saharon shelah- proper and improper forcing second editon: chapter viii: kappa-pic and not adding...

8/3/2019 Saharon Shelah- Proper and Improper Forcing Second Editon: Chapter VIII: kappa-pic and Not Adding Reals

1/33

VIII. ft-pic and Not Adding Reals

0. IntroductionIn the first section we show that we can iterate K 2-complete forcing, and KI-complete forcing which satisfy th e ^2-c.c. in a strong sense.

In the second section we deal with a strong version of the K2-c.c. called ^2-pic. It is useful fo r proving that for CS iteration of length < 2 of proper forcingnotions, th e limit still satisfies the N 2-c.c. This in turn will be used in order toget universes with 2*1 > 2* = N 2.

In the third section we deal again with the axioms; starting with a modelof ZFC (not assuming th e existence of large cardinals) we phrase th e axiomswe can get. There are four cases according to whether 2N is N I or K

2, and 2** 1

is ^2 or larger [ourknowledge on the case 2^ > ^3 is slim].In th e fourth section we return to the problem of when a C S iteration of

proper forcing preserves "not adding reals". We weaken "each Qi (a P^-name)is D-complete fo r some D a (, 1, /)-system", by replacing "each Dx is an NI-complete filter" or even just "each Dx is a filter" by "each D^ is a family ofsets, the intersection of e.g. any two is nonempty". So we can deduce ZFC-fCHV ^. We also try to formulate the property preserved by iteration weakerthan this completeness. See references in the relevant sections.


2/33

404 VIII. -pic and Not Adding Reals

1. Mixed Iteration- ^2-c.c., ^-Complete1.1 Lemma. Suppose Pa (a


3/33

1 . Mixed Iteration-K2-c.c., ^-Complete 405

Proof. Let be large enough, N - < (f(), G),Pao GTV , | T V | | = N I, and everycountable subset of T V belongs to N. Let p GP

oTV , and (ZQ, : < >) be a

list of all maximal antichains of Pao which belong to TV (so (Ia : a < \) Nbut for each * < \ we have that (Ia : a < *) G TV, by the choice of TV).It is trivial that Po is tti-complete. It then suffices to prove the existence ofa p* G Po, p < p* such that p* is (TV, Po)-generic, so proving that Pao is"somewhat proper".

By CH, we can let {(o , A* , ( n : GA n < )) : < } be a listof all triples of the form (,,(7 : GA,n < }} such that < i,A C {2 : 2/3 G0 TV}, \A\ such that:A) Dom(p) {2/3 : 2 < 0} C Dom(p)B) 2 GDom(^) => p(2) = p(2)C) z/ there are pn G PonN(frn< )sucnthat

G^(pn+1r2/J) IK p2/ 3 " (Pn(2/?))-7/3^" and(iii) for some ^ G Ia we have q


4/33


N ow we define p* : if 2/3 G Dom(p), p*(2/3) -p(2/3), and if 2/3 + 1 Ul \ ^4n. So, for some we have a^ = ,

A ^-A, (72/3,n : 2/3 G, n < ) = (7 ^ : 2/3 GA,n < ). As these objectsare countable subsets of JV, by the choice of A / " , they belong to N.

A s N -< ( / f ( ) , G) , fo r this there are #,pn as mentioned in clause (C )above, p > G Za. Again, without loss of generality, pn GPao N andg G J A / " , so < ? , {p^ : n < }, as in (C), are well defined. Now by theproperties of / i 2 / 3 ? we can prove, by induction on 7 < o, that(*) for every < 7, and r GPa such that p \ < r and p^\ < r for n < ,there is an r* GP such that p f y < r*, p^ \j < r* for n < , and r* f = r,and Dom(r*) [C,

7) C (j

n


5/33

1 . Mixed Iteration-^2-c.c., ^-Complete 407

hypothesis we know:

r \(2 + 1) lhPa/J+1 "rf(2/3 + 1) >p*(2 + 1) >%+(2/3+ 1) > p?(2 + 1)",so by the N I -completeness of ?2/m> rexists. Lastly, for 7 = 2/? + 1, thenontrivial case is that 2/3 belongs to (Jn


6/33

408 VIII. K-pic and Not Adding Reals

However as the interaction has no applications now, we shall not discussit further (there are other tries at N 2- c c., see [Sh 80]).

Note also that th e analogous lemma fo r Ni-complete, N I - C . C . forcing holds,but now it has no application.1.3 Claim. If 0^ holds, then in 1.1 we can change the iteration to the usual(< ^2)-support iteration and the conclu sion still holds.Proof. We let N \J


7/33

2. Chain Conditions Revisited 409

2. Chain Conditions RevisitedWe here deal again with problems like those of 1 from VII, but allowing thecontinuum to increase somewhat. Here, K is a fixed cardinal.2.1 Definition. P satisfies the ft-p.i.c. (ft-properness isomorphism condition)provided th e following holds, for large enough:

Supposed < j < , K ; e Ni -< ( j f f ( ) , E , < ) , (


8/33


2.3 Lemma. Suppose (V < ft)**< ft where ft is regular and P satisfies theft-p.i.c. Then P satisfies the ft-chain condition.Proof. Let pi GP for i < ft be given. Let (H(\), G, /(i) < iBy Fodor's Lemma, for some 7 the set {i : /(i) = 7} is stationary. As (V < ft)H < K and ft is regular, for some A C 7, 5 {i : Ni i = A} is stationary.Similarly, we can assume that for some B,i eS=N\N=B (seethe proof of VI.5A). Also C = { < K : (Vi < )(Ni ft C )} is closedunbounded, so S = 5(C is stationary. Now there are K models ( < / V i , p i , i, )e#where Pi,i, ( G B) are individual constants and ft > 2N, and the numberof isomorphism types of such models is 2H, so for some i < j there is anisomorphism h : Ni > A^ (onto), /i(p^) = pj,h\B = the identity. Now applyDefinition 2.1 for N^Nj,h,pi. D2.3

2.4 Lemma. Suppose Q = (Pi,Qj : i < Q:0,j < Oo} is an iteration withcountable support. Suppose further(*) Qa satisfies the ft-pic (for each < QQ) and ft is regular.Then 1) If o < ft, Po satisfies the ft-p.i.c.

2) If 0


9/33

2. Chain Conditions Revisited 411

h(r) G G]'\ there is a q $ G P such that q\ = q , q > p\, q > h(p)\&q is (7Vi,P)-generic and (^P^-generic and q I h p ^ "(Vr T V ; P )[ r G G e i f f / ( r ) G G ] " .Note that (**) with p an element implies the apparently more general

version with p being p, a P-name of a member of N Po, such that g lhp "for some p G Ni Po, p[GpJ = p, an d p[( < < ? an d h(p) \ < q". (Used in theinductive proof fo r limit.)

For a successor, we first, by the induction hypothesis, define q ^_ asrequired (necessarily I G Ni ^ ); then notice that, by the inductionhypothesis, if we force with P _ and get a generic G _ C P ^_ x and q^-i isin this generic set, then h is still an isomorphism etc,so we can use the A c - p . i . c .on Q_[G_].

For cf () = N O , we work as in the proof of properness (III 3.2), using theinduction hypothesis. Noticing that q \\-P "(VrG P$) (r G G iff h(r) G G^)"makes no problem in the limit, we do not have to take special care. For cf() > N I th e proof is similar bu t easier.

2) Trivial by 1) and 2.3 and the proof of III 4.1. D2.42.5 Lemma. If P is proper and K > \P\ then P satisfies the ft-p.i.c.Proof. W e start with i,j,N i, A ^ , ( f f ( ) , E ,


10/33

412 VIII. tt-pic and Not Adding Reals

2.6 Definition. The A c - p . i . c * is defined similarly to ft-p.i.c, but we add oneassumption:for an y a G Ni there is a sequence (aa : < A C ) in Ni A ^ such that ^ = (this implies th e corresponding condition on Nj); equivalently Ni is the Skolemhull of (Ni Nj) U W > an d T V , is the Skolem Hull of (N i Nj) \J{j}.2.7 Lemma.1) The Ac-p.i.c. implies the A c - p . i . c * .2) Lemmas 2.2-2.5 hold fo r /-p.i.c*, (and we call them 2.2*,. . . , respectively).3) P satisfies the ft-p.i.c* if P satisfies the conditions from [Sh:80] which are:

a) P is N I -complete.b) for any pi G P (i < A C ) there are p\ G P, Pi < pj an d pressing down

functions Fn : A C \ {0} > A C , (i.e., Fn() < ) for n < ;, such that: ifi < j and /\nFn(i) = Fn(j), thenp\,pj have a least upper bound inP, called pt A p t .

2.7A Remark. So 2.7(2), (3), 2.4* , 2.4*give an alternative proof of [Sh:80],for the case o < A C . In fact, 2.4*holds for Q Q not necessarily < A C ,when eachQ i is Ni-complete, and this gives an alternative axiom fo r [Sh:80].Proof. 1 ) Trivial.2) The least trivial part is 2.3. Here the extra assumption is the leastobvious. So, by induction, we define N*(k < ). For fc = 0 we choose Nfsuch that: {Pii} G N? - < (H(), G,


11/33

2. Chain Conditions Revisited 41 3

by induction, conditions pn:Po =P,ifp2n is defined, choose >2+ >P2n, such that P2+ P A ^ , P2n+ > (someQn 2n),if P2n+ is defined, consider seqpn+ = (rjn : < K). We can assume w.l.o.g.(V < K ) r>n e P. So there are (Fn : n < ;), (r n : < ft) as mentioned in2.7(3)(b). As {r>n : < K ) G Ni\Nj and n (rememberP2n+ r,n < rJn ) Notice that /(r^n) = r^ , and by the choice of Ni,Njwe have m F W = Fm(j) So rtn = p 2n+ 2 and r]>n = (p2n+2) have a leastupper bound < 7 2 n + 2 = = P2n+2 A (p2n-f2) In the end:

< (p4) qn.Now q is as required. D2.r

2.8 Lemma.1) All forcings used in VII 3 (= applications of Axiom II) satisfyth e K2-p.i .c*, (but of course Levy(K, < K ) if K > H 2 )

2) Moreover for each application we can find a forcing notion doing all theassigments of this kind present in the current universe and satisfies the K2-p.i .c*,and in fact all are (< ;)-proper and D-complete for a simple H O -completenesssystem D.Proof. We elaborate two of them leaving rest to the reader.application F: Let T = ((Ta, /*) : < *) be a sequence of pairs (T,/*), T anAronszajn H i-tree, /* : T > satisfies th e antecendent of (8) in VII 3.8 (in themain case: listing all of such pairs). We define a forcing notion P. A memberp of P has the form p = (i, w, ~ g , C, B), where:

(i) w C * is countable,


12/33

41 4 VIII. K-pic and N ot A dding Reals

(ii) i < c j i , C (Ca : a Gw), Ca the characteristic function of a closed subsetof i -h 1 to which z belongs(iii) g = (ga : a G w), ga a function from.= (T)p.i.c. essentially (but seemingly notformally) implies properness.

2.9 Claim. If o < ft, (Pa,Qa ' & < #0} is a CS iteration, O L Q < K, eachQa satisfies the /-p.i.c* and (V < )H < ft; /ien, in V P( XO we have (V 2

Generally, fo r getting th e consistency of stronger axioms we have to assumeth e consistency of ZFC-h some large cardinal. Here we concentrate on assumingthe consistency of ZFC only.


14/33

416 VIII. >pic and Not Adding Reals

3.3 Notation. , are first order sentences (in the language with , e andone predicate P).

M is a model with universe \ and language of cardinality < N O ,M =(|M|, . . . ,R . . .) i


15/33

3 . The Axioms Revisited 417

N 2-p.i.c.*, \P \ < 2*1, such that \\-P "for every M E V, if (H(K)v, G , M ) N ^,then there is (in T / p) an expansion T V of M satisfying < / ? " .

4 ) 2* = K 2 < 2Hl = anything of cofinality > K 2 +Axiom Schema. I lib- For each (^, y > ) , fo r every M with universe C . f f(N)such that ( / f ( N ) , G , M ) N ^, there is an expansion N of M satisfying ,provided that

( * ) / / / b : the following is provable from ZFC + CH: there is a proper forcing Psatisfying the K 2-p.i .c*., \P\< 2Hl such that:

Ihp "CH and for every M V with universe Ci f ( f f ( N ) v r , e , M ) l = V ' , thenthere is (in Vp) an expansion N of M satisfying " .

Proof. Straightforward by now, when we use the relevant theorems on forcing.3.4

3.4A Remarks on 3.4(3).A ) Notice that we use CH instead of G.C.H, and we here put first 3P andthen V M . We can also assume that P is an iteration with countable supportsatisfying the above conditions.B) If 2N l is such that (V < 2*1) [H < 2N l], we can replace ff(N), by #(2Kl).

O f course, as we use "larger" cardinals K to be collapsed to K 2 , we can getstronger axioms:

3.5 Lemma. If "ZFC -f 3 an inaccessible cardinal" is consistent, then so are"ZFC + each of the following" (separately).

Axiom Schema. Ia: like / & , but we replace (in (*)/ 6 the demand) ". . . P,satisfying the K 2-p. i .c .*" , by ". . . P, |P| < first strongly inaccessible".


16/33

41 8 VIII. K-pic and Not Adding Reals

2) G.C.H. +Axiom Schema. IIa: like IIb, but we replace " . . . , P satisfies the N 2-p.i.c.*,|P| ^whereas in Devlin and Shelah [DvSh:65] (or see somewhat more [Appendix1]) it is shown that CH > ^ But Axiom II does not prove the consistencyof "not ^" with CH. More generally, we can ask whether we can make thecondition on the Qa (in V 7.1, 7.2) weaker.

We saw no point in trying to weaken the assumption "-proper for everyO L < n to e.g. u -proper, as it seemed to us that every natural example of fore-


17/33

4. More on Forcing Not Adding -Sequences and on the Diagonal Argument 419

ing will satisfy it (truly, sometimes we want to destroy some stationary subsetsof \ and under reasonable conditions we can succeed, fo r example, th e proofthe consistency of "the closed unbounded filter on \ is precipitous" (see Jech,Magidor, Mitchell and Prikry [JMMP]), but we can amalgamate such a proofwith our constructions). The hard part seems to be the D-completeness, whereD GV or D is simple, again, do not seem to be a serious obstacle to anything;but the requirement that any finitely many possibilities are compatible (i.e.^(N,p,p) generates a filter) seemed to be an obstacle - e.g. to the natural forc-ing for making ^ false. Remember, B{w,...) was a family of subsets of P(N)with the finite intersection property. We shall try to replace this requirementby the requirement that the intersection of any two is nonempty. As an appli-cation we get consistency with CH of variants of ^ (we hope there will bemore). Note that we replace here 1 ( N , p , p ) by another equivalent formulation.

Note another drawback, which at present is only aesthetical, the D-completeness is not preserved; i.e. we have not stated a natural condition,preserved by CS iterations, that implying that no -sequence of ordinals isadded. Note that we do not use the full generality of Definition 4.2. We treatit in 4.144.22. A minor difference with Chapter V is that we use countablesubsets of some instead of J(), but this is just a matter of presentation.

For another point of view and more results see [Sh:177] or better yet XVIII1 , 2 .4.2 Definition. 1) Let be a cardinal. D is called a (, 1, fc)-system (orcompleteness system) if:

(i) D is a function (where D(x) may be written as D^),(ii) 11^(0,


18/33

420 VIII. K-pic and Not A d d i n g Reals

4.3 Definition. A forcing notion P is called (P, D)-complete, where T > is afilter over S#

0(X ) and D a (, 1,/c)-system, if > 2' p', P is isomorphic to

P* = (P*,


19/33

4. More on Forcing N ot Adding u -Sequences and o n the Diagonal Argument 421

2) Definitions 4.3 and 4.4 (simplicity) are compatible with th e definition ofcompleteness systems from V5.2, 5.3, 5.5.

Specifically:(A) For a forcing notion P, k < K I and a family 8 of subsets of 5 , P is (2


20/33


Note:4.6 Claim. Any simple system in VPa (where P

ais a forcing notion adding

no new -sequences) is in V.Note:4.6A Remark. Every Qa is ;-bounding because it is IF-complete, whichimplies it does not add (to VPa) reals and even does not add -sequences ofordinals.Proof of 4.5. We prove some claims and then the theorem becomes obvious(Claim 4.10 is the heart of the matter).

4.7 Definition. Let A = (Ai : i < /?), countable, each Ai is a countable setof ordinals, Ai(i < ) is (strictly) increasing and continuous. For < , GA O , C A), A as above, we define when A is long for (, ), by induction on :

Case (i). = : A is long for (, ) (under the assumptions above) if > 0.

Case (ii). a successor, > : A is long for (, ) if for some ^ < wehave that (Ai : i < f) is long for (, -).

Case (Hi), a limit: A is long for (, ) if there are i(i


21/33

4. More on Forcing Not A d d i n g -Sequences and on the Diagonal A rgument 423

Proof. 1) By induction on C, and there are no problems.2) Easy. 4.84.9 Claim. Let > , < l and for i < we have Ni X ( f f () , G), and Niare countable increasing continuous an d (N j : j < i) G N i an d < G NQ .Then we can find an a such that < a < , and a countable Ni for / ? < i < such that J V i -< (ff(),e), (^ : j < i) GWm for i < a,Ni are countableincreasing continuous in z an d (^ : i < a) is long for (, " ) .Proof. A gain by induction on .4.10 Claim. Suppose


22/33


r t G G e i+i; but by the assumption (Vg)[f Ge\Ni > g < re] and r^ < re.Hence, by the definition of Z below, we know that r^ Z, so r^ G T1 (seebelow), and necessarily {q G ^ P : q < rf} = Ge A ^. So Ge Ni G W q and G C Ni P such that

(V G G) p' < r and(VJ G T V o ) [J pre-dense in P -> Jn G 0]}J1= { G P : there is a G C A 0 Pe such that (Vp' GG) pr < and

(VJ Wo)[J pre-dense in P -> J G/ 0]}.Case (i): = .

Trivial. Just let G* = G0 N0.Case (ii): a successor ordinal > .

So by Definition 4.7, for some 7 < / 3 , (N; : i < 7) is long for (f ,(- 1). Fore = 0, 1, we can find r^ GGe A7+1 such that r" is above every member ofGe N (see the proof above). Hence, by the induction hypothesis, there is aG* C 0 Pc_ such that:

(a) for every pre-dense X C P ^ _ i , X G A ^ > , the intersection G* Z is notempty,(b) rl lhP "G* has an upper bound in PC-/P",(c) G e A 0 = G*P.

N ow without loss of generality G* is definable (in (ff(), E ,


23/33

4. More on Forcing Not Adding cj-Sequences and on the Diagonal Argument 425

of D^"1) to be in the appropriate closed unbounded subset of i < j, * + + 1 < andfor any (,C) in N. if < < < then (Nj C : i + 2 < j < i+l) is


24/33


long for (,() and (i : i V n T < Cn]

Now we define by induction on n < , kn = k(n) < , G (for G( f c ( n ) ) 2 ) j r * j rjn such that:

1) r* is a member of Pn H T V ^ + ^ + i with domain Cn-W/^+u,, Q < * o , rj


25/33

4. More on Forcing N ot Adding -Sequences and on the Diagonal Argument 4 27

There is no problem for n = 0 (how do we define Q? we can assume thatQ

is closed under disjunction. Let G N ( + 1) be maximal such that

(Vr G N+ P}(r < r0 = r < TI), by the definition of CS iteration it iswell defined. As before we can find r' e GN*++ which is below re and (Vr GN+ Pc)(r < r'e = r < re), r& f-r{ \, < => r' 0 \ \\ -P "r,(), r(() areincompatible in Qe" and define Q as follows: Dom(ro) = Dom(r)Uand

ro(7) (7) otherwise.)So assume we have defined for n and we shall define for n + 1.First we define, by induction on < fc(n), for every G2, the sets G"1"1(see (10) above), and we have to satisfy (9),(7), and r^ should force G1 1 is(bounded by) a condition, if = n\t.

This makes no problem, using the induction hypothesis on and (< \)-properness.Second we want to define k(n + 1),G +1 (for r / G fc(n+1)2). Let Gn C Pn begeneric, r* GG^n and work for a while in V^G^J.

For each p' GN.(Gn] (PCn +i/GCn), there is a G CPn+1/Gn, p' G G,G has an upper bound, and ifp' GNj,j [GCJ. So, in F, we have a PC-name F for it. Asits domain is countable, \\~pn "F GF" (the domain is essentially C 7V^* ) .Also it is clear that, without loss of generality, F GN * + as (Ni : i


26/33


G k(n+ll we can first define G + 1 A^fc(n+i)-f2 so that it depends on \k(n)and not , and then let G,,-F

(/\(G^1

wfcn+i+2)).

Third we define r +1 by V 4.5. EU.ioProo/ o/4.5. Immediate by 4.10. 4.54.10A Remark.

1) So now everywhere we can use 4.5 instead of V 7.1 and strengthen AxiomII, , IIb etc.

2) We could use shorter sequences, e.g. = * + 2 is o.k. (seeimplicitly VI1 , and explicitly XVIII 2.10).

3 ) By easy manipulations, it does not matter in Theorem 4.3, whether D" isa P-name of a member of V, or simply a member of V (i.e. the function _ D < * is in V).

4.11 Conclusion. It is consistent with ZFC-f G.C.H. that:(*) If k < and s is an -sequence converging to for any limit < \ and(A : ^.

4.13 Lemma.1) The demand on each Qa in 4.5 follows from: Qa is 7-proper for every

7 < i, and D-complete for some simple 2-completeness system B.2) We can demand in 4.5 that each Q

ais 7-proper for every 7 < i, and

D-complete for some almost simple 2-completeness system over V.

Remark. See Definition V 5.5.


27/33

4. More on Forcing Not A d d i n g -Sequences and on the Diagonal A rgum ent 429

Proof. Straightforward. H U . i a* * *

Finally, we indicate how to rephrase the proof in the form of a conditionwhich is preserved by iteration.4.14 Definition. We1" call E C S^(X) stationary if for every > X(equivalently, some > N) and for every x G H() there is a sequence f = (Ni : i < a) such that(a) T V = (N i : i < a) is an increasing continuo us sequence of coun table elemen-tary submodels^ of (-ff(), G ,


28/33

430 VIII . -pic and Not A d d i n g Reals

(*) If a = (d i : i


29/33

4. More on Forcing Not Adding -Sequences and on the Diagonal Argument 431

4.19 Claim. 1) Assume E C S 2, we can omit

3) Assume Q = (P,Q : < C*and C < C* } is aCS iteration of properforcings. Assume also that V is a (fine) normal filter on 2, > 6J a cardinal, * < , = ^, and P* has a dense subset of

cardinality < ,


30/33

432 VIII. ft-pic and Not Adding Reals

(d ) is a cardinal, V is a normal filter on S = 2?,B^)-complete (one of them is Q^, th e othersare trivial).)

4.21 Subclaim. Under the assumption in 4.20, for each < * , for some(D , E) GV, D is a (, E, /^-completeness system in V, E e S


31/33

4. More on Forcing N ot Adding -Sequences and o n the Diagonal Argument 43 3

Proof. Let* = { : G S (), tg(a) > 2 and for some -{(, ) : C 0 )

fo r every G0, < < ^g() and ( G7Vc and(a : i = 0 or


32/33


and for each n,m < we have (an}~ (a : m+n + 2 < 7 < /3 m + n + >belongs to "(n) ,(n+i) }

B4.2o

4.24 Concluding Remarks. 1) There is not much difference between using(D, D)-completeness and (F, D)-completeness (where F is a stationary subsetof


33/33

saharon shelah- proper and improper forcing second editon: chapter viii: kappa-pic and not adding...

Documents