holds for all y E c. e

(VY E P E F(O1)) v?, E w4 Vy(VY, E P E F(y + 1)) C C

* V P E F(a) Vy(?, lltY V Y )

* v p E F(a) v y 3y E c ( p u { y > E F(y ) )

* v p E F(O1) 3y E c Vy(P u {y) . E F ( y ) )

C

by lemma 4, ( j? u { y } E F(y ) ) is IT:, so 2: reflection gives

A FORCING APPROACH $0 STRICT-^: REFLECTION AND STRICT-nr = Z;" 475

* v p E F(a) 3y E c(j3 u ( y } E F(a))

3 vp E F(a) ( I s I t w V y )

3 p I P v y .

* Vp E F(a) 3y E c(j3 Itw y ) (by (IH))

C

C

The argument for the cases: 3x E ay(x) , Ay, Vx E ay(x) and i y are similar, except C

that 2: reflection is not used in the last three.

8. Convergence lemma. For any sentences 91 and p E F(a):

a) F(a) = F(a + 1 ) ;

b) V?s E F(a) Vy[P lltY 471 implies VP E F(a) [ P llta TI; c) Vp E F(a) [ p Ita 991 implies [ p Itw TI.

Proof. b) 47 is atomic:

t/l3 b " P lltY 471 * vj3 W P u {47 } E W l 3 VPtF u {47} E Fb)l 3 VFrP llta TI.

VFJ Vy[P lltY VYl =+ V@ v y 3Y E c[P u { Y } E FWl

y is V y : C

C

V@ 3y E c Vy[p u { y ) E F(y ) ] (by z;" reflection)

* V@ 3Y E c[Is u { Y } E W l * V@[P Ilt* V Y I .

C

The cases 3s E ay(z), Ay, Vx E ay(x) , and i y are similar to the above. C

a) is an easy consequence of b).

c) IJI is atomic:

V H P llta qJ) - v m u {T} Fb))

= p IF 119 *plt-'Up,.

=z- Vp(not P I t - 19) (by lemma 7)

is v y : L.

v m llta V Y ) - V F 3Y E c(ls u { Y } E F(a))

* Vp(not p It- i V y ) (by lemma 7 )

=> p IF 17 v y

* P It" V Y . 0

c

C

C

47 6 W. RICHSRD STARK

9. Forc ing l emma: If F,(a) + 0 then 0 Itw T , i n (F,(a) , 2) .

Proof . Vp E F(a) Vy [p ]ItY To] * V p E F(a) Vy[p IIt' VrO(S, z)l

* V P E F ( a ) v y V c [ p IItY O(S> c)l 3 V p E F(a) Vc[p llta 6(S, c )] (by lemma 8b)) s Vc[O 117~' O(S, c ) ] (by lemma 8c) and 0 E F(a))

=2- 0 It VrC6(S, x). The other sentences of T, are treated similarly. 0

now be established from lemma 6 and lemma 9 and the Baire category theorem.

sentence in L+(3), then

The occurrence of stn: reflection and stI7: = z;" in countable admissible set,s can

A. stn; re f lec t ion theorem. I/ 3 is a counttable ad,missible set and @ is a stu:

% b [0 -+ 3a@"]. Proof .

73aO" + Va[3S Vx6l(S, re)]"

+- 0 E I?,(&) + [0 It VxO1(S, r ) ] and V y E dia(3) [0 It&' y:l in (F,(a), 5)

(by lemma 6) (by lemma 9).

Since 3 is countable, both F&) and 1, are countable and therefore the existence of an &-generic filter G in (F,(a), E) is a consequence of the Baire category theorem. From G the model of dia(3) v (Vz6(S, x)} of the form (3; s") can be constructed. Finally 3 I= 38 V r 6 or in other words 3 'F @l.

1 VS 3% i e(s, x, b ) } have 2: definition.$.

B. s t n i = 2; theorem. I n a countabZe admissible set a, strict-n: classes ( b E

Proof. We prove that a st2; cla,ss has a I7: definition.

( b E A I 3x VxO(S, x, 6 ) ) s ( b E A [ 0 E Fe(,,,b,.,)(a)) - c { b E A I 3 8 VxO(S, .r, b ) }

(by lemma 1)

(by Baire category theorem),

so ( b E A 1 3 s VxO(S, z, 6 ) ) = ( b E A I 0 E Fe(,.,b,.,)(a)'}. Using then : lemma we see that {b E A 10 E FO(..,b ,.,(a)} is fl: and therefore the stZ;' set is n,". 0

The machinery just developed can go beyond countable admissible sets in dealing with s t u ? reflection and stn: = 2:. To see this just notice that in developing the first nine lemmas no assumptions on the cardinality of A we made.

5. Applications

We will describe how the algorithm just developcBd, and its generalizations, can be used to prove theorems and develop independence results in the model theory of large fragments of L,, .

Let 3 be any admissible set and let T be a S: theory in L,l. T has a model iff 3 I= 3S(T & X & X is 4-complete). 3S(T & S & S is a 4-completion) is a strict-2; sentence which we shall denote 3S Vx07 ( S , x), where OT is A", ))Every A-finite subset of T has a model (( iff Ve[3S VxOT(S, r)]', where e ranges over rudimentary sets contain- ing coo and O r (and therefore the parameters used in defining T). From this we see

A FORCING APPROACH TO STRICT-ni REFLECTION AND STRICT-n: = 2; 477

that each instance of Z:-compactness, ))every A-finite subset of T has a model*+ + T has a model ((, corresponds to an instance of strict-II: reflection,

1?1 b Ve[3S VdlT(S, x)]" -+ 3s V&?+S, r ) .

How does FeT(a) shed light on the model theory of T? Lemma 1 tells us that if F&) = 0 then T doesn't have a model. ))FeT(a) = 0(( is absolute while ))T has a modela is not absolute, so FeT(a) = 0 implies that T cannot have a model in any unkerse of set theory. This indicates that ))FeT(a) = 0(( is similar to ))T is incon- sistent ((, except that FeT(a) is defined even when 2l cannot be given a complete axio- mitization. FeT(a) offers a generalization of the idea of consistency.

By lemma 6b), if every A-finite subset of T has a model then Fep(a) + 0. A generic filter G through (Fep(a), g) gives a 1-completion, UG, of {VxOT(S, x)} LJ dia(2l). U G defines a subset of A such that (3, s') k 08 is a I-completion of Ta. The basic sentences of S define a model for T. Any model of set theory which contains a generic filter for (FeT(a), s ) will contain a model for T.

Is i t consistent with ZFC to assume that L,,l is L':-compact when A is Lo, (the mi" level of t'he constructible universe)? The theory which says that f is a pairing between coo and o 1 shows us that it isn't. Theories whose models would cause cardinals to move are easily recognized by the fact that (FeT(a) , s ) is either empty or contains an un- countable antichain. Let us say that a theory T is a cc-theoiy if (FeT(a), E) satisfies the countable chain condition. Now if T is a cc-2; theory such that every A-finite subset of T has a model then (Fe,(a), s ) will be a nonempty ccc-partial ordering, and so a model for T would not cause cardinals to move. COHEN'S well-known method yields the relative consistency of the existence of a standard model of ZFC in which L.+ (with A = L,J is ccZ,0-compact.

TO apply the 111 algorithm directly to a theory T s LAI (rather than to Top) whose intended universe may not be A , we need to make a two changes in the definition of 111: replace 5) and 6) by

5' ) p lltz 3mp(z) iff p w { ~ ( c " ) } E X ,

6') iff

Sow that Skolem constants have been introduced, extra measures must be taken to guarantee that " = " is an equivalence relation. Let T* be an expansion of T formed by adding the following sentences to T :

p lltx Vxg;(.z) p w { y (c )> E X for every constant c mentioned in p .

(c = c') & (c' = c") -+ (c = c") & (c' = c ) ,

(c = c') & v ( c ) -+ &').

Y " l Y > for all constants c , c', c", sentences p?(c), p;(c') and atomic sentences y in I T . Let FT(0) be just as before and define FT(y + 1) to be { p E F&) I p l l t Y T*}. For any theory T E Lmm0 the closure ordinal tT of the trimming (i.e., F7.(t1.) = F T ( t T + 1)) is of cardinality 5 IIFT(0)/I. If { F T ( t l , ) , s ) is not empty then the sets necessary for the construction of a 1-completion of T are all open and dense. Consequently, T has a model iff ( F T ( t T ) , s ) has an lT-generic filter. We can say all of this and more if these ideas are 'developed in the presence of an admissible set 1?1.

478 W. RICHARD STARK

FT(a)- theorem. Assume that % is an admissible set and T is a Zf-theory in

a) FT(a) is the fixed point (i.e., zl. 5 ord(A)) . b) If FT(a) = 0 then T cannot have a model in any universe of set theory.

c) If every A-finite subset of T has a model then FT(oI) + 0.

d) If &',(a) $: 0 then the sets D, and { p E FT(0) I p contains either p or i c p } for p E I T

e) In every universe of set theory, T has a model iff (F?(a), s ) has an lT-generic filter.

Assume that {T, 1 a E A} is a uniformly defined family of Z:-theories in Lai , i.e., T , = {p E LA 1 3zp(q c,, c,) is true in %}.

f ) The formula ) ){y} E FT,(a)(( is ZT: in both y and a.

Remark on t h e proof. The recursion theoretic aspects of the algorithm enter through a). a ) is an immediate consequence of GANDY'S lemma and the fact that the trimming operator lltT is I7," when T is Zy. b) is essentially the idea of lemma 1 and the fact that FT(y ) is an absolute function. c) is the subtle point in this theorem, it is proved by the relativizacion arguments (for F",(y)) which were sketched in lemmas 5 and 6. The proof of c ) uses the Iact that A = H(w,) or (u,, E A . d) is a consequence of a). FT(a) is the fixed point is equivalent to ) ) p 111" T, for every p E FT(a)q which (when FT(a) $: 0) gives the density result immediately.

Completeness and provability results are based on f ) . In L,,, , H y is provable from T (( iff OT u { i y } is inconsistent(( iff >){iy} $ F7u(sv7wU)(a)((, and ))y is a theorem(( iff n { i y } $ F{,,,,l(a)((. Part b) of the FI[(ix)-Theorem and the previously discussed absoluteness of F, justify thinking of ) ) { i y } $ F{," irl(a) (( is a generalization of ))y is a theorem(( in the LmWo-case. In this sense, LA would be complete if for every sentence y E LAi: {p} E F,,, Iw) (a ) implies {p} can be extended to a generic filter

Efforts to BARWISE'S Compactness Theorem for B-admissible sets of cofinality wo lead to the recognition of the trimming in its purely recursive form. The use, in ZFC, of this general trimming has made possible a strengthening of this theorem of BARWISE'S and a new completeness theorem, as well as results in ZFC (%, MA (see [15]).

T h e General Recurs ive Trimming. Assume that % is admissible and that we have :

1)

2) 3)

are dense and open over (FT(a ) , E).

through < F [ V V 7 w ] b ) , s) .

( F ( Q ) , s ) in which F(0) is a t worst n:; 2 = ( ( p , a ) 1 p ED, , a E A } is a t worst fly; an %-recursive neighborhood function N : A x A -+ A such that for every y , p and a we have 3 p [ p j3 and 1, E 0, and p E F&)J iff 3 p E N ( a , p ) [ p 5 p and F ED,, and P E F,(y)l.

Under these conditions, llt'2 is called a general recursive trimming of F(0) in 8.

3 p which appears in p IlkY B. p [ItY 3 is equivalent to Va 3 p [ p p E F&)] which, with the help of N , is equivalent to

The importance of N is that it puts an A-finite bound on the existential quantifier p and ( p , a ) E 3 and

Va Vn[n = N(a , p ) -+ 3 p E n ( p 5 17 and ( p , a ) E 's, and ?, E FB(y))].

A FORCING APPROACH TO STRICT-ni' REFLECTION AND STRICT-n: = z: 479

Whenever FB(y ) is IT:, we will have p 1lt-Y 9 is n," and so F& + 1) will be nf. This induction step along with the fact that F(0) is I7," is all that is needed to show that E F&) (( is IT: in both variables. This and 2: reflection yield the existence of a fixed point 5 a.

The General Recurs ive Tr imming Theorem. Assume that (F(O), g) and 3 are defined in '2l so that IItB is a general recursive trimming of F(0) in a. a) If for every b E A , (F(C), s ) contains a (0, E 9 I a E b}-generic filter then 0 E F&).

b) If 0 E FB(a) then each of the sets of 93 is dense and open over (FB(a), s). c) " p E F&)(( is 171".

By generalizing the idea of )) 1-completion (( appropriately, and, when necessary, the assumption of extra recursion-theoretic and set-theoretic properties for a, these methods can be applied to other languages. For example, in order to apply these methods to KEISLER'S LA(&) (where o&x (( means ))there exist uncountably many((), we need only change the idea of 1-completion. The appropriate idea of 1-completion S here uses an indexing proceedure on the constants which guarantees that once we have X in hand we may (outside of A ) stuff uncountably many new constants into S to get a 1-com- pletion, S+, which will yield a standard LAi(Q)-model. It seems reasonable to expect that recursive trimming and forcing methods should play a significant roll in the development of model theory for higher order languages -especially when the interest- ing mcdel theory is independent of ZFC.

3ibliogrsghy [l] BARWISE, K. J., Infinitary logic and admissible sets. J. Symb. Log. 34 (1969), 226-252. [Z] BARWISE, K. J., Implicit definability and compactness in infinitary languages. In: Lecture

[3] BARWISE, K. J., Applications of strict-l7: predicates to infinitary logic. J. Symb. Log. 34

[4] BARWISE, K. J., Admissible sets o5er models of set theory. In: Generalized Recursion Theory (ed. by J. E. FENSTAD and P. HINMAN), North-Holland, Amsterdam 1973, p. 97.

[5] BARWISE, K. J., Notes on forcing and countable fragments. Mimeographed 1970. [6] BARWISE, K. J., Admissible sets and structure. In: Perspectives on Mathematical Logic.

[7] BARWISE, K. J., GANDY, R. O., and Y. N. MOSCHOVAKIS, The next admissible set. J. Symb.

[8] KEISLER, H. J., Model theory of infinitary languages. North-Holland, Amsterdam 1971. [9] KEISLER, H. J., Forcing and the omitting types theorem. In: Studies in Model Theory (ed. by

[lo] KUNEN, B., Implicit definability and infinitary languages. J. Symb. Log. 83 (1968), 446-451. [ l l ] JECII, T. J., Lectures in set theory. Lecture Notes in Math. 217, Springer-Verlag, Berlin-Heidel-

[12] GANDY, R. O., Inductive definitions. In: Generalized Recursion Theory (ed. by J. E. FENSTAD

[13] MAETIN, I). A., and R. SOLOVAY, Internal Cohen extensions. Annals Math. Logic 2 (1970),

[la] STARK, W., Proofs for a forcing approach to strict-l?;-reflection and strict-n: = 2:. Available

[15] STARK, W. R., Martin's axiom in the model theory of LA. J. Symb. Log. (to appear). [I61 STARK, 11'. R., Independence results in the model theory of LA.

Notes in Math. 72, Springer-Verlag, Berlin-Heidelberg-New York, 1968 p. 1 - 35.

(1969), 409-423.

Springer-Verlag, Berlin-Heidelberg-New York 1975.

Log. 36 (1971), 108-120.

M. MORLEY), p. 96.

berg-New York 1971.

and P. HINMAN), North-Holland, Amsterdam 1973, p. 265.

143 - 1'78.

from author.

(Eingegangen am 11. Jiili 1977)