ON REFLECTION OF STATIONARYSETSINP Thomas Jech and Saharon Shelah Department of Mathematics The Pennsylvania State University UniversityPark, PA 16802 Institute of Mathematics The Hebrew University Jerusalem, Israel + Abstract. Let b e an inaccessible cardinal, and let E = fx 2P :cf = x 0 + + cf g and E = fx 2P : is regular and = g. It is consistentthatthe x x x 1 x set E is stationary and that every stationary subset of E re ects at almost every 1 0 a 2 E . 1 Supp orted by NSF grants DMS-9401275 and DMS 97-04477. Typ eset by A S-T X M E 1 2 THOMAS JECH AND SAHARON SHELAH 1. Intro duction. We study re ection prop erties of stationary sets in the space P where is an inaccessible cardinal. Let b e a regular uncountable cardinal, and let A . The set P A consists of all x A such that jxj <.Following [3], a set C P A is closed unbounded if it is -co nal and closed under unions of chains of length <; S P A is stationary if it has nonemptyintersection with every closed unb ounded set. Closed unb ounded sets generate a normal -complete lter, and we use the phrase \almost all x" to mean all x 2P A except for a nonstationary set. Almost all x 2P A have the prop erty that x \ is an ordinal. Throughout this pap er we consider only such x's, and denote x \ = .If is inaccessible x then for almost all x, is a limit cardinal and we consider only such x's. By [5], x the closed unb ounded lter on P A is generated by the sets C = fx : x \ 2 F <! <! and F x xg where F ranges over functions F : A ! A. It follows that a set S P A is stationary if and only if every mo del M with universe A has a submo del N such that jN j <, N \ 2 and N \ A 2 S . In most applications, A is identi ed with jAj, and so we consider P where is a cardinal, >.For x 2P we denote the order typ e of x. x We are concerned with re ection of stationary sets. Re ection prop erties of stationary sets of ordinals have b een extensively studied, starting with [7]. So have b een re ection principles for stationary sets in P , following [2]. In this pap er we ! 1 concentrate on P where is inaccessible. De nition. Let b e an inaccessible and let a 2P be such that is a regular a uncountable cardinal. A stationary set S P re ects at a if the set S \P a is a a. a stationary set in P a The question underlying our investigation is to what extent can stationary sets re ect. There are some limitations asso ciated with co nalities. For instance, let S and T b e stationary subsets of such that every 2 S has co nality ! ,every ON REFLECTION OF STATIONARYSETSINP 3 2 T has co nality ! , and for each 2 T , S \ is a nonstationary subset of 1 b b b cf. [4]. Let S = fx 2P : sup x 2 S g and T = fa 2P : sup a 2 T g. Then S b do es not re ect at any a 2 T . + Let us consider the case when = . As the example presented ab ove indicates, re ection will generally fail when dealing with the x's for whichcf < , and so x x we restrict ourselves to the stationary set fx 2P :cf cf g x x + + Since = ,wehave foralmostallx. x x Let + E = fx 2P : is a limit cardinal and cf =cf g ; 0 x x x + + E = fx 2P : is inaccessible and = g : 1 x x x + The set E is stationary,andif is a large cardinal e.g. -sup ercompact then 0 E is stationary; the statement\E is stationary" is itself a large cardinal prop erty 1 1 cf. [1]. Moreover, E re ects at almost every a 2 E and consequently, re ection 0 1 of stationary subsets of E at elements of E is a prototyp e of the phenomena we 0 1 prop ose to investigate. Belowweprove the following theorem: 1.2. Theorem. Let beasupercompact cardinal. There is a generic extension in which + + a the set E = fx 2P : is inaccessible and = g is stationary, 1 x x x and b for every stationary set S E , the set fa 2 E : S \P a is nonstationary 0 1 a in P ag is nonstationary. a A large cardinal assumption in Theorem 1.2 is necessary.Asmentioned ab ove, a itself has large cardinal consequences. Moreover, b implies re ection of sta- + :cf <g, whichisalsoknown to b e strong tionary subsets of the set f < consistency-wise. 4 THOMAS JECH AND SAHARON SHELAH 2. Preliminaries. We shall rst state several results that we shall use in the pro of of Theorem 1.2. We b egin with a theorem of Laver that shows that sup ercompact cardinals have a }-like prop erty: 2.1. Theorem. [6] If is supercompa ct then there is a function f : ! V such that for every x there exists an elementary embedding j : V ! M with critical point such that j witnesses a prescribeddegreeofsupercompactness and j f =x. Wesay that the function f has Laver's property. 2.2. De nition. A forcing notion is <-strategical ly closed if for every condition p, player I has a winning strategy in the following game of length : Players I and I I take turns to play a descending -sequence of conditions p >p > >p > 0 1 ; < , with p>p , such that player I moves at limit stages. Player I wins if 0 for eachlimit<, the sequence fp g hasalower b ound. < It is well known that forcing with a <-strategically closed notion of forcing do es not add new sequences of length <, and that every iteration, with <-supp ort, of <-strategically closed forcing notions is <-strategically closed. + 2.3. De nition. [8] A forcing notion satis es the <-strategic- -chain condi- tion if for every limit ordinal <,player I has a winning strategy in the following game of length : + Players I and I I take turns to play,simultaneously for each < of co nality >p > >p > ; < , with , descending -sequences of conditions p 0 1 player I I moving rst and player I moving at limit stages. In addition, player I + cho oses, at stage , a closed unb ounded set E and a function f such that + for each < of co nality , f < . : < i has a lower Player I wins if for each limit <,each sequence hp T b ound, and if the following holds: for all ; 2 E ,iff =f for all < ON REFLECTION OF STATIONARYSETSINP 5 : <i and hp : <i have a common lower <, then the sequences hp b ound. + It is clear that prop erty 2.3 implies the -chain condition. Every iteration + with <-supp ort, of <-strategically -c.c. forcing notions satis es the <- + strategic -chain condition. This is stated in [8] and a detailed pro of will app ear in [9]. In Lemmas 2.4 and 2.5 b elow, H denotes the set of all sets hereditarily of cardinality <. 2.4. Lemma. Let S be a stationary subset of E .For every set u there exist a 0 + regular > , an elementary submodel N of hH ; 2; ;ui where is a wel l + ordering of H such that N \ 2 S , and a sequence hN : <i of submodels S + + N \ and for al l <, of N such that jN j < for every , N \ = < hN : < i2N . + + Proof. Let > b e such that u 2 H , and let =2 ; letbeawell or- dering of H . There exists an elementary submo del N of hH ; 2; containing + u, S and hH ; 2; H i such that N \ 2 S and N \ is a strong limit + cardinal; let a = N \ . Let =cf . As a 2 S ,wehave cf sup a= , and let , <,bean a + increasing sequence of ordinals in a , co nal in sup a.Lethf : < i2N b e such that each f is a one-to-one function of onto . Thus for each 2 a, f maps a \ onto . There exists an increasing sequence , <, of ordinals a co nal in ,such that for each < , f < . a For each <,letN be the Skolem hull of [f g in hH ; 2; H ; hf ii. N is an elementary submo del of H of cardinality <, and a N 2 N . Also, if < then 2 N b ecause f < andsoN N . As N \ is a strong limit cardinal, it follows that for all <, hN : < i2N . 6 THOMAS JECH AND SAHARON SHELAH S Also, N N for all < , and it remains to prove that a N . < S As supf : <g = ,wehave N . If 2 a, there exists a a a < < < such that < and f < .Then 2 N and so 2 N . 2.5. Lemma. Let S be a stationary subset of E and let P bea<-strategical ly 0 P closed notion of forcing. Then S remains stationary in V . + _ Proof. Let C be a P -name for a club set in P , and let p 2 P .We lo ok for a 0 _ p p that forces S \ C 6= ;. 0 Let b e a winning strategy for I in the game 2.2. By Lemma 2.4 there exist a + _ regular > ,anelementary submo del N of hH ;;;P;p ;;S;C i where 0 + is a well-ordering suchthat jN j < and N \ 2 S , and a sequence hN : <i S + + of submo dels of N such that jN j < for every , N \ = N \ and < for all <, hN : < i2N .