Homogeneous sequences of cardinals for ordinal definable partition relations Citation Kafkoulis, George (1990) Homogeneous sequences of cardinals for ordinal definable partition relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hchw-ca34. https://resolver.caltech.edu/CaltechETD:etd-06132007-073731 Abstract Error parsing XML in render_xhtml_field: :1: parser error : xmlParseEntityRef: no name <br />In this dissertation we study the consistency strength of the theory ZFC & ^ :1: parser error : xmlParseEntityRef: no name ] + [...] strong limit + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & ^ :1: parser error : xmlParseEntityRef: no name t + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & [...] inaccessible & ^ Item Type: Thesis (Dissertation (Ph.D.)) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Kechris, Alexander S. (advisor) Woodin, W. Hugh (advisor) Thesis Committee: Unknown, Unknown Defense Date: 16 May 1990 Record Number: CaltechETD:etd-06132007-073731 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-06132007-073731 DOI: 10.7907/hchw-ca34 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2572 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 03 Jul 2007 Last Modified: 16 Apr 2021 22:32 Thesis Files Preview PDF (Kafkoulis_g_1990.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

HOMOGENEOUS SEQUENCES OF CARDINALS FOR ORDINAL DEFINABLE PARTITION RELATIONS Thesis by George Kafkoulis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1990 (Submitted May 16, 1990) ii Acknowledgements | would firstly like to thank Professor Leonard Lipshitz at Purdue University who was my first teacher of Mathematical Logic and Professor Jean E. Rubin for introducing me to axiomatic set theory. Also, | would like to thank my fellow graduate students at Caltech, David De Latte and Tan-Chu Shih, for many interesting conversations and especially Carlos Uzcategui for his friendship and all the exciting discussions on Descriptive Set Theory. Two people at Caltech influenced me the most. | am indebted to Professor Alexander Kechris and to my thesis advisor Professor W. Hugh Woodin. Professor Kechris showed me, through his teaching, the beauty of Set Theory. Professor Woodin guided me in the solution of my thesis problem and was patient enough to answer all of my questions. | am privileged to be a student of these two fantastic mathematicians. | am grateful to the Caltech Mathematics Department for the financial sup- port, the NSF Grant DMS-8718847 for partial support and the Caltech Campus Computing Organization for giving me full access to their mighty computers and thus helping me typeset this thesis. | also would like to thank my parents, Aristomenis and Penelope Kafkouli, for waiting stoically for my return to Athens. The final remarks are reserved for my love, Kiriake Xerohemona Kafkoulis. She was my invaluable companion during these six years away from Greece. | dedicate this thesis to her. i if I | ili Contents Acknowledgements 2.0.0.0... ccc ence cece tence eect en te nee enseeeeneeues ii FO 6 Vv NOtGAHON. cence cence een e teen eee tent ee eeseutennens vii Introduction and Statement of Results..........00 000 cece e cece eee e eens viii 1 OD partitions in a Prikry model 1 1.A Prikry forcings: Basic facts................0000, 1 1.B Prikry generic sequences and their models ............ 4 2 The inner Model V(H) 31 2.A The power set ofkin V(H) ............. 0200004 31 2.B The Prikry-generic sequences in V(H)............... 49 3 The partition relation in V(H) 67 3.4 The way from V(H) to VIG] ..............2205., 67 3.B From V(H) to V[G]: Another Forcing .............. 86 3.C OD partitions in the model V(H) ................. 94 iv 4 The partition relation in a ZFC model 101 4.A <A-—DC inthe inner model V(H) ................ 101 4.B A ZFC model of the partition relation .............. 120 | Abstract In this dissertation we study the consistency strength of the theory ZFC & (dk strong limit)(Vu < K)(K ob (w)¥,) (*), and we prove the con- sistency of this theory relative to the consistency of the existence of a su- percompact cardinal and an inaccessible above it. If U is a normal measure on P,(A), then Py denotes the Supercompact Prikry forcing induced by U. K oF (w)¥, is the partition relation kK > (wy, except that we consider only OD colorings of [k]*. Theorems 1,2 are the main results of our thesis. Theorem 1. If there exists a model of ZFC in which x is a supercompact cardinal and A is an innaccessible above k, then we can construct a model V of the same properties with the additional property that if U is a normal P,.(A)-measure and G is Py — generic over V, then V[G] does not satisfy the (Vu < k) (« oD (o)%, partition property. Oo If G is a Py-generic over V filter, then we define H to be the set H: ULP(«) NV{Gf a] | a< at, and we consider the inner model V(H), which is the smallest inner model of ZF that contains H as an element. We prove that V(H) satisfies the above partition property (+). Moreover, V(H) satisfies < 4 — DC and using this fact we define a forcing P, which is almost-homogeneous, < — closed forcing that forces the AC over V(H) and does not add any new sets of rank < k. Theorem 2. If G is P -generic over V(H) and V,:% V(H)(G], then Vi — [ZFC + « strong limit + Vu < K(k ob (w)¥,)]. Therefore i t vi Con(ZFC + (AK, A)[K supercompact & 2 inaccessible & « < ]) > Con(ZFC + (3x strong limit)(Vp < K)(x ob (WX, )). 0 So vii Notation Our Set Theoretic notation follows that of [Jech 78] and [Kunen 80]. ZFC denotes the theory of Zermelo Fraenkel Set Theory with the Axiom of Choice. Ordinals and cardinals are denoted by the small letters of the Greek alphabet. OD denotes the class of all ordinal definable sets. If X is a set, then P(X) denotes the power set of X. If k, are cardinals, then P,(A) is the set of all subsets of A of cardinality < x. If a is an ordinal, then V, denotes the set of all sets of rank < a and for any class M, VM is the relativisation of V, to M. [k]* denotes the set of all subsets of « of order-type a which is identified with the set of all (strictly) increasing a — sequences from «. For the forcing terminology we also follow [Jech 78] and [Kunen 80]. Given a forcing P, I denotes the canonical P — name for the P — generic filter. The notation x of the canonical P — name for an element x of the ground model is supressed. We talk of generic filters as if they exist since all those forcing arguments can be easily translated to rigorous arguments about countable transitive models of finite fragments of ZFC. viii Introduction and Statement of Results In this dissertation we prove that an upper bound of the consistency strength of the theory (1) ZFC & (3k strong limit)(Vu < x) (« x (wy, } is the existence of a supercompact cardinal and an inaccessible above it. « ob (w)%, is the partition relation Kk — (w)¥, except that we consider only OD colorings of [x]”. This answers an open question proposed by Professor W. Mitchell. The theorem is proven after the series of the following results: Let U be ao—complete ultrafilter on a cardinal 6 and P%, be the Prikry-tree forcing induced by YU on 6. A condition of this forcing consists of a UU — splitting tree on P,,(A) and a stem which is a node of this tree. The partial order is as in the Prikry forcing. First, we prove that if o and 7 are two P7, — generic over V sequences with the property V[o] = V[r], then they are tail — equivalent. (1.B.9) Using the above fact about the Prikry-tree forcing (i.e., 1.B.9), we prove the following: If U is a normal measure on P,,(A) and if o and 7 are two Py — generic over V sequences with the property V[o] = V[z], then they are tail — equivalent. (1.B.10) 1.B.11 Theorem. We assume that V is a transitive model of ZFC in ix which « is a supercompact cardinal and A is an inaccessible above x. Also, U is a normal P(A) — measure and Py is the Supercompact Prikry forcing associated with U. If G is Py — generic over V, then there exists in V[G] a coloring F of []* into V, colors for some pz: < k, with the property that F is ordinal definable over V[G] with parameters from V and has no homogeneous set, i.e., V[G] FE a(Vu < k) (« vob (WX, ). Oo 1.B.13 Theorem. If V’ is a transitive model of ZFC in which k is a supercompact cardinal and A is an inaccessible above «, then there exists a generic extension V of V’ in which « remains supercompact, \ remains inaccessible and the following holds: If U is a normal P,(A) — measure, Py is the Supercompact Prikry forcing associated with U and G is Py — generic over V, then a sufficient large rank initial segment of the ground model V is OD over the Prikry generic extension V[G] and V[G] - 7>(Vp < x) (« ob (wi, ). Oo The Theorems 1.B.11 and 1.B.13 suggest that a Supercompact Prikry generic extension of the Universe is not the right model for the theory (9). Instead, we consider an inner model of such generic extension. If G is a Py — generic over V filter, then we define H® to be the following set: HO: U{ P(r) NV[Gha] | a < at, and we consider the inner model V(H), for H: © HS, which is the smallest inner model of ZF that contains H as an element. (2.A.1) Then we prove that H = P(k)M V(H). Moreover V(H) G V[G] and V(H) FA=«T. (2.4.16) The next two results show that there exists a plethora of Prikry generic se- quences of height < X in V(H). We let Py be the P,,(3) — Supercompact Prikry forcing with respect to U/ and K <8 <1. We take QC.r.o.(Py) and g € V(H) be a Q — generic over V filter. We prove that there exists in V(H) a K which is Py — generic over V such that KN Q = g. (2.B.7) We suppose that (z;A) € Py and that x € V(H) isa Py, —generic sequence, with the property that a < @ and z}a Cx C zZf}aUAfa. Then, there exists ay € V(H) such that y is Py, — generic over V , 2 C y CZUA and yfa =x. (2.B.8) We define, in V(H), a forcing Q with the following property: If G* is a Q -generic filter over V(H) and G* is the Py — generic over V filter induced by G*, then H = H®". (3.A.1, 3.A.8) The abundance of generic sequences in V(H) is the key factor in the proof of the latest property of Q. This result is used, in the proof of the partition relation (J) inside the model V(H). (3.C.1) 3.C.1 Theorem. We assume that ¢ is a formula and py < « so that V(H) & Vs € [k]”A!x € V, G(s, x, 6, H). Then, there exists an s € [k]* M V(H) and an xp € V, such that V(H) — Vt € (s]°4(t, x0, 5, H). In particular, V(H) — (« is astrong limit) & Vu<k(k ob (wy). Oo Assume that b € H. For every formula ¢ and parameters a € V, there iN ii Hf s xi exists a formula ¢* such that V(H) & ¢(a, b) <=> V[b] & ¢*(a, U,«, A, b). The latest (3.C.3) is used in the proof of < \— DC inside the inner model The model V(H) satisfies the partition relation (() but unfortunately may not satisfy the Axiom of Choice. Thus we want to force Choice over V(H) and still preserve the partition relation. We force AC over V(H) with < — sequences of H as conditions and we show that this forcing, say it P, is homogeneous and < \ — closed. < 1 — DC helps in the proof that P does not add generically any new < Kk — sequences of k. (4.4.11) 4.B.2 Theorem. Let V be a transitive model of ZFC, in which « is a supercompact cardinal and A is an inaccessible cardinal above x. Let G be a fixed P — generic filter over V(H). If Vi: V(H)[G], then Vi & [ZFC +« is a strong limit + Vu < K(K ob (w)¥, DI. Oo Chapter 1 OD partitions in a Prikry model 1.A Prikry forcings: Basic facts We begin with some basic facts about the Supercompact Prikry forcing and the Prikry-tree forcing. If U is a normal measure on P,(A), then Py denotes the Supercompact Prikry forcing induced by U. The version of this forcing which we use is as follows: _ 1.A.1 Definition If U is a normal measure over P,(), then the supercom- pact Prikry forcing induced by U is denoted by Py and (P1,...,PajA) € Py t PiC,.5CPrEP (A) & ACU & & (vPEA\(P, CP), | where Q C P means that Q ¢ P & |Q|<|PNMx&|. The sequence (P),...,Pn) is called the stem of the condition (P}, ...,PyjA). 2 Chapter 1. OD partitions in a Prikry model 1.A.1 Definition 0 As in the Prikry forcing on a measurable cardinal, it has been proven that Py satisfies the so called Prikry property, i.e., for any condition and any for- mula of the forcing language we can shrink the measure one set, leave the stem unchanged and with the new stronger condition we can decide the formula. In addition, as in the Prikry forcing, Py has the so called geometric property, i.e., a sequence of P,,(A) — sets is Py — generic iff eventually belongs to every U-measure one set. The nontrivial aspect of this result and its corollary-that every subsequence of a Py — generic sequence is Py — generic—has been proven in [Mathias 73], for the Prikry forcing. However, the same arguments apply for the Supercompact Prikry forcing Py. 1.A.2 Definition IfU/ is a o — complete ultrafilter on a cardinal 6, then the conditions of Pj, (the Prikry-tree forcing induced by U on 6) are of the form (s;T) for some U-splitting tree on 6 and s € T, and the order is defined as follows: (s:T) < (s:T'): S$ s'Cs & TCT. 1.A.2 Definition O The Prikry-tree forcing PZ, satisfies the Prikry property and a variation of the geometric property, which is the following: for every Pj, — generic 1.A. Prikry forcings: Basic facts 3 sequence g (g € “6), and for every F : <“6 + U, in V, there exists a natural number k € w such that (VN > k)(VM > N)g(M) € F(g[N). These results belong to set-theoretic folklore. inna SS 4 Chapter 1. OD partitions in a Prikry model 1.B Prikry generic sequences and their models In this chapter, we are going to show it is consistent that for any G Py —generic filter over V, the model V[G] does not satisfy the partition relation. Towards this result, first we prove that any two Py — generic sequences o and 7 with the property V[a] = V[r] are tail — equivalent. All the arguments that we are going to use do not depend on the fact that U is a normal measure on P(A). Hence we are going to show the above result for the Prikry tree-forcing induced by a o — complete ultrafilter U/ on a cardinal 6. We start with a o — complete ultrafilter U/ on a cardinal 6. The Prikry-tree forcing induced by U on 6 is denoted by Pj, and is defined as follows: 1.B.1 Definition A set T is called a U — splitting tree on 6 iff T C 6<” and (VsET)(Vt)(tcsSteT) & (WseE T)({a E6|stae T} EU). The conditions of Pj, are of the form (s;T) for some U — splitting tree T on 6 and s € T. In addition if < denotes the P7, — order, then (s;T) < (s';T’): €$ s! C s & TCT". As usual, for s,s’ € 6<”, s’ C s means that there exists some k < lh(s) such that s' = sf k. 1.B.1 Definition 0 1.B.2 Definition Two sequences x,y € [k]* are said to be tail — equivalent iff for some m’,n' € w and for allk € w, x(n’ +k) = y(m’ +k). 1.B.2 Definition O i i | I ! 1.B. Prikry generic sequences and their models 5 The first result of this thesis is the following theorem: Theorem A: Ifo and 7 are two Pi, — generic sequences with the property that V[o] = V[r], then they are tail — equivalent. Oo This theorem is proven in 1.B.9. For the following results and throughout this part we fix two Pj, — generic over V sequences o and 7 which satisfy the property V[o] = V[7]. 1.B.3 Lemma (H. Woodin) There exist two reals a and ( (in the ground model V) and two sequences (f, | n € w) and (g, | n € w) of functions in V such that f, : [6]°™ — [5]" & gy: [6]9™ — [6]" with the property that f,(of a(n)) = 7fn and g,(7f B(n)) = of n. Proof: We pick a Pj, — name 7 so that for some kg € w and some T* which is a U-splitting tree on 6, (of ko;T*) belongs to the P¥, — generic filter G induced by o and r=(7)%) & (ofko;T*) | [7 is a Px, — generic sequence]. Pi, Let us define in V[G] a real a € [w]” as follows: a(n) =m - €% m is the least integer > kg such that AT |(of m;T) (Wk < n)(7(k) = (7(k)y )]. u Notice that a is well defined, since (7)Vlel =r & (ofko;T*) |E [7:0 > 6 is a P¥, — generic over Vv] Pi, 6 Chapter 1. OD partitions in a Prikry model hold. Moreover, as V and V[G] have the same reals, then a € V. Next for every n € w we define functions f, : [6]°() — [6]" as follows: Let (of I*;S*) € G such that (of 1*;S*) < (of ko;T*) and (ot ss") — [yaa T[(tt a(n):2) I (vk < a) (C700) = (FIT) i Piy Then, for every ¥ € [6]°), we define {% if (of 1°3S*) |-— [AT [(&;T) |E— Vk < n(#(k) = (F(K) I] f,(X) = Ps Ps (0, if otherwise. Each f, is well defined and by the definability of the forcing relation PZ, in V we have that each f, isin V. Moreover, (f, | n € w) € V. By the choice of a we conclude that (Vn <w)[fi(ofa(n)) =7fn] & (Vx)[fn4i(x)[n = fr(xf a(n))). The finding of the real a and of the sequence (f, | n € w) in V with the above properties consists of the analysis of the term 7 in V[o]. Similarly, if we find a P}, — name @ so that o = (o)¥"l and we interchange the roles of o and 7 in the previous argument, then we will be able to find another real @ (in particular @ € [w]”) and another sequence (g, |n € w) € V of functions such that g, : [6}%™ — [6]" with the property that g,(rf G@(n)) = ofn & (Vx)[gn4i(x)fn = gn(xf B(n))]. 1.B.3 Lemma O 1.B.4 Proposition Suppose that o and 7 are not tail — equivalent. There exist an n* € w, a tree T*, a Pf, — name #, two reals @, 3 and two sequences 1.B. Prikry generic sequences and their models 7 (Fr | n € w),(G, | n € w) with the following properties: For all n € w, F, : [6J@™ — [6]" & Gy : [6/8 = [5]" and (0;T*) I [o;, P are not tail — equivalent] & & (Vn <w)[Fi(Gai(af B(G(n))))) = etn & (4) & Ga(Fam)(sel &(G(n))))) = ophn & & F, (of @(n)) = pfn & G, (pf B(n)) = apf a}. In addition, for every P*, — generic K which contains (0;T*) and for s:@ of n* we have that (p)¥'**] = (7)Vex), Proof: We start with the sequences (f,, 2, | n € w) found in 1.B.3. Then we find a condition (s;T) € G, where as before G is the Pj, — generic filter induced by a, with the property (s;T) |F- — [o%, 7 are not tail — equivalent] & Piy & (vn <w)[f(opfo(n))=7fn & galt} B(n)) = spf n). Let n*:% Ih(s) and (T), be the part of the tree T below the node s, i.e., (T),: 2 {' € 6“ |ste th. For all n < w we define a(n): @ a(n* +n) —n*, B(n): B(n* +n) and for all x € [6]®) N(T),, y € [68 N(T), Fa(x): faean(87x)fn, and G,(y):© gnean(y)f [n*,n* +n). Let K be any Px, — generic filter such that (0;(T),) € K. Then s~ox is a P¥, — generic sequence, and if K* denotes the corresponding P7, — generic filter, then we have that (s;T) € Kk". 8 Chapter 1. OD partitions in a Prikry model Next we observe that (0;(T),) I-— [(ax)(x = (7)VEel)], where T is the canonical name for the PZ, — generic fiter and dg; is the canonical name for the generic sequence induced by I’. Then the set A = {p € Py, | (37)(p I v= i (Ve) )} is P7, —dense below (0;(T),). Let Z be a maximal antichain below (0;(T),) included in A. For every p € Z choose a 7 such that p av [> = (+) el)]. Let p be the Py, — name which is the mixture of {7, | P € T}. Since K is a PZ, — generic filter which contains the condition (;(T),), then we have that (6)¥'*] = (7,)VIKl, whenever p € ZMK. Moreover, s~ox is a Pz, — generic sequence, and if K* denotes the corresponding Py, — generic filter, then we have that (s;T) € K*. By the definition of p, we conclude that for the unique p € NK, (p)VKI = (7,) VK = (7)VO ee! = (7) Vl = 7, Since (6:1) [LK [(Va < w)(fa(d¢ta(a)) = Fn & ga(At An) = oytn)] holds, then for every n<w F((op)¥™Pa(n)) = fyean(s7(6p) If a(n" +n) = n*)P a = fyea((80K)f a(n" + n))f 0 = (7)¥e «lp n* +nfn = (AVE) Pn = (iV, Thus (0;T*) (Vn < w)[F,(6;[ @(n)) = pf n]. Similarly, for all n < w we i have that Gu((A)""P Bn) = gneen((7)Y™P Bn" + n))f fn" n* +n) 1.B. Prikry generic sequences and their models 9 = ((s*ox)[n* + n)f [n", n* +n) = (6¢)*™P. Thus (0;T*) [o;, 6 are not tail — equivalent] & "— Fa(dpha(a)) = Ata & Galt n(n) = sph This implies ({). Now the proposition follows if we set T* = (T),. 1.B.4 Proposition 0 The notation and the assertions of 1.B.4 are used in the next proposition. 1.B.5 Proposition IfU/* denotes the (finite) k-th iterate of the ultrafilter U, then (dan) €w)(Vn >n1) (Vk Ew)(VA €U*)(SB € Ut) (Vt € B)[Fazn(t)f[nyn+k)e A] & (Ang €w)(Vn >ne) (Vk Ew)(VA €U*)(SB € 8+) (Vt € B)[Gnix(t)f [n,n +k) € Al. Proof: We assume towards a contradiction that such n,; does not exist. Then we choose an infinite sequence (nj, k;, Aj, B; | i € w) such that for allie w 1. ny + k; < Di+1; 10 Chapter 1. OD partitions in a Prikry model 2. Ai € Ui, 3. B; € Ucn tki) 4. (vt € Bi)[Faites(t)f [mi, ni + ki) ¢ Ai]. Using the (B; | i € w) we construct a U — splitting tree over 6, T* such that T* c T* and (V1 € w)(Vt € T*)(Ih(t) = &@(n) +k) > t € B)). Next, we consider the set 5: {p € Piy | i)( I~ et bum +i) € Ad} iu and we show that S is Pj, — dense. Towards this end, we take (u;T) € Py and we assume that lh(u) = mo. Then, we pick i € w such that nj > mg and we shrink appropriately the tree T to another tree T’ with the properties that (Vt € T’)(Ih(t) > nj +k; => tf [m, nj + k;) € Aj) and (u;T’) < (u;T). Obviously (u;T") € S, since for every generic K containing (u;T’), the induced generic sequence ox is a branch through T’. Hence, S is indeed Py, — dense. If K is any Py, — generic filter that contains (@;T*), then (f)"*! is a P¥, — generic sequence and so the filter induced by (p)¥'*] meets the set S. Thus there exists an i € w such that (@)¥/KIf [n;, nj + k;) € Aj. In addition the choice of K implies that F,(oxf @(n; + kj))f {ni,nj + kj) ¢ Aj. Since (@;T’) < (O;T*), then F,, (ox @(nj + ki)) = (6)Y* fn; +k; and consequently (6)¥ I} [n,,nj +k) ¢ A;. But the latest is a contradiction. Similarly, we argue and prove the existence of ng. 1.B.5 Proposition O 1.B. Prikry generic sequences and their models 11 Let us fix the two natural numbers nj,n2 given by 1.B.5 and let us take def mo: 1+max{n;,n9}. Then, we consider 6:2 o*f mow) & 7:2 rf [mo,). Since V[o*] = V[r], then V[a] = V[7]. 1.B.6 Lemma Suppose that the two generic sequences 0,7 are not tail — equivalent. Then there exist an’ € w, aU — splitting tree T”, two sequences of functions (f, | n € w), (gn | n € [n',w)), two reals 7,7 and a Py, — name fo which have the following properties: (6;T”) I [o;, Po are not tail — equivalent] & u & (Wn <w)|fa(gymy(Pol n(y(n))) = potn & — (de) & gal(fny(Gel y(n(n))))) = op & & £(epha(a)) = pola & ga(fol n(n) = sel a]. In addition, for every P%, — generic K which contains (0;T”) we have that rh mg (fg) VIR) = (p)Vrtme ol, fi(oPy(n))=7[n & ga(7fn(n)) = apn, and for every k € w the following hold: (Acut= {yee |AG)eayeu™), (1) (a EUk = fy €5™ | gly) EA} un). (tt) 12 Chapter 1. OD partitions in a Prikry model Proof: We are using the techniques of 1.B.4 for (0;T*), o*,6,@,8,mo and (F,,G, | n <w) as follows: let n’:“ mg and T’: & (T*)e*tm, be the subtree of ef T* below o*f mp; for n € w, 7(n): © &(n-+mo)—mg and n(n): B(n+mo)—mo; we also define for x € [6"(™ NT”, y € [6]"™ NT” that f,, (x): def Fing+n(o” | mo ~x)f [mo, mo + n) nly): Gino+n(7f mo ~y)f mo, mo + n). In addition, we construct a name fp as an appropriate mixture in a way anal- ogous to the construction of p in 1.B.4, so that for every Pz, — generic K which contains (0:T”) we have that rf mo ~(jp)V(@*! = (p) Viet mo"exl, If K is a Pi, — generic filter which contains (9;T”), then K* denotes the P#, — generic induced by the generic sequence o*{ mp “ox. Thus £.((o¢)¥ "IP y(m)) = Frno4n(o*f mo ~(oxl &(mo +n) — mo))f [mo, mo + n) = Frptn(0K-| @(mo + n))f [mo, mo + n) - ((p) Vet o"exIr mo + n)f [mo, Mo + n) = (A)VieTmo eI fimo, mo + n) = (in) VP Similarly, for alln < w ex(— (fo)¥"Itn(n)) = | = Gmo+n(tf mo ~(fo) "ft B(me + n) — mo)f [mo, mo + 2) = Ging-+n((p) Yt 7*)F B(my + n))f [mo, mo + n) 1.B. Prikry generic sequences and their models 13 = ((a*f mo ~ox)f mo + n)f [mo, mo + n) = (6;)¥ pn. Hence, ga((Go)¥!f n(n)) = (6;) Vl n. Now it is straightforward to show that (fn; Gn | n € w) satisfy (de). Using the trees T* (as in 1.B.4) and T” we construct another U/ — splitting tree over 6, which we denote by T**, such that 1. T** c T* and 2. (vtE T*)(Ih(t) > mo > t} [mo, Ih(t)) € T"). Then, it is easy to show that (0;T™) (Vn € w)[Frnotn(Opf @(mo +n)) = pfmotn & & Ginotn(Pf B(mo+n))=6;fmo tn & & f,(opf [mo0,mo + 7(n))) = Af [mo,mo +n) & & gn(of [mo, mo + n(n))) = apf [mo, mo + n)). Therefore, (H:T™) [= (Wn € w)[Fnotn( Spl An + mo))f [ano,n + m9) = = f,(def [momo +7(n))) & & Gmo4n(At B(mno + n))f fimo, n + mo) = = gn(f [mo, mp + n(n)))]. 14 Chapter 1. OD partitions in a Prikry model Next, we observe that (Vn € w)(Vk <1 < n)(VA)(A € UW" = {xf [k, 1) € &-* | x € A} €U'*). Using the above fact we conclude that for all k < w, A €uks> 7 (s E 6atmo+h) | F4(x)f [mo, mo +k) € a} E U4a(mo+k) => {x € 62(mo+k) | £ (xf [mo, mo + 7(k))) € a} & Yalmotk) => {x1 [mo, mp + 7(k)) € 6%) | £.(xf [mo, mo + 7(k))) € at ey) > iy eh | f(y) € a} € Ur), Thus (Vk € w)(A €uk iy € 6”) | f(y) € a} € Ux). Similarly, we prove that (Vk € w)(A €EU* => iy € 6") | p(y) € a} Ee Un), Moreover, we observe that on T” the sequence (f, | n < w) is coherent in the following sence: For all x € T”, fy41(x)[n = f,(xfy(n)) and in a similar fashion the sequence (g, | n < w) is coherent. 1.B.6 Lemma O 1.B.7 e In the following, we repeat some well-known facts about iterated ultrapowers. We consider the iteration of ultrapowers of V using the measure U and we denote the n — th iterate of V by M, and the corresponding embeddings by Jon, i.e., My = V M,, = Mostowski collapse of V™ /U/* for n < w and eae 1.B. Prikry generic sequences and their models 15 M., = dirlimit{M, | n < w}. jas : M, — Msg is elementary embedding, where a < 8B < w and joa = idf Ma. We define a sequence (6, | 1 <n < w) as follows: For every 1 < n < w, bn jnw(Yn) Where Yq: [idl jon—1(8)) sta and [sy denotes the element of M, represented in the corresponding ultrapower of M,_1 by the function f. Using the elementarity of j’s we can prove that (VA) (a EU* <> (6),...,64) € jos(A)). (*) In addition, we can show that M, = {jou(F)((61,-..,6n)) | 1 <n<w & FeV}. To prove the above characterization of M,,, we show by induction on n < w, that M41 = {ona (£)(Giinsa(m4) «+e Jntin¢1(In41))) | f € v}. Since (*) holds, then using the “geometric property” of the Prikry-tree forcing, we show that (6, | 1 <n < w) is a P3,, y) — generic sequence over M.,, below Jow(6). Our next step is to use all the properties of the sequences (f,,g, | n < w) to define another sequence of ordinals (e, | 1 < n < w) below jo,(6). For n <w let us define s,:2 Jo..(fr)((61,.--,6y(n))), and by the coherence of the functions f,’s we conclude that U s, is a well-defined sequence. Hence, we let l<w (En |n< w):2 Usp, ie., 1<n<w (V1 <n <w)|(e1,..- En) = Joulfa)((515--- 54))]- () ee 16 Chapter 1. OD partitions in a Prikry model 1.B.70 Under the above notation we prove the following lemma. 1.B.8 Lemma For every natural number n, we have that €, = 6. Proof: The basic idea of the proof appears already in the case n = 1. The same argument works for the general case, except that the notation becomes quite involved. Assume, towards a contradiction, that ¢; < 6,. Then by () of 1.B.7, we have that Jo. (fn )((61, --- 64(1))) < 61. (T) Before we proceed, we establish the following notation: For a sequence of functions (g; | 1 < i < w) we define I gi ‘ef id and for 1<i< m>til1 JI gi: ed g,0...0g8,. Also, if g is a function, we define gO. id 1<i<m — m—~times and g®):€ g¢ and for m > 0 g™+D; & g(™) og and ifx = (x; | 1 <i<n) then (x); = xj. Using (}) of 1.B.6, (*) of 1.B.7 and (9) of 1.B.7, we define the following sequence of sets: Au {x € 6 | f(x) < (x); }. By (*) of 1.B.7 we conclude that A; € U7). Then, by induction on 1 <n Ew and by using (+), we define A, € U7) An {xe o@]( TT toa) <(( TL £00q))(x)):}- 0<k<n-1 1<k<n-1 1.B. Prikry generic sequences and their models 17 Since the sequence (A, | n < w) belongs to V, then by the o — completeness of the measure U/ we can find a sequence z € 6” such that (Vn < w)[zn: 2 zf-y™(1) € Ag. Let us define 6,:% ( ITI — f,c”a))(Zn). Then, due to the coherence of the O<k<n-1 functions f,’s, we observe that for all n € w On = (( I £,00(1))(Zn41))1- 1<k<n Hence, (n € w)(0, > 9,41) which contradicts well — foundedness. Therefore, Ey x 6}. On the other hand if 6; < €;, then 61 < jou(fi)((1, - - - 64¢1))). (111) Thus by (*) of 1.B.7 we conclude that {x € 6 | (x), < f,(x)} €U™. Next by applying (fT) of 1.B.6, we get that By: {xe 6 |xeT’ & (gyay(x))r < filgyay(x))} EU, Since for all x € T” we have that f,(g41)(x)) = (x);, then B, = {x € gn) | (g41)(x))1 < (x)1} €e unr), Then, by induction on n > 1, we define Bi {xe 6M) | (( I Ey((4(1)) (X))1 < (o. Fn ((1)) (X))a Fe 0<k<n-1 18 Chapter 1. OD partitions in a Prikry model Using (*) of 1.B.7, we get that B, EU"), Again, by the —completeness of the measure U/ we can find a w € 6” such that (Vn < w)[wa: = why(7(1)) € B]. As before we set (,:%2 (( II — &,00@1)))(Wn))1- But the coherence of the 0<k<n-1 functions g,’s, implies that Gy = (( II &,001)))(Wn41))1. Hence, (Vn € 1<k<n w)(Cn > Gn41) which contradicts wellfoundness. Thus ¢; # 6,. Therefore, y= 6}. Using exactly the same arguments, we show that (Vn < w)(é, = 6). 1.B.8 Lemma O 1.B.9 Theorem Let U be ao—complete ultrafilter on a cardinal 6 and Pj, be the Prikry-tree forcing induced by U on 6. If o and 7 are two Pj, — generic over V sequences with the property V[o] = V[r], then they are tail — equivalent. Proof: Suppose that the generic sequences 0,7 are not tail— equivalent. Then the generic sequences G,7 are not tail — equivalent (in the notation of 1.B.6). Then, by (4) of 1.B.7, we conclude that (V1 <n <w)[(51,-..,6n) = jJou(fn.)((51,... 64(n)))]- Hence for every n such that 1<n<vy, there exists a set D, € U™ such that D,: {x € 6™™ | f(x) = x[n}. From the sequence (D, | 1 <n < w) and using the o — completeness of the measure U we construct aU — splitting tree J over 6 such that (Vn<w)[xeT & Ith(x) > 7(n) > xfy(n) € D,]. 1.B. Prikry generic sequences and their models 19 Therefore, for all n < w and all x, ifx € T and lh(x) = 7(n), then f(x) = xfn. Let 7*:° T’:NT. Then, for alln <w (OT) I= [f(eet-1(a)) = holm & gursn(Gat nla! + n)) = of], Uu But then the choice of the tree T* obligates that for alln <w (0;7*) | off n = pofn (x) Pi, and since (@;7*) < (0;T”), then (O;T*) |F-— [6%, fo are not tail — equivalent). (xx) Pi Finally, it is obvious that («) and (««) lead to a contradiction. Therefore, the sequences 6,7 are tail — equivalent and the same is true for the original sequences o and T. 1.B.9 Theorem 0 1.B.10 Corollary Let U be a normal measure on P,,(A) and Py be the Su- percompact Prikry forcing induced by U. Ifo and 7 are two Py — generic over V sequences with the property V[a] = V[r], then they are tail — equivalent. Proof: The normality of the measure U implies that the forcing Py is equiva- lent to the forcing Pj,. To show this equivalence we use the following fact: As it was proven in [Mathias 73], for every sequence (Ax | x € [P,(A)]<”) of sets in U, there exists a set B € U such that (Vx € [P,(A)]<“) I{P EB|xc p} C Ay]. Then, we project the measure U on X and we use 1.B.9 to complete the proof. 20 Chapter 1. OD partitions in a Prikry model 1.B.10 Corollary 0 1.B.11 Theorem We assume that V is a transitive model of ZFC, in which kK is a supercompact cardinal and X is an inaccessible above k. Also, U is a normal P,,() — measure and Py is the Supercompact Prikry forcing associated with U. If G is Py — generic over V, then there exists in V[G] a coloring F of [k]* into V, colors for some 4p < k, with the property that F is ordinal definable over V[G] with parameters from V and has no homogeneous set, ie. V[G] F AV < K)(K VoD (wy) Proof: First, we choose in V a sequence (F, | a < «), with the property that for every a < k, F, : [a]* — 2 and each F, has no homogeneous set in V. Since for every a < « there are no new w-sequences of a in the Prikry extension V[G], then we conclude that in V[G] there are no F,-homogeneous sequences. Let us fix x € [k]” M V[G]. If x is cofinal in «, then we define F(x): 2 {@ € (w<”)*) | AG’ fg is Py — generic over V & & VIG]=ViG] & & Yn< w( a(n) = (k €w | of(n) <x(k) < o%,(n + »))] \ If x is bounded below x, then we define F(x): @ Fyup(x)(X). It follows that there exists a < « such that F : [k]* > V, and that F is VOD over V[G]. In fact, the above coloring F is OD with parameters from Ve of the ground model, for some large enough £. This is true because the 1.B. Prikry generic sequences and their models 21 relation V[G’] = V[G] is equivalent to the fact that the P(£) C V[G’] for some € large enough, so that we can code in the usual absolute way the Py — generic filter G as subset of €. In order to express the fact that P(€) C V[G'], we need to say that every subset of € is the realization through the generic G’ of a nice Py — name of a subset of £, which is obviously an Ordinal Definable property with parameters from V¢. We claim that F has no homogeneous set. Let us suppose towards a contradiction that F has an homogeneous set namely h € [«]” such that |F”[h]*| = 1. By the choice of F,’s we conclude that such an h must be cofinal in x. First, we observe that if a, belong in F(x) with witnesses ( two Py — generic filters) G., Gg then by virtue of 1.B.10 and since V[Ga] = VIG] = V[Gg], (Ano, ni < w)(Vi < w)[oG, (no +i) = oG,(mi + i)]. Thus a,@ are tail — equivalent. Let us define S:@ {2 € w | |ran(h) n [oG(n), o6(n +1))| = 1}. Then there are two cases: Case 1. S is infinite. We set h’ € [h]” such that for all n € w |ran(h’) N [oG(n),o&(n + 1))| < 1. Then we consider the even subsequence h’, of h’, ie., h(n) = h/(2n), and two reals a,a’ with the property that a € F(h’), a’ € F(h{) with witness G. The latest implies that for all n < w, we have that a(n) = (k € w | o€(n) < h’(k) < of(n +1)) and a(n) = (k € w | of(n) < hi(k) < oG(n +1)). Since F(h’) = F(hi) then a, a’ are tail — equivalent and so 22 Chapter 1. OD partitions in a Prikry model |ran(a)Aran(a’)| < No. However, this is a contradiction since a, a’ imply that |ran(a@)Aran(a’)| = Xp. Case 2. S is finite. We set h*(n):@ h(n + 1) and we consider a € F(h) and a’ € F(h*) with witness G, as in the previous case. Then, as a,a’ € F(h), we can find no,n; < w such that (Vi < w)[a(no +i) = a(n, + 1)]. Since S is finite, then we can find a large enough i < w, such that |a(ng + i)| > 2. Let ko: mina(ng + i) and kj: maxa(ng + i). Since a(np + i) = a/(n; +i) and ko, ko + 1 € a(no +i), then o@(np + i) < h(ko) < o&(no +i+1) and oG(mi +i) < h(ko +1) < o&(n; +141). But the latest forces that ng = nj. Moreover, as k; € a(np + 1), we have that o§(no +i) < h(ki) < 0&(no +i+ 1) and o@(no+i) < h(ki+1) < 0§(no+i+1), which implies that k; +1 € a(no+i), an obvious contradiction to the maximality of k). Thus h is not F — homogeneous. Therefore, the VOD over VG] coloring F has no homogeneous set in V[G]. This completes the proof of the theorem. 1.B.11 Theorem OQ 1.B.12 e Before we exhibit an OD over V[G] partition with no homogeneous sequence, we are going to show that under some special circumstances a rank initial segment of the ground model is OD over the Prikry generic extension. 1.B. Prikry generic sequences and their models 23 To accomplish the above result, we consider the cardinal € which is men- tioned in the proof of 1.B.11 (without loss of generality we can assume that € is strong limit and thus |V,| = €) and had the property that the coloring F Before we force over V that V¢ of the ground model V is OD over any Supercompact Prikry generic extension, we have to make sure that « remains a supercompact cardinal in the generic extension. In order to accomplish that goal, we use R. Laver’s forcing Q of cardinality «. As in [Laver 78], Q is kK —cc and in V®°, « remains supercompact upon forcing with any < « — directed closed forcing. Since Q is Kk — cc, then forcing with Q does not change the behavior of the power-set function above k. Hence, starting with G.C.H in our ground model, we still have G.C.H above « in any of the Laver models which we get via Q and moreover \ remains an inaccessible cardinal above x. We pick a Q-generic extension of the ground model of the supercompact and the inaccessible above it and we consider this generic extension as our new ground model which is denoted by V. The following arguments are simple variations of those in [Menas 75]. We code V¢ by a subset A of € using a bijection f of € onto V¢ in the following standard way: For a, € € we set aE@: f(a) € f(@). Thus EC Ex €isa well — founded binary relation on €. Using the canonical absolute bijection of € x € onto € we code E by E* C €. Then, it is easy to show that (V¢,¢) is the Mostowski transitive collapse of (€,E) and f is the collapsing map. We call E* 24 Chapter 1. OD partitions in a Prikry model to be the code of V¢ and we observe that Vz can be decoded from E* in an absolute and ordinal definable way. At this point we recall that our ground model V satisfies GCH above x. Let v = 22’. By using Easton forcing, we force the continuum function to behave in the following fashion: If a € E*, then 2¥»+e+1 = 049 and ifa ¢ E*, then 2®+e+1 = Ni oi3. Let V* denote the generic extension of V via the Easton forcing. Then for every a € E* a € EX > V* & tot = Ri ayo. Since the Easton forcing we use is < v-directed closed, then U remains a normal measure over P,,(A) in V*. Thus V* thinks that « is a \-supercompact cardinal and that is an inaccessible cardinal above x. Let us take G to be a Py — generic filter over V*. Our goal is to show that E* remains OD over V*[G]. Since the forcing Py satisfies the v—cc, then V* and V*[G] agree on cofinalities above v, and so they have the same cardinals above v. Hence, there exists a @ such that Vv" _ yV"IGl Rv any. In addition, the fact that V* and V*[G] have the same cardinals implies the following: 1.B. Prikry generic sequences and their models 25 * V*(G (Va) [ORY aia) = Op )]. Then (QReta+1 )V"Is] = (Qhereth yv"ls) = (Qe )V"IGl, At this point, we use a standard fact about v — cc forcings. Since Py has size <vin V*, then (26a VIG < JyXMon|Y < (gen )V, Thus (2Retats )¥"IG] = (Qed )v" . Consequently, * Vic Ve * * (2¥etat1 VIS] — Prete => (2Retati)V" = res This implies that for all a < € a € EX => V"[G] — (24441) = Negras. Therefore E* is OD over V*[G]. Since E* is ordinal definable over V*[G], then the collapsing map of (€,E) is ordinal definable and consequently every element of VY is OD over V*[G]. In particular, the set of all Py — dense sets is OD over V[G], since it is an 26 Chapter 1. OD partitions in a Prikry model element of VY. The latest is essential for the next theorem. As a result of the above considerations, we adopt as our ground model the model V* and we denote it again by V. 1.B.12 0 Hence, all the above arguments prove the following theorem. 1.B.13 Theorem If V' is a transitive model of ZFC in which k is a super- compact cardinal and X is an inaccessible above k, then there exists a generic extension V of V' in which % remains supercompact, \ remains inaccessible and the following holds: If U is a normal P,,(A) — measure, Py is the Supercompact Prikry forcing asso- ciated with U and G is Py — generic over V, then a sufficient large rank initial segment of the ground model V is OD over the Prikry generic extension V[G] and V[G] F 7(Vu < K)(« xt (Wy, ). 1.B.13 Theorem 0 1.B. Prikry generic sequences and their models 27 The following result is not really crucial for our thesis, even though it uses extensively the methods of [Menas 75] to produce a stronger form of the previous theorem. However, the technology which is used is appealing to the author and thus we make a sketch of the proof. 1.B.14 e As we proved in 1.B.11, there exists in V[G] a VOD partition with no ho- mogeneous sequence. Our goal is to show that it is consistent for the ground model to be OD over the Prikry generic extension, thus making the partition that appears in 1.B.11 OD over the Prikry extension V[G], which proves that the partition relation (Vu < K)(« oD (o¥, } fails in V[G]. Towards this goal, we follow almost identically the proof of a theorem of T. Kk. Menas [Menas 75] (20. THEOREM p.89) in which he proves that starting with a model of a supercompact cardinal « and an inaccessible cardinal above kK, there exists a model M and a generic extension N of M with the property that « is supercompact in both M and N and moreover N F ZFC + V = HOD. We fix \ to be a regular cardinal above x. The proof, in [Menas 75], involves a reverse Easton forcing construction in which he codes each e(() rank initial segment of the Universe into the power-set function, where e is an enumeration of all the Beth fixed points. The only modification that we are making in that proof is that we take the enumeration e to start from the first Beth fixed point above A, say it vy. Hence we construct the model N in which « remains 28 Chapter 1. OD partitions in a Prikry model supercompact. Moreover, as in [Menas 75], if 6 is an ordinal, then we code in the usual absolute way VN6) by a subset A of e(6) and the forcing construction guarantees that NF (Wy < €(5))[y € A <> 2Me4741 = Noss 743]. Since the decoding of VN, from A is OD and absolute, then this shows that every element of Vx) is HOD over N. Thus N — V = HOD. Let us consider V: © N to be our ground model before we force with the Supercompact Prikry forcing. We recall that v denotes the first Beth fixed point above \ and e(0) = v. Let U be a normal measure over P,,(A) in V. We take G to be a Py — generic filter over V. Our goal is to show that V remains OD over V[G]. Let 6 be an ordinal. As before, we code in the usual absolute way VX5) by a subset A of e(5). Since the forcing Py satisfies the v — cc, then V and V[G] agree on cofinalities above v, and so they have the same cardinals above v. Hence there exists a @ such that Ve _ yVIG) RY) = Ry In addition, the fact that V and V[G] have the same cardinals implies the following: V[G (Va)|(Yoyo4r) = (SEA). Then (QRete+1 )VIGI = (heck )VIGl = (28Xs+o41 VIG], 1.B. Prikry generic sequences and their models 29 At this point, we use a standard fact about v-ce forcings. Since Py has size <vyin V then (28s)4041) VICI < Xoorot [V < (QNoyto4s )V. Thus, (QRetatt ) VIG] = (QNoten )¥. Consequently, (28e+a+1) VIG] — xy = (2Niose1)V = Vv at3 v+e+3: Lhis implies that for all 7 < e(5) ye A => VIG] F (29+) = Raa. Therefore, every element of Vs) is HOD over V[G]. This shows that V is HOD over V[G]. Thus the VOD over V[G] coloring F, which appears in 1.B.11 is OD over V[G] and has no homogeneous set. 1.B.14 0 Therefore, all the above arguments prove the following theorem. 1.B.15 Theorem If there exists a transitive model of ZFC in which k is a supercompact cardinal and there exists an inaccessible above k, then for \ any regular cardinal above k, there exists another model of ZFC V in which k is supercompact and the following holds: If U is a normal P,,(A) — measure, Py is the Supercompact Prikry forcing associated with U and G is Py — generic 30 Chapter 1. OD partitions in a Prikry model over V, then the ground model V is OD over the Prikry generic extension VIG] and VIG] K 7(¥n < x) (« st (wi, ). 1.B.15 Theorem O It is an open question whether the above partition relation fails in any Prikry generic extension of any model of ZFC with a supercompact cardinal. 31 Chapter 2 The inner Model V(H) 2.A The power set of « in V(H) In the first chapter we proved that if the ground model is prepared appro- priately, then the partition relation fails in its Supercompact Prikry generic extension. In order to find a model of the partition relation, we consider a specific inner model of the Prikry extension and we are going to show that it is a Symmetric model with a lot of Prikry generic sequences. First, we define the inner model. 2.A.1 Definition Let U be a fixed normal measure on P,,() and Py be the corresponding Supercompact Prikry forcing. If K is a Py — generic over V filter, the we define H¥ to be the following set: HK. Uf{P(«) AV[Kfa] |o< a}. 32 Chapter 2. _The inner Model V(H) Whenever the generic is denoted by G, then H® is denoted by H. For a fixed Py — generic over V filter G we consider the inner model V(H) which is the smallest inner model of ZF that contains H as an element. 2.A.1 Definition O Our aim in this section is to show that the power set of « in the model V(H) is H. We start with a definition that appears in [Magidor 77]. 2.A.2 Definition Let h be a permutation of X that is the identity on some ordinal a € [k, X). We define an automorphism h of Py as follows: If (Pi,...,Pn3A) € Py let h((Pi,...,Pa;A)) = (h”Py,...,h”P,sh”A) where h”A = {h”Q | Qé€ A}. 2.A.2 Definition 0 2.A.3 Lemma (M. Magidor [Magidor 77]) If h is a permutation of X such that for some a € [k,\) hf a = idf a then h extends to an automorphism of r.0.(Py) . 2.A.3 Lemma O 2.A.4 Definition If h is a permutation of such that hf « = id} a for some a € [k, A) then we call the support of h to be the largest 3 such that hf 6 = idf 6 and we denote it by suppt(h). We set A = {h € Aut(Py) | suppt(h) € [«, A)}. 2.A. The power set of « in V(H) 33 2.A.4 Definition O 2.A.5 Definition We define g to be the following Vs — name: gs ={(¥.p) | SPi,...,Pn € P.(A)SA € USx (p= (Pi,...,Pn;A) & & y=xnB)}. Ifx = P; for somei € {1,...,n} such that p = (Pi,...,Py;A), then we say that x appears in p. 2.A.5 Definition O Obviously gg is the canonical Py — term for the restriction of any Py — generic sequence on {, where / is an ordinal above « and below i. Next we show that all these names are invariant under the automorphisms in A. 2.A.6 Lemma Lethe A& BE [k,A). Assume that suppt(h) = 7 > £. Then, h(g) = &. Proof: Since h is an automorphism of Py, then h is extented on VP by recursion on the rank of the boolean-valued terms as follows: (for simplicity we use the same symbol for h and its extension on vPv) h(t): { (6(6) hey) [Me p) € +}, 34 Chapter 2.__ The inner Model V(H) for any 7 € VPv, In order to show that h(g,) C &@ we take a (¥ ,p) € &g where y = x for some x that appears in p and we recall that h((¥ ,p)) = (A(¥) ,h(p)). Assume that p = (P),...,Pn;A) and x = P, where 1 <k <n. Then, h((¥ ,p)) = (¥ ,h(p)) - Since h(p) = (h”P,... wh” P,;h”A) and suppt(h) = y > @ then h"P,.NB =P,NB =xnf. Therefore, (¥ sh(p)) = (bP. 8) h(p)) and we observe that h”P, appears in h(p). Hence, h((¥ sp)) = ((h" Py 9 8)"h(p)) € ge. Conversely, we consider (¥ ,p) € 83, where y = x @ for some x that appears in p and x = P, for some k, 1 < k <n, and p = (P),...,Pn;A). Then we let q: (h-'P,,...,h7!P,;h7!A). Hence, xNB =P,NPB =h!P,nB 2.A. The power set of « in V(H) 35 since suppt(h) > . Also h(q) = p and (¥ Pp) = (¥ h(a) = ((h-'P,N B),h(q)) = (h((h7*Pi, 9 BY) ,h(q)) = h(((h-*P,N B)",9)) and h7'P, appears in q. Hence, ((h7!Py 9 B)",q) € Ss and so (¥ ,p) € h(g,). Therefore, » C h(gg). This shows that h(gs) = g@. 2.4.6 Lemma O However, the previous result can be improved in terms of the following lemma. 2.A.7 Lemma Assume that ho € A and suppt(ho) = ¥. If 6 = sup{hj@ Uhp1GU {G,7}} +1, then for any hy € A with suppt(h;) = 6 the following holds: hy(hogg) = hogs. Proof: We take a ((xA BY ,hiho(p)) € hyho(g,), where x appears in p. Assume that p = (P1,...,P,;A) and x = P, where 1 < k < n. We show that there exists a q such that (x BY, Biho(p)) = ((y A B)’Jho(a)) for some y that appears in q. By the definition of ho, h; we conclude that hyhop = (bvhUP,,... .hYn"P, shiv Ay. 36 Chapter 2.__ The inner Model V(H) (Without loss of generality we assume that for all P € A P=h/P = hjP.) Let us fix i € {1,...,n}. Then hyhoP; = hy[((hpPi) 1.) U ((hgPi) \ 4)] = hy((hoPi) M6) U by ((hoPi) \ 6) = ((hgPi) 1 6) U hy ((hoPi) \ 4). Notice that the second equality follows from the first, since suppt(h,) = 6. Then, we find P; € P; such that (hjP;)N6 = hgP!. If ¢ € P|N BG, then h¢ < 6 and so¢ € PiN@. Hence, PIN G=P;NZ. Since suppt(hi) = 6 and (hgP;) \6 C A\ 6, then h7(hgp; \ 6) CA\ 6. Assume that h/(hgP; \ 6) = hgWj. If ¢ € W;N Pi, then ho¢ > 6 and ho¢ < 6 which is a contradiction. Thus W;M P/ = 0. Moreover, if ¢ < @, then h€ < 6 so ¢ ¢ Wj, ie, WiN GB = 0. We set P* = PLU Wi. Thus, hyhoP; = hoP], where P?N 6 = P}NG. We set q = (Pi,...,P*;A). Then hoq = hyhop and if x appears in p as Py, xNB=PENGZ. Consequently, (xB) hihop) € hogs. Therefore, hy hogs C hogs. Conversely, if ((x N B) shop) € hoga, where p = (Py,...,P,;A) and x = P, for some k € {1,...,n}, then we have to find q € Py such that (x B) hop) = (yn B) Jhyhop), where y appears in q. As before, without 2.A. The power set of « in V(H) 37 loss of generality, we assume that for every P in A hjP = h/P = P. We set PY C P; such that hgP? = (hoP;)6. Then, hyhjP; = hgP;. Let W; be such that (hgP;) \ 6 = h{hgWj. Then, every ¢ € P/N W; satisfies ho¢ < 6 and so hiho¢ = hog < 6 and as ¢ € Wi, then hyho¢ € (hGP)) \ 6, ie., hyh¢ > 6 which is a contradiction. Moreover, if ¢ € 8M W; has the property h¢ < 6 and as 6 = suppt(h;), then hiho¢ < 6 and since h{hgwW; C A\ 6, then hyho¢ > 6. Thus, W,NG = 0. Next, we set Pf? = Pi UW;. Then, P7}N 8 = P}NB and hoP; = h{hgPY. Let q = (Pj,...,P*;A). So hp = hyhoq and if x is P;, then P?>NGB=xN QZ. Therefore, ((x M 8)’,hop) € hihogg and so hogy C hy hog. Hence, hyhogs = hogs. 2.A.7 Lemma O 2.A.8 Definition Let (x;A) € Py. A subset A’ of P,(6) is called 6 — good for (x;A) iff A'eUs & Wee (A (xt Ca Jw € [AJ*(x Cw & wld =2)). 2.A.8 Definition 0 2.A.9 Lemma Let (x;A) € Py. Then, for every 6 € [k,X) there exists A’ which is 6 — good for (x;A). 38 Chapter 2. _The inner Model V(H) Proof: We assume that x = {P,...,Pn}. Let us take a B € U such that B C A and VP € B(P, C P). We define the following partition of Bf 6(€ Us) f: [Bf 6] + 2 such that 1, if dw € [B]“(wf 6 =z) f(z) = 0, if otherwise. Since U; satisfies the usual partition property, then there exists an A’ C Bf 6 such that A’ € Us which is homogeneous with respect to f, ice., (vn <w)((e"[A/P| = 1). We show that (Vn < w)(|f”[A’]"| = {1}). We fix n € w and without loss of generality we assume that 1 <n < w. Since gs and g are the Py, and Py — generic sequences respectively induced by the Py — generic filter G, then 4m;, mp2 < w such that gst [mi,w) C A’ and gf [mj,w) C B. Notice that gs = (g(i)N 6[i < w). Let mp = max{m), mg}. Then, gsf[m,w) CA’ & gf[m,w) CB. Consider x = (gs(m),...,gs(m + n)). Then, x € [A] and for w = (gs(m),...,gs(m+n)), w € [B]“ and wf 6 =x. Hence, f(x) = 1 and by the homogeneity of A’ we conclude that f(A? = {1}. Therefore, f”[A’]* = {1} and this shows that A’ is 6 — good for (x;A). 2.A. The power set of k in V(H) 39 2.A.9 Lemma O 2.A.10 Definition A condition (x;A) € Py is called 6 — nice iff 6 € [k, A) and Af 6 is 6 — good for (x;A). 2.A.10 Definition O 2.A.11 Corollary For every (x;A) € Py there exists a condition {x;A*) < (x;A) so that (x;A*) is 6— nice. This shows that D, = {p € Py | p 6 — nice} is dense. Hence (Vp € G)(Jq < p)(q is 6 — nice & q € G). Proof: We are using 2.A.9 to find a set A’ which is 6 — good for (x;A). Then, we define A* = {P eA|Pnée At. By definition of A* we have that A*t6 ¢ A’. In addition, if P € A’, then JQ € B such that QN6 =P, ie., Q € A* and so P € A*}$. Hence, A*[6 = A’ and A* C BC A. Obviously, A*} 6 is 6 — good for (x;A*) . Therefore, for every 6 € [k, A) and (x;A) € Py there exists A* C A such that (x;A*) is 6 — nice. 2.A.11 Corollary O By taking A to be the least innaccessible above x, it is easy to show that the measure U concentrates on a set XY with the following property: For every P,Q E # PNK =QNK=> [P| = [QI]. Then a priori we take the forcing Py to have all the conditions below (0;4). 40 Chapter 2. _The inner Model V(H) 2.A.12 Lemma Assume that h € A with suppt(h) = 7, F is a Py — name which is invariant under A, 6 = sup(h"BUh1GU{G,y})+1 and p is a 6 —nice condition. If p I d(&, ti, hg, F) and q € Py such that qf 6 = pf 6, then u q IK 9(4, 4, hgs, F). Pou Proof: Assume that p = (Pi,...,Py;A) and q = (Qj,...,Qn;B). Since p is assumed to be 6 — nice then A[6 is 6 — good for p. Assume towards a contradiction that a ILA 4(4, 4, hg, F). Pu Then, there exists such that q’ |HK— 39(&, ti, hés, F). Since q! <q, then Pu and so (R,N6,...,R,6) € [Bf 4]. Since Bf 6 = A] 6, then (Ri N6,...,R,6) € [Af 6]. By the assumption that Al 6 is 6— good for A, then there exists (T),...,T,) € [A] such that P, C T, and (Ri N6,...;Rs 6) = (¥1N6,...,Y,6). We set A= A\ {Tj,...,T;} and C = ANB’. Then, set 2.A. The power set of « in V(H) 41 and qi = (Qi,.-.,Qn,Ri,.-. ,Rs3C). Since p; < p and q, <q, then Pi o(&, ,hgg, F) and qi I 3¢(&, i, h hg, F) and pif é = qil 6. At this point we define a permutation h, of \ such that suppt(h,) = 6 and h{S; = 5; where p; = (Si,...,Sn45;C) and qy = (S'1,.--,S’n+s3C). The con- struction of such an automorphism has been influenced by similar arguments that appear in [Magidor 77] and [Apter 85]. The definition of such an h, is as follows: We recall that the measure U has the property which implies that for every P, Q appearing in some condition for which PX k = QNk, then |P| = [Q]. Since 5316 = 5S; 6 and 6 > x, then |S;| = |S!|. Therefore, |S; \ 5| = |S \ 6] and we set hii: = % (idf 5) Uk; where ky: $1 \6 3 $1 \ 6. If ki41 : Siga \ Si i Si44 \ Si, then we define as def hiiga? = hij U Kian and we observe that hii ©... C hrigt A2 Chapter 2.__ The inner Model V(H) and hy j41 : Si41 — St,, is a bijection with h,;,,/6 = idf 6. Finally, we set hi n+s+1 : (A \ Sn+s) \ 6 — (A \ Sits) \ é and bh: Uo hy. 1<i<n+s+1 By the construction of the hj,’s, hy is a permutation of A with hi[6 = idfé and h/S; = Si. Hence, we consider the automorphism h, of Py induced by h; and we observe that hi(p1) = (8'1,--- 8’n¢sih{C) is compatible with q,. Let r < hi(p1),q1- As Py then hy (p1) I o(&, ti, hi (hg), F). U By 2.A.9 h;(hgs) = hg, and by the definition of F we conclude that hi(F) =F due to the fact that suppt(h; oh) = min{suppt(h;), suppt(h)} > « for any h with suppt(h) > « (i.e, hoh € A). Therefore, U 2.A. The power set of k in V(H) 43 and asr < hi(p1), a1, then Pu and at the same time r |LK 7¢4(4, i, hg, F) Pu which is a contradiction. Therefore, q |K— ¢(4, a, hg, F). Pu 2.A.12 Lemma O The next result is in the spirit of [Apter 85]. 2.A.13 Lemma Let H: = U{P(«) 1 V[(hgs)”l] | a € [x, A)}. Ifz € V and x € P(z) V(H), then there exists a 6 € [k,) such that x € P(z)N V[Gf 6]. Proof: Let x € P(z)N V(H). We fix F to be a Py — name for H in V[G] which is invariant under A. Then, there exists a formula ¢ ,v, 6 € [k, A), and heA with suppt(h) = y and v € P(k) N V[(hgs)¥'l] such that for all a € z, a€xo V(H) FE d(a,u,v,(F)Y). Using the formula ¢ we can find another formula W such that for all a € z a€x V[G] F U(a,u,(hgs)", (F)Y'). Let x € VPU such that xV!Sl = x and we fix a condition Po € G such that Po |F- Va € Z(a EX U(a, i, h(g,), F). Py 44 Chapter 2.__ The inner Model V(H) We let 6 = sup(h”G Uh “!8U {8,7}) +1. Next, we define the following subset of z: y = {a € z|Fp < po(pisé — nice & pf € Gf 6 &p |L— U(4, t, hgs, F))}. Pu We observe that y € V[Gf 6] and we claim that y = x. Proof of “Vy Cx”: Let a € y. We fix a 6 — nice p such that p[é € Gf 6 and Pu Since p[ 6 € Gf 6, then there exists a q € G such that p[ 6 = qf 6. By 2.A.7 we get that q |JFFX Ud, i, hes, F). Since q € G, then Pu V[G] F ¥(a, u, (hg), (F)V'S), 1.e.,a@ €x. Soy Cx. Proof of “«K Cy”: Let a € x. Then we can find p = (of n;A) € G such that p<p: and p I WG, i, hg, F). U By the 2.A.11 we can find aq € G, q < p such that q is 6 — nice. Hence, q - D(a, a, hg, F) and as q € G this implies that ql é € Gfé. Thus, a € y. Hence, x Cy. This concludes the proof that x = y. Consequently, x € V[Gf 6]. 2.A. The power set of « in V(H) 45 2.A.13 Lemma O 2.A.14 Corollary Let H: U{P(k) N V[(hgs) Sl] | a € [K,A)}. Ifz € V[Gf a] for a € [K,) and x € P(z)N V(H), then there exists a 6 € [K, ) such that x € P(z)N V[Gf 6). Proof: We repeat the proof of 2.A.9 with the following changes. We find the Py — names 2, x such that (2)¥(Gl = z and (x)V'@l = x and a condition pp € G such that Do |K- Va € (a € ¥ © VG, ti, h(gs), F). Pu We define 6 = sup(h”@ Uh'8U {a, 2, 7}) +1 and the set y is defined exactly in the same way. y = {a €z| Sp < po(p is 6 — nice-& pf6 € Gfé & p |L— U(4, ti, hg, F))}. Pu Then, the proof of y = x goes through and as z € V[G[6], then x = y € V([Gf 6], i-e., x € V[Gf 6]. 2.A.14 Corollary O 2.A.15 Lemma If H= U{P(«) N V[(hgs)¥l] | h € A & B € [k, \)}, then H= H and H = P(k)N V(H). Proof: Since H= U{P(«) 1 Vi(hga) YS] | h € A & B € [,d)} C P(x) NV(H), 46 Chapter 2. _ The inner Model V(H) then H CH is obvious. Conversely, if x €H then x € V(H)N P(x) and by 2.A.13 there exists some 6 € [K, A) such that x € P(k) A V[Gf 4] = V[(hg)¥'S)}. Therefore, x € H. So H =H and H = P(x) V(H) = P(x) N V(H). 2.4.15 Lemma 0 2.A.16 Corollary If G is the fixed Py — generic over V and H = H° = U{P(«) N V[Gf a] | a € [K, A}, then V(H) G V[G] and moreover, V(H) FA=at. Proof: We have already seen that for every 3 € [k, A) V[GT 6] F [6] = x. So in each V[Gf 4] there exists the collapsing map fg of \ onto k, i.e., fg: K— A. By the usual arguments there exists Eg C x x k, Eg € V[Gf 4] such that fg is the Mostowski collapsing map of («, Eg). 2.A. The power set of « in V(H) A7 Using the bijection of k onto Kk x « in V Ez is coded by a subset of « EZ which is of course in V[Gf 6]. Similarly, the sequence gg itself is coded in a subset of « in V[Gf 4]. Therefore, V[GI 4] ¢ V(H) and V(H) E VB < (6 =x). Since ) is inaccessible and |Py,| < A, then VIG} 6] E A is inaccessible. If V(H) FA = 6, then we could code the collapsing map of X on « by a subset E} of & as above. But by 2.4.13 for some ( € [k, A) we would have that EX € V(Gf 4]. Since the coding is absolute, then that would imply VIGt A] EAI =x. But this is contrary to the fact that V[G[ @] F A isinaccessible. Therefore, V(H) — A is a cardinal & (V8 < A)(|B| = &). 48 Chapter 2. _The inner Model V(H) Moreover, as V[Gf 6] ¢ V(H) ¢ VIG], then Vie < «(Vp C (V,)¥% C VVIGI = V,). Since V[G] [k is strong limit], then V(H) E « is a strong limit & «kt = X. But VIG] E [Al = and consequently, V(H) ¢ V[G]. 2.A.16 Corollary 0 To summarize, in this section we establish that V(H) is a proper inner submodel of V[G] which believes in the regularity of 4. Moreover, the power set of K is H itself. 2.B. The Prikry-generic sequences in V(H) 49 2.B The Prikry-generic sequences in V(H) In this section we will show that there exist many canonical forcings in V[Gf 4] so that V[G] is a generic extension via these forcings of V[Gf @]. We recall that A’ is called 6 — good for (x;A) € Py iff V2 € [A] (xf 6 C2 Sy € [AJP(K Cy &z=yff)). Also we recall that a condition (x;A) € Py is called @ — nice iff Af is 8 — good for (x;A). 2.B.1 Definition Ifa € [k,\) andr is a Py, — generic sequence over V, then we define Q,(7) to be the following forcing: For any (x;A) € Py (x:A) € Q,(r): € (4B C A) (x;B) is 6 — nice & (xf a;Bf a) € Fe where Furs {(z;C) € Py, |zc7Cz2zuC} is the Py, — generic filter over V that induces the Py, — generic sequence T (r € [P,.(6)|*). The order on Q,(7T) is < NQ.(T) X Q(T) where < is the Py — order. 2.B.1 Definition O 2.B.2 Proposition Let o* be the Py, — generic sequence induced by G} a, Le, Fo. = Gla. If Ha = Qa(oa)NG, then Hy is Q,(0%) — generic over V[Gf a] and V[Gf a][H.] = V[G]. Ea SESE Sea se Sa SS 50 Chapter 2. _ The inner Model V(H) Proof: We denote by o the Py — generic sequence by G, i.e., F4 = G. First, we show that H, is a Q,(0°) — filter. Assume that qi, qo, € Ha. Since qi, q2 € G, then Jq € G(q < qu, qo). By 2.A.11 (dq’ € G)(q' is a — nice & q’ < q < qi, q2) and so q’ € H,. If qé€Ha &q<p& pe Q,(0%), then p € G as q € G and so p € Hy. Therefore, Hy is a Q,(0%) — filter. It remains to be seen that H, is Q,(o0%) — generic over V[Gf a]. Let D be Q,(0%) — dense in V[Gf a]. Then there exists VP¥« — name 7 such that D = (+)VIGal, We define D C Py such that D = {pe Py | (37> p)(p Fe (7)Vl6-1)}, We observe that if qéEDNG, then (Sr > p)( p |E- te (7) Ve! Pu and so ViG] Ere DNG, ie, DNH, £9. 2.B. The Prikry-generic sequences in V(H) 51 Hence, it suffices to show that DM G # 0. In order to show that, we prove that for every p € G there exists q < p such that q € D. So let p = (of K;A) € G. We find p* = (of m;A*) which is a —nice and belongs in Hy such that p* < p. Since D is Q,(o%) — dense, then we can find r= (ofm~(Py,...,Pn);B) € D such that r < p* and r is a — nice. Let s = (of[m~(o(m+1),...,0(m+n))~(o(m+n+41),...,¢(m+n+]1));B’) EG such that s < p* and s|k—re (7) Vee], Pu Without loss of generality, we assume that s is a ~ good ands € G. Since tla € Fe. = Gha, then P;\Na=o(m+l1)nNa for all i such that 1 < i < n and since rfa € gfa, sfa € gf a and lh(sf a) > lh(zf a), then o(m+n+1)NaeéBla for 1<i<n. But then, as r is a@ — nice, there exists (Pi,.--,Pa41) € [B]@ 52 Chapter 2. The inner Model V(H) so that P, C P,4; and PrayiNa=o(m+n+i)Na for 1<i<l. We let E={PeBNB'|Pay cP & o(m+n+1)C P}. Next we find E’ a — good for E and EX ={PeEE|PnaeF}. Then, E* C E and s* = (of (m+n+1+1);E*) <s and s* is a —nice. So s* ||-— ¢ € (7) Vel Pu and q = (of m~(Py,...,Pn,...,Pau)j3E*) <r and qha=s*fa. Since $* is a — nice and qf a = s*fa, then by 2.A.12 qlk-Fe(7)¥8! & gsr. Pu Therefore, q € D. Since r < p* < p, then q < p. Hence, we have shown that (Vp € G)(Aq € D)(q < p). Therefore, DM G # @ and as we have shown this implies that DN H, # 0. Thus, H, is a Q,(0%) — generic filter. Obviously, H, € V[G] and so V[Gf a][H.] C V[G]. Since (Vp € G)(Aq € Ha)(q < p), eS 2.B. The Prikry-generic sequences in V(H) 53 then o’ =({x| 3A(x;A) € He} =o and so G € V{Gf a][H,]. Therefore, V[G} a]{H.] = V[G]. 2.B.2 Proposition 0 2.B.3 Lemma Let U be a P,(G) — normal measure for @ € [K,) and for a € [k,B) we let U. = {Afa | A € U} and Py, Py, be the corresponding Supercompact Prikry forcings. Then there exists a complete embedding i: Py, > r.0.(Py). Proof: We define i: Py, — r.o.(Py) as follows: For p € Py, we set, i(p): = S{q € Pu | dha <p}. Recall that if q = (x;A), then ql a = (xfa;Afa). We set X(p) = {q € Pu | aha < p}. Hence, Vp € Pu, [i(p) = D7 ¥(4)]. Next, we show that i is a complete embedding. (1) We assume that qi <a qo where <, is the Py, — order and < is the Py — order. If r € ¥(q1), then rfa <a qi <a qe. Hence, rfa < qa, ie., 54 Chapter 2. _The inner Model V(H) r€ X(qe). As X(q1) C ¥(q2), then >> ¥(ai) < YS X(a2) and so i(qi) < i(qg). (2) Assume that q, 1 qo in Py,. We takes € ¥(q1) and t € V(qo). Ifs' <s,t, then s'fa <q tha <a qa, ie., q1 £ qe. So (Vs € V(q1))(Vt € ¥(q2))(s Lt). Hence, i(qi) 1 i(qz). This shows that if i(q,) 1 i(q2) in Py,, then i(q,) L i(q2) in r.o.(Py). (3) Next we consider p = (z;A) € Py and we have to find a reduction q of p in Py. For that we find A’ a — good for (z;A) and we set (z{a;A’). We claim that q € Py, is a reduction of p in Py,. Let t= (zhacwA") <q. Since w’ € [A’]*”, then we find w € [A]“ such that w’ = wl a. We set A“={peA|PCw & PnNaeA"} and p’ = (z7w;A*). Then p’ < p and p'fa <r. Hence, ax € ¥(r)(x || p) 2.B. The Prikry-generic sequences in V(H) 55 which implies that p || i(x). This shows that q is indeed a reduction of p in Py,. The parts (1),(2),(3) complete the proof that i: Py, — r.o.(Py) is a complete embedding. 2.B.3 Lemma O 2.B.4 Lemma Let U be a P,(() — normal measure for 8 € [K,A) and for a € [k,) we let U. = {Afa | A € U} and Py,Py, be the corresponding Supercompact Prikry forcings. Assume that h is a Py — generic sequence and FP is the corresponding Py — generic filter. If K = {p € Py, | i(p) € Fi}, then K is Py, — generic and its corresponding Py, — generic sequence ox satisfies the property ox = hf a. Proof: By 2.B.3 K is a Py, — generic filter, as i is a complete embedding. Next we fix p € K. Since i(p) = ©4(p) € K and K is Py — generic, then there exists an r ré X(p) Nk. Let r= xf A. Since h = Ors, then x Ch C xUA and asr € X(p), then rf a = (xfa;Afa) < p = (z;B). Therefore, xfa ChfaCxfaUAfa 56 Chapter 2. _The inner Model V(H) and so (z;B) € Fita: Hence, K C Fi}. Since K and Fita are Py, — generic filters over V and K C Fitter then Therefore, ox = hf a. 2.B.4 Lemma O The following theorem was proved by K. Prikry [Prikry 70] for the Prikry forcing on a measurable cardinal x. Exactly the same arguments work for any P,,(6) Supercompact Prikry forcing. So, we mention the result without a proof. 2.B.5 Theorem We let 8 > « where k is a Supercompact cardinal, 3 is a regular cardinal, U is a normal measure on P,(() and Py is the corresponding Supercompact Prikry forcing and G is a Py — generic over V. 1. If7 isa Py — name such that i [7: ftv & range(7) =E£] where pt < « and (E)¥IGl = E is a set of ordinals, then there exist sets {E, | n € w} C V such that (|E,| < w)¥ andE=U Ey. n<w 2. If X € V and E € V(G], E Cc & such that (|E| < «)Y'Sl, then there exists a set {E, | n € w} C V such that (|E,| < «)Y and E=U E,. n<w 2.B. The Prikry-generic sequences in V(H) 57 3. If X E€V,EC &, E € VG and (|B| = «)Y!Gl, then there exist setsE, €V_ (|E,| <«)¥ such that E=U Ey. n<w 2.B.5 Theorem O 2.B.6 Theorem We let \ > « where Kk is a Supercompact cardinal, 3 is a strong limit cardinal, U is a normal measure on P,,(A) , Py is the corresponding Supercompact Prikry forcing and G is a Py — generic over V. Let U be any normal measure on P,,() for 6 € [k,A). Then, in V(H), we can find a Py — generic. Proof: Since @ € [k,A) and X is strong limit, then we can find a cardinal 7 € [k,A) such that (\U| = 7)¥. Since ViGhy F bl =«, then we conclude that VIGh K Gi(f:« SU) & UC (P.(A))¥ and (P,((8))¥ € V. According to 2.B.5 we can find sets E, € V for n € w such that IE,|]<« with U=(|J E,. n<w 58 Chapter 2. _ The inner Model V(H) By taking then, we have that {E,,|mew}eVi[Gfy] & EReEV & |BR|<«x & U=\| EX. m<w We let Ah =NE =N{A eu] Ae EX}. By the x-completeness of the measure U and as ER, EV & (|ER| <x), then each AX, € U. We set Po € A*. Then we find P, C Pyyi such that Poy € AZ, and Vmew {P,|«>m}c A*. Therefore, (P, |n € w) € V[Gfy] and (P, | n € w) is a Py — generic sequence over V. Since (P, | n € w) € V[G +] c V(H), then we found a Py, — generic filter in V(H). 2.B.6 Theorem OG 2.B. The Prikry-generic sequences in V(H) 59 2.B.7 Lemma We let Py be the P,,() — Supercompact Prikry forcing with respect toU. We take QC.r.o.(Py) and g € V(H) be a Q — generic over V filter. Then there exists in V(H) a K which is Py — generic over V such that KN Q=g. Proof: We use the notation of 2.B.6 and for some ,7¥ € [{k, A) we find (Aj |n € w) € V[GP 4] such that for every A € U (an <w)(A D A*). We set Vi; = V[G[ 7]. Then, by recursion on w, we define (7,* | n € w) € V; as follows: (Po,---;Pm) € To: SS (Po... Pm) €[Pe(S]Y & & (Wk<m)(P,¢ At) & PheA® & & (Vx € g)((Po,-..,Pm—1;A"*) || x). If we assume that T* has been constructed, then we define (Po,.--5Pm) € Ty, : S (Po... Px) EF" or & Py...Pm1€ A, & PheAt, & & (Vx € g)((Po,...,Pm—1;A741) || x). 60 Chapter 2, _ The inner Model V(H) Then (7,* | n €w) € Vj. We set TY 7 n<w and we define a tree T C [P,(6)]“ such that Vx € [P,(5)}* (x eT: 2% By ET(xC y)). Obviously, T € V}. In Vi, we construct the quotient forcing P,/g and we take K to be Py/g — generic over V;. Then K is Py — generic over V and KN Q = g. Let a be the Py — generic sequence induced by K. Then, by induction on n, we can show that (Vn)(4kn (ol kn € T,"). Hence, o is a branch through the tree T, i.e., Vi[K] — [T is not well — founded]. But by absoluteness of the “well — foundedness” Vi F [T is not well — founded] and as V; C V(H), then V(H) F [T is not well — founded]. Hence, 7 has a branch in V(H). Obviously, any branch b through T is a Py — generic sequence and if Fe is ig 2.B. The Prikry-generic sequences in V(H) 61 the generic filter induced by the sequence b, then FP € V(H) and Fon Q=g. 2.B.7 Lemma O 2.B.8 Lemma We assume that (7;A) € Py and that x € V(H) is a Py, — generic sequence, where U is a P,,(3) — normal measure measure in V, a < £, Uu. =Ufa={Afa]AEu} and Z}a C x C Z}aUAfa. Then, there exists a y € V(H) such that y is Py — generic over V, ZC y CZUA and yfa =x. Proof: We use 2.B.3 to find the canonical embedding i: Py, Ccr.o.(Py). If Fe is the Py, — generic filter induced by x, then we let Q be the forcing i’Py, and obviously i”Py, C.r.o.(Py). Also, we consider g to be the Q — generic filter over V induced by i”F2 in Q. We have proved in 2.B.4 that if K is Py — generic over V such that QN K = g, then ox[ @ =x. Next, we do the construction of (T% | n € w) in some V[G[ 4] for 7 > @ where Ag is replaced by Aj N A and 7,* is defined as follows: (Po,-.-sP1) € Ty: — (Poy... Pi) € [Pe(BY & 62 Chapter 2. _The inner Model V(H) (Vi\(Ih(2) <i<] PEA) & Vi<l P\GAQ & P\EAU & ee BF RB & If we assume that T? has been constructed, then we define (Po,--+sPm) € Ti: ~— (Po, .-. Pk) ET," or [Pk,---;Pm—1 € A* & & PreAt,, & (Wxeg) ((Po,--->Pm—1An4s) | xD]. As before, we find a Py — generic K in V(H) and if ox is the induced Py — generic sequence which is a branch through T, then oxfa =x and ZC ox C zUA. This concludes the proof of the lemma. 2.B.8 Lemma O 2.B.9 Lemma Let U be our fixed P,,(A) — normal measure in V,Py is the corresponding Supercompact Prikry forcing and Py, where U, = Ufa. As- sume that y is a Py, — generic sequence and (z;A) € Py for some (3 € [k, 4). Then there exists a forcing in V|y] of size < + that forces a Py — generic filter K such that ox[ 8 =y andZC ox CZUA. Proof: We have shown that since is an inaccessible above the Super- compact «, then |U] < At. Let C = Coll(w,\*) and L be C — generic over 2.B. The Prikry-generic sequences in V(H) 63 Viy]. Then ViyIE] FU] = 4 and let (A, | n € w) be the enumeration of U in V[y][L]. We set A*=AoNA & At=f) Aa, n<k ie., Aj D... 2D Ag D... and each A* € U. Then we construct the tree 7, as in 2.B.8, where @ = \ and g = Fé. Then, by DC, we can find in V[y][L] a branch b through T. Let K be the Py — generic F}. Then zCbCZUA and Df f@=y. We consider the forcing Q3(y) as we defined it in the 2.B.1. Since bf 6 = y, then we let Hg = Qa(y) NK. Hence, by 2.B.2 Hg is Qg(y) — generic over V and V{y|[Hs] = V[b] and bff =y. Moreover, Qs(y) C Pu and |Qp(y)| < Ar. A different way to obtain this result is as follows: We let Q = i"Py,C.r.o.(Py) where i is the canonical complete embedding defined in the proof of 2.B.3 i: Pu, C.r.o.(Py). 64 Chapter 2. _The inner Model V(H) We let g be the Q — generic over V filter induced by i" FP . Then, we consider the forcing r.o.(Py)/g € V[G] = Vip]. We let K be a r.o.(Py)/g — generic filter over V. Then K is r.o.(Py) — generic over V such that KN Py =g. Therefore, ox[ @ = y. Now again, Hg is Qg(y) — generic over V and so ViyllHs] = Vbv1,.o.p,,,,(K] = VIN. 2.B.9 Lemma O 2.B.10 Lemma Let 7 be a Py, — name such that |K— [(7 is Pu, — generic over V) & (x*[n C7 Cx*[nUAfa)] Pu, andk <a<f<.. If G* isa Py — generic filter such that V[G*] = V[G] and {x} n;A) € G*, then in V(H) there exists a Py, — generic sequence h such that 7VI" = G*fa. Proof: Without loss of generality we identify the forcing Py, with (Pu, )qx#rnaray = {t € Pu, |r < (xP n;AP 6)} and Py, with (Pu, )(xepn;Afa)- Then we still have that IK-~ [(7 is Py, — generic over V]. Pu, 2.B. The Prikry-generic sequences in V(H) 65 We use the standard argument (see [Jech 78]) to show that there exists a Po € Py, such that Vp S Polb € 7], , op, #9 o(Py, Following that argument we define Y={sé€ro(Py,)| [Be Do, = O}. 0(Py, We let so = supY € r.o.(Py,). But then, [5 € 7] = [Ax e Wx e7)] r.0.(Py,) r.0.(Py,) =) {ke Lowe, |x € Y} = 0. We let po € Py, such that pg < 759. Then Vp S pol € 7] # 0. r.0.(Py,) The above argument shows that the map e: (r.o.(Py, )) 7 r.o.(Py,) defined by e(p) = Ip € Toy ) .0.(Py, is a complete embedding. Without loss of generality we assume that pp = 1y.0.(Pu,): Then, we get that G*fa is Py, — generic over V. We set Q:@ e"r.o.(Py,) and k™ to be the Q — generic filter over V generated in Q by e”(G*} a). By 2.B.7 we find a Py, — generic sequence in V(H) such that FP NQV=k*, 66 Chapter 2.__ The inner Model V(H) i.e., peGla << e(p)eFP ~ B <> [pe A, o.Pu,) € Fi => pe (7)VH, Therefore, 7¥l4] = G*f a and h € V(H). 2.B.10 Lemma O 67 Chapter 3 The partition relation in V(H) 3.A The way from V(H) to V[G] In the following section we are going to prove that for every Py, — generic sequence T V with 7 € V(H) we can find a Py, — generic filter G* over V such that og-[ 2 = 7 and H = U{ P(K) N V[G*f a] | @ € [«, A) }. In this first section we are defining the following forcing Q. 3.A.1 Definition We define, in V(H), the following forcing Q: (x,n,a,A)EeQ: 4S (xf nA)EPy & (x*fn)-xf[n,w) is a Py, —genericoverV & xe€V(H) & & |F- [Ay(y is Py — generic over V & Q» & xfncycxfnUA & yla=xfa) where Q): © Coll(w, (2?")+) and x°f n(x(k) Na|k <n). The partial order on 68 Chapter 3. The partition relation in V(H) the set OQ is defined as follows: For (x,n,a,A), (y,m,@,B) E Q (x,n,a,A) <g (y,m,@,B): el, (xfnjA) < (yfm;B) & xfB=yfZ. 3.A.1 Definition 0 3.A.2 Proposition Let (x,n,a,A) € Q and 8 € (a,X). Then, there exists (y,m,@,B) € Q such that (y,m,G,A*) <g (x,n,a,A). Proof: Since (x,n,a,A) € Q, then Vix] lK-— [Ay(y is Py — generic over V & xfncycxfnUA & Q) & yfa=xfa)]. As in the proof of 2.B.7 we find some y € [@, A) and we construct a tree Tg on P,.(Z) in V[G[ 7] such that every infinite branch of Tg is a Py, — generic sequence z and this branch z has the following properties: 1) x*fn Cx*fnU Af @ and 2) zf a = xf a(= x*fn Ux [n,w)). Since x, 73 € V[Gf 7] and V[G[ 4] is a generic extension of V[x] via a forcing R of size < (2?")+, then R x Q) collapses (22")*+ when we force with it over V[x] and R x Qy has size < (22")*. Hence, by Solovay’s result R x Q) is equivalent with Q) in Vx], i-e., r.o.(R x Q)) = r.o.(Q)). = 3.A. The way from V(H) to V[G] 69 We let K* be a Q) — generic over V[G[ 4]. Then G[y x K* is R x Q) — generic over V[x] and so there exists a K which is Q, — generic over V[x] and satisfies the following: V[GP y][K*] = VixI[K]. Since Vix) lK-~ [Ay(y is Py — generic over V & xfnCyCxfnUA Q) & yla=xfa) holds, then V[x][K] & dy (y is Py, — generic over V & yla=xfa & xfn cy cx tnUAls). Thus V[x][K] — [TZ is not well — founded]. By the absoluteness of the well — foundedness property, we conclude that V{GT 7] & [Zs is not well — founded], i.e. V(G} y] = [Zs has an infinite branch]. Let y be such a branch. As we have seen in 2.B.8 the construction of the tree 7g forces that y is a Py, — generic sequence such that ylo=xfa & x*fncycx nua. “Sa scare 70 Chapter 3. The partition relation in V(H) In addition, y is in V(H) as V[G[ 4] c V(H). Next we have to show that (xf nyf [n,w),n,3,A) is a condition of the forcing Q. In 2.B.9 we defined a forcing Qg(y) C Py and Qg(y) € V[y] so that if we force with Q,(y) over V[y], then we add generically to V[y] a Py — generic sequence w with the property that xfnCxfnUA & yfB=wf PZ. Then, in V[y] the forcing Qg(y) x Q) has size < (2?")+ and collapses (22")* so it is equivalent to Q). If K* is a Q) — generic filter over V[y][w], then we can find a K which is Q) — generic over V[y] such that Viy][w]["] = VIy][K]. Therefore, Viv] |-K-— Aw(w is Py — generic over V & xfnCwcxfnUA Qa & wlS=yf 8). Set ¥ =x[ny) [n,w). Then, (¥,n,3,A) € Q and (¥,n,0,A) <o (x,n,a,A). Hence, the proposition has been proven. 3.A. The way from V(H) to V[G] 71 3.A.2 Proposition 0 The next result shows that any Prikry generic sequence in V(H) can be lifted high enough in V(H) so that it captures any fixed restriction of the generic sequence induced by G. 3.A.3 Lemma Let a’ € [k,A) and (x,n,a,A) € Q. Then, there exists a 6 € [k, A) and ay* € V(H) so that xtn=y*fn & yfa=xfa & xtncy*cxfnuAfé & & Gha’ € Vly’). Proof: We assume towards a contradiction that the conclusion of the assertion of the Lemma fails, i.e., we assume that for some fixed (x,n,a,A) € Q and a’ € [k,A) V(H) & may*(x*fn C y* Cx nU APs & yta=xfa & Gfha’é Vly’). Then, we make the following claim: 3.A.3.1 Claim V{[x, Gf a’] FE (VR non — atomic forcings) 7(4p € R) [p IK 464g¢(5, g, Gf a, x, a) ] 72 Chapter 3. The partition relation in V(H) where ¢ is the formula o(6,2,G a’,x,n): gis Py, — generic ovr V & Gla’ Ee Vig] & 6 & gla=xfa & & xtncgcxTnUAle. Proof of the Claim: If the claim fails, then there exists a forcing R € V[x, Gf a] and for some K which is R — generic over V[x, Gf a] there exist 6, g so that V{[x, Gf a’][K] - (4, g, Gf a’, x, n). Since Gla’ € V[g], then we find a Py, — name 7 such that (7)VIGI =Gha’. We let Ao = {[p € 7 P o= {[b¢ They, |p € Pu} and B; the Boolean subalgebra of r.o.(Py,) generated by 4%. Then B; g is B; — generic over V and moreover, if h is any Py, ~ generic over V such that B} Ng = B; Mh, then (1) = Gh al = (VE. 3.A. The way from V(H) to V[G] 73 Let i: Pu, — r.o.(Py,) be the canonical embedding which we defined in 2.B.3. Since xfa is Py, — generic over V, we let k* be the i’Py, — generic filter generated by i” Fy, in i"Py,. By the definition of i, if h is a Py, — generic w — sequence over V such that k* = i’Py, NF, then hfa =xf a. In addition, we find n € (6, A) such that V[GPm] [Us| = &. Hence, as in 2.B.6 we can find in V[G[ 7] a sequence (A, | n € w) such that (Vn<w)(A, EV & (|An| <«)¥) and Us=U An. n<w We assume that A[6 € Ap and we define Ay = N{B | 3k < n(B € A,)} and so, (AF |n €w) € UZ NV[Gf y] and moreover, VB € Usan < w(A* CB). 74 Chapter 3. The partition relation in V(H) Then, in V;: © V[Gf 1] we define by the recursion on w a sequence (T,* | n € w) as follows: (Po,---;Pur) € To’ 4S (Po, ... Par) € [P(X & & (Wk <n’)(Px ¢ Ag) & xn (Po,...,Py-i1) & PLEeAs & &Vz € k"Vw € Br N g((Po,.--,Pa—13Ap) || z, w) and (Po,---;Pm') € Trg, : SS (Poy... Pm) € Tt or or [(Sk < m’)((Po,...,Px) € 7,7) & &PyyesPmié At & PLeAt,, & &Vz €k*Vw e BNg We set T* = U 7," and we define a tree T on P,(6), n<w ie. T C [P,.(6)]@ and Vz € [Px(6)] [le eT: 4% sy eT (2c y)]. Obviously, T € V;. Moreover, if y is a Py, — generic sequence over V such that Ga’ € V[y] and yf a=xfa, then FINB;= ENB. & k*=Feni’Py, Sees 3.A. The way from V(H) to V[G] 75 and y is a branch through T. Conversely, if y is a branch through T, then y is a Py, — generic sequence and in addition F? 1 B;, F?ni’Py, are B; and i"Py, — generic (respectively) filters over V with the property: (Vw € EN B;)(W2 € k*)(Wa! € FS) [le || w, 7). Therefore, gO B; = FEN B; and k* = FP ni'Py,. This implies that (#)¥l = Gha! € Vi] and yla=xfa & x*fncycxfnuAa. If we have chosen K to be R — generic over V,, then we would have V[K] F T is not well — founded and so Vi F T is not well — founded. Hence, V[Gf 7] & [TZ has an infinite branch] See roe Se. 76 Chapter 3. The partition relation in V(H) and since V[G[ 7] C V(H), then V(H) & ay [xh cycx}nUA & yfa=xfa & Gfha' € VIy]|. But the latest contradicts our assumption that no such y exists in V(H). Therefore, our claim has been proven. 3.A.3.1 Claim O Since our latest claim holds, if x, Gla’ € V[Gf @] for some 3 > a,a’, then we can find a Py, —name 7 and a Py —generic filter G* such that (xf n;A) € G* and V[G] = V[G*]. This is possible by the homogeneity of the Supercompact Prikry forcing Py. Moreover, the following holds: Ir G is Py, — generic over V) & (x*fnc7tcxnUAfa) & & (viF.the] k wera’, *))| where wp a’,7): S$ — (VR non — atomic forcings)=(4p € R) P I~ 363y9(5,y, (It vy (zy a) and ¢ is the formula which appears in 3.A.3.1. We apply 2.B.10 to find a Pu, — generic sequence h € V(H) such that (7)¥"l = Gha. 3.A. The way from V(H) to V[G] 77 We let 6 be an ordinal in (8,A) such that h € V[G*[ 6]. We consider R to be the forcing so that V[G*} 4] is an R — generic extension of V[G*} a, hf al. Since V[G*T 6] = V[G*} a, bf a’) [L] and ViG*ta,bfa'l[L] F (G*d)fa=Gta & & hla eViGts & & (xP n;Ad) € G*P6, then Vib} [V[(7)"™, bf a’) & Y(hf a! (7)¥™)], V[G*fa,hfa’][L] | 3éAy(y is Py, — generic over Ve & & yfa=Gta & xfncycx fnuAfé & & hfe’ € Viy)). But then, there exists a p € Py, such that p — (Vir, Tfa’] EK (AR anon — atomic forcing)(Jq € R) Up a | 363va(6,y, (Pray (#3), which is contrary to IK [(+ is Py, — generic over V) & (x*ncrcxsnuAfa) & Pu, 78 Chapter 3. The partition relation in V(H) & (Vi7,rpe] Fut a’,7))). The above argument shows that for any (x,n,a,A) € Q and any a’ € [k, A) there exists a 6 € [k, \) and ay* € V(H), y* which is a Py, — generic sequence over V such that y‘ta=xfa & (x*fn;Af6) € Ff. with Gf a’ € V[y*]. Therefore, the proof of our lemma is complete. 3.A.3 Lemma 0 However 3.A.3 can be generalized in terms of the following result: 3.A.4 Corollary Let (x,n,a,A) € Q and Y € H. Then, Aé4y* € V(H) such that x{ a = y*/a and xn cy* cx fnuafé and Y € V[y*]. Proof: Since Y € H, then there exists an a! € [k, X) such that Y € P(k)N V[Gf a’). By 3.A.3 we can find 6, y* such that 6 € [K,A) & y* isa Py, — generic sequence over V & y*eEV(H) & x fncy*cxfnuAfé & & xfa=y*fa & Gha' Ee V[y’*]. 3.A. The way from V(H) to V[G] 79 But then, P(«) N V[Gf a’] € V[y] and so Y € V[y*]. 3.A.4 Corollary O 3.A.5 Proposition Let (x,n,a,A) € Q and Y € H. Then, the following holds: 46 € [x,A)dy € V(H)(Y € Vly] & (y,n,6,B) <9 (x,n,a,A)). Proof: We use 3.A.4 to find a y* € V(H) so that y* is a Py, — generic sequence over V and yfn=xtn & y*fa=xfa & y*f{nw)cAlé & Ye VIy’]. Then, as in 3.A.2 we consider the forcing Q5(y*) in V[y*] and by forcing with it we add generically to V[y*] a Py, — generic over V sequence w such that whd=yf6 & (x,n,a,A) € F?. Then, in V[y*] we consider the forcing Q) and the product forcing Q;(y*) x Qy in V[y*]. As before, Qs5(y*) x Q) is equivalent to Q) and every K* which is Q, — generic over V[y*][w] induces a K which is Q, — generic over V[y*] so that Viy"IIw]EK"] = Viy"][b). 80 Chapter 3. The partition relation in V(H) Moreover, Viy")[K] & daw wis Py — generic over V & & wlhé=y td & & (xfn;A)é€ Fi). Hence, Viy*] [-K— Aw [w is Py — generic over V & wi5 = y*f6 Qr This shows that (y,n,6,A) € Q where y = xf n~y*f [n,w) and in addition (yn,6,A) <q (xn,a,A) & Y € Viy|(= Vly’). This completes the proof the above proposition. 3.A.5 Proposition O 3.A.6 Corollary Let Y € H and Dy = {(x,n,a,A) € Q| Y € V[x}}. Then, Dy is a dense subset of the forcing Q and Dy € V(H). Proof: It follows easily from 3.A.5. 3.A.6 Corollary 0 3.A. The way from V(H) to V[G] 81 3.A.7 Proposition Let G* be Q — generic over V(H). Then G* induces a Py — generic filter over V. Proof: Let G* = {(w;A) € Py | 4x,n,a, A’((x,n,a,A’) € G* & xflh(w)=w & & A'CA)}. This is the subset of Py that G* induces. We claim that G* is Py — generic over V. We define o* = (J{w | 3(x,n,0,A) € G(w = xfn)}. Then, we make the following claim: 3.A.7.1 Claim G* = {(w;A) |w C o* CWUA} and o= U{w | SB((w;B) € G*)}, 1.e., o* is the Py — generic sequence induced by G* and G* is the Py — generic filter induced by o*, provided that G* is Py — generic over V. Proof of the Claim: We assume that (w;A’) € G*. Then there exists a condition (x,n,a,A) € G* such that (xfnj;A) < (w;A’). We fix B € U and mcéw. Let Dnp={(x,n,a,A)EQ|/ACB & n>m}. 82 Chapter 3._The partition relation in V(H) We show that Dyn is a Q — dense set. Let (y,k,a,A) € Q and k < m. Since then Viy] |-— [Ay* (y* is Py — generic over VV & Qo & (yfkA)ey* & y*fa=yfa). We fix a Q) — name y* such that Viyj IK [y* is Py —genericoverV & (yfksA) ey" & yfoa=(yfa). Q) Then dn > m such that Viv] IF [y* is = Py — generic over V & (y[k;A) € y* Q) & yfa=(yfay & y*f[nw) CBNAI. Let us find a q € Q) and a w € [P,(A)]"~* such that Vil q [-— [y"Tlk,n) = w). Qr Then, we set ¥ =yfk-weyf [n,w). Obviously, (¥,k,a,A NB) € OND ap. Therefore, Dp is a Q -dense set. Hence, if (w;A’) € G* and (x,n,a,A) € G* such that (xf[n;A) < (w;A’), then we can find a (z,m,a,C) € G* such that m>n & CCA’ & (z,.m,a,C), (x,n,a,A) € G*. 3.A. The way from V(H) to V[G] 83 Since (z,m,a,C), (x,n,a,A) € G* and as m > n, then z[n = xfn and zfa = xfa. However, as (Vm’)(Di,a: is Q — dense), then wco*cwUA’. Conversely, if w C o* C wUA, then 3(x,n,a,A’) € G* with n > lh(w) such that w C xfn and A’ C A. Then, (xf n;A’) < (w;A), 1.e., G* = {(w;A) | w C o* CwUA}. So o* = og according to the notation established in Chapter 2. Next we will show that G* = F.. Towards this goal, we take w C o*. Hence, d(x,n,a,A) € G* such that w C xfn. Then (xfn;A) € G* > wc U{w | SB((w;B) € G*)}, 1.€., o* C U{w | JB({(w;B) € G*)}. Since dom(o*) = w = dom(a¢:), then o* = og+. This completes the proof of the above claim. 84 Chapter 3. The partition relation in V(H) 3.A.7.1 Claim O In order to finish the proof of our proposition, we have to show that o” is a Py — generic over V sequence in [P,,(A)]”. So we let A € U. Since (Vm)(Dm,a is Q — dense), then dk(o*} [k,w) Cc A) and so o* is a Py — generic sequence and G* is the corresponding Py — generic filter induced by G*. 3.A.7 Proposition O 3.A.8 Proposition Let G* be a Q — generic filter over V(H) and G* is the Py — generic over V filter induced by G* as it was defined in 3.A.7. We set H& =(}{P(«) A V[G*f a] | a € [k, A)}. Then, H = H® where H = HS. Proof: Let Y € H. By 3.A.6, we have that Dy = {(x,n,a,A) € QO] Y € Vix]} is Q — dense and Dy € V(H). So by the genericity of G* over V(H) we have that G*ND,y £0. ive 3.A. The way from V(H) to V[G] 85 If (x,n,a,A) € G* ND, then xf a = o*fa and Y € Vix] = Vio"ba] = Vict al. Therefore, Y € HS”. Hence, H C H@. Conversely, as QO € V(H), then Va € [k, \)(G*} a € V(H)) and so P(k)N V[G*fa] C H. Thus H® CH and so H = H®. 3.A.8 Proposition G fa fig ina on 86 Chapter 3. The partition relation in V(H) 3.B From V(H) to V[G]: Another Forcing The purpose of this section is to demonstrate that there exists another, more combinatorial forcing, which can produce the same effects as the forcing Q of the previous section. As usual, we begin with a definition. 3.B.1 Definition A’ be a set in Us where U is a P,.(3) normal measure and Us: Sf Ut 6 = {AP 5 | A EU} andk <6<£8<X. Then, A’ is 6 — very good for A aon Vx € [A’]S°VP € A(PN6 Cx dwe [A]J““(PCw & whd=%)). 3.B.1 Definition 0 3.B.2 Definition LetU be aP,(@) normal measure. A condition p = (x;A) € Py is called 6 — very nice iff A[ 6 is 6 ~ very good for A. 3.B.2 Definition O Remark: We recall that the forcing Py is defined as follows : p=(sA)ePy: 4% xe [P.(B)]<* & AEu & (WPEA)(XCP), where x C P means that if Q = x(Ih(x) — 1), then Q CP, i.€., Q¢eP & |Q| <|PNk|. 3.B. From V(H) to V[G]: Another Forcing 87 3.B.3 Lemma Let U be a P,,() normal measure. For every 6 € [k,) and for every A €U there exists a set A’ which is 6 — very good for A. Proof: We recall that the measure U; satisfies the following partition property: For every coloring £: P,(5)<" > 2 there exists a set A’ € Us such that Vn € w|f”[A]"| = 1 where So we define the following coloring of Af 6(€ Us). For x € [Af 6]<* 7 0, if V(P) € [A]<'((PN 6) Cx— dwe [A]<*(P Cw& wd = x)) w= 1, if otherwise. Since f is a coloring of [Af 6]<* in V and A} 6 € Us, then there exists a set A” € Us such that A’ C Af 6 and A’ is homogeneous for f. We show that (Vn < w)(f"[A’]" = {0}). Let us fix n € w. In addition we fix a Py — generic filter over V. Then, as A’ € Us and o@ is the Py, — generic over V sequence induced by Gf 6, we can find m € w such that ob f [m,w) cA’. Let pal = o%,f [m,m +n]. 88 Chapter 3. The partition relation in V(H) We are going to show that f(x) = 0. Let (P) € [A]<! such that PN 6 C x, ie. P16 C X(0). Then, (P64) “ogf [m, w) is a Py, — generic over V sequence. We set A={QeEA|P CQ} and we consider the following condition: ((P);A) € Pu. By 2.B.10 we can force to find a z which is Py -generic over V sequence such that zf 6 = (P16) ~o2} [m, w) andPCzUPCA. Next, we let w = z[[1,1+k). Since w € [P,(@)]<”, then Ww € V and w é€ [A]<* c [A]<* and P Cwk w{é =x. Thus VP) € [A]“'((P6) CR AWE [ATS*(PCW & wh6 =). In particular, f(x) = 0. Since X € [A’]" and A’ is f-homogeneous, then f”[A‘]® = {0}. As n was arbitrary then, f”[A’]<* = {0}. This shows that A’ is 6 — very good for A. Therefore, the lemma has been proven. SSS SSeet 3.B. From V(H) to V[G]: Another Forcing 89 3.B.3 Lemma O 3.B.4 Corollary Let U be a P,,(3) normal measure. For every 6 € [k, 3) and for every p € Py there exists aq € Py such that q < p and q is a 6-very nice condition. Proof: Assume that p = (x;A). By the previous lemma we find a set A’ which is 6 -very good for A. Then, we consider the set A*={PEA|PNSEA}. Obviously, for q = (x;A*) we have that q € Py and q < p. Moreover, we claim that A*[6 = A’ and that A’ is 6 -very good for A*. First we show that ATO= Al Obviously, A*f 6 C A’. Conversely, if Q € A’, then @ € [A]<' and since ON6 C Q, then SP € A such that PM 6 = Q and in particular P € A*. So A*} 6 = A’. Similarly, if (P) € [A]S" and x € [A‘]<* and (PN 4) CX, then as A’ is 6 — very good for A, there exists a w € [A]<* such that whé=x & (P)Ccw. Since wf 6 = X € [A’]<*, then w € [A*]<” and therefore, A’ = A*} 6 is 6 — very good for A*. This shows that q is a 6 — very nice condition of Py, below p. 3.B.4 Corollary O 90 Chapter 3. The partition relation in V(H) 3.B.5 Definition Let U be our fixed normal measure over P(A) and Py be the corresponding supercompact Prikry forcing. We define the following forcing Q* in V(H) (x,n,a,A) € Q*: 4% (xfn;A) isa—very nice & & x°fnxf[n,w) is Py — generic & & xf[n,w) c Afa. 3.B.5 Definition O 3.B.6 Proposition The following set Dg is Q* — dense. De = {(x,n,a,A) € Q*|a > B}. Proof: Let (x,n,a,A) € Q and assume that a < 3. We let C; be 8—very good for A and Cz be a-very good for C, and we find m > n such that xf [m,w) C Co. Since xf [n,m) c Ata and Af a is a -very good for A, then we can lift xf [n,m) to some x’ C A such that xfa=xf[n,m) & xCx’. We set AY ={PeEA|x' CP} 3.B. From V(H) to V[G]: Another Forcing 91 and Ch = {P eC |xP8C PH. We set y = xf{n7x’ xf [m,w). Then (y?f'm;Cy) € Fe, and we can find in V[h] a z which is Py, — generic over V such that ata =y* & (y"fm;C}) € z. We have to show that Cj is @ — very good for A*. For that, we let x € [Ci]<° and F¥ € [Ay]<” such that y[ 9 CX. Since x € [CI]<* & ¥e [Aj]<”, then dw e[A]““(whé=x & FCW). Since ¥ € [A*]<”, then’ CV>X CW & We [A]<*, ie, we [A*]<. Hence, Cy is 8 — very good for A*. Set C={PeA*|PNBE CH}. Then, C{ @ = Cj and (z,m,f,C) is in Q* and (z,m,,C) < (x,n,@,A). 92 Chapter 3. The partition relation in V(H) 3.B.6 Proposition 0 Next we prove a proposition analogous to 3.A.5 for the forcing Q*. 3.B.7 Proposition Let (x,n,a,A) € Q* and Y € H. Then, the following holds: 56 € [k,A)dy € V(H)(Y € Viy] & (y,n,6,B) <g« (x,n,a,A)). Proof: We use 3.A.4 to find a y* € V(H) so that y* is a Py, — generic sequence over V and yfn=xtn & y*ta=xfa & y'f[nw)cAfé & YEVIy’). Then, we observe that (y*,n,6,A) <g+ (x,n,a,A). This completes the proof of this proposition. 3.B.7 Proposition 0 After this result we can duplicate the arguments of the previous section, in order to show the analogues of 3.A.6, 3.A.7, 3.A.8. We mention these results without proofs. 3.B.8 Corollary Let Y € H and Dy = {(x,n,a,A) € Q* | Y € V[x]}. Then, Dy is a dense subset of the forcing Q* and DY, € V(H). Shas 3.B. From V(H) to V[G]: Another Forcing 93 3.B.8 Corollary 0 3.B.9 Proposition Let G* be Q* -generic over V(H). Then, G* induces a Py — generic filter over V. 3.B.9 Proposition 0 3.B.10 Proposition Let G* be a Q* — generic filter over V(H) and G* is the Py — generic over V filter induced by G* as it was defined in 3.A.7. We set H® = U{P(«) NV[G*f a] | a € [«, A)}. Then, H = H® where H = HS. 3.B.10 Proposition O 94 Chapter 3. The partition relation in V(H) 3.C OD partitions in the model V(H) 3.C.1 Theorem We assume that ¢ is a formula and p < k so that V(H) FE Vs € [x] A!x € V,,0(8,x, 6, H). Then, there exists an s € [K]*” N V(H) and a xp € V, such that V(H) — Vt € [s]*4(t,x0, 6,H). In particular, V(H) F (x is a strong limit) & Vu< «K(k oD (wy, ). Proof: Since vi =V¥ev, and |V,,| < «, then for every x € V,, there exists an A, € U such that (OAs) Ilpo [V(H") F o(PT Rx, 6, H") where I is the canonical Py — name for the Py — generic filter. The above is possible since Py satisfies the Prikry property. As {A,|xeV,} EV and |{Ax [x¢€ VuSl <K, 3.C. OD partitions in the model V(H) 95 then the « — completeness of the measure U implies that A: S$ F{A, | xe V,} € U. If for allx € V, (0:8) I= VCH) o(b T,X, 5,01), then we get a contradiction as follows: By V(H) F Vs € [k]*Alx € V,.4(s, x, 6,H) there exists a p € Py such that p < (0;A) and xo € V,, such that ml: PIL [V(H") £ d(EP %, Xo, 6, H*)]. Also, since (Vx € V,)[(0;A) I SV(H") & o(PP%, x, 6, BY] holds, then we conclude that PIL VCH) & oft ,x, 6, HP), But this is a contradiction. Thus for some xg € V, such that ~ | = (O:A) I~ [V(H") F oP, 50, 6H"). U We use the homogeneity of the forcing Py to find a Py — generic G such that (Q;A) EG 96 Chapter 3. The partition relation in V(H) and V[G] = V[G]. We let T= 06 =0¢)K = c@e- We claim that it is homogeneous for the OD coloring ¢. In order to verify the above claim, we let s € [7]”. By the geometric property of the Prikry type forcings, s is a Py, — generic over V sequence with s C A[«. Then, we consider G* a Q — generic over V such that (s,0,k,A) € G*. Notice that (s,0,k,A) belongs to Q since for some a € |u|’ with s = Toa, then y = 0g 0 is a Py — generic over V sequence with (0;A) € y and with y[« =s. If G* is the Py — generic filter induced by G*, then G*fe=s and (0;A) €G"*. Consequently, VIG] E [VHS ) & ("TP , 0, 6, HY, 1.€., V(H°") & d(s, Xo, 6, H“). But then, by 3.A.8 H@ = H. Therefore, for all s € [7]* V(H) FE 4(s,x0, 6, H) eee OTT 3.C. OD partitions in the model V(H) 97 and so we have proven that V(H) EF [Wu < K(k 58 (w)v,) & «& is strong limit]. 3.C.1 Theorem 0 3.C.2 Lemma Assume that ¢ is a formula a € [k,X) and u € V. Then, [V(H) — ¢(u,Gfa,H)] = = [(Ap € Py)(p is an @ —nice condition & & plaeGla & & pik [VHT) F o(i,Pha,H*)] ]. In particular, there exists a formula ¢ such that [V(H) F (u, Gla, H)] => [V[Gfa] & J(u, Gf a, v)] for some v € V. Proof: Assume that V(H) E ¢(u, Gf a, H). Then, there exists p € G and p is an a — nice condition such that U But then, pfa € Gfa and so dp € Pu(pisana —nice condition & pfaeGhla & & pik [V(H) & 4@,Tfa,H")} 98 Chapter 3._ The partition relation in V(H) Conversely, if there exists a p € Py such that p is an a ~ nice condition and placGla & & pi-—[V(HT) o(u,tpa,H")), U then there exists a q € G such that ql a = pfa. Consequently aI [V(B*) & o(u, TP a, HY and so V[G] F V[H°] F 4(u, Gla, H®), V(H) & ¢(u, Gf a, H). Obviously, if we set o(x,y,z):€& (ape Py)[p is an a —nice condition & & placeyfa & & plk— VE") & o@,ta, Hy), then V(H) F (u, Gla, H) => V[Gf a] F d(u, Gf a, (a,U,Py,A)). 3.C.2 Lemma O 3.C. OD partitions in the model V(H) 99 3.C.3 Lemma Assume that b € H and H = HS where G is our fixed Py — generic filter over V. For every formula ¢ and parameters a € V there exists a formula ¢* such that V(H) F ¢(a, b) => V[b] — ¢*(a, U,«, A, b). Proof: We define the formula ¢* as follows: ¢*(a, U, kK, A, b): eek, I dy(y is a Py — generic over V A [V(H”) F ¢(a, b)].) We claim that V(H) — ¢(a,b) <> V[b] — ¢*(a, U,«, A, b). Assume that V[b] — ¢*(a,U,x«,4,b). Let K be Q) — generic over V[b] and we find G Py — generic such that V(H®) — ¢(a,b). Since b € HS, then we find € («,) such that b € V[G[n]. So we find a Py, — name 7 such that (7)¥(Gtnl = b and we consider the Boolean algebra B; generated by 7. Then, for ¥ = {EF roe la <b} we know that V[b] = V[G[ 7N4]. Since Gf nN& = {|| & € 7 I[r.o.Pu,)| @ € b}, then G[nN X € V(H). Since B; C, r.0.(Pu,), then by the 2.B.7 we can find a Py, — generic over V filter Z such that Hence, (7)¥4l = (7)VIGI = b. Then, as in the proof of 3.C.1 we find a Q — generic filter G* over V(H) such that for the induced Py — generic filter G* we have that G*fn = Z. 100 Chapter 3. The partition relation in V(H) Therefore, (+)V[G*T"l = (7)VIGT"l = b. Since V(HS) § ¢(a,b), then we might choose the Py, — name 7 such that IL V(H E (a, (7)V'OT mI) and consequently, V(H™) F ¢(a*,b). Now we apply 3.A.8 to conclude that V(HS") = V(H) and so V(H) & ¢(a,b). Conversely, if V(H) - ¢(a,b), then there exists a generic extension of V[b] namely V[b] such that V(H°) - ¢(a,b). But then, using the collapsing algebra Q)(= Coll(w,((2?")*+)) we should have that V[b] - ¢*(a,U,,b). This completes the proof of the lemma. 3.C.3 Lemma O 101 Chapter 4 The partition relation in a ZFC model 4.A <.A—DC in the inner model V(H) In this section we are going to prove that the inner model V(H) satisfies <rA-DC. < A — DC is the following statement: “For every cardinal u < X and every partial order R either there exists an R-descending chain of length y or there exists a maximal R-descending chain of length < 1.” Obviously, the Axiom of Dependent Choices (DC) is the statement < w; —DC. If A is a set, then < \ — DC differs from < 4 — DC only in the requirement that R should be a partial order on A,ie., RC Ax A. First, we are going to show that for any ordinal @ the model V(H) satisfies <A— DCoxu. In the following arguments we fix an ordinal 6 and a partial order R, where 102 Chapter 4. The partition relation in a ZFC model R C (@ x H) x (0 x H) and R € V(H). As we have seen, R is definable over V(H) by some formula ¢ with parameters u and Gfag where u € V, ag € [K, ) and G is our fixed Py — generic over V filter with H = HS. We recall that we have already proven that H = HS = P(x) M V(H). Using 3.C.3, we fix a formula ¢* such that pRq => V[Gf a0, p,q] F 4*(p,q, Gf ao). Having R, Gf ao, @* and H fixed, we are preparing for the proof of < A—DCoy4 by giving the following definitions. 4.A.1 Definition Let z € V(H) be a Py, — generic sequence over V such that zl ag = Gf ag and a € (ao, 4). If < X, then we define, in V(H), the set M¢(z) of all R— maximal descending chains of length < € which are elements of V{z], i.e., xEMe(z) :<$xEVE] & x€(OxP(K)NVE)E & (Vn <8 & (V[Gf a0, x(7 + 1),x(1)] F o"(x(7 + 1), x(n), Gf a0) & «(dp €@ x P(K)N V[z]) (Vq € ran(x))(V[G[ ao,p,q] F $°(p,4, Gf ao)). 4.A.1 Definition O 4.A.2 Definition Ifz¢€ V(H) is a Py, — generic sequence over V such that z{ a = Gf ao for a € (ao, ) and if M,(z) is the set defined in 4.A.1, then we 4.A. <2A— DC in the inner model V(H) 103 define 6(a,z,) as follows: If Dia,z,£) :% {1 E (a, d) | Sw(w is Py, — generic over VV & & weV(H) & wla=z & & Wx € Me¢(z)(dp € 8 x P(k) N V[w])(Vq € ran(x)) [V{Gh 0, p.4] F 6°(p,4, Gl ao)], then we let ND(a,z,£), if D(a,z,£) #0 0, if otherwise. 6(a,z,€) = 4.A.2 Definition 0 In the following two propositions we will show that under some appropriate assumptions D(a, z, €) is non-empty and that (a, z, €) is independent from the Py, — generic sequence z. 4.A.3 Proposition Assume that z € V(H) is a Py, — generic sequence over V such that zf a9 = Gfao for a € (a,A) and that in V(H) there exists neither an R — descending sequence of length € nor a maximal (in V(H)) R — descending sequence of length < £. Then D(a,z, €) is non-empty and so 6(a,z,€) € (a, A). Proof: First we observe that M¢(z) is non-empty. The reason is the following: 104 Chapter 4. The partition relation in a ZFC model If in V[z] there exists an x € (8 x P(K) N V[z])§ N V[z] such that (Wn < €)(VIGT a0, x(7 + 1),x(m)] K 4 (x(n + 1),x(n), Gh a0), then by definition of ¢* this will imply that | (Vn < €) [x(n + 1)Rx(n)] and as x € Vz], then x € V(H). But this proves that V(H) contains an R — descending chain of length €, contrary to our hypothesis. Therefore, in V[z], there is no R — descending chain of length €. As V[z] — AC, then V[z] contains an R — descending chain x of length < €, | so that V(H) thinks that x is maximal for R[@ x P(«) N V[z]. Hence, M¢(z) ! is non-empty. Moreover as M¢(z) C (@ x P(k) N V{z})<é and since } V([z] & [A is inaccessible], then there exists a cardinal p < X and a bijection 7 so that Since, according to our hypothesis, every x € M,(z), although a maximal R — descending sequence of length < € in V[z], cannot be maximal in V(H), then there exists a Py, — generic sequence y € V(H), for some 7 € (a, A), so that yfa =z and (Sp € 8 x Pl) AV{y})(Va € ran(x))] VICE a0, p, 4] F 4°(P,4, Gh a0)]. 4.A. <2— DC in the inner model V(H) 105 Hence we can define in V(H) a map 7 : yt — 4 as follows: a(B):2 Mn €[«,A) | (Ay € V(H))(y is Py, — generic over V & & yla=z & (Ap € @ x P() N V[y])(Va € ran(a~"())) [VIGt a0,P.4] F 4*(P,4, Gh a0)] By the above remarks, we conclude that 7 is a well-defined map in V(H) with ran(7) © A. Since yz < and V(H) § [A is regular], then sup7#”u < X. We set y = sup7” ps < X. Next, we show that if w is any Py, — generic in V(H) such that wf a =z, then (Vx € Me(z))(Ap € P(«)NV[w])(Vq € ran(x))|VIGP00,p,4] F "(p,4, Gf ao). In order to prove the above claim, we fix an x € M¢(z). By definition of 7, we know that there exists a Py, — generic y such that ylo =z, and moreover, Viy] F Gp € 8 x P(n))(V¥q € ran(x))| VIGE a0,p,4] F #*(P,4, Gl 20)}. Let us fix w an arbitrary Py, — generic in V(H) such that wf a =z. Also we fix a Py, —name x so that (x)¥@! = x. Then, we can find an a—nice condition 106 Chapter 4._ The partition relation in a ZFC model (as we defined it in 2.4.10) q € Fj such that q LT [Ap € 8 x P(n)¥q € (x) IV [ET a9, p,q] E o(p.a, Pf a0)]. Uy Since qhaeF] & Fifa=F?, then there exists a condition s € F7 such that qia =sfa. At this point we use the fact that q is an a — nice condition and we apply 2.A.12 to conclude that s IL— [Bp € 8 x PlnWa € (x)¥ IV FT a0, p,q] E "(p,q Pf a0). Since s € FY, then Vw] F [(Ap € 0x P(x))(Vq € ran((x)¥Te!))[V[wl a0, p,q] F $°(p, 4, wf a0). But then we recall that (x)Vlel = x and that F7f ag = Gf ag. Therefore, (Sp € 4 x Pl) M-VIw)(¥a € ran(x))|VIGT a0, p,4] F 4°(P.4, Gh 20)| and this shows that the claim holds, i.e., if w is any Py, — generic in V(H) such that wf a =z, then (Vx € Me(z)) (Sp €@x P(K)N Viw})(Vq € ran(x)) [VICI 20,p,4] F #°(p,4,Gha0)]. In particular, we have shown that ayaw(w is Py,-—generic & we V(H) & wla=z) & 4.A. <—DC in the inner model V(H) 107 & Vx € M,(z)(Ap € 6 x P(K) N V[iw])(Vq € ran(x)) [Vict a0,p.4 F $'(p,4,Gha0)}), and ayVw[(wis Py,—generic & we V(H) & wla=z)=> => (Vx € M,¢(z))(Ap € 8 x P(K) N V[w])(Vq € ran(x)) V[GT ao, Pp, a] = $"(p, 4, Gf a0). Therefore, the set D(a, z, €) is non-empty and consequently 6(a, z, €) is a well— defined ordinal in (a, ). 4.A.3 Proposition 0 4.A.4 Proposition Assume that z € V(H) is a Py, — generic sequence over V such that zf ag = Gf ao for a € (ao, A) and that in V(H) there exists neither an R-descending sequence of length € nor a maximal (in V(H)) R— descending sequence of length < £. Let o* = oc a is the Py, — generic sequence generated by Gla. Then 6(a,z,€) = 6(a,0%, €). Proof: We fix an ordinal 6 € (a, ) such that 8 € D(a,z,€) with witness w, i.e., w is a Py, — generic in V(H) such that w/a =z and | Wx € Me(z)(Sp € 8x P(K)NV|w])(¥q € ran(x))|V{GP ao, p,q] E 2"(P, 4, Gha0)}. Since wf a = z and zf ap = o°f ao, then we can find a Py, —name w such that lIK-- [wis a Py, — generic sequence & Vx € M¢(wf a)(Ap € 8 x P(K) A V) [vw] Pu, (Wq € ran(x))[V [wf ao, p,q] F $"(p, 4, wf ao)]. 108 Chapter 4. The partition relation in a ZFC model Then, we use 2.B.10 to find a Py, — generic filter K in V(H), so that (VIN) = arg. Therefore, (VxEe Me((GTA)fa)) (Ap € @ x P(K) N VIGT A])(Vq € ran(x)) Hence, Gf @ is a Py, — generic such that (Gf @)f a = Gf a and (Vxe M,.(Gfa)) (dp €@x P(K)N V[Gf 6])(Vq € ran(x)) This implies that 6(a,z,€) = 6(a, 0%, €). 4.A.4 Proposition O 4.A.5 Theorem For every ordinal 0, the inner model V(H) satisfies <A—- DCoxn. Proof: Let R be a partial order on 6 x H, ie., R C (89 x H) x (9 x H). Let us fix € a regular cardinal below \. As in the beginning of this section, we fix an ordinal a € [K, A) and a formula ¢* such that pRq => V[Gf ao, p,q] F $*(p,q, Gf ao). We assume towards a contradiction that in V(H) there exists neither an R — descending sequence of length € nor a maximal (in V(H)) 4.A. <A— DC in the inner model V(H) 109 R — descending sequence of length < €. We set 65 to be the least ordinal 7 above ap so that (Ap,q € 8 x P(K) N V[Gf 7])VIGT a0, p, a] F *(P, 4, Gf a0). Then, we define by recursion on ¢ < € inside the model V(H) a & — sequence of ordinals (6¢1¢ < &) as follows: Set 69 = 6. If (5¢ | ¢ < 8) has been already constructed, for some 2 < €, then to define 6g we consider two cases. CASEI: B=at+l1 In this case we set 63 = 7: a, dye V(H) (yisa Py, —genericoverV & & 7 = 6(ba,Y,€)). CASE IT: £ is a limit ordinal. Since the construction of (6¢ | ¢ < 3) took place in V(H), then by the regularity of \ in V(H) we conclude that (6; | ¢ < 8) is bounded below \. Assume that eZ sup{d¢ | ¢ < 8}. Then we set 6g = 7: & aye V(H)( yisa Py, — generic over V & & 7=6(6",y,§)). 110 Chapter 4._ The partition relation in a ZFC model This completes the recursive definition of (6 | ¢ < £). Since the construction of this sequence took place in V(H), then it belongs to V(H) and by the regularity of \ we conclude that (& | ¢ < €) is bounded below . Let 6*: 2 sup{é¢ | ¢ < €}. Next, we consider the Py,, — generic over V filter G[6* and in V[Gf 6] we construct an R — descending chain of length € to get a contradiction. Using 4.A.3 and 4.A.4 we conclude that for any @ < & fast = 6(5as0%, €)), | and for any limit ordinal @ < & 5g = 6(5",0° ,8)), where 6* = sup{é¢ | ¢ < SB}. In V, we fix a wellordering < of all Py, — good names (for a < 6*) of elements of (0 x P(«))<¢. By recursion on ¢ < &, we construct a € — sequence (x¢ | ¢ < €), which is an D-descending chain of R — descending chains of length < €. Our goal is to make the above construction in V[Gf &*]. We set Xo :2 4-—least Py,, — good name x for an element of (8 x P(K))<§ N V[Gf do] such that (x)¥!ST%) € Ad_(Gf d). 4.A. <.A—DC in the inner model V(H) 111 Then, we define x9: SE (sip) VIG, We assume that (xc | ¢ < ) - for some 2 < €- has been already constructed in V[Gf 6], and that it satisfies the following inductive hypothesis: 1) (VC < B)(An)(xe € Me(Gf4,)) 2) (Vy <7’ < B)(x, C x). In order to define xg, we consider two cases. CASEI: B=a+t+l1 By the inductive hypothesis, x, belongs to some M,(Gf6,). Using the defini- tion of 6,41 we set , def Xg :=<4-—least Py, | — goodname x for an element of (9 x P(K))< N VG 6,41] such that (x)V!Ghém+] © 4 where A: © {x € Me(Gf 641) | Xa C x}. Then, we set x9: (%_) Vita, CASE IT: G is a limit ordinal. By the choice of < and the above construction, we conclude that (Xa | a < B) € VIGf &], eas 112 Chapter 4. The partition relation in a ZFC model where 6* = sup{6,, | a < ¢}. So, we consider x =u, x¢. According to the inductive hypothesis, z is a well — defined R — descending chain in V[Gf 6]. Let ng: = sup{n¢ | ¢ < B}. Using the definition of (6 | ¢ < €), we can define the following: XB :& a least Py, — goodname x for an element of ing (8x P(K))En V([Gf 6,,] such that (x) V(GP engl € B, where B: © {2 € Me(Gf 6,,) | x C 2}. Then, we set X@: def (xg) V IGT onal It is easy to verify that in both cases the xg satisfies the inductive hypothesis. Therefore, we can construct in V[G] 6*] the sequence (x¢ | ¢ < €) with the following two properties: (A) (VO < €)(An)(x¢ € Me(Gf 4,)). (B) (Wy <7 < )(x, C xy). Thus, if we set 4.A. <X—DC in the inner model V(H) 113 using the properties (A) and (B) we would have that w is an R — descending chain of length € in V[G[ 6*] and consequently is in V(H). But the latest is contrary to our hypothesis that no such chain exists in V(H) and so we get a contradiction. Therefore, < 4 — DCgx holds in V(H). 4.A.5 Theorem O 4.A.6 Theorem The model V(H) satisfies < \ — DC. Proof: Let R be a partial order in V(H). If sg: field(R), then by the definition of V(H) we conclude that for every element x of S there exists an element z of V, a member w of H and a formula ¢ such that x is definable in V(H) by ¢ with parameters z and w. Then, using Replacement in V(H), we can find an ordinal a with the property that each element of S is definable over V(H) with parameters from V, which appear in VY"). But then all the V — parameters necessary for the definitions of the elements of S appear in VY. Using this fact we define in V(H) a surjection 7 : VY x H 2"8 S. Since V — AC, then we find in V a well — ordering of VY, say (x, | a < 6) is an enumeration of VY in V for some ordinal 6. Next, we define a partial order R* on 6 x H by lifting R via z, i.e., (y,a)R*(6, b): m((X,, a))Ra((xs,b)). Since m is a surjection, then for every pt < A, every maximal R* — descending chain of length < yu induces via 7 a maximal R — descending chain of length < x. Then, we apply < A — DCgy.4 114 Chapter 4. The partition relation in a ZFC model in V(H) for R* to prove that < \— DC holds for R in V(H). This is possible, since the 7 —image of an R* —descending chain of length us is an R—descending chain of length p. Therefore, < 4 — DC holds in V(H). 4.A.6 Theorem 0 Next, we are going to generically force choice over the model V(H). Since this is equivalent to adding generically a well — ordering of the set H, we use in V(H) the following canonical forcing P. 4.A.7 Definition For all A, we define in V(H) AéP: &% Aisafunction & dom(A)<A & ran(A) ¢ H. The order on the set P is the reverse inclusion, i.e., A< BS BC A, and we denote the poset (P , <) again by P. 4.A.1 Definition 0 4.A.8 Lemma Let x € P(k) NM V(H)(= H). We set D, = {A € P| x € ranA} and for B < Eg ={A€ P| 6 > lh(A)}. Then, D, and Eg are P — dense in V(H). Proof: Let B € P and assume that x ¢ ran(B). Since a: dom(B) < 4, we set A= BU {(a,x)}. Obviously, A € P, A < B and x € field(A). Therefore, D, is P — dense. 4.4. <X-—DC in the inner model V(H) 115 In order to show that Eg is P — dense for a fixed @ below , we consider a condition B = (x¢ | € < a). Since a < A then Ja] < «. We pick an injection g from a into x. Using g we code B by the set a = VU. {g(€)} x xe. Then a C « X K and a € V(H). This implies that a € V[G[¥] for some 7 € [k,). But then we can find a large enough 6 € [k,A) above y such that V([Gf 4] — (|@| = «). Therefore, in V[G[ 6] we can find a sequence b of sets in P(«) \ ran(B) of length 6. Then B~b defines a condition A in P of length > 8. Moreover, A < B and A € Eg. Hence, Eg is P — dense. 4.A.8 Lemma 0 Let G be a P — generic filter over V(H). Then, (Vx € P(«)N V(H))(Dx NG #9). We set g:@ UG. Since G is a filter, then g is a well-defined map with ran(g) C P(«) and domain contained in 4. By 4.A.8, we have that ran(g) = P(«k)N V(H) & dom(g) = A. Thus any P — generic over V(H) filter G induces a surjection g such that g : \ — H. Therefore, H is well — ordered in V(H)[G]. Next, we will show that P is an almost homogeneous forcing. If A,B € P with domains yz and v respectively, then A~B denotes the concatenation of the two sequences, i.e., A(a), ifa<y A~B(a) = BB), ifa=B+u. 4.A.9 Proposition The forcing P is almost homogeneous. 116 Chapter 4. The partition relation in a ZFC model Proof: The almost homogeneity of the forcing P is equivalent to the following fact: For every condition p € P and every P—generic filter G there exists another P — generic filter G*, with the property that V(G| = V[G*| and p € G*. Following the above characterization of the homogeneity, we consider A to be a condition in P and we fix G to be a P — generic filter over V(H). Let g denote the generic surjection g : \ “8 P(«)NV(H), induced by G. Obviously, G is the set of all initial segments of g. Let 4p = dom(A). Using g and A, we define a surjection g* of length \ onto H, as follows: A(a), ifa<yp g(a) = g(8), ifa=B+yp. Since g is a surjection of A onto H, then so is g* and moreover if G* is the P — filter induced by g*, then A € G*. It remains to be seen that G* is P — generic over V(H). We argue towards a contradiction and we assume that G* is not P — generic. First, we define 7:42 {iA ~BY ,B) | Be PI. Then, (7) YO] = g*, If D is a P — dense set, such that G* MD = @, then we can find a condition B€G*, such that B I 7D = @. Then, we consider the condition A~B € P and by the density of D we can find a sequence C, with the property that A~B~C €D. Using the definition of 7, we verify that for every P — generic filter G’ that contains B~C we have that A~B-~C € (7)¥™)IG1 M4 D. Hence, 4.A. <— DC in the inner model V(H) 117 BC |K-- 7 ND 49. However, as B~C < B then B~C |K- 7 ND = 9, which P P is a contradiction. Therefore, G* is P — generic and A € G*. In addition, V[G] = V[G*] because G and G* easily describe one another. Thus P is an almost homogeneous forcing. 4.A.9 Proposition 0 Next we show that forcing with P over V(H) does not add any new < « — sequences of «. First, using the regularity of A in V(H) and the definition of the P — forcing conditions, we prove the next proposition. 4.A.10 Proposition The forcing P is < X — closed in V(H). 4.4.10 Proposition O 4.A.11 Proposition Let G be P—generic over V(H) and set Vi: V(H)[G}. Then, for every regular cardinal ju < « we have that kK” MQ V(H) = K#N V;. Proof: Let f € k*MV,. We show that such f belongs to the ground model V(H). Let 7 be a P -name for f and x* be a condition such that x" |L-— [7 is a function from y into «]. Pp We define R C (P x 1) x (P x p) to be the following relation: R((x,a), (y,8)):#$ x<y & Aly<x<x") & adB & & (Wn < a)(Fy < K)(x IE 7(7) = 9). raat: ee 118 Chapter 4. The partition relation in a ZFC model By the definability of the forcing P in V(H), we conclude that R € V(H). We also claim that there exists no maximal R — descending chain of length < p. To verify this claim, we argue as follows: If a = (ag | € < ) is any R — descending chain for n < yu and if ag = (X¢, ae), then by the regularity of js we find 6 = sup{ag | € < n} below yp. Also, since P is < —A closed, then there exists a condition x in P below every x¢ for £ < 7. Since x* om [7 is a function from p into x], then there exists ay € Py <x and a 6 <« such that (y |/K-~ 7(@) = 4). Hence, P (VE < n)(R({y, 2 +1), ae). This shows that a = (ag | € < 7) is not an R-descending chain. Therefore, R has no maximal descending chain of length < y. Since < \ — DC holds, then in V(H) there exists an R — descending chain of length yp. Let D = ((xe, a¢) | € < us) be any such chain. Since \, are regular in V(H), then the sequence (a¢ | € < jz) is cofinal in p. Then there exists a sequence (fx | € < ys) of ordinals below « in V(H), such that (WE < 11) (Xe I 7() = Be). If x is a condition of P below every x¢ (it is is possible to find such x since P is < A-closed), then (VE < W)(x IE 7() = Be). 4.A. <A—DC in the inner model V(H) 119 If we repeat this argument below any condition of P, then we show that Vp <x*dqeP(q<p & She «*NV(H)q|--7=h). ? Therefore, {alalk- +e V} is pre — dense below x* in P and this implies that 7VIG ¢ V(H), ie., f € V(H). Hence, x“ V(H) = «#1 V, for every regular cardinal below 4.A.11 Proposition O 4.A.12 Corollary If V;:% V(H)[G], then for every uu < k vyH) = vu. Proof: This follows by induction on € < k, by using 4.A.10 and the fact that (V6 < K)[V¥ = V2]. 4.A.12 Corollary 0 120 Chapter 4. The partition relation in a ZFC model 4.B A ZFC model of the partition relation In the following section we are going to show that for any G which is P — generic over V(H), V(H)[G] is a model of ZFC which satisfies the partition relation: Vue < K(k 58 (w)v,) & 4K is strong limit. 4.B.1 Proposition Let V be a transitive model of ZFC, in which k is a supercompact cardinal and X is an inaccessible cardinal above k. Let G be a fixed P — generic filter over V(H), where P is the forcing defined in 4.A.7. Let Vi: V(H)[G]. Then, Vi — ZFC + « is a strong limit. Proof: Since V, is a generic extension of a model of ZF, namely V(H), then V, is itself a model of ZF. Moreover, as we have shown in 4.A.12 (Vu< KV = vy) and as « is a strong limit cardinal in V(H), then (Vu < K)(IV.| <r), 1.e., K is a strong limit cardinal in V}. So it remains to be seen that V; — AC. The proof of V; — AC is based upon the following three claims. 4.B. A ZFC model of the partition relation 121 4.B.1.1 Claim If 7 € V(H)” and S = dom(7) € V(H)?, then efS:S > e(7) is onto and belongs to V,, where e is the interpretation map e(x) = xV(IG] — Vi for any x € V(H)?. Proof of the Claim: This is a well-known fact that appears also in [Jech 78]. 4.B.1.1 Claim 0 4.B.1.2 Claim If M is a transitive model of ZF and f is a surjection f : S + T and §,T € M with S being well — ordered, then T is well — orderable. Proof of the Claim: For any x € T let Y, = f~!{x}. Since f,T € M then, (x |x € T) € M. Let « be a well — ordering on S. Let z%:% the << —least element of Yx. So (z, | x € T) € M. Next, we well — order T by < such that for x,x’ € Tx <x: 4% 2, 4 z,. Obviously, < is a well — order of T. 4.B.1.2 Claim O 4.B.1.3 Claim For every S € V(H), Vi ES is well — orderable. Proof of the Claim: We fix an S € V(H) and by the definition of V(H) we conclude that for every element x of S there exists an element z of V, a member w of H and a formula ¢ such that x is definable in V(H) by ¢ 122 Chapter 4. The partition relation in a ZFC model with parameters z and w. Then, using Replacement in V(H), we can find an ordinal a with the property that each element of S is definable over V(H) with parameters from V, which appear in VY), But then all the V — parameters necessary for the definitions of the elements of S appear in VY. Using this fact we define in V(H) a surjection t: VY x H 3 S. Since V satisfies the Axiom of Choice and since H is well — ordered in Vj, then we conclude that VY x H is well — ordered in V;. Then, by 4.B.1.2, we get that S is well — ordered in Vi. 4.B.1.3 Claim O To finish the proof of the proposition, we consider an X € Vj, and we show that ¥Y can be well — ordered. We let Y be a V(H)?-name for %, i.e., e(X) = XV! = & and we set S:2 dom(#). Then, S € V(H) and by 4.B.1.1 we have that ef S € V; and eS: S > & is onto. Since V; — ZF and S € V(H) , then by 4.B.1.2 we get that S is well—ordered in V; and thus according to 4.B.1.3 % is well — ordered in Vj. So, this concludes the proof of V; - AC. Therefore, Vi — ZFC +« is a strong limit cardinal. 4.B.1 Proposition O 4.B. A ZFC model of the partition relation 123 4.B.2 Theorem Let V be a transitive model of ZFC, in which x is a super- compact cardinal and X is an inaccessible cardinal above k. Let G be a fixed P — generic filter over V(H), where P is the forcing defined in 4.A.7. If Vi: < V(H)IG), then Vi - [ZFC + « is a strong limit + Vu < K(K ob (w)¥, I]. Proof: We fix < « and we consider an OD coloring F € V, F : [k]° > vu. We have already seen that [A]? N'Vy = [Kk]? N V(A) and Vi _ 1 V(H) V¥i = VV), So we assume that there exists a formula ¢ and ordinal parameters s such that for all s € [«]* and x e VY) F(s) =x + V(H)[G] F 4(s,x, 8). Since s,x € V(H) and P is homogeneous, then F(s)=x + V(H)IG] F 4(s,x, 6) = Ip IL 46%, 5). 124 Chapter 4. The partition relation in a ZFC model Since the forcing P is definable by H in V(H) and by the definability of the forcing relation |/—, then there exists a formula ¢* such that P F(s)=x = Ip E- 46%, 5) <> V(H) K 6"(s,x, 6,H) and V(H) F (Vs € [«]”)(AIx)$*(s, x, 6, H). By 3.C.1 there exists o € [k]* and xp € VY) such that Vr € [o]” F(r) = xo. This shows that F has an homogeneous set in V,. Hence, ViE«K ob (wy, Then by 4.B.1 we get that Vi — ZFC + « is strong limit + Vu < K(K« ob (wy). 4.B.2 Theorem O 125 Bibliography [Apter 85] A. Apter. An AD-like Model. The Journal of Symbolic Logic, 50, (1985): 531-543. [Jech 78] T. Jech. Set Theory. New York: Academic Press, 1978. [Kunen 80] K. Kunen. Set Theory. North Holland, 1980. (Laver 78] R. Laver. Making the supercompactness of « indestructible under « — directed closed partial orderings. Israel Journal of Mathematics, 29, No 4, (1978): 385-388. [Magidor 77] M. Magidor. On the Singular Cardinals Problem I. Israel Journal of Mathematics, 28, (1977): 1-31. [Mathias 73] A.R.D. Mathias. On Sequences Generic in the sense of Prikry. Journal of the Austarlian Mathematical Society, 15, (1973): 409-414. [Menas 75] T.K. Menas. On strong compactness and supercompacteness, An- nals of Mathematical Logic, 7, (1975): 327-359. 126 BIBLIOGRAPHY [Prikry 70] K. Prikry. Changing Measurable into Accessible Cardinals. Disser- tationes Mathematicae, LXVII, (1970): 1-55.