Iterated Forcing and Normal Ideals on <Emphasis Type="Italic

Sh:253

ISRAEL JOURNAL OF MATHEMATICS, Vol. 60, No. 3, 1987

ITERATED FORCING AND NORMAL IDEALS ON COl

BY SAHARON SHELAHt Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel; Department of Mathematics and EECS, University of Michigan, Ann Arbor, Michigan USA; and Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA

ABSTRACT We prove that suitable iteration does not collapse R, [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets of to~ is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + "for some stationary set S c_ to1, ~(tol)t(D,o,+ S) is the Levy algebra~ (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (Re, < R2) (and we can add "a variant of PFA") and the consistency of the same, with "Ulam property" replacing "Levy algebra"). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration).

§0. Introduction

By Foreman, Magidor and Shelah [FMS 1 ], CON(ZFC + r is supercompact) implies the consistency ofZFC + "Do,, is .R2-saturated" [i.e., if~ is the Boolean algebra P(ogt)/Do,,, "Do, is R2-saturated" means "~ satisfies the R2-c.c."]; see there for previous history. This in fact was deduced from the Martin Maxi- mum by [FMS 1] whose consistency was proved by RCS iteration of semi- proper forcings (see [Sh 1]). Note that [FMS] refutes the thesis: in order to get an elementary embedding j of V with small critical ordinal, into some transitive class M of some generic extension Vp of V, you should start with an elementary embedding ofj of V' into some M' and then force over V'.

t This research was partially supported by the NSF. This paper was largely written during the author's visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality. Received March 7, 1986 and in revised form December 16, 1986 345 Sh:253

346 s. SHELAH Isr. J. Math.

This thesis was quite strongly rooted. Note that it is closely connected to the existence of normal filters D on 2 which are 2 +-saturated or at least precipitous (use for P the set of non-zero members of ~(2)/D ordered by inverse inclusion, j the generic ultrapower). See [FMS 1] for older history. In fact, it was proved directedly that MM ÷ ~SPFA +. Much later we prove that MM is equivalent to the Semi-Proper Forcing Axiom (in ZFC) [Sh 5]. Following [FMS 1, §1, §2] much activity follows. Woodin proves from CON(ZF + ADR + 0 regular) the consistency of ZFC + "~ t S is R~-dense", for some stationary S _c o91. By Shelah and Woodin [ShW], if there is a supercompact cardinal, then every projective set of reals is Lebesgue measurable (etc.). This was obtained by combining (A) and (B) below which were proved simultaneously: (A) The conclusion holds if there are weakly compact cardinal x and a forcing notion P, I el = x, satisfying the x-c.c., not adding reals and I~-e "there is a normal filter D on o9~, B = ~(t~)/D satisfying the ~2-C.C." (B) There is a forcing as required in (A) (see [FMS 1, §3]). This was improved to using cardinals x satisfying: Pr~(x) def= X is strongly inaccessible, and for every f: x ~ x there is an elementary embedding j : V--* M (M a transitive class), x the critical ordinal ofj and H(j(f)(x)) c_ M, or at least Prb(r)* do__=fx is strongly inaccessible, and for every f: x ~ x there is x~ < r, ('¢ct

t Now usually called Woodin cardinals. After this work was written, the results of Martin Steel and Woodin clarify the connection between determinacy and large cardinals and Woodin has some consistency proofs for larger ideals on w~; very interesting among them is ~D,o, + S is R~-dense~ from a huge cardinal. Sh:253

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supercompacts, that we can force, for a stationary co-stationary S __ tot, that GCH + ~(tol)/(Do,, + S) is the Levy algebra, i.e., the completion of Levy (•0, ~ R2) (or S -- tot, but 2 ~0 = R2) and related results. The method is (RCS) semi-proper iteration, hence it can be used, e.g., to prove consistency with additional statements. Relative to the progress of [ShW], we improved some results to the use of K's satisfying Prb instead of supercompact; this is not written now as [ShW] hasn't materialized yet, hut as a result we do not try to "save" in the use of large cardinals (one such 2 suffices for D~ol R i-saturated), t Note that the r supercompact is seemingly necessary if we want a suitable variant of MA. Those points (i.e., using smaller large cardinals and getting also variants of MA (mainly without adding reals)) will be dealt with in a sequel paper. Around the time this was reasserted Woodin proved, from some r's satisfying Prb, that CON(ZFC + 2 ~o = R 2 + ~'(toi)lDo,,) is the Levy algebra, by methods related to Steel forcing. Also in Spring 1983, Foreman proved (in [FMS 2]), from the consistency of almost huge cardinals, CON(ZFC + GCH + on 3. + + there is a layered ideal) for arbitrary regular/l. This was interesting as by [FMS 2], we can get a uniform ultrafilter D on 3. which not only is not regular but even 3.~*/D -- 3.+. Note that some years ago Magidor [Mg] proved the consistency of ZFC + GCH + "for some uniform ultrafilter on N2, N~o2/D -- R2". Another result of [FMS 1] is that we can get "for every 3., D~ is precipitous" and we can get this even to Chang filters. Later Gitik, for 3. a supercompact cardinal, gave a proof which does not use a supercompact above 3.. Much later the author proves, for D a layered filter on 3., that (if 2 x = 3. +, 3. =3.<~) there is a homomorphism h from ~(3.)/D into ~'(2), A/D = [h(A/D)]/D; see [Sh 6]. Note that S-completeness is used for convenience; weaker notions can be used as well (see [Sh 1, VIII, §4], [Sh 4, §2]). We can also make that set of ordinals in which something nice occurs, a name. Note that instead of one S, we can change it getting the result for a maximal antichain of S's. Note that we can use 2.13 (3) much more extensively, e.g. in 2.19. Suppose s: is strongly inaccessible (~:i, i < r } increasing continuous, ~i + t is supercompact

* For "ZFC + D~ is Rz-saturated + MA(semi-proper)" one 2 suffices. For ~ZFC + GCH + (D,~ + S~) is layered + MA~ (S2-complete semi-proper forcing of power < R2)~ ($1, $2, $3 a partition of to~ to a stationary set) it suffices to use {2 : A < ~, A satisfies Prb} is stationary. Similarly for the Levy algebra (with a suitably stronger assumption). Sh:253

348 S. SHELAH Isr. J. Math.

for i non-limit, x = t..Jt<~x~ (P, Q~, t;, B~: i

NOTATION AND BASIC FACTS. (I) ~(A)isthepowersetofA,Sset theory, i.e., if V~B~ ,~B2, V c_'Vt then V 1~B~ ,~B2. (d) By Solovay-Tenenbaum [ST] < is transitive, and if (B~ : i < a) is <-increasing and continuous then Bi < 1,1J~<,B~. Also, if (Be : ~ < ~) is a _ -increasing sequence of Boolean algebras and B0 < B~ for ~ < ~, then B0 "~ ~ <¢ Be. (5) Also, if in V Pt ,~/)2,~/)3, in V e~ $e, ,~ ~e,, and in V e, ~8t', < $P, then in V t',

(6) For a set a and forcing notion P, .Ge is the P-name of the generic set and a [.Ge] = a U {.x[.Gp]: x ~ a a P-name}. So a [.Gel is a P-name of a set, and for G c_ P generic over Vits interpretation is a[G] = a U {x[G]: x~a a P-name} (x[G] is the interpretation of the P-name V). (7) We sometimes do not strictly distinguish between a model and its universe. Sh:253

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(8) P, Q, R denote forcing notions, ~p denotes the minimal element of P (i.e., I k--e". • ." iff ~ Ik-e "'''); without loss of generality it exists. (9) If2 > Ro is a cardinal, N a countable elementary submodel of (H(2), E ), PEN, G c. P generic over V, then N[G] < (H(2) v', E) (as H(2) v" = {[[G]: ~H(2) a P-name} and if tk-r"(H(2) v', E)~xC(xa)" then for some P-name r EH(2)I k--e"(H(;t) v,, ~) ~ ¢(~, a)"). See [Sh 1]. Also (N, G) -~ (H(A)v,~, G) (i.e., G is an extra predicate, so you may write (N, G n IN I)). Also, ifR is any relation (or sequence of relations) on H(2) v, N < (H(A)v,~, R) (and PEN, G ___ N generic over V) then (N, G) -~ (H(2)v,~, R, G). Usually we use a well ordering <* of H(2). (10) Let N <~M mean M __. M and N O x is an initial segment of M n x and M < N; if we use it for sets (rather than models), the last demand is omitted. Note that ifN -~ M < (H~), ~), x <~, N n x = M n xthen N <,,+ M.

§1. Preliminaries

1.1. DEFINITION. (1) A forcing notion P is semi-proper if: for every 2 regular> 2 lel, any countable N -~ (H(2), E) to which P belongs, and p E P n N there is q, p < q EP, q (N, P)-semi-generic (see below). (2) For a set a, a forcing notion P and q EP, we say q is (a, P)-semi-generic if: for every P-name aEa of a countable ordinal, q II---e".aEa"; i.e., if qlt--"a[qe] n cot =an cot". (3) We call W c_ S<~,(A) (where cot C_ A) semi-stationary in A if for every model M with universe A and countably many relations and functions, there is a countable N < M, such that ( 3 a ~ W)[N n co~ c. a c_ N] [equivalently, {a ES

1.2. CLAIM. (1) If W C_ S<~,(A) is stationary then it is semi-stationary. (2) Ifah c_ A _C B, and W _c S<~,(A) then: Wis semi-stationary in A iff Wis semi-stationary in B (so we can omit "in A'). (3) If Wt C_ W2 C_ S

350 s. SHELAH Isr. J. Math.

(5) If pEP is (b, P)-semi-generic, b O 0.) I ~. a C_ b then p is (a,P)-semi- generic. (6) If Wc_ S<~,(2), #>,l, N < (H~u), E), WEN, and for some aEW, NAo91C_aC_N then W is semi-stationary [if not, some M= (2 .... , F, .... ) exemplify W is not semi-stationary, so some such M belongs to N, hence N n ,t is a submodel of M, contradiction].

1.3. CLAIM. A forcing notion P is semi-proper iff

We = {a ES<~,(P U e(o91 + 1)): for every p EP n a there is q, such that p < q EP and q is (a, P)-semi-generic}

contains a club ofS

ot°[G] = Min{h(r): rEG},

a h[G ] is a°[G] ifa°[G] < ol and zero otherwise.

1.4. CLAIM. The following are equivalent for a forcing notion P: (1) P is semi-proper. (2) P preserves semi-stationarity. (3) P preserves semi-stationarity of subsets of S<~,(2 tet).

PROO~. (1)=*(2). Let o91 _ A, W C_ S<~,(A) be semi-stationary. Suppose p E P, p I F-e" W is not semi-stationary". So there are P- names of functions F, (n < o9) from A to A, 17 n-place, and p IF- "ifa c_ A is countable closed under F, (n <09) then -1(3 b)[a n wl ___ b c_ a ^bE W]". Let 2 be regular large enough. Let N < (H(2), E) be countable, let A, (F, : n < co), p, P belong to N, and there is b E W such that N n o91 __. b __. N (which is possible as W is semi-stationary). Let q be (N, P)-semi-generic, p < q EP. So q l~e"N[G] n o91 = N n oJl and N c_ N[G]" hence for the b above

q I t---e "N[G] n o91 __. b ___ N[G]".

Also q I ~j."N[G] o A is closed under the F,'s" (as N[G] -~ (H(A)[G],E), see Notation (9) in §0), contradictory to the choice of the F,'s. (2)=, (3). Trivial. 7(1)=*n(3). Let W = S

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Hence W~ is semi-stationary. But easily p(.) I F-- "W~ is not semi-stationary" so (3) fails.

1.5. DEFINITION. (1) Rss(x, ).) (reflection for semi-stationarity) is the assertion that for every semi-stationary W C_ S x.

1.6. CLAIM. (1) In Definition 1.5(1) we can replace 2 by B, when t B I = 2, col_ B. (2) If x < xl ___<;h < 2 and Rss(x, 2) then Rss(x~, 20; and if x < ;h < 2, Rss+(x, 2) then Rss+(x, 20; lastly if Rss+(x,-,2) (i

PROOF. (1) Trivial. (2) Use 1.2(2). (3) Let x __ A, W C_ S<~,(A), W _C S

1--__ l"a uI'b ol"c

where

F a = {c~ 4:c2 : q, c2 are distinct members of A },

F ~ = (R(co, C, ..... c, .... )~<~: c~A, (ct:l

352 s. SHELAH Isr. J. Math.

f ~[(V Xo, x, .... , x. .... ).<~,

if {x0, x, .... } is closed under F,(n to) then

7( :1 y0, yl .... )(R(Yo ..... Y, .... )A{XI:I

Every subset of F of power

F, = (R(c0, c, .... ): c, A, (c,: 1 < to} W,}.

Now F a U r v u LJi<~ F~ has no model, hence (using the Los theorem) for some a < x, F a U F c Ui<~ F/b has no model. (6) Easy.

1.7. CLAIM. (1) If Rss(x, 2 let), p not semi-proper, then P destroys the semi-stationarity of some W___S<~,(A), IAI dr [use (1)*~*(3) from 1.4, then 1.5(1), 1.2(2)1. (2) If P destroys the semi-stationarity of W c S<~,(A), IAI = R ,, then P destroys the stationarity of Sw c_ to, [Sw defined in 1.2(4), which says that it is stationary in Vbut not in VP]. (3) If Rss(R2, 2 le~) and P does not destroy stationarity of subsets of to,, then P is semi-proper [by parts (1), (2) above]. (4) If W c_ S<~,(A) exemplifies the failure of Rss(R2, IA I), then there is a forcing notion P of power IA I~o, not semi-proper but not destroying stationary subsets of R,. (5) Rss(R2) is equivalent to: every forcing notion preserving stationarity of subsets of to, is semi-proper.

1.8. DF.FINITION. (P~, Qj: i <= a,j

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(2) if i

1.9. THEOREM. Suppose 2 is measurable, (P~, Q~ : i < 2) is a semi-proper iteration, IP;I <2 for i <,;t, and {i<2: Q~ semi-proper} belong to some normal ultrafilter D on 2. Then in Ve`, Player H wins Gm= Gm({Rt}, to, R2) (see below).

1.9A. REMARK as in [Sh 1, Ch. XII, Def. 1.1]. (1) The game last to moves; in the nth move Player I chosef~ : R2--" co~ and Player II chose ~, 2 ~, N a countable elementary submodel of (H(2), E, <*), then for arbitrarily large i < o92, there is N' < (H(2), E, <*), N' countable, N _ N', i EN~ and N O co~ = N' n o9~ (hence N ~,, f,(g)

PROOF Or 1.9. Let D be a normal ultrafilter on 2 (in V), A ~D a set of (strongly) inaccessible cardinals, such that: ( ~' r CA) [( V i < r) IPi [ < ~:) A Q~ is semi-proper]. Sh:253

354 s. SHELAH Isr. J. Math.

For each r EA, Px/P~ (in V e,) is a semi-proper forcing, hence in the following game, PG'°(p, PflP~, Ri) Player II has a winning strategy which we call F(PflP~)(E Ve.): [by [Sh 1], Ch. XII, 2.7(3), p. 403, Definition 2.4, p. 401] a play of the game lasts to-moves, in the n th move Player I chooses a P~/P~-name ~. of a countable ordinal and Player II chooses a countable ordinal ~.. PlayerlIwinsaplayif(3q)(p~qCPflP~^qIF-"A.[~.< U m a(.) the Sh:253

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sequence (f~(x), ~t) : 1 < co is a play of PG ~'( ~ p, P,JPa, R ~) where Player II uses his winning strategy (this is a P~-name, but fortunately (~t:l < co)~ V[G,~.)]). So there is q,~ EPa/P,~ a P~-name, so that

q,ct[--ede, "Aft(r)< U ~m" I ~ n

(more exactly:

q~ It-(e,~,.,y~ej~,.,~" A#(x) < U ~" I n

q~ a P~/G,~,)-name of a PJP~-condition). We can consider q~ as a P~-condition with Dom q~ __ [x, 2), because we use RCS-iteration. Now easily (q~" r ~A,o )~ V[G~,)], II-p,/~..,"{x~A" q EG~} is unbounded in 2" as every rEPa/G~,) has domain bounded in 2, so q~, for x large enough, is possible, i.e., compatible with it. 1.10. CLAIM. Suppose x is measurable, (~ a semi-proper iteraction, lg((~) = x, lea < x for i < x and {i: Qi semi-proper} belong to some normal ultrafilter on x. Then: (1) Rss+(x, 2) implies IF-p Rss(x, 2). (2) If Q is a P~-name of a semi-proper forcing notion, I F-p,+, "(PJPi +t * Q) is semi-proper for i < x" then I F-p. "Q is semi-proper". (3) We can replace measurability of x by: x strongly inaccessible IF-p, "Player II wins Gm({Rt}, col, R2)" and P~ satisfies the x-c.c. PROOF. (1) Let W be a P~-name, p~P~, plF-p ".W___S<~,(2) is semi- stationary". Let for i

356 s. SHELAH Isr. J. Math.

some i < x, N, a are P~ -names, and q EP~. Now by 1.9, in Ve., for arbitrarily large 0 < x, N ~°l n to~ = N ¢3 o9~, and Qo is semi-proper, where we let:

Nt01 def Skolem Hull (N O (0}) (in (H(/~)z,E, <*, G~) working in the universe V[G~]).

Choose such a 0 > i. Now as 0 EN t°], (N t°], Go) < (H(~) V, ~, <~', Go). Clearly W.o[Go]EN t°l, a.[Go]E .W0[G0], tOl tq N 1°1 C_ a.[Go] c_ Nl01, hence by 1.2(6), V[Go] ~ "W.°[Go] is a semi-stationary subset of S<~,(2)". As Rss+(x, 2) dearly V[Go] ~ Rss(r, ;t), hence in V[Go] for some A c ;t, IA I sup(N[G] n x) be such that N[G t°l N o9~ =N[G~] O oJ~ (and N is a P0-name). Now work in V[G,~ N Po+ i] and use I ~e,+, (P,,/Po+1) * Q is semi-proper. (3) In the proof of (2) we use this only. In the proof of (1) choose 0 a successor ordinal (so Q~ is semi-proper). It preserves the semi-stationarity of A, hence V[Go] ~ "A is semi-stationary".

1.11. CLAIM. Suppose Rss(x, 2~), x regular and: x--R2 or (V/z 2 ~ for every countable N < (H(2),E, <~*) to which x belongs, for arbitrarily large i

1.12. REMARK. (1) The "x = R2" ." can be omitted if we replace "for arbitrarily large i" by "for some i < x, i > sup(N O x)'. (2) We can replace "x = R2, or ..." by "ifa < x, then there is CES

PROOF. Let

W = { INI: N < (H(x+),~, <*+), Ncountable and

for some iN < x for no i E [iN, x), N <~/~il}. Sh:253

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Assume W is a stationary subset of H(x÷). So, as Rss(x, 20 holds (and In(x+)l =20 there is A C_H(x+), ogiC_A, Ial

M clef (A, ~ tA, <*÷ tA) < (n(x+), ~, <*÷).

As for N~ _ N2 countable elementary submodels of (H(x +), E,

Now for some club C 1 ~ C, for every a ~ q, a tq : Skolem Hull of a U {(}, satisfies a tq n A -- a, hence a <~ at¢], but some a ~ q n Wa, contradiction. So W is not stationary and let C* _ S<~,(H(x+)) be a club disjoint to W. Let 2 > 2 ~, so H(r+), WEH(2), and let xEN < (H(2), E, <*) be countable. So H(x+)~Nhence W~Nand without loss of generality C* ~N. Hence N n H(K+)~C *, and for arbitrarily large i < x there is NI < (H(x+),E, <*÷), countable, i ENI, N n H(x +) <~ NI. Let N / be the Skolem Hull of N U (Nt n x). We can easily check that N i n x : N~ n x, so N ~is as required.

1.13. DEFINITION. A forcing notion P satisfies the (S, S)- condition (S a set of regular cardinals, S __ to1 stationary) if there is a function F (domain implicitly defined in (c)) so that: Suppose (a) T is an S-tree, f: T--- P, g: T-* tot. (b) v <~ ~/in T implies f(v) < f(ri) in P, and g(v) < g(rl) ( < toO. (c) There are fronts J, (n < to) of T such that every member of J.+l has a proper initial segment from J. and t/EJn implies (~/is splitting node of Tand)

(Sucr(tl),((f(v),g(v)) : v ~sucr(r/))) = F(r/, w[r/], ((f(v),g(v)) :v <~l))

where w[t/] = {k: ~/lkE Ut<,o Ji}. Then for every T', T <* T' there is p ~ P such that p I F-e "there is ~/~ lim T' such that if sup{g(~/I k) : k < to} ~S then { f(t/I k) : k < to} c_C_.Ge".

1.14. NOTATION. We say "P is (*, S)-complete" if f satisfies the ({R : regular > R t}, S)- condition. Sh:253

358 s. SHELAH Isr. J. Math.

1.15. THEOREM. The natural generalization of the theorems from [Sh 1, ch. X1] (and Gitik-Shelah [GSh]) holds for the (S, S)-condition.

1.16. REMARK. We use 1.13, 1.14, 1.15 only in 2.16; we can alternatively use pseudo-(S, S)-completeness (see [Sh 1, X]) but use Nm(D), or use 2.16 A(3).

§2 2.1. DEFINITION. We say Q=(P~,Qj, tj:i<-_a, j

NOTATION. Ct 0 -----or, e~ = Pi, Qjatja = tj.

2.2. CLAIM. (1) Suppose Q = ( P~, Qj, tj : i < a, j < a) is a semi-proper iteration (see 1.8 for definition). Then: (a) If i < a is non-limit or Q~ is semi-proper or Q~ preserve stationarity of subsets of tot from Ve,'then every stationary subset of co~ in V e, is stationary in Ve- too (i.e., ~[P~] is a subalgebra of $[P~]). (b) (c) If a is strongly inaccessible ( > co), and I Pil < a for i < a, then P~ satisfies the a-c.c, and so

~(toO V''= [.J ~(co0 Vr', Ve-~"2 ~ = R2". i

(d) If cot - Sis stationary, each Q~ is (cot - S)-complete [Sh 1, oh. V], then so is P~, hence forcing by Pa does not add co-sequences of ordinals (hence V r. ~ CH). (e) If (_)ENt ~ N2 < (H(2), E), N 2 countable, Nl <~N2, a inaccessible, -> Ie, I for i

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(N2, Pi)-semi-generic where i = Min(a ¢q N2 - N~) is strongly inaccessible. (2) Any S-suitable iteration Q is a semi-proper iteration and [t~ = 1 =* ~[P~] I S ,~ ~8[Pj] l S] when: j > i, i successor, j successor or strongly inaccessible. (3) If (in (1)) x < a is strongly inaccessible, levi< x for i < x, and IF--e, "Rss(R2)" then Q~ (and P/P~ when x

PROOF. Left tO the reader. For instance: (1)(c) If/~ N2 is a maximal antichain ofP~, then by [Sh 1] X 5.3(3) for some j < i, I __. Pj, hence there is suchj in N2, hencej ~N~ and also the rest is easy. (3) By 1.7(3) it is enough to prove that forcing with Q~ does not destroy the stationarity of any A c_ col, A ~ V e,. However, by 2.2(1)(c) (and 2.2(2)) for some//< a,A EVe,. CleaflyA E Ve, and is a stationary subset of cot in Ves+,. As P~ +~/Pp +1 is semi-proper, A is stationary also in (VPP+')e'~+'le'+l = V P~+, ~- (VP, c)q ,~ as required.

2.2A. REMARK. So if x is strongly inaccessible, and IP~ I < x for i < x, then ifA is a stationary subset of COl in V e. then A is a stationary subset ofco~ in Ve. for every large enough a < x.

2.3. CLAIM. Suppose Q = (Pj, Qi :J <= a, i IPi I for i < a, and a is strongly inaccessible, then ~a = ~e.

PROOF. (1) For (D) use the semi-proper iteration lemma. The others are obvious too. (2), (3), (4) Easy, too.

2.4. DEFINITION. Let ~[ = (92¢: ~ <~) be a sequence of subalgebras of ~( = ~v), S C_ col stationary. (1) Sm(~, S) = {A _C S: for some ~ < ~, {x ~ 9/¢ : 92~ ~ x ~ 0, x N A = Sh:253

360 s. SHELAH Isr. J. Math.

modDo,,} is predense in PI¢} (we should have written x/D,o,~9@ x C O)l}. (2) We define the sealing forcing Seal (~, S) = {e : e a partial function from Sm(~l, S), with countable domain, and for A E Sm(~, S), ifA = N then G is a function from some u < o9~ to 2 ~,, and if A ~ ~ then dA is a continuously inceasing function from some countable 7 + 1 to o)t - A }, the ordering is defined by:

(3) If~ = (~) we write in (1), (2) above ~ instead of~. (4) We define, for x strongly inaccessible ( > R0), the strong sealing forcing SSeal(~I, S, x) as P~, where ( Pi, Qj" i < x, j < x) is an RCS iteration, with Qj = Seal(~I, S)e,, (5) For I c ~v let seal(I) = {(ai: i < a) :ai~S<~,(H((2~,)+)), ai (i < a) is increasing continuous and a~ N o9, is an ordinal which belongs to I,.J aetna, A } and is order by the inverse of being an initial segment. (6) We call I ___ ~v semi-proper if seal(I) is a semi-proper forcing notion. (7) WSeaI(S) is the product, with countable support, of seal(l), I semi- proper, O~l - S E I. (8) We define, for x not strongly inaccessible, but (,) (V/t < x)[#~0< x], x = cfx, xl~d = xfor ~ <~, and ~ -<_ x, x > Rl the strong sealing forcing SSeal(~/, S, x) as P~ where (P~, Qj" i < xj < x) is an RCS iteration; Qj = seal(/~, S) v'~, L is a maximal antichain of 9~w) for some (U) < ~ (in Vr) and every maximal antichain / of some ~¢ from V e. is 6 for some j < to. We call x ~-inaccessible if it satisfies (,) above, and call it R0-inaccessible if (V/t < x)(p~0< x = cfx). (9) If I _ {I: I c_ ~} then seal(I) is the product, with countable support of seal(l) for I E I.

2.5. REMARKS. (1) We could have used CS iteration for SSeal. (2) If every maximal antichain of~ v is semi-proper, the difference between WSeal(S) • Levy(R, 2~,) and Seal(~ v, S) (defined in 2.4 (7), (2), respec- tively) is nominal. (3) If 92¢ tsar8 v FS/br ~ <~, then Seal(~, S) is equivalent to the Levy collapse of 2 ~, to R, by countable conditions. Sh:253

Vol. 60, 1987 ITERATED FORCING 361