The Hebrew University of Jerusalem Einstein Institute of Mathematics On The LST Number Of The Logic L(I) by Omer Shimon Zilberboim A thesis submitted

The Hebrew University of Jerusalem Einstein Institute of Mathematics On The LST Number Of The Logic L(I) by Omer Shimon Zilberboim A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics supervised by Prof. Magidor Menachem Omer Shimon Zilberboim I.D.: 061017406 Submitted on June 9, 2011 עומר שמעון זילברבוים ת.ז.: 061017406 הוגש בתאריך: ז' סיון תשע "א

Acknowledgments I wish to thank my parents, for instilling in me that education is a higher value, that it is the key to better life and that I should always pursue it. For their support in me over the years and for always having faith and belief in me. To Prof. Menachem Magidor for his dedicated teaching and tutoring, for his guidance and his insightful views. For the chance to work on an interesting problem in a field of mathematics reserved solely for the most outstanding mathematicians. To Prof. Saharon Shelah for introducing me to the fields of logic and set theory, and for the long and many hours of teaching he invested in me. To Professors Zlil Sela, Ehud De-Shalit, Emmanuel Farajoun and Ehud Hrushovski, each is responsible personally for my success and for showing me hidden paths in the academic and mathematical world.

תקציר מורחב: בעבודה זאת נשתמש בשיטות של שיכונים אלמנטריים וכפייה על מנת להוכיח כי אם ההנחות הבאות הינן עקביות: μ + "קיים מונה על קומפקטי " + "קיים מונה μ שהוא על קומפקטי" ZFC אזי, עקבית גם הטענה הבאה: + ZFC "קיימים בדיוק שני מונים אי נשיגים, הראשון מביניהם הוא מספר לוונהיים-סקולם-טרסקי של הלוגיקה " כאשר הלוגיקה הינה הרחבה של לוגיקה מסדר ראשון על ידי כמת שפירושו )בהינתן מודל (: אם ורק אם עוצמה. הקבוצות ו הינן בעלות אותה העבודה ממשיכה את מאמרם של פרופסור מנחם מגידור ופרופסור יוקו ואאנאנן, המוכיחה כי מתיישב )ביחס להתיישבות קיום מונה על קומפקטי( שקיים מודל של תורת הקבוצות בו ישנו מונה אי נשיג אחד בלבד, המהווה באותה עת מספר לוונהיים-סקולם-טרסקי של הלוגיקה. בדומה לעבודתם, גם כאן נשתמש בכפייה מטיפוס פריקרי על מנת להפוך מונים על קומפקטיים, נניח λ, )שהינם סדירים מהגדרתם( למונים חריגים בעלי סופיות, π תוך כדי שאיננו מפילים מונים ואיננו מוסיפים קבוצות חסומות של λ. מטרתנו תהיה ליצור שתי חזרות של כפיות, אותן נבנה במהלך העבודה, על מנת להשיג עולם של תורת הקבוצות בו קיימים שני מונים אי נשיגים בלבד, ועליו נוכל להוכיח את טענתנו. בנוסף, מסתמן משיטת ההוכחה כי ניתן להרחיב את התוצאה לכל מספר סופי של אי נשיגים. לאורך העבודה אנו נעזרים בשיכונים אלמנטר יים בעלי תכונות סגירות מסויימות על מנת להוכיח טענות עזר, ולמעשה הוכחת המשפט המרכזי תיעזר גם היא בשיכון אלמנטרי לתוך מודל נוסף של תורת הקבוצות. שילוב זה בין שתי השיטות מעניין במיוחד, מכיוון שהרחבות המתקבלות באמצעות כפייה הינן באופן גורף לא הרחבות אלמנטריות.

Contents 0.1. Introduction 1 1. Preliminaries 3 1.1. The Logic L(I) and the LST number. 3 1.2. Large Cardinals. 6 1.3. Forcing. 13 2. Basic Denitions and Tools 16 3. Forcing Construction 23 4. Iteration 39 4.1. First Iteration 44 4.2. Second Iteration 45 5. The Final Model 46 6. Discussion 56 6.1. Extending the Theorem. 56 References 56 0.1. Introduction. In this work we investigate the Löwenheim-Skolem- Tarski (LST) number of the logic L(I), introduced in [5]. This logic is an extension of rst order logic by a quantier I, which allows to compare cardinalities of denable sets. As it is with any abstract logic, we wish to determine how far is it from rst order logic, by exploring how small can its LST number be. This number is also important when it comes to constructions of models for this logic since the smaller it is, the more economical we can be when considering a continuous sequence of such models. In [15] it is shown that LST(L(I)) exists only if inaccessible cardinals exist, and then LST(L(I)) is at least as large as the rst of them. The proof is actually quite straightforward, using mostly the denition of the LST number for this logic. The most natural question is then: Is it possible that LST(L(I)) is the rst inaccessible cardinal? This question was open until answered positively in [11], where it was proven that it is consistent, relative to the consistency of a supercompact cardinal, that LST(L(I)) can indeed be the 1

rst inaccessible cardinal. Furthermore, the assumption of a supercompact cardinal was well argued in [11] to be essential, i.e.: it seems that nothing short of a supercompact cardinal suces for this type of result. The model constructed in [11] contained only one inaccessible cardinal (which is proved to be LST(L(I))). The question is whether or not this is an unavoidable result, namely: can one obtain a model with arbitrarily many inaccessible cardinals, the rst of which is LST(L(I))? Furthermore, can (essentially) the same methods of [11] be used to obtain such a result? This work is a rst step to solving these questions. We show that using similar methods, one can obtain a nal model which is not restricted to just one inaccessible cardinal, and where the rst inaccessible is LST(L(I)). The model we construct will contain two inaccessible cardinals, where we conjecture that using similar methods one can obtain a model with nitely many inaccessible cardinals, the rst of which is LST(L(I)). We assume G.C.H. and the following large cardinal assumptions: κ is a supercompact cardinal, µ > κ is a µ + supercompact cardinal and (without loss of generality) is the last inaccessible. We show that based on these assumptions, a set theoretic universe can be constructed in which κ < µ are the only inaccessible cardinals, and κ is the LST number of the logic L(I). The use of the assumption that µ is µ + supercompact (as opposed to, for example measurable) will be pointed out during the work, as it is required in proving somewhat deeper technical properties which exceed the scope of the introduction. It should be noted, however, that an attempt to assume just measurability of µ was made in an early version of this work, and seems insucient in the current version. Our primary tools for the task in hand are forcing and elementary embeddings. Once we have stated that forcing will be used, we have to address the issue of the meaning of the symbol. We take the non- Jerusalem approach, namely that q p will denote that q is a stronger condition than p (so in particular, 1 P denotes the greatest condition of the forcing P ). This approach is taken for the ease of reference to [11], 2

which this work essentially continues, and where the following theorem can be found: Theorem. If ZFC+There is a supercompact cardinal is consistent, so is ZFC+There is an inaccessible cardinal+lst(l(i)) is the rst inaccessible cardinal. In Ÿ1 the necessary concepts in abstract logic, large cardinals and forcing (specically tree Prikry forcing) are introduced. In Ÿ2 we review some relatively known notions of forcing and relevant properties, which will be used in constructing the forcing notions of Ÿ3 (these resemble the forcing notions constructed in [11]). In Ÿ4 we iterate these notions of forcing in two iterations to obtain the desired set theoretic universe, on which the main theorem of this work is proved in Ÿ5, namely: Theorem 1. If ZFC+There is a supercompact cardinal κ+µ > κ is µ + supercompact is consistent, then so is: ZFC+There are two inaccessible cardinal+lst(l(i)) is the rst inaccessible cardinal. Ÿ6 is reserved to a discussion on generalizing the result even further. 1. Preliminaries In this section we bring the denitions and theorems necessary in order to begin our work. After we address the denitions regarding the logic L(I), we record a short review of the large cardinals we use and some of their properties. Finally, we describe tree Prikry forcing and conclude with a lemma for the so called Master Condition. 1.1. The Logic L(I) and the LST number. Since the only logic interest in this work is the equicardinality logic L(I), there is no need to give the most general denition for an abstract logic, and so we decided to refer to a slightly dierent denition than what can be found, for instance in [8]. In fact, for the scope of this work, an abstract logic can just be regarded as a logic between rst and second order. Furthermore, we only deal with vocabularies τ that contain relation and constant symbols (so no function symbols). So the following denition will be used: 3

Denition. Let τ be a xed vocabulary. An abstract logic L consists of (1) A set of formulas of L (which by abuse of notation we also denote by L), elements of which will be denoted by φ, Φ, ψ, ϕ etc., such that if ϕ L, then there is a natural number n ϕ, called the length of the sequence of free variables. (2) A relation = M = ϕ[a 0,...a nϕ 1] between M which is a structure for τ, φ L and (a 0,...a nϕ ) which is a sequence of elements from M. It is assumed that this relation satises the isomorphism axiom, i.e.: if π : M = N then M = ϕ[a 0,...a nϕ ] if and only if N = ϕ[(a 0 ),...π(a nϕ 1)]. 4 For a xed vocabulary τ, we say that M is a substructure of N, if both are models for τ, M N and for every n-ary relation R τ and (a 0,...a n 1 ) M n we have (a 0,...a n 1 ) R M if and only if (a 0,...a n 1 ) R N, and for every constant symbol c τ we have c M = c N. So the usual denition of an elementary submodel for rst order logic, can be used to dene an elementary submodel in the case of an abstract logic (following our denition). More precisely: Denition. Let M, N be two structures for a xed vocabulary τ in the abstract logic L. We say that M L N (i.e.: M is an L elementary substructure of N) if M is a substructure of N, and for every formula Φ L and a sequence of elements on M of length n Φ : (a 0,...a nφ 1) we have M = ϕ[a 0,...a nϕ 1] if and only if N = ϕ[a 0,...a nϕ 1]. 1.1.1. The Logic L(I). The logic L(I) is dened to be the extension of rst order logic by a quantier Ixy. If Φ(x 0,...x n 1 ) and Ψ(y 0,...y n 1 ) are two formulas, then Ix 0 y 0 Φ(x 0,...x n 1 )Ψ(y 0,...y n 1 ) has the meaning: For given a 1,...a n 1 and b 1,...b n 1 the cardinality of the set of elements x 0 which satisfy Φ(x 0, a 1...a n 1 ) is the same as the cardinality of the set of elements y 0 which satisfy Ψ(y 0, b 1...b n 1 ).

5 Clearly, this extension of rst order logic should be capable of better expression of the set theoretical universe, as can be seen in a later discussion. 1.1.2. LST Number. A known theorem which can be found for instance in [2], and is attributed to Löwenheim, Skolem and Tarski gives us the following result for rst order logic: Theorem. If A is a structure in the vocabulary τ for rst order logic, then there exists a structure B A such that B ℵ 0 + τ. This theorem motivates us to dene the following Denition. Let τ be a xed vocabulary. The Löwenheim-Skolem- Tarski number LST(L) is the smallest cardinal such that if A is any τ structure, then there exists a substructure B of A of cardinality < κ+ τ such that B L A. So for example, the LST number of rst order logic is ω 1. Next we show that assuming that the LST number of the logic L(I) exists, then it must be a xed point of the ℵ function. We also pay the debt of giving an example of the expression capability of the logic L(I). We leave it as a discussion rather than a theorem, so that not all details are made formal. Suppose that LST(L(I)) exists and denote it by κ = ℵ δ. If (towards contradiction) δ < κ, then since δ is not the LST number, then there exists a theory T in L(I) which models are of cardinality no less than δ. We add to the vocabulary of this theory a new unary relation R, a new binary relation < and a new binary relation F. R will distinguish the elements that form a model of T, < is to be interpreted as a full ordering of the structure, such that every initial segment of it is not the the whole structure and F is to be interpreted as a partial function, its domain is the set of elements which are in the relation R, it is one to one and furthermore: if x y are in the relation R, then the sets {z : z < t 1 F (x, t 1 )} and {z : z < t 2 F (y, t 2 )} are of dierent cardinality. A model M for this new theory which we denote by NT (which is satisable), houses within itself a model R M for the theory

T, so it is of cardinality no less than δ, but by the requirements on the partial function F, a model for this new theory must contain at least δ sets of dierent cardinality, which means that the cardinality of M is at least ℵ δ. So in particular, the theory NT admits no models of cardinality less than ℵ δ, a contradiction. We also have the next theorem from [15]: Theorem. The existence of LST(L(I)) implies the existence of an inaccessible cardinal, and that LST(L(I)) is at least as large as the rst inaccessible. In response, we obtain from [11], which is the ground for the present work, the following Theorem. If ZFC+There is a supercompact cardinal is consistent, so is ZFC+There is an inaccessible cardinal+lst(l(i)) is the rst inaccessible cardinal. 1.2. Large Cardinals. 6 1.2.1. Inaccessible cardinals. Denition. For an ordinal α, let cf(α) be the minimal length of an increasing sequence α i : i < β such that for every i < β we have: α i < α, and if γ < α then there exists an i < β with γ < α i. Easy conclusions are then, that for every ordinal α: (a) cf(α) α. (b) cf(α) is a cardinal and: (c) cf(cf(α)) = cf(α). In the case where cf(α) = α we say α is regular, otherwise we say α is singular. We assume G.C.H., namely that for every cardinal α we have α + = 2 α and so we can use the following Denition. A regular cardinal λ such that α < λ implies 2 α < λ is called an inaccessible cardinal.

Remark. It is customary to distinguish between cardinals that suce just: α < λ implies α + < λ and cardinals that suce our denition of an inaccessible cardinal, but since we assume G.C.H. these denitions coincide. We also remark that the existence of an inaccessible cardinal cannot be proved in Z.F.C.: as it implies the consistence Z.F.C., by Godel's incompleteness theorem its existence cannot be proved from it (assuming Z.F.C. is consistent). 1.2.2. Mahlo cardinals. Denition. Let λ be a regular cardinal. (1) A closed unbounded (club) set C of λ is a subset of λ such that: (a) If β i : i < δ is an increasing sequence of ordinals of length δ (where δ < λ), and for every i < δ we have β i < λ, then sup( β i : i < δ ) C (in which case we say that C is closed). (b) If γ < λ then there exists an η C with γ < η (in which case we say that C is unbounded). (2) We say that S λ is stationary if for every club C we have C S. Remark. An easy result is that if C and D are clubs, then so are D C and lim(c) - the set of limit point of C. Denition. Let λ be a regular cardinal. We call λ a Mahlo cardinal if the set of regular cardinals in λ is stationary. An easy observation is that a Mahlo cardinal must be inaccessible by itself. 1.2.3. Supercompact Cardinals. Denition. A lter F P(P(I)) on a set I is a set such that: (1) / F and I F. (2) If a b I and a F then b F. (3) If a, b F then a b F. 7

A lter U is called an ultralter if in addition to the above, for every a I we have either a U or I\a U. A lter F is called σ complete if whenever a i : i < γ < σ is a sequence of length γ < σ such that a i F for every i < γ then a i U. Finally, an ultralter U is i<γ called principal if there exists a I such that A U a A, and non-principal if there is no such a I. Given a set I and an ultralter U on I, the ultraproduct V I /U is a model of Z.F.C., with the natural embedding: dened by: ι : V V I /U ι(a) = [c a ] U i.e.: ι(a) is the equivalence class (modulo U) of the constant function c a (x) = a for any x I. If in addition U is ω 1 complete, then the Mostowski collapse can be used to form a transitive model of Z.F.C.. So we have: π ι : V M = V I /U, where π is the Mostowski collapse. Various arguments in this work will rely on embeddings j : V M where M is closed enough, in the sense that M contains all sequences of a certain length. This desire for a closed enough world could be viewed as the motivation for the next denition: 8 Denition. Let κ λ. We say that κ is λ supercompact if there is an elementary embedding j : V M such that: (1) j has critical point κ (i.e.: j(α) = α for every α < κ and j(κ) > κ). (2) λ M M. We call such j a λ-supercompact embedding. We say that κ is supercompact if κ is λ supercompact for all λ κ. It is shown, for example in [14], that without loss of generality we can assume that j(κ) > λ, which is an assumption we will make throughout this work, even if not

so stated. In that case, we have an equivalent, combinatorial, denition of κ being λ supercompact, but we need to introduce some more concepts rst. If κ λ then P κ (λ) is the set of subsets of λ of cardinality < κ (namely: {x : x P(λ), x < κ}). We say that an ultralter U on P κ (λ) is normal, if every function f : P κ (λ) λ that satises {A : f(a) A} U, is constant over a set D U (we might call such a function a choice function on P κ (λ)). We say that such an ultralter U is ne, if for every β < λ the set B β = {A P κ (λ) : β A} is in U. We also have the following theorem, which can be found for instance in [14]: Theorem. If κ λ, the following are equivalent: (1) κ is λ supercompact. (2) There is a κ complete, non-principal, ne, normal ultralter on P κ (λ). Remark 2. We will frequently switch from using either characterization of κ being λ supercompact. Also note that whenever an ultralter U on a set I is mentioned, the resulting embedding j : V M = V I /U will not always be stated explicitly, namely: it will only be written implicitly as j : V M, where M is to be understood as the Mostowski collapse of V I /U. Remember that throughout this work we assume G.C.H.. We will be interested in the following question: Suppose that η µ V is η + supercompact and that j : V M is a µ ++ supercompact embedding (and remember that we assume without loss of generality that j(κ) > µ ++ ) with critical point κ. Is η still η + supercompact in M? What about the other way around? namely, assume in that η µ is η + supercompact in M. Is it η + supercompact in V? The answer to both these questions is yes, and is proven in the following Lemma 3. Assume G.C.H. and that κ < µ. Assume also that there is an embedding j : V M such that M µ++ M and that j(κ) > 9

µ ++. Then: η µ is η + supercompact in V if and only if η is η + supercompact in M. 10 Proof. Assume rst that η µ is η + supercompact in V. We will nd a ne, normal η complete ultralter on P η (η + ). We do not intend to go far, in fact we prove that if U is an η complete, non-principal ne, normal ultralter on P η (η + ) then U is a ne, normal ultralter in M. Since M µ++ M then P η (η + ) is the same in M as it is in V. Now, U is a set of cardinality η ++, then we can enumerate its elements by a sequence Û: Û = A i : i < η ++ every such A i is in M and M µ++ M, so Û M and every i < η++ M so U M. U is easily seen to be an η complete ne, non-principal ultralter on (P η (η + )) M. We only note that every f : (P η (η + )) η + which is a choice function in M, is also a choice function from (P η (η + )) to η + in V. So if f is such choice function, then there is a δ < η + such that {A P η (η + ) : f(a) = δ} U, which shows normality. Now assume that η µ is in M an η + supercompact cardinal. Then there exists a ne, non principal, normal, η complete ultralter on (P η (η + )) M (where (P η (η + )) M denotes what M computes as P η (η + )), and denote it by U. We claim that U is by itself in V, a ne, normal η complete non-principal ultralter on (P η (η + )) V. We note as before, that since M µ++ M then (P η (η + )) M = (P η (η + )) V which implies that U is indeed a non-principal ultralter on P η (η + ). It is also easily seen to be η complete and ne. To see that it is normal, we only need to remind that (using G.C.H. and that η µ) if f is function f : P η (η + ) η + in V, then it is a subset of P η (η + ) η + (and in particular of cardinality η + ) so we can enumerate its elements by an η + length sequence. Since M µ++ M this sequence is already in M, which implies f M as well. In particular any choice function on P η (η + ) is already in M, and so if f : P η (η + ) η + is such that f(a) A, the there exists a δ < η + such that {A : f(a) = δ} U, since U was assumed to be normal in M.

Notice that in fact, we only needed M µ++ M for the case that η = µ, and only for one direction, i.e.: to obtain that if U is in V, then the sequence enumerating it Û is in M (since it is of length µ++ ). Since we intend to use this lemma when the embedding in question, j, may well be far beyond µ ++ supercompact, we see no point in splitting it. As a nice result that we will later use, we obtain: Corollary 4. Suppose κ is a supercompact cardinal, then the set {η : η is η + supercompact} is unbounded in κ. Proof. Let δ < κ. Choose an embedding j : V M such that j(κ) > κ and M κ++ M. Then according to lemma (3) κ is κ + supercompact in M. Since δ < κ then j(δ) = δ and we have: M = x[j(δ) < x < j(κ) and xis x + supercompact] since j is elementary, we have: 11 V = x[δ < x < κ and x is x + supercompact] as required. The following lemma will also be helpful when in Ÿ4: Lemma 5. Assume α V is α + supercompact, then there is an α + supercompact embedding, j, α = crit(j), j : V M, such that in M α is no longer α + supercompact. Proof. Choose an embedding j : V M = V Pα(α+) /U for which j(α) is minimal. We claim that α is no longer α + supercompact in M. For suppose to the contrary that there is a ne, η complete, non-principal, normal ultralter U on (P α (α + )) M. Since M α+ M we have that (P α (α + )) M = (P α (α + )) V and U is, in V, an α complete, ne, nonprincipal ultralter on P α (α + ). To see that U is also normal, let f : P α (α + ) α + be a choice function in V. Then f is a subset of P α (α + ) α +, its cardinality is α +, so we can enumerate its elements by a sequence of length α +. Since M α+ M, then f M as well. Since U was assumed to be normal (in M) and f is a choice function,

then there exists a δ < α + such that the set {A P α (α + ) : f(a) = δ} is in U. This set witnesses in V that f is constant over a set in U, which shows that U is normal in V as well. So we now have two α + supercompact induced embeddings that we can discuss: i : V M = V P α(α +) /U 12 and: i : M M = M P α(α +) /U Our claim now is that if δ is any ordinal, then i(δ) = i (δ). This is because if f : P α (α + ) δ is a function in V, then by the same argument as before f M. Since i(δ) is represented in V Pα(α+) /U by the equivalence class of the constant function c δ (A) = δ, then (without loss of generality) [c δ ] U = {f/f : P α (α + ) δ}. Since every such function is also in M (by our last argument) and since surely if g : P α (α + ) δ is a function in M then it is in V, then we have: i(δ) = i (δ). Now, for η which is inaccessible and satises η > P α (α + ) = α +, we have i (η) = η. This is because the cardinality of the set {f/f : P α (α + ) η} is η, and if g : P α (α + ) η is a member of this set, then {f < U g : f : P α (α + ) η} is of cardinality less than η, i.e.: every initial segment of [c η ] is of cardinality less than η. So in the Mostowski collapse we have i (δ) = δ. Now we reach a contradiction. Since for any ordinal δ i(δ) = i (δ) then i(α) = i (α) < i (j(α)) = i(j(α)) and since α is α + supercompact in V, it is inaccessible, and hence M = j(α) is j(α) + supercompact. Since j(α) > α + = P α (α + ) we have i (j(α)) = j(α) so in total i(α) < j(α)

13 a contradiction to the choice of j as an α + supercompact embedding for which j(α) is minimal. Remark. Comparing lemma (3) with lemma (5), we obtain that the embedding j in lemma (5) cannot satisfy M α++ M. This would be a good point to say a few words on the amount of maneuverability that we have (or need), when dealing with supercompact cardinals, and it also relates to Prikry forcing. Loosely speaking, we have on one hand lemma (3), which assures us that we can keep some supercompactness properties if we allow enough closure. On the other hand lemma (5) assures us that (if we wish) we are able to loose some supercompactness properties while keeping a fair amount of closure. This is, in a sense, will turn out to be the underlying conict in this work: we want to have enough closure so that we can keep some properties, but we cannot expect too much closure, or we will not be able to obtain other, new properties. So a main issue will be to get the balance right. Although not very high on our priority list, we should still mention the denition of a measurable cardinal: Denition. A regular cardinal α is called a measurable cardinal, if it satises one of the following equivalent conditions: (1) there exists a κ complete, non principal ultralter U on α. (2) There is an embedding j : V M = V α /U, such that α = crit(j) and M α M. It follows in particular, that every cardinal η which is at least η + supercompact is also measurable. A crucial point of this work will rely on a result by Laver which is phrased in the following theorem and can be found in [7] Theorem 6. Let κ be supercompact. Then there exists a function h : κ V κ such that for every x and every λ κ there is a λ- supercompact embedding j : V M such that j(h)(κ) = x. 1.3. Forcing. Following [4], with the necessary change of reversing the sign (namely that q p reads q is a stronger condition than p) we bring here the denitions of a Prikry forcing notion:

Denition 7. Suppose λ is a regular cardinal. A forcing notion P, is said to be a λ-prikry forcing notion if there is a partial order such that: (1) Every decreasing sequence of length α < λ has a bound (namely: if q i : i < α < λ is such that j < i implies q i q j then there exists q such that q q i for every i < α). (2) For every statement Φ in the forcing language for P and for every p P there exists q P such that q p and q Φ or q Φ (i.e.: q decides on Φ, which we will sometimes denote by q Φ). We call the direct extension relation. The prototype for λ Prikry forcing notions is given in the following denition. Assume that λ is measurable and that U is a λ complete ultralter on λ.we have the following denition of tree Prikry forcing, which we intend to incorporate into the forcing we construct in this work: Denition 8. P is called a U-tree with a trunk t (in notation P, t ) if: (1) P is a set of nite increasing sequences of ordinals less than λ. (2) P, is a tree, where is the partial order dened on λ <ω by a b if and only if a is an initial segment of b. (3) t is a trunk of P, i.e.: t P and for every s P we have s t or t s. (4) For every t s the set Suc P (s) = {α < λ : s α P } is in U. (5) For every n < ω we dene: Lev n (P ) = {s P : length(s) = n}. 14 Denition 9. For r, R, r, R P T (λ) we dene r, R r, R ( r, R is stronger than r, R ) if and only if R R (and note as before that this implies r r and that r R). We say that r, R r, R ( r, R is a direct or Prikry extension of r, R ) if R R and r = r.

So we can dene P T (λ) to be the set of all pairs t, P such that P is a U tree with trunk t, with the partial orders. P T (λ) turns out to be a λ Prikry forcing notion, in fact, the very proofs given in this work can be easily altered to prove this fact. Forcing with P T (λ) introduces an ω sequence conal in λ, while not collapsing any cardinals and not adding bounded subsets of λ (more facts which can be derived from later proofs). We intend to incorporate the forcing P T (λ) into the forcing which we construct in Ÿ3, along with another notion of forcing which we explain in Ÿ2. Another key point in this work will depend on the answer for the following question: Suppose j : V M is an elementary embedding. Suppose that P V is a notion of forcing, then also j(p ) M is a notion of forcing. Let G be P generic over V and let H be j(p ) generic over M. Under what circumstances can j be extended to j : V [G] M[H] which is also elementary? One possible answer is given in the following Lemma. In the above settings, suppose that j(q) H for every q G, then there exists such a j. Proof. Let Re(G, t) V [G] be the realization of a term t in the forcing P. Dene j (Re(G, t)) = Re(H, j(t)) (which is in M). We have to verify that j is well dened, meaning that it does not depend on the choice of t. So let s be another term, and let p G force that t = s. So we have that j(p) j(t) = j(s), and since j(p) H we have that M[H] = Re(H, t) = Re(H, s). Once we have dened j, it is easily veried that it is an elementary embedding. We shall use two kinds of realizations of this lemma, one will be in the form of choosing H as an end extension of G, the other will be by a use of a so called master condition, namely, we will nd a condition h which is j(g) than every g G. So that h H implies that j(g) H for every g G. Lastly for this section, we dene the Easton product for a sequence of forcing notions 15

Denition 10. Let P i : i < λ be a sequence of forcing notions (where we can assume without loss of generality that each P i has a maximal element 1 Pi ), then we dene the Easton product to be the set of sequences τ = τ i : i < λ such that: (1) For every i < λ: τ i P i. (2) For every α λ such that α is regular we have: support(τ) = {i < λ : τ i = 1 Pi } is bounded in α. 2. Basic Definitions and Tools In this section we will establish the ground from which we derive the forcings Q λ and δ Q λ built in section (3), the various types of forcing notions that will be incorporated into it as well as into the iterations in section (4), and phrases and denitions that will be used throughout this work. For an ordinal α, let us denote by µ α the rst η > α such that η is η + supercompact, and let us also denote by i(α) the rst inaccessible cardinal above α (assuming they exist). So in particular, in the universe V that we assume: for every α < κ we have µ α, i(µ α ) < κ (according to lemma (4)) and also µ κ = µ. Lemma 11. Assume V = GCH and κ is supercompact, and also that µ is the rst µ + supercompact cardinal above κ and the last inaccessible. Then there exists a function f : κ µ ++ such that for every α < κ we have α < µ α f(α) < i(µ α ) and such that for every λ > κ there exists an embedding j : V M such that M λ M,crit(j) = κ, j(κ) > λ and j(f)(κ) > λ. Proof. Without loss of generality, we can assume λ to be greater than 2 µ+ = µ ++. Dene f to be h(α) + if h satises: α < µ α h(α) < i(µ α ), otherwise dene: h(α) = µ + α. According to theorem (6) applied toλ and x = λ +, we obtain a λ supercompact embedding j : V M that satises j(h)(κ) = λ +, M λ M and j(κ) > λ. We show that indeed j(f)(κ) > λ: since M λ M we have at least below λ, that j(f) is a function that satises either: α < µ α j(f)(α) < i(µ α ) or α < f(α) = µ + α. Since λ > µ ++, then M µ++ M and we have according to lemma 16

(3) that in M, µ is still µ κ (in the sense of M, namely that in M, µ is the rst cardinal η > κ which is η + supercompact). Having said that, we obtain that κ < µ < j(h)(κ) = λ +, and since there is no inaccessible cardinal between µ and λ we have that the rst case applies for the denition of f and so κ < µ κ < λ j(f)(κ) < i(µ κ ). Remark 12. From this point on we x such a function f. We will frequently address two special types of cardinals: the cardinals α which resemble κ, in the sense that they are closed under f, they are at least α + supercompact, and are also a limit of cardinals η which are η + supercompact. The other type of cardinals resemble µ, in the sense that they are cardinals β such that there exists α < β, α is α + supercompact (or 0) and no η with α < η < β is η + supercompact. In the latter case we say β is of type µ and that α is the corresponding witness. Denition 13. A cardinal λ which is both λ + supercompact and a limit of cardinals η which are η + supercompact will be further called a κ type cardinal. A cardinal λ which is λ + supercompact and is not of κ type will be called a µ type cardinal. Remark 14. Notice that if a cardinal λ is closed under f, then it has to be a limit of µ type cardinals. If it is also λ + supercompact, then by our denition it is a κ type cardinal. Finally, notice that κ is of κ type and µ is of µ type, as witnessed by κ (a trivial, yet worth mentioning point). Throughout this work, we give a slightly dierent treatment for µ type cardinals than κ type cardinals. We intend to use the same ideas for both, but apply them a bit dierently. Every forcing that will be used in conjunction with κ type cardinals will also have a variant that will be used on µ type cardinals, so the same name (P for instance) might be used in both cases to denote two slightly dierent notions of forcing, while the context (whether it is a κ type cardinal or a µ type cardinal) will determine the exact version of the forcing that is used. We will do our best to avoid any confusion, however, we might not 17

declare which specic variant of the forcing is used if only properties shared by both variants of the forcing are used. For instance, both the notions of forcing in (15),(17) are called NM(λ), where in (15) it is a notion of forcing dened on a type κ cardinal, and in (17) it is dened on a µ type cardinal. Denition 15. Suppose λ is a κ type cardinal (so it is λ + supercompact, closed under f ). (1) Dene NM(λ) to be the set of all closed bounded subsets C, of λ, such that: (a) Every member of C is a cardinal. (b) C contains some point below the rst inaccessible cardinal. (c) If β is a limit point of C, then β is singular. (d) If β C, and β is the rst point of C above β, then β is an inaccessible cardinal and f(β) < β. (2) Dene the partial order on NM(λ) by: D C if and only if D, C NM(λ) and D is an end extension of C. Remark 16. So the successor members of C are all regular and limit points of C are closed under f. It is easy to see that if C NM(λ) and C contains a point above β < λ then {D : D C} is β closed (with the natural bound, namely D = D α {sup( D α )}). It follows that α<β forcing with NM(λ) introduces no new β sequences of ordinals when β < λ. So λ remains regular, and since no new bounded subsets of λ are added then λ remains inaccessible. Also, it is easy to see that if G NM(λ) is a generic lter then G is a closed unbounded subset of λ. Every limit point of G is singular, so in the generic extension λ is not a Mahlo cardinal. We dene a slightly dierent non-mahlo forcing notion for cardinals which are of µ type. By abuse of notation we denote this notion of forcing NM(λ) as well. The exact type of non-mahlo forcing will be clear from the context, since a cardinal cannot be both of type µ and of type κ at the same time. α<β 18

Denition 17. Assume that δ < λ, λ is a µ type cardinal as witnessed by δ. Dene NM(λ) to be the following forcing notion: (1) NM(λ) is the set of all closed bounded subsets C of λ such that: (a) Every member of C is a cardinal greater than δ +++. (b) C contains some point below the rst inaccessible cardinal greater than δ. (c) If β is a limit point of C, then β is singular. (d) If β C, and β is the rst point of C above β, then β is inaccessible. (2) The partial order on NM(λ) is dened by D C if and only if D, C NM(λ) and D is an end extension of C. 19 As before, it is easy to see that if C NM(λ) and C contains a point above β < λ then {D : D C} is β closed. It follows that forcing with NM(λ) introduces no new β sequences of ordinals when β < λ. So λ remains regular, and since no new bounded subsets of λ are added then λ remains inaccessible. Also, it is easy to see that if G NM(λ) is a generic lter then G is a closed unbounded subset of λ. Every limit point of G is singular, so in the generic extension λ is not a Mahlo cardinal. Let α < λ where λ is a type κ cardinal, and δ α < λ where λ is a type µ cardinal and δ is the corresponding witness. We shall also be interested in the forcing NM α (λ) which is just the subset of conditions C of (the relevant) NM(λ) such that min(c) > α +++, C contains a point below the rst inaccessible cardinal greater than α, with partial order dened by C NMα(λ) C if and only if C NM(λ) C. Next we have a Claim 18. Assuming the settings of denition (17), forcing with N M(λ) leaves δ δ + supercompact in the generic extension. Proof. The fact that NM(λ) adds no new bounded subsets of λ implies that the power set of δ + remains unchanged, and so, if U was a δ

complete, ne, normal, non-principal ultralter on P δ (δ + ) in V, then it is still such an ultralter in V NM(λ). Given two regular cardinals α < β, Col(α, < β) is the usual Levy collapse of all the cardinals γ where α < γ < β to α. It is an α closed forcing notion, and if β is inaccessible or the successor of a regular cardinal, the forcing notion Col(α, < β) satises β c.c.. We dene two types of collapses to be used. The rst is given in Denition 19. Let C be a closed set of cardinals. For β C\{sup(C)} let β be the next point of C after β. We assume that if β C then β is inaccessible and so β + < β. The forcing notion Col 1 (C) is dened to be the Easton product of Col(β +, < β ) for β C\{sup(C)}. The second collapse we shall use is given in Denition 20. Let C be a closed set of cardinals. For β C\{sup(C)} let β be the next point of C after β. We assume that if β C then β is inaccessible and f(β) < β. The forcing notion Col 2 (C) is dened to be the Easton product of Col(f(β) +, < β ) for β C\{sup(C)}. We intend to use the collapses Col 1 (B) and Col 2 (D) with sets B, D that were produced by NM(λ), where λ is of type µ or of type κ correspondingly. As stated before, we may drop the excess notation and write just Col(C) instead of Col 1 (C) or Col 2 (C) when it is to be understood that according to the type of λ, the corresponding type of NM(λ) forcing is used and the corresponding type of Col(C) is used. We have the following lemma which applies in both collapses: Lemma 21. Assume λ is regular and C is a NM(λ) generic set, then forcing with Col(C), leaves λ regular in the generic extension. Proof. Suppose to the contrary that λ becomes singular of conality γ < λ. Let η be some inaccessible cardinal greater than γ, in C. Then we can break the Easton product before and after η, namely: Col(C) = Col(C η + ) Col(C\η). 20

Then Col(C η + ) is of cardinality η, while Col(C\η) is η closed (so in particular no sequences shorter than η are added). Then λ cannot become of conality γ < η, a contradiction. Remark 22. Assume λ is of type µ and δ is the corresponding witness for it, and δ < i < λ where i is the rst inaccessible above δ. Then if C is NM(λ) generic, and C contains a point below i, then after forcing with Col 1 (C), λ becomes the rst inaccessible above δ (note that if C is NM(λ) generic, then every α C satises δ + < α). We intend to look at pairs (c, h) such that c NM α (λ) and h Col(c). If c c (in the non-mahlo sense) then if h Col(c) and h Col(c ) then h can be considered as a member of Col(c ). So we can dene for two such pairs (c, h ) (c, h) if c c and h c = h. We denote this notion of forcing by P (α, λ) (so we actually have two such notions of forcing, P 1 (α, λ) in case λ is of type µ and P 2 (α, λ) in the case that λ is of type κ). It is also apparent that if p = (c, h) P (α, λ) and c contains a point above β < λ and p i : i < β < λ is a decreasing sequence, then there is a lower bound for the sequence, specically c = c i and h = h i satisfy that (c, h ) P (α, λ) is a i<δ i<δ lower bound. So we record a Denition 23. Let λ be a λ + supercompact cardinal. (1) If λ is of type µ, the forcing P (α, λ) is dened to be the set of pairs (c, h), such that c NM α (λ) (where NM α (λ) here corresponds to denition (17)) and h Col 1 (c). The partial order is dened by (c, h ) (c, h) if and only if c c and h c = h. (2) If λ is of type κ, the forcing P (α, λ) is dened to be the set of pairs (c, h), such that c NM α (λ) (where NM α (λ) here corresponds to denition (15)) and h Col 2 (c). The partial order is dened by (c, h ) (c, h) if and only if c c and h c = h. We observe that the set of open and dense sets in P (α, λ) is a λ complete lter. To see that, let D α : α < η < λ be a sequence of 21

dense open sets of length η. We show that the intersection D α is a α<η dense open set: let p = (c, h) P (α, λ), and without loss of generality assume that c contains a point above η. Dene by induction a decreasing sequence of length η: p 0 = p. If α = β + 1, then since D β is dense and open then there is a condition p α D β such that p α p β. If α is a limit, take a bound p α of the sequence p i : i < α which is possible since c contains a point above η > α. So the sequence is dened, and now we can (by the same argument as before on p) take a lower bound for it p. since p p α for every α < η we have that p D α for every α. This proves that the intersection is dense, it is also clearly open (since if q D α and q q then q D α for every α<δ α, by D α being open, and so q D α ). α<δ The next lemma shows that if λ is λ + supercompact, then we can extend the open-dense lter to a λ complete ultralter on P (α, λ) (since both µ and κ type cardinals have this property, then the proof applies for the kinds of P (α, λ) notions of forcing). Lemma 24. Assume that λ is λ + supercompact, then the open-dense lter on P (α, λ) can be extended to a λ complete ultralter O on P (α, λ). Proof. Let j : V M be a λ + supercompact embedding with critical point λ (M λ+ M and j(λ) > λ + ). The forcing j(p (α, λ)) is in M the forcing P (j(α), j(λ)). Let D α : α < λ + count the dense-open subsets of P (α, λ), then each j(d α ) is dense and open in P (j(α), j(λ)) and j(d α ) : α < λ + is a sequence of dense and open subsets of P (j(α), j(λ)). Furthermore, M contains the sequence j(d α ) : α < λ + since M λ+ M. The intersection α<λ +j(d α) is dense and open according to the preceding discussion, and in particular is not empty, and let q be a witness. We dene the ultralter O by A O if and only if q j(a). It is clear that O is an ultralter, it contains all the sets which are dense and open (since q j(d α ) by choice of q), and it is λ complete. Remark 25. We point out that the last lemma is precisely why we needed µ to be µ + supercompact (as opposed to just measurable). 22

23 3. Forcing Construction We assume that λ is a λ + supercompact cardinal (so λ could be for instance of type µ or of type κ), and that U is a λ complete ultralter on λ. We also assume that O s : s (λ α<λ P (α, λ) <ω is a sequence of λ complete ultralters, such that if s = a 0,...a k 1 and a k 1 is an ordinal then O s is an ultralter on the forcing P (α k 1, λ) that extends the λ complete lter of dense open subsets of P (α k 1, λ) (if s =, the empty sequence, then O s is a λ complete ultralter that extends the dense open lter on P (δ, λ) in case λ is of type µ, and where δ is the corresponding witness as in denition (13), and in case λ is of type κ then O s is a λ complete ultralter that extends the dense open lter on P (0, λ)). Our next order of business is to combine the Prikry tree forcing notion with the forcing NM(λ) into a single notion of forcing. We will, however, have to introduce another object - a condition in the forcing Col(D) where D is the set that will be introduced by the NM(λ) part of the tree. For the sake of this discussion, we only address the Prikry and non-mahlo part of the tree, while the role of the Col(D) condition will be addressed later. We are going to dene a tree, along which progression will be made both in the Prikry front and the non-mahlo front. The tree will be made of nite sequences a 0,...a k 1 such that at level 0 of the tree, we can choose an element of NM(λ). The set of candidates for such an element will be of measure one relative to the ultralter O. At the next level of the tree, the sequence can be extended by an ordinal α < λ from a set of measure one relative to the ultralter U. Generally, at even levels, we can extend the NM(λ) part of the sequence by a condition in NM α (λ) where α is the last ordinal that was chosen, and where the set of such candidates is in O s (s being the sequence so far). At odd levels we can extend the sequence by an ordinal β < λ which is greater than all the ordinals that appeared in the ordinal part of the sequence, but also greater than the supremum over the NM(λ) part

of the sequence. The set of candidates for such an extension is in the ultralter U. Let us assume now that λ is a µ type cardinal, and δ < λ is the corresponding witness. We x a λ complete non-principal ultralter U on λ and O s : s (λ P (α, δ<α<λ λ)<ω which is a sequence of λ complete ultralters, as explained above. Denition 26. We say that T is an interlaced tree with trunk t if T consists of nite sequences s = a 0,...a k 1 such that: (1) If i = 0 then a 0 = (c 0, h 0 ) P (δ, λ) such that c contains a point below the rst inaccessible cardinal above δ. (2) If i = 1 then a i is an ordinal < λ such that a i > max(c 0 ) (where a 0 = (c 0, h 0 )). (3) If i is even greater than 0 then a i = (c i, h i ) P (a i 1, λ). (4) If i is odd then a i is an ordinal < λ such that a i > max(c i 1 ) (where a i 1 = (c i 1, h i 1 )). (5) T, is a tree, where is dened by s t if and only if t Dom(s) = s (so s is an initial segment of t). (6) t is a trunk of T, i.e.: t T and for every s T we have s t or t s. (7) If t s = (a 0,...a k 1 ) T (so s is of length k) then: (a) If k is even then the set Suc T (s) = {a : s a T } is in O s, i.e.: Suc T (s) P (a k 1, λ) and Suc T (s) O s. (b) If k is odd then the set Suc T (s) = {a : s a T } is in U, i.e.: Suc T (s) {sup(c k 1 ) < α < λ} (where a k 1 = (c k 1, h k 1 )). (8) For every n < ω we dene: Lev n (T ) = {s T : length(s) = n}. Next we dene the forcing δ Q λ : Denition 27. In the above settings, we dene δ Q λ to be the set of all pairs t, T such that T is an interlaced tree with trunk t. For p = t, T δ Q λ and p = t, T δ Q λ we say that t, T is a stronger 24

condition than t, T and denote it by p p if T T. We say that p is a direct extension of p and denote it by p p if also t = t. We also record some notation: Denition 28. If s T and s = a 0, a 1,...a k 1 then: (1) We dene a(s) to be the subsequence of ordinals of s namely a(s) = a 1, a 3, a 5,.... (2) We dene c(s) to be the set {c : 0 i < ω such that for some h : a 2i = (c, h)}. (3) We dene h(s) to be the set {h : 0 i < ω such that for some c : a 2i = (c, h)}. (4) If a i = (c i, h i ) then we dene c(a i ) = c i and h(a i ) = h i. (5) If p = t, T then a(p), c(p), h(p) are dened by a(t), c(p), h(p) correspondingly, and length(p) := length(t). The forcing δ Q λ is actually a variant of the forcing Q λ, which is dened for λ which is of type κ, namely: Let us now assume that λ is a κ type cardinal. We x a λ complete non-principal ultralter U on λ and O s : s (λ P (α, λ) <ω which is a sequence of λ complete α<λ ultralters, as explained before. We then dene an interlaced tree with trunk t in the same way as in denition (26), but this time the notion of forcing P (α, λ) has a dierent meaning, since it is now made of pairs (c, h) such that c NM α (λ) for the NM forcing notion that ts λ, and h Col 2 (c). We also have to change in (1) of denition (26), that a 0 = (c 0, h 0 ) P (0, λ) (rather than P (δ, λ)), and c 0 contains a point below the rst inaccessible. The rest is essentially the same. We can then dene the forcing δ Q λ in the same manner, namely: Denition 29. In the above settings (where now λ is a κ type cardinal), we dene Q λ to be the set of all pairs t, T such that T is an interlaced tree with trunk t. For p = t, T Q λ and p = t, T Q λ we say that t, T is a stronger condition than t, T and denote it by p p if T T. We say that p is a direct extension of p and denote it by p p if also t = t. 25

Remark 30. We have, as usual that if G is Q λ generic, then the set {a(s) : s p G} is a Prikry sequence of length ω, conal in λ, and if H is δ Q λ generic then {a(s) : s p H} is a Prikry sequence of length ω which is lays between δ and λ. We turn to investigate the set D G = {c(s) : s p G} which is generated by the NM part of δq λ. We wish to show that this set generates a NM(λ) generic lter, namely that Lemma 31. If G is generic for δ Q λ then the set D G = {c(s) : s p G} NM(λ) generates a generic lter for NM(λ) over V. Proof. First we note that the set D G is closed and unbounded in λ, every limit point of which is singular. Let p Q λ and let E be a dense and open set in NM(λ). We show that there is an extension p p such that p E D G. Suppose p = t, T and t = a 0,...a k 1, and without loss of generality k is even which means that it is time to extend t by a P (a k 1, λ) condition (c, h) such that c NM αk 1 (λ) (otherwise extend by any ordinal from the set of successors). The set Suc T (t) is in the ultralter O t, and contains pairs of the form (c, h). The projection on the c coordinate intersects every dense and open set in NM ak 1 (λ). We look at E = {(c, h) Suc T (t) : c c(t) E}, which is again in O t, and in particular is non empty and let (c, h) be a witness. Dene t = t (c, h) and T = (T ) t (where (T ) t is the reduced interlaced tree (T ) t := {s T : s t } with trunk T ). Then the pair p = t, T is an interlaced tree with trunk t, and p D G E. Remark 32. We note that with a slight change of notation, lemma (31) states that if H is δ Q λ generic, then the set D H = {c(s) : s p H} NM(λ) generates a generic lter for NM(λ) over V, but also that D H lays between δ and λ. Next we wish to show that Q λ and δ Q λ introduce no new bounded subsets of λ, and do not collapse any cardinals. Lemmas (33) - (41) 26

apply both to Q λ and δ Q λ, with the necessary changes in notation. For that reason we state and prove them only for the case δ Q λ, while pointing dierences when it is required. Lemma 33. Assume that t, T α : α < δ < λ is a sequence of interlaced trees with the same trunk t, then t, T is an interlaced tree with trunk t, where T = α<δ T α (and in particular a common lower bound for the sequence). Proof. Assume that t, T α : α < δ < λ is a sequence of conditions in δ Q λ (so they all have the same trunk t). Then T = T α is an α<δ interlaced tree with trunk t. For suppose that a 0,...a k 1 = s t, s T, then: (1) If k is even, then the set Suc Tα (s) is in O s and also a α<δ Suc s (T ) implies min(a) > a k 1. (2) If k is odd then the set Suc Tα (s) is in U and also a Suc s (T ) α<δ implies a > sup(c(a k 1 )). (3) T T α for every α < δ. So t, T is an interlaced tree T with trunk t, which is a lower bound for the sequence. As a corollary we obtain: Corollary 34. Every decreasing sequence of conditions in δ Q λ of length δ < λ has a bound (namely: if q i : i < δ < λ is such that j < i implies q i q j then there exists q such that q q i for every i < α). So if we can prove the following lemma: Lemma 35. For every statement Φ in the forcing language for P and for every p δ Q λ there exists q P such that q p and q Φ or q Φ (i.e.: q decides Φ, which we will sometimes denote by q Φ). then it will follow that δ Q λ adds no new bounded subsets of λ, as proved in the following lemma: 27