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,, such that for every n, <$„ satisfies <«0"4,-i>lhp H MJh Q l I Vm>n3fc a ^ a j " 1 ) . We claim that q = <„ satisfies ^lhVn3/c
an =
ft
(5.15)
To verify (5.15), let n
The iteration in general The generalization of the above proof from co to arbitrary limit ordinals is not trivial. To illustrate the difficulties, let y be an ordinal of cofinality co, say y = limnyn. By induction, we assume that each iteration Rn = PJn i7n is proper. Then the proof given in 5.8 shows that the ^-iteration R of {Rn}n is proper. However, the iteration R need not be in general equivalent to the y-iteration P. Given a sequence {fn)neR, each fn is forced to be a function in Py _ y with countable support; but there is in general no reason why there should exist a countable set S c y such that Vn the support of fn is included in S (so that (fn}n can be represented as a function in P, with countable support). We prove Theorem 5.1 by induction on the length of iteration. Simultaneously, we prove the following generalization of the Factor Lemma 7.13, Part II. 5.17
The 6>-Factor Lemma Let {yn}n be an increasing sequence of ordinals with limny,, = )>. Let P be an iteration of length y of proper {Qa}. Let Ro = P yo , and for
5 Iteration of proper forcing each n let Rn + 1=Pyy
107
, and let R be the (full) iteration of
{AH}m
a = j3k
(5.18)
It is not difficult to see that this version (seemingly more difficult for II) is equivalent to the proper game. To formulate the right induction hypothesis, let Py be an iteration of proper forcings, and consider the proper game ^ on P y . Let o be a strategy for II. We say that a is a good winning strategy if for every sequence of moves p, Ao,... ,An,... of player I, o produces a sequence {/?„}„ such that there exists a q ^ p that satisfies (5.18), and, moreover. support(q)^{Pn}n<(O
(5.19)
We shall prove Theorem 5.1 by induction on y. The inductive hypothesis (somewhat stronger than 'P is proper') reads as follows: VT/
(5.20)
We prove (5.20) by induction on y. If y is a successor ordinal then (5.20) follows easily from the earlier proofs. Thus, we assume that y is a limit ordinal. If cf y = a>, then we first prove Lemma 5.17. The proof of (5.20) incorporates the procedure used in the proof of Lemma 5.17. 5.21
Proof of the 6>-Factor Lemma Let y = limMyn, and let Py be an iteration of length y, of proper {<2Js< r For each n, let An = Pjm_l7u (and R0 = Pyo); l e t R b e the co-iteration of {Rn}. We wish to show that B(R) = B(Py). For any pePv let r = (rn}n<(O, where rn = p \[yn_ uyn). Thus, P y embeds in R; it suffices to show that Py embeds in R densely. Thus, let r = „ be a condition in R. We wish to find pePy so that p^r. By the inductive hypothesis (5.20), each Rn has a good winning strategy on. We use these good strategies to produce p. We play the proper games &(Rn), simultaneously for all n. The game „) is initiated with the condition fn. The moves of player I are names
108
/ / / Proper forcing
for countable sets of ordinals; the moves of player II are according to the strategy 6n. At step 0, we start &(R0). Player I plays r0 and a name A% for the support of fl. Player II responds po. At step 1, we start ^(Rx) in VRo. Player I plays rx and a name A\ for the support of r 2 . Player II responds a?. Then we continue the game ^(R0Y player I plays a?, and II responds /?!. Generally at step n, we start %(Rn) in yR°*-*Rn-K Player I plays fn and a name i4J for the support of fn+1. Player II responds a"" 1 . Then, playing &(Rn- J , player I plays a"~x and II responds a"" 2 . And so on, until II plays pn in the game 9(R0). Since the 6n are good winning strategies, there exists a condition q = (qn}nER, stronger than
(5.22)
and the support of qn is included in the set of all ordinals played by II (5.23) Let S = {pn}n<(O. It follows from (5.22) and (5.23) that, for every n,
q\n\\- support(qn)^S We shall conclude the proof by constructing a condition pePy so that p = q (under the embedding of P in R). This we do by induction on £ < y: if ^ S we let p(£) = 1, and if £eS, then we let p(^) be the condition ieQ^ so that p\£\[-i = qn(£\ where n is the unique n for which yn_x ^ £
Proof of the inductive condition (5.20) Let 7 be a limit ordinal, and let Py be an iteration of proper
We want to show that (5.20) is true for y. It suffices to prove it for rj = 0: when Y\ < y is arbitrary, we do the same proof inside Vp", because by the Factor Lemma, P^ is in Fp* a countable support iteration (of length y - rj) of proper forcings, and by the induction hypothesis, |— P Vfy Vy such that rj ^ fj < y < y( \\- p_ II has a g.w.s. on P--) where g.w.s. = good winning strategy. Thus, we want to show that II has a good winning strategy in the proper game ^(P y ). We shall follow closely the proof of 5.8 (iteration of length a>). If cf y = o>, we choose beforehand a cofinal sequence {yn}n with limit y; if cf y > a) then we define a certain increasing sequence {yn}n below y in the course of the proof.
5 Iteration of proper forcing
109
Player I starts the game ^(Py) by selecting pePy and by choosing a Pyname a 0. If cf y > co, we choose some y0 < y (if cf y — a>, y0 is already given), and consider Ro = Pyo. Let po = p \y0', by Lemma 5.6 there are jR0-names &% and s0 such that p0 \\- RJ0 ^ p \ [y0, y) and (p0, s0) |h ag = a 0 . We start the game ^(Ro) by letting I play p0 and an # 0 -name for the countable set jag} usupport (s0). Player II uses a good winning strategy a0 to return /?0. We let this j80 be player IPs first move in &{Py). At the nth move, I chooses a Py-name an. If cf 7 > co, we choose some 7n
and
Sn^§n-X\lyn9y)
and We start the game ${&„) by letting I play pn and X", where A"n = {dnn}v support (sn).
II uses a good winning strategy 6n to play d^" 1. Then we continue &(Rn-1) by letting I play ann~ \ to which II responds d^"2. And so on, until II plays (by o0 in ^(Ro)) an ordinal fin. It remains to show that the strategy described above is a good winning strategy for II in &(Py). Let y^ = limnyn and S = {/?„}„. As in the proof of 5.8 we obtain a sequence q = <„ in the co-iteration R of {K,,},, such that
Since the an are good winning strategies, it follows, as in the proof of the co-Factor Lemma, that B{R) = B(Py ) and that q is a condition in P and support(^f)cS. Let us identify the condition geP ]oo with (jl l...ePy (if 7^ <7). Since S^y^ and for every n, g fn||- support (sn)^5, we have
It follows that ^f^p, q\\- Vn3/c otn = pk (and support(^f) ^ 5). Hence, the strategy is a good winning strategy. • 5.25
Iteration of length co2
By Theorem 5.1, iteration of proper forcing preserves K^ Assuming the CH, iteration of length at most co2 preserves other cardinals as well, provided the forcing notions iterated are of size at most Kx. This has applications when one adds iteratively generic reals, such as K2 Laver reals. We conclude this chapter by stating (without proof) the following lemma, the consequence of which is (see Corollary 7.10, Part II) that an iteration of
110
/ / / Proper forcing
length co2 of proper forcings of size Kx has the X2-chain condition and therefore preserves all cardinals: 5.26
Lemma Assume the CH. Let OL
6
The Proper Forcing Axiom
We consider a powerful internal forcing axiom that generalizes the version of Martin's Axiom (Part II, Section 3.2). 6.1
Proper Forcing Axiom (PFA) If P is a proper notion of forcing and if {Da:a < a^} are dense subsets of P, then there exists a filter G on P that meets all the Da. The aim of this chapter is to prove the following consistency result, due to Baumgartner and Shelah:
6.2
Theorem If there exists a supercompact cardinal then there is a generic model that satisfies PFA and 2N° = K2. 6.3 A few remarks before we start the proof of Theorem 6.2. Every c.c.c. notion of forcing is proper, and so PFA implies MAXX. Consequently, PFA implies 2 K °>K 1 . At present it is unknown whether PFA is consistent with 2S° > K2, but a stronger axiom than PFA, the so-called Martin's Maximum, implies that 2X° = K2. Unlike (the general version of) Martin's Axiom, PFA only deals with Xx dense sets at a time. It is impossible to generalize PFA to deal with X2 dense sets, as is easily seen when one considers the co-closed notion of forcing P that adjoins a collapsing map of co2 onto cot (with countable conditions): for each a < a>2, let Da = {pePiaerange (p)}; no filter on P can meet all the dense sets D a ,a < co2. The consistency proof given below uses a supercompact cardinal. There are consequences of PFA showing that some large cardinal assumption is necessary for the consistency of PFA. At present it is not clear exactly how strong large cardinal assumption is needed. We recall the definition of a supercompact cardinal: 6.4
Definition An uncountable cardinal K is supercompact if for every X ^ K there 111
112
/ / / Proper forcing
exists an elementary embedding with critical point K (i.e. K is the least ordinal such that ;(/<;) > K) such that J(K) > X and Mk a M, i.e. every A-sequence {aa:a < X) c= M is in M. A major tool in the proof of Theorem 6.2 is a Laver function: 6.5
Lemma Let /c be supercompact. There exists a function/: /c -• VK such that for every set x, for every sufficiently large X there exists an elementary embedding;: K-»M with critical point K such that j(k) > X, MA c M and (7(/))M = x
(6.6)
'Sufficiently large X" means X^K and X^ |TC(x)|, where TC(x) is the transitive closure of x. The proof of Lemma 6.5 can be found in Laver [1978]. The proof of Theorem 6.2 follows loosely the proof of the consistency of Martin's Axiom (see Theorem 3.5, Part II). We construct a notion of forcing by countable support iteration of length K. Each notion of forcing used in the iteration is proper, and so Kx is preserved. The iteration uses only forcings of size < /c, and so K and cardinals above K are preserved. Cardinals between K t and K are collapsed, and K becomes K2; and the model satisfies 2N° = K2. The crucial property of the model is of course that it satisfies PFA. This cannot be arranged as easily as in the case of MA, because we do not have an analog of Lemma 3.4, Part II (equivalence of MA with MA*). Instead, we use a Laver function, which makes it possible to handle all potential proper forcings in only K steps. 6.7
Proof of Theorem 6.2 Let K be supercompact, and l e t / b e a Laver function. We construct a countable support iteration PK of {(5a:a < K} as follows: At stage a, consider the set/(a). In Vp* we define Qa as follows: if/(a) is a pair (A Qj) of Pa -names such that P is a proper notion of forcing and @ is a ysequence of dense subsets of P, for some y
6 The Proper Forcing Axiom
113
2*° ^ K2, we shall be done as soon as we prove that K[G] satisfies PFA and ic = K2. We shall prove that K[G] satisfies If P is proper and 2 = {Da:a < y}, y < /c, is a sequence of dense sets then there is a filter F on P that meets each Da (6.8) The condition (6.8) clearly implies PFA in K[G]; let me show that it also implies K = K2: if y < K, let P be the notion of forcing that collapses y onto co1 with countable conditions. P is proper. For each a < y let Da = {/?eP:aerange (p)}. Applying (6.8), we obtain collapsing map of y onto a*!. Thus, K = X 2 . In order to prove (6.8) in K[G], let P be a proper forcing in F[G] and let 2 = {Da:oc < y} be a collection of y < /c dense subsets of P. Let P and ^ be PK-names for P and 3). Let A be a sufficiently large cardinal (say X > 2 ); we may also assume that P e l Since / is a Laver function, there exists an elementary embedding j : V-+M with critical point K such that j(k) > A, Mx a M, and (6.9) 6.10
Lemma P is a proper notion of forcing in M[G~\.
Proof P is proper in V[G]\ using Definition 3.7 (and the remark following it), this is witnessed by some club set C aiy^]™ of countable models, where t] > 2 |P| and r\ < k. Since Mk a M and because G is generic on PK that has the fc-chain condition, every >l-sequence of ordinals in V[G] belongs to M[G~\. It follows that the club C is in M[G]. Hence, C witnesses that P is proper in the model M[G\ • Now consider the notion of forcing j(PK) in M. It is an iteration (with countable support) of length J(K), using the Laver function^/). Since; \ VK is the identity, the first K stages are the same as in PK, i.e. PK = (j(PK)) \K. At stage K:, we havey(/)(/c) = (P, &) and P is proper in M[G\ and therefore P is the forcing used at stage K of j{PK). Thus, for some A, j(PK) = PK*P*R
(6.11)
Let H*K be a K[G]-generic ultrafilter on P*R9 and let us work in K[G*//*K]. We extend the elementary embedding j.V^M to an elementary embedding 7*:K[G]->M[G*iJ*K]
(6.12)
114
/ / / Proper forcing
This is done as follows: for every PK-name x, let j*(i/G)=j{i)/(G*H*K)
(6.13)
The definition of j * does not depend on the choice of the name x, for \\x = y\\eG implies \\j(x)=j{y)\\eG*H*K. (Because j(p) = p for every pePK.) Similarly, j * is elementary, since ||(p(x)||eG implies \\(p(j(x))\\e G*H*K. Also,;* =3;. The filter H on P is K[G]-generic and thus it meets every D a, a < y. Let
F = {j*(p):peH}
(6.14)
As we assume that P c A, we have F = {j(p):peH}, and because; f AGM, the set F is in M[G*H*K~]. Moreover, F generates a filter on;*(P) that meets every j*(Dx\ a < 7. Thus, M[G*/f *K] |h 3F on j*(P) that meets every Dej*{@) and, since ;* is elementary, ^[G] Ih 3F on P that meets every Z)e^ which proves (6.8).
•
References Baumgartner [1983] gives a number of applications of the Proper Forcing Axiom. Laver, R. (1978). Israel J. Math. 29, 385-8. Baumgartner, J. (1983). In Surveys in Set Theory (Mathias, A.R.D., ed.), Cambridge University Press, 1-59. Shelah, S. (1982). Proper Forcing. Lecture Notes in Mathematics 940, Springer-Verlag, NY.
7
Martins Maximum
The present chapter deals with a generalization of the Proper Forcing Axiom that, in a sense explained below, is a maximal consistent generalization of Martin's Axiom MAXX. The axiom PFA of 6.1 is an internal forcing axiom that generalizes MANj. The generalization is obtained by considering a more extensive class of forcings to which the axiom applies. To explore further generalizations, let ^ be an arbitrary class of forcing notions, and let us consider the principle 7.1
MA(^): If P is a notion of forcing in # and if {Da:a < a*!} are dense subsets of P, then there exists a filter G on P that meets all the Da: Thus MAX is MA(c.c.c) and PFA is MA (proper). An important generalization of PFA is Martin's Maximum. We say that P is stationary preserving if every stationary subset S of co1 remains stationary in K[G], for any generic filter on P. Martin's Maximum is the principle MA (stationary preserving): 7.2
Martin's Maximum (MM) If P is a stationary preserving notion of forcing and if {Da:a < a^} are dense subsets of P, then there exists a filter G on P that meets all the Da. Since every proper notion of forcing preserves stationary sets, the class of all stationary preserving P is at least as large as the class of all proper P, and so MM implies PFA. The maximality of the principle MM follows from this observation: 7.3
Lemma Let P be a notion of forcing with the property that there is a stationary set S e a>1 such that || S is not stationary || = 1
Then there are Kx dense sets such that no filter G on P meets them all. Proof Let C be a name such that || C is a club and CnS = 01| = 1
(7.4) 115
116
/ / / Proper forcing
We define dense sets Da and £ a , a < a^, such that no filter G can meet all the Da and all the Ea. For each a, let
= {p: eitherp |— aeC, or 3y < a such that for all ^ between 7 and a, p | | The sets Da and £ a are dense, because C is, respectively, unbounded and closed. Let G be a filter on P that meets all the Da and Ea and let C = {(x:3peG
p\\-oceC}
The set C is unbounded because G n Da ^ 0 for all a. Also, C is closed: Let a be a limit point of C and let peGc\Ea. It must be the case that p \\- aeC, because otherwise there is y < a such that p \\- £$C for all ^ between y and a, while some other qeG forces £eC for some £ between 7 and a. But p and g are compatible. Now S is stationary, and so there is some ccGSnC. Hence, some condition forces cceC for an oteS which contradicts (7.4). • The following theorem of Foreman, Magidor and Shelah establishes the consistency of Martin's Maximum (relative to a supercompact cardinal): 7.5
Theorem If there exists a supercompact cardinal then there is a generic model that satisfies MM. I will not give a proof of the consistency of MM here. Instead, an outline of the proof, as well as some discussion of the technique, is given below. In the second half of this chapter, we give three applications of MM. One consequence of MM is that 2X° = K2. This underscores the still open problem whether PFA is consistent with 2N° > K2. Another consequence of MM is that the club filter o n K i is K2-saturated. This is known to imply strong large cardinal properties, thus justifying the use of a supercompact cardinal in the consistency proof of MM (and possibly of PFA as well). 7.6 An obvious attempt at a generalization of the consistency proof of PFA given in Chapter 6 would be to iterate notions of forcing that are stationary preserving. Such an approach does not succeed, for the following reason. There is a stationary preserving notion of forcing Pf, that for a given sufficiently fast growing f u n c t i o n / : ^ -^cou adjoints generically a fast growing function g which is eventually below / . It is clear that such
7 Martin's Maximum
117
forcing cannot be iterated o times without collapsing Kx. (For this example, consult [Shelah, 1982], p. 255.)
Semiproper forcing We consider a property of forcing intermediate between proper and stationary preserving. For motivation, we refer the reader to Chapter 3, particularly to Theorem 3.13 showing that Definitions 3.1, 3.2, 3.7 and 3.11 are equivalent. 7.7
A game theoretic definition Consider the semiproper game on P. Player I selects a condition p, and chooses a name d0 for a countable ordinal. Player II chooses an ordinal /?0. At the nth move, I plays a name an for a countable ordinal, and II plays an ordinal /?„: p; <*o,/Mi> 0i, •.•>*,., ft,. •»
(7-8)
II wins the game if and only if Iq^p
Vn 4lh3/c
an =
fik
(7 9)
7.10
Definition A notion of forcing (P, < ) is semiproper if player II has a winning strategy in the semiproper game for P. 7.11
A model theoretic definition Let X be sufficiently large and let M be an elementary submodel of (VX9 e, P, <). A condition q is (P, M)-semigeneric if for every name a e M for a countable ordinal, q\\-3PeM
a=£
(7.12)
7.13
Definition A notion of forcing (P, < ) is semiproper if for some (for all) sufficiently large X there is a club set C ^ [VxY" °f countable elementary submodels M<(K A ,e,P, < ) with the following property: VpeM3q ^ p
q is (P, M)-semigeneric
(7.14)
Definitions 7.10 and 7.13 are weaker versions of the corresponding definitions of properness. Instead of arbitrary ordinal names, the definitions refer only to names for countable ordinals.
118
/ / / Proper forcing
A careful perusal of the equivalence proof in Chapter 3 will lead the reader to the following conclusion: 7.15
Theorem (a) The two definitions of semiproperness given above are equivalent. (b) If P is semiproper then P is stationary preserving. • The last part of the proof of Theorem 3.13 does not go through in the semiproper case, as the argument uses preservation of stationary subsets of [X]03 for some large A, not just X = X x . 7.16 It is consistent, relative to a large cardinal, that every stationary preserving P is semiproper (the converse of 7.15 (ft)); we discuss this later in some detail. As shown by Shelah, a large cardinal assumption is necessary for this implication. There is a notion of forcing (the Namba forcing) which is stationary preserving, but not semiproper unless 0 # exists.
Semiproper Forcing Axiom and revised countable support iteration As 'semiproper' is a property intermediate between 'stationary preserving' and 'proper', the forcing axiom MA (semiproper) is stronger than PFA and a consequence of MM: 7.17
Semiproper Forcing Axiom (SPFA) If P is semiproper and if {Da:a < a>1} are dense in P, then there is a filter G on P that meets all the Da. For the record*: MM-•SPFA-•PFA
(7.18)
The axiom SPFA is consistent relative to a supercompact cardinal. Its consistency (due to Shelah) is established in a manner similar to the proof of Theorem 6.2, namely by iteration of semiproper forcings. The iteration used in the consistency proof of SPFA is called revised countable support iteration. If one iterates proper forcing (with countable support), every countable set of ordinals at any stage a of the iteration is included in a countable set in the ground model. This fact is essential in the proof of the Factor Lemma, which enables us to replace a condition with countable support in K[G a ] by a condition with countable support in V. * S. Shelah proved recently that SPFA and MM are equivalent.
7 Martin s Maximum
119
When forcing with a semiproper partial order, however, the cofmality of an ordinal may change to co. With this in mind, the method of revised countable support uses a different kind of limit at the limit stages of the iteration, in a way anticipating when an ordinal changes its cofmality to co. Roughly speaking, the support of a condition is not a countable set of ordinals, but rather a countable set of potential names for ordinals. The execution of this idea is technically quite involved, and can be found in [Shelah, 1982]. An analog of Theorem 5.1 states that semiproperness is preserved under revised countable support iteration. This fact, along with the technique used in the proof of Theorem 6.2, establishes the consistency of SPFA relative to a supercompact cardinal. Forcing principles MA + ( We shall now consider a stronger version of the internal forcing axioms (7.1):
7.19
If P is a notion of forcing in V9 if {D^.OL < w1} are dense subsets of P and if S is a name such that || S is a stationary subset of col || = 1 then there exists a filter G on P that meets all the D a , and such that the set is stationary.
(7.20) +
For the particular classes # we use the notation MA^, PFA , SPFA + and M M + . Baumgartner has shown that MA^ = M A X ; the plussed version of the other forcing axioms is strictly stronger than the axioms themselves. The consistency proof of PFA in Chapter 6 can be modified in the obvious way to yield the consistency of PFA + . Similarly, the consistency proof of SPFA can be modified to give the consistency of SPFA + . Thus: 7.21
Theorem SPFA + is consistent relative to a supercompact cardinal.
We shall conclude our discussion on the consistency of MM by proving the following: 7.22
Theorem MA + (co-closed) implies that every stationary preserving forcing P is semiproper.
120 7.23
/ / / Proper forcing Corollary
And consequently, Theorem 7.21 yields Theorem 7.5, the consistency of Martin's Maximum. Note that by 7.16, the axiom MA + (co-closed) implies (at least) 0#. Toward the proof of Theorem 7.22, let P be a notion of forcing, let X be a sufficiently large cardinal, and consider the club set F of all countable elementary submodels of the model (F A ,e,P, < ) . An elementary chain is a sequence <Ma:a<0>
(O^coJ
(7.24)
of elements of F such that M a < Mp whenever a < /?, and My = \Ja< yMa for every limit y. 7.25
Lemma MA + (co-closed) implies that for every stationary set X ^ F there exists an elementary chain <M a :a < cox > such that M a => a for all a, and that the set {a:MaeX} is stationary. Proof Let Q be the following co-closed notion of forcing: a condition q is an elementary chain <M a :a < 6} for some countable 0, such that M a 2 a for all a; the ordering is by extension. A generic filter G on Q produces an elementary co^chain CG = <MG(a):a
7 Martins Maximum Xp = {MeT: peM
121 and Vg ^ p q is not (P, M)-semigeneric}
is stationary.
(7.26)
By Lemma 7.25 there exists an elementary chain <M a :a
(frM is an element of B(P% and is possibly 0). Let G be a generic filter on P with peG; we shall show that the set {oc:bM eG} contains a club. Let $£9 £
and
is a club (in V). Let, in K[G],
C is a club, and if a e C then for every SeMa there is geG such that q |h (3PeMa)S = p. Hence, bMaeG. Consequently, C c {a:fcM G G}. D
Consequences of Martin's Maximum Of the numerous applications of Martin's Maximum, we present three here: the continuum is equal to K2 , the Singular Cardinals Hypothesis holds, and the club filter on Kx is K2-saturated. 7.27
Theorem Assume MM. If K ^ K2 is a regular cardinal then K*° = K.
7.28
Corollary MM implies 2*° = X2.
7.29
Corollary MM implies the Singular Cardinals Hypothesis: for every singular
122
/ / / Proper forcing
(By Silver's theorem, it suffices to prove Corollary 7.29 for the case cf X = a>; this follows from Theorem 7.27 by letting K = A+.) Theorem 7.27 uses the following lemma: 7.30
Lemma Assume MM, and let K ^ X2 be regular. Let E = {
First we prove that Lemma 7.30 implies the theorem: let {Sf:i < K} be a pairwise disjoint collection of stationary subsets of K. Given a countable set X CZK, Lemma 7.30 provides an ordinal 9X
veAn
then f(v)eSn
(7.31)
The ordering of P is by extension. Once we prove that P is stationary preserving, and that for every a < co1 the set Da = {/eP:dom / ^ a + 1} is dense in P, Martin's Maximum yields a continuous increasing function f:co1^S with property (7.31), and 9 = sup v /(v) is as desired. In both proofs we shall make use of the following: 7.32
Lemma Let A c a>1 be stationary, and let T be a stationary subset of K such that cf £ = co for all £eT. Every model (KA ,e,...), where A^/c, has a countable elementary submodel M such that Mnco1eA and sup(Mn/c)GT.
Proo/ Since T is stationary in fc, there exists an elementary submodel N (of size K J such that co1czN and sup (AT nk)eT. N is the union of an elementary a>x -chain of countable models M a such that sup(M a n/c) = sup(iVn/c) for each a. Since ,4 is stationary in X l9 there is M = M a such
7 Martin's Maximum
123
We prove first that each Da is dense. We do it by induction on a. Assume that it is true for all ordinals less than a. Let peP; we find q 3 p of length at least a + 1. Let n be such that cceAn, and let T=Sn. By Lemma 7.32 there is a countable elementary submodel M of (KA,P,p, a) for sufficiently large X such that y = sup(Mnfc)eT. Let {an}n be a sequence of countable ordinals with limit a, and let {yn}n be a sequence converging to y. We construct a sequence of conditions p — p0 c px cz • • • c pn a - • as follows, such that pneM for each n: given pn, let prt +! be, inside M, an extension of pn such that a n edom(p n+1 ) and that At + i(°O ^ 7nJ s u c h Pn+i exists because M, an elementary submodel of KA, satisfies the induction hypothesis that Dan is dense. The function q = (J^ = o p n u{(a,y)} is a condition of length a + 1. Finally, we prove that P is stationary preserving. Let A be a stationary subset of col9 let peP, and let C be a name such that p \\- C is a club subset of a)x
(7.33)
We want a stronger g and some <xeA such that q\\-aeC. Let n be such that AnAn is stationary, and let T = Sn. By Lemma 7.32 there is a countable elementary submodel M of (KA,P,p, C) such that oc = Mr\co1eAnAn and y = sup(Mn/c)eT. Let {an}M be a sequence with limit a, and let {yn}n be a sequence with limit y. We construct conditions p = p0 <= ••• c P n <= •••, each in M, as follows: given pn, let pn+1eM be such that anedom(pn+1\ that pn + ^ a j ^ yn9 and that for some /^M ^ an in M (therefore
Definition The club filter on Kx is K2-saturated if there is no collection of stationary sets {At:i < K2} such that Ain Aj is nonstationary for all i ^j.
By the work of Solovay, Kunen and Mitchell, K 2-saturation of the club filter has very strong large cardinal consequences. By the following theorem, so does Martin's Maximum: 7.35
Theorem MM implies that the club filter on K t is K 2-saturated.
We start the proof by reviewing some facts on stationary sets of countable ordinals.
124
/ / / Proper forcing
7.36
Lemma For any collection {Ai'.ieZ} of stationary sets, with |Z| = X1? there is a stationary set A such that (i) At — A is nonstationary, for all ieZ, and (ii) every stationary subset of Z has stationary intersection with some A>v The set A is unique modulo the club filter, and is denoted A = £ i e Z At. Proof Assuming Z = cou we let A be the diagonal union:
(xeA^xxe ( J ^
D
(7.37)
Let S be a stationary set and let Ps be the notion of forcing that adds a club subset of S (Example 1.2). 7.38
Lemma If A c 5 is stationary, then A remains stationary in VPs.
Proof We follow closely the proof of co-distributivity (Lemma 1.4). Let p \\- C is a club we shall find q < p and xeA such that q \\- XeC. As in Lemma 1.4, we construct a chain {Aa}a of countable subsets of Ps and an increasing sequence ya. Given >4a, for each qeA^ we find P = f$(q) > ya and some r = r(g) ^ q so that r |— jSeC, and max (r) > ya. Then we let Aa+1 = Aav{r(q):q
Lemma Assume MM, and let {At:ieW} be a collection of stationary sets with the property that every stationary set has stationary intersection with some Av Then there is Z c w of size < Xx such that T.iez^i ^s i n the club filter; hence, every stationary set has stationary intersection with some Ah ieZ. The lemma clearly implies the theorem.
7 Martin's Maximum
125
Proof We apply MM to the following forcing notion P: first let Q be the forcing that collapses \W\ onto Xx with countable conditions. In F Q , consider S = Y.iew^i a n d l e t P = Q*P§Equivalently, let P (a dense set in Q*P$ be the set of all pairs (q,p) such that q: y + \->W for some y < CD1 , and p is a closed countable subset of col such that ocep implies ae (J A ^ }
(7.40)
A condition (q\ p') is stronger than (q, p) \iq' ^.q and /?' is an end-extension of p. Note that P is stationary preserving. If A is stationary, then for some ie W, A n Ai\s stationary. A n Ai remains stationary in VQ, and so An A(nS is stationary in VQ. By Lemma 7.38, AnA(nS remains stationary in Q P p V * *. Hence, A is stationary in V . For each a < col, let Da = {(q,p)eP:(x ^ max(p)}. Each Da is dense in P. By Martin's maximum, there is a filter G on P that meets all the Da. Let F = [J{q:(q,p)eG for some p] Z = range (F) and C = {J{p:(q,p)eG for some q] The set C is a club, and by (7.40)
In other words, C = Y^iez^i-
•
References Foreman, M , Magidor, M. and Shelah, S. (1986). Ann. Math, (in press). Shelah, S. (1982). Proper Forcing. Lecture Notes in Mathematics 940, Springer-Verlag, NY.
8
Well-founded iteration
In this final chapter, we discuss some of the ramifications of the iteration methods described in earlier chapters, as well as questions raised by the consistency proof of the Proper Forcing Axiom. One still unresolved question is whether the proper forcing axiom is consistent with 2*°>X 2 . ^ n v * e w °f Martin's Maximum, which implies 2K° = K2, this question may not be just a technical problem resulting from the particular method used in the consistency proof. Nevertheless, the inability to construct a model with 2 X °>X 2 by the method of Chapter 6 is caused by the failure of the countable support iteration to preserve cardinals above Xx. Indeed, it has been observed that countable support iterations of length > co2 do not necessarily preserve K2. One effect of this phenomenon is that in all applications of the countable support iteration one ends up with a model in which the continuum is exactly X2. This is in contrast with the finite support iteration which, as in the case of Martin's Axiom, enables us to construct models with the continuum arbitrarily large. We shall describe a more general method of iteration that makes it possible to preserve cardinals while adding a large number of generic reals. The method is a generalization of both iterated forcing and product forcing. In particular, it is then possible to give a partial answer to the question of consistency of PFA with 2N° > K 2 . Without going into details, let me state the result as follows. There is a class $ of proper forcing notions that includes, among others, Cohen forcing, Sacks forcing, Prikry-Silver forcing, Mathias forcing and Laver forcing, and for which the analog of PFA is consistent with 2N° > K2. 8.1
Theorem The axiom MA(^) is consistent with 2N<> > X 2 .
The class ^ is considerably smaller than the class of all proper forcings. Note that there is no mention of large cardinals in Theorem 8.1, as the consistency is relative just to ZFC. This is unlike the consistency of PFA for which large cardinals are necessary. 126
8 Well-founded iteration
127
We omit the proof of Theorem 8.1 but outline the method of wellfounded iteration which the proof uses. Well-founded iteration The following definition is a generalization of iteration of forcing described in Definition 7.1, Part II. Let (W, <) be a well-founded partial order. For xeW, let W[x] denote the initial segment {yeW:y < x}. In general, an initial segment of W is a set X c W with the property that xeX and y < x then yeX. Definition 8.2 is by induction on the height of W\ 8.2
Definition Let (W9 <) be a well-founded partial order. A forcing notion Pw is a (well-founded) iteration along W if it is a set of functions on W with the following properties: (i) For every xeW, PW[X] = Pw \ W[x] = {p \ W[x]\p£Pw} is an iteration along W[x] (ii) For every x e W there is a forcing notion &x e VPw^x\ and for every p e Pw \hPw[x]P(x)eQx (iii) For all
p,qePw iff for all xeW
and
(iv) lePW9 where l(x) = 1 for all xeW. If X is an initial segment of W, let Px = PW\X = {p\X\pePw). (v) If X is an initial segment of W and if pePw and qePx are such that q^p p\X then the following function r is a condition in Pw\ r{x)
\q{x) \p(x)
if if
xeX xeW-X
If the well-founded partial ordering Win Definition 8.2 is a well ordering (of length a) then Definition 8.2 is just Definition 7.1, Part II, of iteration of length a. The analog of Lemma 7.2, Part II, holds for well-founded iterations as well: 8.3
Lemma If X is an initial segment of W and Px = Pw \ X, then Vp* c Vpw.
•
128
/ / / Proper forcing
Note that well-founded iteration is also a generalization of product forcing: a product of forcing notions {P f :/e/} is in fact an iteration along /, where / is given the discrete partial order. As in the case of well-ordered iteration, we can define the forcing notion Pw by giving the names Qx9 xeW, and specifying the support of each condition. In particular, we may consider well-founded iterations with finite support, with countable support, etc. To illustrate the use of well-founded iterations, let us consider the problem of preserving X2. Suppose that Pw is such that the support of every pePw is countable, and is included in an initial segment X of w of size ^ K x . (Note that there are arbitrarily large partial orders with this property; only the height of W is limited to ^ X2.) Assuming that each PW[X] has the K2-chain condition, a standard A-system argument shows that Pw has the K2-chain condition as well. The preservation of Xx by well-founded iteration is an entirely different problem. In some cases, it suffices to use countable supports. Such is the case when iterating Sacks forcing, for example; a somewhat complicated fusion argument shows that Pw is proper (in fact, satisfies Axiom A). However, when applying well-founded iteration to other forcings, such as in the proof of Theorem 8.1, one needs a more general iteration. Note that in view of Propositions 5.16 and 5.17, Part I, and because iteration along W subsumes product forcing, neither finite support iteration nor countable support iteration is suitable when iterating Laver or Mathias forcing. The iteration used in the proof of Theorem 8.1 uses mixed support. We omit the (rather technical) definition of mixed support iteration, and instead give an example, a special case of mixed support iteration that is a prototype of the method. 8.4
Example (Baumgartner) Mixed support product of Mathias forcings Let us consider the following product P of a collection {P^.iel}, where each Pt is Mathias forcing (Section 3.11, Part I). Condition in P are certain functions p = (p(i):iel} such that each p(i) is a Mathias condition (sf, Sf). The support of p is the set
(Here 1 = ( 0 , co).) The root of p is the set root«p i :i6/» = {i:stem(pI.) * 0 }
8 Well-founded iteration
129
8.5
Definition The mixed support product of Mathias forcings {Pt:iGl} is the set of all functions p = (pt:ieI} whose support is countable and whose root is finite. The mixed support product of Mathias forcing preserves K^ 8.6
Proposition If p \\- X\CD -• V then there exists q ^ p and a countable A such that q \\- X c A. Moreover, we can find the q ^ p with the same root as /?, and, in fact, stem(^) = stem(pt) for every is I. Proof The proof is a variation on the proof of Lemma 3.12, Part I, or Theorem 5.12, Part I. To simplify matters, assume that p has empty root; we shall find q as desired, with empty root as well. Let {un}n be a sequence of natural numbers such that each u appears infinitely often. We construct a sequence p = p0 ^px ^p2 ^ •••; let Wn denote the support of pn. Let W= \J?=0Wn; by a suitable enumeration we also construct a sequence of finite sets F Q C F J C ^ C . . . with {J?=0Fn = W. For every n, we make sure that
This guarantees that for each ie W, p^i) = \imnpn(i) is a Mathias condition (with empty stem), and so p^ = (Poo(i)}i is a condition with support W and empty root. We also produce a sequence of finite sets {An}n with the iteration that p^ \\- X c A, where A = {J?=0An. At stage n, we already have pn = <(0,S,$)):ie/>. For each ieFm let Kn(i) be the set of first n elements of Sn(i). Let r l 9 ..., TZ be all the functions T on Fn such that T =
fl;
(8.7)
for some akn then we put akn into Xw, and let qk +1 be the condition (with empty root) <(0,S(fc+1)(O):*'<EO, where
[ 7(0
otherwise
130
/ / / Proper forcing
The sequence {pn}n converges to a condition p^. The rest of the proof is a verification of the fact that p^W-X ^ A. The argument follows closely the conclusion of the proof of Lemma 3.12, Part I. I omit it, as I believe that the readers who have been able to bear with me up to this point will be able to complete the proof on their own. • Reference Groszek, M. and Jech, T. Generalized Iteration of Forcing (1986) (in press).
Bibliography
Baumgartner, J. (1983). Iterated forcing. In Surveys in Set Theory (Mathias, A.R.D., ed.), pp. 1-59, Cambridge University Press. Baumgartner, J. (1984). Applications of the proper forcing axiom, in Handbook of Settheoretical Topology (Kunen, K. and Vaughan, J.E., eds.), pp. 913-59, North-Holland, Amsterdam. Borel, E. (1919). Sur la classification des ensembles de mesure nulle, Bull. Soc. Math. France 47, 97-125. Cohen, P. (1966). Set Theory and the Continuum Hypothesis, Benjamin, NY. Dales, H.G. (1979). A discontinuous homomorphism from C(X\ Am. J. Math. 101,647-734. Dales, H.G. and Woodin, W.H. (1986). An Introduction to Independence for Analysts, London Mathematical Society Lecture Notes 115, Cambridge University Press. Devlin, K. and Johnsbraten, H. (1974). The Souslin Problem. Lecture Notes in Mathematics 405, Springer-Verlag, NY. Easton, W. (1970). Powers of regular cardinals, Ann. Math. Logic 1, 139-78. Eklof, P. (1976). Whitehead's problem is undecidable, Am. Math. Monthly 83, 775-88. Ellentuck, E. (1974). A new proof that analytic sets are Ramsey, J. Symb. Logic 39, 163-5. Esterle, J. (1978). Sur l'existence d'un homomorphisme discontinu de C(K), Proc. London Math. Soc. 36, 46-58. Fleissner, W. (1984). The normal Moore space conjecture and large cardinals, in Handbook of Set-theoretical Topology (Kunen, K. and Vaughan, J.E. eds.), pp. 733-60, North-Holland, Amsterdam. Foreman, M., Magidor, M. and Shelah, S. (1986). Martin's maximum, saturated ideals and non-regular ultrafilters, Ann. Math, (in press). Fremlin, D.H. (1984). Consequences of Martins Axiom, Cambridge Tracts in Mathematics 84, Cambridge University Press. Gray, C.W. (1982). Iterated forcing from the strategic point of view, Thesis, Berkeley, CA. Grigorieff, S. (1971). Combinatorics on ideals and forcing, Ann. Math. Logic 3, 363-94. Groszek, M. and Jech, T. (1986). Generalized iteration of forcing (in press). Jech, T. (1971). Trees, J. Symb. Logic 36, 1-14. Jech, T. (1973). Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5, 165-98. Jech, T. (1978). Set Theory, Academic Press, NY. Jech, T. (1984). More game theoretic properties of Boolean algebras, Ann. Pure and Appl. Logic 26, 11-29. Kueker, D. (1977). Countable approximations and Lowenheim-Skolem theorems, Ann. Math. Logic 11, 57-103. Kunen, K. (1980). Set Theory, North-Holland, Amsterdam. Laver, R. (1976). On the consistency of Borel's conjecture, Acta Math. 137, 151-69. Laver, R. (1978). Making the supercompactness of K indestructible under /c-directed closed forcing, Israel J. Math. 29, 385-8. Martin, D.A. and Solovay, R. (1970). Internal Cohen extensions, Ann. Math. Logic 2, 143-78. Mathias, A.R.D. (1977). Happy families, Ann. Math. Logic 12, 59-111. Menas, T. (1976). Consistency results concerning supercompactness, Trans. Am. Math. Soc. 223,61-91.
131
132
Bibliography
Rudin, M.E. (1969). Souslin's conjecture, Am. Math. Monthly 76, 1113-19. Rudin, M.E. (1977). Martin's axiom, in Handbook of Mathematical Logic (Barwise, J., ed.), pp. 491-501, North-Holland, Amsterdam. Sacks, G. (1971). Forcing with perfect closed sets, in Axiomatic Set Theory (Scott, D., ed.), pp. 331-55, American Math. Society, Providence, RI. Shelah, S. (1974). Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18, 243-56. Shelah, S. (1982). Proper Forcing. Lecture Notes in Mathematics 940, Springer-Verlag, NY. Shoenfield, J. (1971). Unramified forcing, in Axiomatic Set Theory (Scott, D., ed.), pp. 357-82, American Math. Society, Providence, RI. Shoenfield, J. (1975). Martin's axiom, Am. Math. Monthly 82, 610-17. Solovay, R. (1970). A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math. 92, 1-56. Solovay, R. (1971). Real-valued measurable cardinals, in Axiomatic Set Theory (Scott, D., ed.), pp. 397-428, American Math. Society, Providence, RI. Solovay, R. and Tennenbaum, S. (1971). Iterated Cohen extensions and Souslin's problem, Ann. Math. 94, 201-45. Souslin, M. (1920). Probleme 3, Fund. Math. 1, 223. Woodin, W.H. (1987). Set theory and discontinuous homomorphisms of Banach algebras, Memoirs Am. Math. Soc. (in press).
Index of symbols
MA*
(P, <) 2 G 2 F 2 F[G] 2 a 2 Ih 2 fl(P) 3
pk 4 P-L
vp^yQ
53
f<9 61 f
8
sat(P) 11 ^ n 15, 17, 18, 20, 100 Sit) 19 «n 20 PxQ 23 Uipi 24 sip) 24 PK(/1) 40 [ A ] < K 40
P*Q 44 G*// 44 D\B 48 MA 53 MAX, 53
Ultygf 61 supp (p) 68 Pj,a) 70 ^a/, 70 A<(O 80 C F 80 Ps 81 [/!]" 83 ACa 83 a
&0 89 ^! 90 ^» 91 ^ 92 PFA 111 MM 115 SPFA 118 MA+(
MA (a) 126
133
Subject index
K2-saturated filter, 123 amalgamation, 16, 18, 20 antichain, 4 Axiom A, 100 Boolean-valued model, 3, 5 Boolean values, 5 branch, 57 branching point, 15 canonical embedding, 51 canonical names, 5 canonical projection, 51 cardinal, measurable, 39 real-valued measurable, 39 strongly compact, 40 supercompact, 111 cc.c, 11 chain condition, /c-, 11 closed, H>, 11 closed unbounded, 80, 83 Cohen real, 13 collapse, 35 compatible, 4 complete Boolean algebra, proper, 97 conditions, 2 costationary, 81 countable chain condition, 11 countable support iteration, 68 decides, 6 A-system, 24 dense, 2 dense set of conditions, 2 directed, 85 direct limit, 68 distributive, K>, 10 distributive laws, 10 dominates, 14 Easton product, 26 Easton support, 26 extends, 13 extension, generic, 2
134
Factor Lemma, 70 co-, 106 filter, K2-saturated, 123 fine ultrafilter, 40 finite support iteration, 50 forcing, 2 forcing conditions, 2 forcing relation, 2, 6 Forcing Theorem, 2 free abelian group, 59 fusion, 15, 17, 18,20 fusion sequence, 15 game, countable choice, 91 cut and choose, 90 descending chain, 89 proper, 92 semiproper, 117 gap, 64 strong, 64 generic, 2
(P,M)-, 96 generic extension, 2 Grigorieff real, 21 ground model, 2 group, abelian, 59 Krfree, 60 free, 59 torsion free, 59
W-,59 height, 56 homogeneous, weakly, 7 incompatible, 4 indecomposable, 102 initial segment, 127 inverse limit, 68 iteration, 50, 67 countable support, 68 Easton support, 72 finite support, 50 /-support, 68
Subject index revised countable support, 118 two step, 44 well-founded, 127 /c-chain condition, 11 K-distributive, 10 /c-product, 25 /c-saturated, 11 Laver function, 112 Laver real, 19 Laver tree, 19 uniform, 21 Levy collapse, 35 limit, direct, 68 inverse, 68 Martin's Axiom, 53 Martin's Maximum, 115 Mathias real, 17 matrix of partitions, 96 maximal antichain, 4 measurable cardinal, 39 measure algebra, 38 measure space, 38 measure, product, 39 mixed support, 129 model, Boolean-valued, 3 ground, 2 monotone function, 61 names, 5 normal Moore space conjecture, 40 notion of forcing, 2 co-Factor Lemma, 106 open dense, 3 p-point, 21 partition, 4 partitions, matrix, 96 perfect tree, 15 uniform, 17 predense, 3 Prikry-Silver real, 17 product, 23, 24 Easton, 26 infinite, 24
135 product forcing, 23 product measure, 39 proper forcing, 95-7 Proper Forcing Axiom, 111 proper game, 92 property (K), 23 pure subgroup, 60 random real, 13 real, 13 Cohen, 13 Grigorieff, 21 Laver, 19 Mathias, 17 Prikry-Silver, 17 random, 13 Sacks, 15 real-valued measurable cardinal, 39 refinement, 10 revised countable support iteration, 118 root, 24, 128 Sacks real, 15 saturated, K2-, 123 K-, 11 semigeneric, (P,M)-, 111 semiproper forcing, 117 Semiproper Forcing Axiom, 118 semiproper game, 117 separable, 13 separative, 4 separative quotient, 4 stationary, 80, 83 stationary preserving, 115 stem, 18 strong measure zero, 73 stronger than, 2 strongly compact cardinal, 40 sufficiently large, 95 supercompact cardinal, 111 support, 24, 68, 128 Suslin line, 56 Suslin tree, 56 tree, 15 Laver, 19 perfect, 15 Suslin, 56 two step iteration, 44
JC-,25
mixed support, 129 a-, 25
W-group, 59 well-founded iteration, 127
Author index
Baumgartner, J., 33, 100, 101, 111, 114, 128 Martin, D.A., vii, 53, 115, 121 Mathias, A.R.D., 17, 32, 126, 128 Borel, E., 71, 73 Mitchell, W., 123 Cohen, P., vii, 6, 13, 27, 126 Namba, K., 118 Nyikos, P., 41 Easton, W., 26, 28, 72 Foreman, M., 90, 116
Prikry, K., 17, 30, 126
Gray, C , vii, 92, 110 Grigorieff, S., 21, 34 Groszek, M., 130
Sacks, G., 15, 17, 30, 126 Scott, D., 38 Shelah, S., vii, 59, 71, 95, 110, 111, 116-19 Silver, J., 17, 30, 72, 121, 126 Solovay, R., vii, 37, 39, 48, 61, 123
Jech, T., 90, 130 Kaplansky, I., 61 Kunen, K., 41, 123
Suslin, M , vii, 53, 56 Tennenbaum, S., vii
Laver, R., 19,32,73,75, 112, 126 Levy, A., 35, 49 Magidor, M , 116
136
Velickovic, B., 90 Whitehead, J.H.C., 59 Woodin, W.H., 61
