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Thomas Jech and Saharon Shelah 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 .
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