1 The closed unbounded filter Closed unbounded sets Stationary sets in generic extensions... 9

Size: px
Start display at page:

Download "1 The closed unbounded filter... 3. 1.1 Closed unbounded sets... 3. 1.4 Stationary sets in generic extensions... 9"

Transcription

1 Contents I Stationary Sets 3 by Thomas Jech 1 The closed unbounded filter Closed unbounded sets Splitting stationary sets Generic ultrapowers Stationary sets in generic extensions Some combinatorial principles Reflection Reflecting stationary sets A hierarchy of stationary sets Canonical stationary sets Full reflection Saturation κ + saturation Precipitousness The closed unbounded filter on P κ λ Closed unbounded sets in P κ A Splitting stationary sets

2 2 CONTENTS 4.3 Saturation Proper forcing and other applications Proper forcing Projective and Cohen Boolean algebras Reflection Reflection principles Nonreflecting stationary sets Stationary tower forcing

3 I. Stationary Sets Thomas Jech 1. The closed unbounded filter 1.1. Closed unbounded sets Stationary sets play a fundamental role in modern set theory. This chapter attempts to explain this role and to describe the structure of stationary sets of ordinals and their generalization. The concept of stationary sets first appeared in the 1950 s; the definition is due to G. Bloch [16], and the fundamental theorem on stationary sets was proved by G. Fodor in [24]. However, the concept of a stationary set is implicit in the work of P. Mahlo [71]. The precursor of Fodor s Theorem is the 1929 result of P. Alexandroff and P. Urysohn [2]: if f(α) <αfor all α such that 0 <α<ω 1,thenf is constant on an uncountable set. Let us call an ordinal function f regressive if f(α) <αwhenever α>0. Fodor s Theorem (Theorem 1.5) states that every regressive function on a stationary set is constant on a stationary set. As a consequence, a set S ω 1 is stationary if and only if every regressive function on S is constant on an uncountable set. In this section we develop the theory of closed unbounded and stationary subsets of a regular uncountable cardinal. If X is a set of ordinals, then α is a limit point of X if α > 0and sup(x α) =α. AsetX κ is closed (in the order topology on κ) ifand only if X includes Lim(X), the set of all limit points of X less than κ. 3

4 4 I. Stationary Sets 1.1 Definition. Let κ be a regular uncountable cardinal. A set C κ is closed unbounded (or club for short) if it is closed and also an unbounded subset of κ. AsetS κ is stationary if S C for every closed unbounded C κ. It is easily seen that the intersection of any number of closed sets is closed. The basic observation is that if C 1 and C 2 are both closed unbounded, then C 1 C 2 is also closed unbounded. This leads to the following basic property. 1.2 Proposition. The intersection of less than κ closed unbounded subsets of κ is closed unbounded. Consequently, the closed unbounded sets generate a κ-complete filter on κ called the closed unbounded filter. The dual ideal (which is κ-complete and contains all singletons) consists of all sets that are disjoint from some closed unbounded sets the nonstationary sets, and is thus called the nonstationary ideal, denoted I NS. If I is any nontrivial ideal on κ, theni + denotes the set P (κ) I of all I-positive sets. Thus stationary subsets of κ are exactly those that are I NS -positive. 1.3 Definition. Let X α : α<κ be a κ-sequence of subsets of κ. Its diagonal intersection is the set α<κ X α = {ξ <κ: ξ α<ξ X α} ; its diagonal union is Σ α<κ X α = {ξ <κ: ξ α<ξ X α}. The following lemma states that the closed unbounded filter is closed under diagonal intersections (or dually, that the nonstationary ideal is closed under diagonal unions): 1.4 Lemma. If C α : α<κ is a sequence of closed unbounded subsets of κ, then its diagonal intersection is closed unbounded. This immediately implies Fodor s Theorem: 1.5 Theorem (Fodor [24]). If S is a stationary subset of κ and if f is a regressive function on S, then there exists some γ<κsuch that f(α) =γ on a stationary subset of S. Proof. Let us assume that for each γ<κthere exists a closed unbounded set C γ such that f(α) γ for each α S C γ.letc = γ<κ C γ.asc is closed unbounded, there exists an α>0ins C. By the definition of C it follows that f(α) α, a contradiction.

5 1. The closed unbounded filter 5 A nontrivial κ-complete ideal I on κ is called normal (and so is its dual filter) if I is closed under diagonal unions; equivalently, if for every A I +, every regressive function on A is constant on some I-positive set. Thus Fodor s Theorem (or Lemma 1.4) states that the nonstationary ideal (and the club filter) is normal. In fact, the nonstationary ideal is the smallest normal κ-complete ideal on κ: 1.6 Proposition. If F is a normal κ-complete filter on κ, thenf contains all closed unbounded sets. Proof. If C is a club subset of κ, let a α : α<κ be the increasing enumeration of C. Then C α<κ {ξ : a α+1 <ξ<κ} F, because F contains all final segments (being nontrivial and κ-complete). In other words, if I is normal, then every I-positive set is stationary. The quotient algebra B = P (κ)/i NS is a κ-complete Boolean algebra, where the Boolean operations α<γ and α<γ for γ<κare induced by α<γ and α<γ. Fodor s Theorem implies that B is in fact κ+ -complete: if {X α : α<κ} is a collection of subsets of κ, then α<κ X α and Σ α<κ X α are, respectively, the greatest lower bound and the least upper bound of the equivalence classes X α /I NS B. This observation also shows that if X α : α<κ and Y α : α<κ are two enumerations of the same collection, then α X α and α Y α differ only by a nonstationary set. The following characterization of the club filter is often useful, in particular when used in its generalized form (see Section 6). Let F :[κ] <ω κ; an ordinal γ < κ is a closure point of F if F (α 1,...,α n ) <γwhenever α 1,...,α n <γ. It is easy to see that the set Cl F of all closure points of F is a club. Conversely, if C is a club, define F :[κ] <ω κ by letting F (e) be the least element of C greater than max(e). It is clear that Cl F =Lim(C). Thus every club contains Cl F for some F, and we have this characterization of the club filter: 1.7 Proposition. The club filter is generated by the sets Cl F, for all F : [κ] <ω κ. AsetS κ is stationary if and only if for every F :[κ] <ω κ, S contains a closure point of F Splitting stationary sets It is not immediately obvious that the club filter is not an ultrafilter, that is that there exist stationary sets that are co-stationary, i.e. whose comple-

6 6 I. Stationary Sets ment is stationary. The basic result is the following theorem of Solovay: 1.8 Theorem (Solovay [85]). Let κ be a regular uncountable cardinal. Then every stationary subset of κ can be partitioned into κ disjoint stationary sets. Solovay s proof of this basic result of combinatorial set theory uses methods of forcing and large cardinals, and we shall describe it later in this section. For an elementary proof, see e.g. [49], p To illustrate the combinatorics involved, let us prove a special case of Solovay s theorem. 1.9 Proposition. There exist ℵ 1 pairwise disjoint stationary subsets of ω 1. Proof. For every limit ordinal α<ω 1, choose an increasing sequence {a α n } n=0 with limit α. We claim that there is an n such that for all η<ω 1,there are stationary many α such that a α n η: Otherwise there exists, for each n, someη n such that a α n η n for only a nonstationary set of α s. By ω 1 - completeness, for all but a nonstationary set of α s the sequences {a α n} n are bounded by sup n η n. A contradiction. Thus let n be such that for all η, thesets η = {α : a α n η} is stationary. The function f(α) =a α n is regressive and so by Fodor s Theorem, there is some γ η η such that T η = {α : a α n = γ η} is stationary. Clearly, there are ℵ 1 distinct values of γ η and therefore ℵ 1 mutually disjoint sets T η. Let κ be a regular uncountable cardinal, and let λ<κbe regular. Let E κ λ = {α <κ:cf α = λ}. For each λ, E κ λ is a stationary set. An easy modification of the proof of 1.9 above shows that for every regular λ<κ, every stationary subset of E κ λ can be split into κ disjoint stationary sets. The union λ Eκ λ is the set of all singular limit ordinals. Its complement is the set Reg of all regular cardinals α<κ. The set Reg is stationary just in case κ is a Mahlo cardinal Generic ultrapowers Let M be a transitive model of ZFC, and let κ be a cardinal in M. LetU be an M-ultrafilter, i.e. an ultrafilter on the set algebra P (κ) M. Using functions f M on κ, onecanformanultrapowern = Ult U (M), which is

7 1. The closed unbounded filter 7 a model of ZFC but not necessarily well-founded: f = g {α : f(α) =g(α)} U, f g {α : f(α) g(α)} U. The (equivalence classes of) constant functions c x (α) =x provide an elementary embedding j :(M,ɛ) (N, ), where j(x) =c x, for all x M. An M-ultrafilter U is M-κ-complete if it is closed under intersections of families {X α : α<γ} M, for all γ<κ; U is normal if every regressive f M is constant on a set in U Proposition. Let U be a nonprincipal M-κ-complete, normal M- ultrafilter on κ. Then the ordinals of N have a well-ordered initial segment of order type at least κ +1, j(γ) =γ for all γ<κ,andκ is represented in N by the diagonal function d(α) =α. Now let κ be a regular uncountable cardinal and consider the forcing notion (P, <) wherep is the collection of all stationary subsets of κ, andthe ordering is by inclusion. Let B be the complete Boolean algebra B = B(P ), the completion of (P, <). Equivalently, B is the completion of the Boolean algebra P (κ)/i NS. Let us consider the generic extension V [G] givenby a generic G P. It is rather clear that G is a nonprincipal V -κ-complete normal ultrafilter on κ. Thus Proposition 1.10 applies, where N = Ult G (V ). The model Ult G (V ) is called a generic ultrapower. There is more on generic ultrapowers in Foreman s chapter in this volume; here we use them to present the original argument of Solovay s [85]. First we prove a lemma (that will be generalized in Section 2): 1.11 Lemma. Let κ be a regular uncountable cardinal, and let S be a stationary set. Then the set T = {α S : either α/ Reg or S α is not a stationary subset of α} is stationary. Proof. Let C be a club and let us show that T C is nonempty. Let α be the least element of the nonempty set S C where C =Lim(C ω). If α is not regular, then α T C and we are done, so assume that α Reg. Now C α is a club subset of α disjoint from S α, andsoα T. We shall now outline the proof of Solovay s Theorem: Proof. (Theorem 1.8.) Let S be a stationary subset of κ that cannot be partitioned into κ disjoint stationary sets. By 1.9 and the remarks following

8 8 I. Stationary Sets its proof, we have S Reg. Let I = I NS S, i.e. I = {X κ : X S I NS }.TheidealI is κ-saturated, i.e. every disjoint family W I + has size less than κ; equivalently,b = P (κ)/i has the κ-chain condition. I is also κ-complete and normal. Let G I + be generic, and let N = Ult G (V ) be the generic ultrapower. As I is κ-saturated, N is well-founded (this is proved by showing that every name f for a function in V on κ can be replaced by an actual function on κ). Thus we have (in V [G]) an elementary embedding j : V N where N is a transitive class, j(γ) =γ for all γ<κ,andκ is represented in N by the diagonal function d(α) =α. Note that if A κ is any set (in V ), then A N: this is because A = j(a) κ; infacta is represented by the function f(α) =A α. Now we use the fact that κ-c.c. forcing preserves stationarity (cf. Theorem 1.13 below). Thus S is stationary in V [G], and because N V [G], S is a stationary set in the model N. By the ultrapower theorem we have V [G] S α is stationary for G-almost all α. This, translated into forcing, gives {α S : S α is not stationary} I but that contradicts Lemma Another major application of generic ultrapowers is Silver s Theorem: 1.12 Theorem (Silver [84]). Let λ be a singular cardinal of uncountable cofinality. If 2 α = α + for all cardinals α<λ,then2 λ = λ +. Silver s Theorem is actually stronger than this. It assumes only that 2 α = α + for a stationary set of α s (see Section 2 for the definition of stationary when λ is not regular). The proof uses a generic ultrapower. Even though Ult G (V ) is not necessarily well founded, the method of generic ultrapowers enables one to conclude that 2 λ = λ + when 2 α = α + holds almost everywhere. Silver s Theorem can be proved by purely combinatorial methods [10, 11]. In [30], Galvin and Hajnal used combinatorial properties of stationary sets to prove a substantial generalization of Silver s Theorem (superseded only by Shelah s powerful pcf theory). For further generalizations using stationary sets and generic ultrapowers, see [51] and [52]. One of the concepts introduced in [30] is the Galvin-Hajnal norm of an ordinal function. If f and g are ordinal functions on a regular uncountable

9 1. The closed unbounded filter 9 cardinal κ, letf<gif {α <κ: f(α) <g(α)} contains a club. The relation < is a well-founded partial order, and the norm f is the rank of f in the relation <. We remark that if f<g, then in the generic ultrapower (by I NS ), the ordinal represented by f is smaller than the ordinal represented by g. By induction on η one can easily show that for each η<κ + there exists a canonical function f η : κ κ of norm η, i.e. f η = η and whenever h = η, then{α : f η (α) h(α)} contains a club. (Proof: Let f 0 (α) =0, f η+1 (α) =f η (α) +1. If η<κ + is a limit ordinal, let λ =cfη and let η = lim ξ λ η ξ. If λ < κ, let f η (α) = sup ξ<λ f ηξ (α) andifλ = κ, let f η (α) =sup ξ<α f ηξ (α).) A canonical function of norm κ + may or may not exist, but is consistent with ZFC (cf. [53]). The existence of canonical function f η for all η is equiconsistent with a measurable cardinal [50] Stationary sets in generic extensions Let M and N be transitive models and let M N. Let κ be a regular uncountable cardinal and let S M be a subset of κ. Clearly, if S is stationary in the model N, thens is stationary in M; the converse is not necessarily true, and κ may even not be regular or uncountable in N. It is important to know which forcing extensions preserve stationarity and we shall return to the general case in Section 5. For now, we state two important special cases: 1.13 Theorem. Let κ be a regular uncountable cardinal and let P be a notion of forcing. (a) If P satisfies the κ-chain condition, then every club C V [G] has aclubsubsetd in the ground model. Hence every stationary S remains stationary in V [G]. (b) If P is λ-closed for every λ < κ, then every stationary S remains stationary in V [G]. Proof. (outline) (a) This follows from this basic fact on forcing: if P is κ-c.c., then every unbounded A κ in V [G] has an unbounded subset in V. (b) Let p Ċ is a club; we find a γ S and a q p such that q γ Ċ as follows: we construct an increasing continuous ordinal sequence {γ α } α<κ and a decreasing sequence {p α } of conditions such that p α+1 γ α+1 Ċ,

10 10 I. Stationary Sets and if α is a limit ordinal, then γ α = lim ξ<α γ ξ and p α is a lower bound of {p ξ } ξ<α. There is some limit ordinal α such that γ α S. It follows that p α γ α Ċ. We shall now describe the standard way of controlling stationary sets in generic extensions, so called shooting a club. First we deal with the simplest case when κ = ℵ 1. Let S be a stationary subset of ω 1, and consider the following forcing P S (cf. [9]): The forcing conditions are all bounded closed sets p of countable ordinals such that p S. A condition q is stronger than p if q end-extends p, i.e. p = q α for some α. It is clear that this forcing produces ( shoots ) a closed unbounded subset of S in the generic extension, thus the complement of S becomes nonstationary. The main point of [9] is that ω 1 is preserved and in fact V [G] adds no new countable sets. Also, every stationary subset of S remains stationary. The forcing P S has the obvious generalization to κ>ℵ 1, but more care is required to guarantee that no new small sets of ordinals are added. For instance, this is the case when S contains the set Sing of all singular ordinals <κ. For a more detailed discussion of this problem see [1] Some combinatorial principles There has been a proliferation of combinatorial principles involving closed unbounded and stationary sets. Most can be traced back to Jensen s investigation of the fine structure of L [59] and generalize either Jensen s diamond ( ) or square ( ). These principles are discussed elsewhere in this volume; we conclude this section by briefly mentioning diamond and club-guessing, and only their typical special cases Theorem ( (ℵ 1 ), Jensen [59]). Assume V = L. There exists a sequence a α : α<ω 1 with each a α α, such that for every A ω 1,theset {α <ω 1 : A α = a α } is stationary. (Note that every A ω is equal to some a α,andso (ℵ 1 ) implies 2 ℵ0 = ℵ 1.) 1.15 Theorem ( (E ℵ2 ℵ 0 ), Gregory [40]). Assume GCH. There exists a sequence a α : α E ℵ2 ℵ 0 with each a α α, such that for every A ω 2,the set {α <ω 2 : A α = a α } is stationary Theorem (Club-guessing, Shelah [82]). There exists a sequence c α : α E ℵ3 ℵ 1,whereeachc α isaclosedunboundedsubsetofα, such that for every club C ω 3,theset{α : c α C} is stationary.

11 2. Reflection 11 Unlike most generalizations of square and diamond, Theorem 1.16 is a theorem of ZFC but we note that the gap (between ℵ 1 and ℵ 3 )isessential. 2. Reflection 2.1. Reflecting stationary sets An important property of stationary sets is reflection. Itisusedinseveral applications, and provides a structure among stationary sets it induces a well founded hierarchy. Natural questions about reflection and the hierarchy are closely related to large cardinal properties. We start with a generalization of stationary sets. Let α be a limit ordinal of uncountable cofinality, say cf α = κ>ℵ 0.AsetS α is stationary if it meets every closed unbounded subset of α. The closed unbounded subsets of α generate a κ-complete filter, and Fodor s Theorem 1.5 yields this: 2.1 Lemma. If f is a regressive function on a stationary set S α, then there exists a γ<αsuch that f(ξ) <γ on a stationary subset of S. If S is a set of ordinals and α is a limit ordinal such that cf α>ω,we say that S is stationary in α if S α is a stationary subset of α. 2.2 Definition. Let κ be a regular uncountable cardinal and let S be a stationary subset of κ. If α < κ and cf α > ω, S reflects at α if S is stationary in α. S reflects if it reflects at some α<κ. It is implicit in the definition that κ>ℵ 1. For our first observation, let α<κbe such that cf α>ω. There is a club C α of order type cf α such that every element of C has cofinality < cf α. Thus if S κ is such that every β S has cofinality cf α, thens does not reflect at α. In particular, if κ = λ + where λ is regular, then the stationary set E κ λ does not reflect. On the other hand, if λ<κis regular and λ + <κ,thene κ λ reflects at every α<κsuch that cf α>λ. To investigate reflection systematically, let us first look at the simplest case, when κ = ℵ 2. Let E 0 = E ℵ2 ℵ 0 and E 1 = E ℵ2 ℵ 1. The set E 1 does not reflect; can every stationary S E 0 reflect? Let us recall Jensen s Square Principle [59]: ( κ ) There exists a sequence C α : α Lim (κ + ) such that

12 12 I. Stationary Sets (i) C α is club in α, (ii) if β Lim(C α ), then C β = C α β, (iii) if cf α<κ,then C α <κ. Now assume that ω1 holds and let C α : α Lim (ω 2 ) be a square sequence. Note that for each α E 1, the order type of C α is ω 1. It follows that there exists a countable limit ordinal η such that the set S = {γ E 0 : γ is the η th element of some C α } is stationary. But for every α E 1, S has at most one element in common with C α,andsos does not reflect. Thus ω1 implies that there is a nonreflecting stationary subset of E ℵ2 ℵ 0. Since ω1 holds unless ℵ 2 is Mahlo in L, the consistency strength of every S E ℵ2 ℵ 0 reflects is at least a Mahlo cardinal. This is in fact the exact strength: 2.3 Theorem (Harrington-Shelah [41]). The following are equiconsistent: (i) the existence of a Mahlo cardinal. (ii) every stationary set S E ℵ2 ℵ 0 reflects. Theorem 2.3 improves a previous result of Baumgartner [6] who proved the consistency of (ii) from a weakly compact cardinal. Note that (ii) implies that every stationary set S E ℵ2 ℵ 0 reflects at stationary many α E ℵ2 ℵ 1. A related result of Magidor (to which we return later in this section) gives this equiconsistency: 2.4 Theorem (Magidor [70]). The following are equiconsistent: (i) the existence of a weakly compact cardinal, (ii) every stationary set S E ℵ2 ℵ 0 reflectsatalmostallα E ℵ2 ℵ 1. Here, almost all means all but a nonstationary set. Let us now address the question whether it is possible that every stationary subset of κ reflects. We have seen that this is not the case when κ is the successor of a regular cardinal. Thus κ must be either inaccessible or κ = λ + where λ is singular. Note that because a weakly compact cardinal is Π 1 1 indescribable, every stationary subset of it reflects. In [68], Kunen showed that it is consistent that every stationary S κ reflects while κ is not weakly compact. In [76] it is shown that the consistency strength of every stationary subset of κ reflects is strictly between greatly Mahlo and weakly compact. (For definition of greatly Mahlo, see Section 2.2.)

13 2. Reflection 13 If, in addition, we require that κ be a successor cardinal, then much stronger assumptions are necessary. The argument we gave above using ω1 works for any κ: 2.5 Proposition (Jensen). If λ holds, then there is a nonreflecting stationary subset of E λ+ ℵ 0. As the consistency strength of λ for singular λ is at least a strong cardinal (as shown by Jensen), one needs at least that for the consistency of every stationary S λ + reflects. In [70], Magidor proved the consistency of every stationary subset of ℵ ω+1 reflects from the existence of infinitely many supercompact cardinals. We mention the following applications of nonreflecting stationary sets: 2.6 Theorem (Mekler-Shelah [76]). The following are equiconsistent: (i) every stationary S κ reflects, (ii) every κ-free abelian group is κ + -free. 2.7 Theorem (Tryba [90]). If a regular cardinal κ is Jónsson, then every stationary S κ reflects. 2.8 Theorem (Todorčević [88]).If Rado s Conjecture holds, then for every regular κ>ℵ 1, every stationary S E κ ℵ 0 reflects A hierarchy of stationary sets Consider the following operation (the Mahlo operation) on stationary sets. For a stationary set S κ, thetrace of S is the set of all α at which S reflects: Tr(S) ={α <κ:cfα>ωand S α is stationary}. The following basic properties of trace are easily verified. 2.9 Lemma. (a) If S T,thenTr(S) Tr(T ), (b) Tr(S T )=Tr(S) Tr(T ), (c) Tr(Tr(S)) Tr(S), (d) if S T mod I NS,thenTr(S) Tr(T ) mod I NS. Property (d) shows that the Mahlo operation may be considered as an operation on the Boolean algebra P (κ)/i NS.

14 14 I. Stationary Sets If λ<κis regular, let M κ λ = {α <κ:cfα λ}, and note that Tr(Eκ λ ) = Tr(M κ λ )=Mκ λ +. The Mahlo operation on P (κ)/i NS can be iterated α times, for α<κ +. Let M 0 = κ M α+1 = Tr(M α ) M α = ξ<κ M αξ (α limit,α= {α ξ : ξ<κ}). The sets M α are defined mod I NS (the limit stages depend on the enumeration of α). The sequence {M α } α<κ + is decreasing mod I NS,andwhen α<κ,thenm α = M κ λ where λ is the αth regular cardinal. Note that κ is (weakly) Mahlo just in case M κ = Reg is stationary, and that by Lemma 1.11, {M α } α is strictly decreasing (mod I NS,aslongasM α is stationary). Following [13], κ is called greatly Mahlo if M α is stationary for every α<κ +. We shall now consider the following relation between stationary subsets of κ Definition (Jech [47]). S<T iff S α is stationary for almost all α T. In other words, S < T iff Tr(S) T mod I NS. As an example, if λ<µare regular, then E κ λ < Eκ µ. Note also that the language of generic ultrapowers gives this description of <: 2.11 Proposition. S<T iff T S is stationary in Ult G (V ). The following lemma states the basic properties of < Lemma. (a) A<Tr(A), (b) if A<Band B<Cthan A<C, (c) if A A and B B mod I NS,andifA<B,thenA <B. By (c), < can be considered a relation on P (κ)/i NS. By Proposition 1.11, < is irreflexive and so it is a partial ordering. The next theorem shows that the partial ordering < is well founded Theorem (Jech [47]). The relation < is well founded. Proof. Assume to the contrary that there are stationary sets such that A 1 > A 2 >A 3 >. Therefore there are clubs C n such that A n C n Tr(A n+1 )

15 2. Reflection 15 for n =1, 2,...Foreachn, let B n = A n C n Lim(C n+1 ) Lim(Lim(C n+2 )) Each B n is stationary, and for every n, B n Tr(B n+1 ). Let α n =min(b n ). Since B n+1 α n is stationary, we have α n+1 <α n, and therefore, a decreasing sequence α 1 >α 2 >α 3 >. A contradiction. As < is well founded, we can define the order of stationary sets A κ, and of the cardinal κ: o(a) =sup{o(x)+1:x<a}, o(κ) =sup{o(a)+1:a κ stationary}. We also define o(ℵ 0 ) = 0, and o(α) =o(cf (α)) for every limit ordinal α. Note that o(e κ ℵ 0 ) = 0, and in general o(e κ λ )=o(mκ λ )=α, ifλ is the α th regular cardinal. Also, o(ℵ n )=n, o(κ) κ +1 iff κ is Mahlo, and o(κ) κ + iff κ is greatly Mahlo Canonical stationary sets If λ is the α th regular cardinal, then E κ λ has order α; moreover, the set is canonical, in the sense explained below. In fact, canonical stationary sets exist for all orders α<κ +. Let E be a stationary set of order α. IfX E is stationary, then o(x) o(e). We call E canonical of order α if (i) every stationary X E has order α, and (ii) E meets every set of order α. Clearly, a canonical set of order α is unique (mod I NS ), and two canonical sets of different orders are disjoint (mod I NS ). In the following proposition, maximal and is meant mod I NS Proposition (Jech [47]). A canonical set E of order α exists iff there exists a maximal set M of order α. Then(modI NS ) E M Tr(M),M E Tr(E), and Tr(E) Tr(M). One can show that the sets M α obtained by iterating the Mahlo operation are maximal (as long as they are stationary). Thus when we let E α = M α Tr(M α ), we get canonical stationary sets, of all orders α<κ + (for α<o(κ)). The canonical stationary sets E α and the canonical function f α (of Galvin- Hajnal norm α) are closely related:

16 16 I. Stationary Sets 2.15 Proposition (Jech [47]). For every α<κ +, α<o(κ), E α {ξ<κ: f α (ξ) =o(ξ)} Full reflection Let us address the question of what is the largest possible amount of reflection, for stationary subsets of a given κ. As A<Bmeans that A reflects at almost all points of B, we would like to maximize the relation <. But A<Bimplies that o(a) <o(b), so we might ask whether it is possible that A<Bfor any two stationary sets such that o(a) <o(b). By Magidor s Theorem 2.4 it is consistent that S<E ℵ2 ℵ 1, and therefore S<Tfor every S of order 0 and every T of order 1. However, this does not generalize, as the following lemma shows that when κ ℵ 3, then there exist S and T with o(s) =0ando(T ) = 1 such that S T Lemma (Jech-Shelah [54]). If κ ℵ 3, then there exist stationary sets S E κ ℵ 0 and T E κ ℵ 1 such that S does not reflect at any α T. Proof. Let S γ, γ<ω 2, be pairwise disjoint stationary subsets of E κ ℵ 0,and let C α, α E κ ℵ 1, be such that for every α, C α is a club subset of α, oforder type ω 1. Because at most ℵ 1 of the sets S γ meet each C α,thereexistsfor each α some γ(α) such that C α S γ(α) =. There exists some γ such that the set T = {α : γ(α) =γ} is stationary; let S = S γ. For every α T, S C α = and so S does not reflect at α. This lemma illustrates some of the difficulties involved when dealing with reflection at singular ordinals. This problem is investigated in detail in [54], where the best possible consistency result is proved for stationary subsets of the ℵ n, n<ω. Let us say that a stationary set S κ reflects fully at regular cardinals if for any stationary set T of regular cardinals o(s) <o(t ) implies S<T, and let us call Full Reflection the statement that every stationary subset of κ reflects fully at regular cardinals. Full Reflection is of course nontrivial only if κ is a Mahlo cardinal. A modification of Theorem 2.4 shows that Full Reflection for a Mahlo cardinal is equiconsistent with weak compactness. The following theorem establishes the consistency strength of Full Reflection for cardinals in the Mahlo hierarchy:

17 3. Saturation Theorem (Jech-Shelah [55]). The following are equiconsistent, for every α κ + : (i) κ is Π 1 α-indescribable, (ii) κ is α-mahlo and Full Reflection holds. (A regular cardinal κ is α-mahlo if o(κ) κ + α; κ is Π 1 1 -indescribable iff it is weakly compact.) Full Reflection is also consistent with large cardinals. The paper [57] proves the consistency of Full Reflection with the existence of a measurable cardinal. This has been improved and further generalized in [38]. Finally, the paper [91] shows that any well-founded partial order of size κ + can be realized by the reflection ordering < on stationary subsets of κ, in some generic extension (using P 2 κ-strong κ in the ground model). 3. Saturation 3.1. κ + saturation By Solovay s 1.8 every stationary subset of κ can be split into κ disjoint stationary sets. In other words, for every stationary S κ, theideali NS S is not κ-saturated. A natural question is if the nonstationary ideal can be κ + -saturated. An ideal I on κ is κ + -saturated if the Boolean algebra P (κ)/i has the κ + -chain condition. Thus I NS S is κ + -saturated when there do not exist κ + stationary subsets of S such that the intersection of any two of them is nonstationary. The existence and properties of κ + -saturated ideals have been thoroughly studied since their introduction in [85], and involve large cardinals. The reader will find more details in Foreman s chapter in this volume. We shall concentrate on the special case when I is the nonstationary ideal. The main question, whether the nonstationary ideal can be κ + -saturated, has been answered. But a number of related questions are still open. 3.1 Theorem (Gitik-Shelah [37]). The nonstationary ideal on κ is not κ + - saturated, for every regular cardinal κ ℵ Theorem (Shelah). It is consistent, relative to the existence of a Woodin cardinal, that the nonstationary ideal on ℵ 1 is ℵ 2 -saturated.

18 18 I. Stationary Sets The consistency result in Theorem 3.2 was first proved in [87] using a strong determinacy assumption. That hypothesis was reduced in [92] to AD, while in [27], the assumption was the existence of a supercompact cardinal. Shelah s result (announced in [81]) is close to optimal: by Steel [86], the saturation of I NS plus the existence of a measurable cardinal imply the existence of an inner model with a Woodin cardinal. All the models mentioned in the preceding paragraph satisfy 2 ℵ0 > ℵ 1. This may not be accidental, and it has been conjectured that the saturation of I NS on ℵ 1 implies that 2 ℵ0 > ℵ 1. In fact, Woodin proved this [94] under the addional assumption that there exists a measurable cardinal. We note in passing that by [27], 2 ℵ0 = ℵ 1 is consistent with I NS S being saturated for some stationary S. Woodin s construction [94] yields a model (starting from AD) in which the ideal I NS is ℵ 1 -dense, i.e. the algebra P (ω 1 )/I NS has a dense set of size ℵ 1. This, and Woodin s more recent work using Steel s inner model theory, gives the following equiconsistency. 3.3 Theorem (Woodin). The following are equiconsistent: (i) ZF + AD, (ii) there are infinitely many Woodin cardinals, (iii) the nonstationary ideal on ℵ 1 is ℵ 1 -dense. As for the continuum hypothesis, Shelah proved in [80] that if I NS is ℵ 1 -dense, then 2 ℵ0 =2 ℵ1. We remark that the mere existence of a saturated ideal affects cardinal arithmetic, cf. [63] and [52]. Let us now return to Theorem 3.1. The general result proved in [37] is this: 3.4 Theorem (Gitik-Shelah [37]). If ν is a regular cardinal and ν + <κ, then I NS E κ ν is not κ+ -saturated. The proof of 3.4 combines an earlier result of Shelah (Theorem 3.7 below) with an application of the method of guessing clubs (as in 1.16). The earlier result uses generic ultrapowers and states that if κ = λ + and ν cfλ is regular, then no ideal concentrating on E κ ν is κ + -saturated. The method of generic ultrapowers is well suited for κ + -saturated ideals. Forcing with P (κ)/i where I is a normal κ-complete κ + -saturated ideal makes the generic ultrapower N = Ult G (V ) well founded, preserves the cardinal κ +, and satisfies P N (κ) =P V [G] (κ). It follows that all cardinals

19 3. Saturation 19 <κare preserved in V [G], and it is obvious that if E κ ν G, thenn (and therefore V [G] as well) satisfies cf κ = ν. Shelah s Theorem 3.7 below follows from a simple combinatorial lemma. Let λ be a cardinal and let α<λ + be a limit ordinal. Let us call a family {X ξ : ξ<λ + } a strongly almost disjoint (s.a.d.) family of subsets of α if every X ξ α is unbounded, and if for every ϑ<λ + there exist ordinals δ ξ <α,forξ<ϑ, such that the sets X ξ δ ξ, ξ<ϑ, are pairwise disjoint. Note that if κ is a regular cardinal than there is a s.a.d. family {X ξ : ξ<κ + } of subsets of κ. 3.5 Lemma. If α < λ + and cf α cf λ, then there exists no strongly almost disjoint family of subsets of α. Proof. Assume to the contrary that {X ξ : ξ<λ + } is a s.a.d. family of subsets of α. We may assume that each X ξ has order type cf α. Let f be a function mapping λ onto α. Since cf λ cfα there exists for each ξ some γ ξ <λsuch that X ξ f γ ξ is cofinal in α. There is some γ and a set W λ + of size λ such that γ ξ = γ for all ξ W. Let ϑ>sup W. By the assumption on the X ξ there exist ordinals δ ξ <α, ξ<ϑ, such that the X ξ δ ξ are pairwise disjoint. Thus f 1 (X ξ δ ξ ), ξ W,areλ pairwise disjoint subsets of γ. A contradiction. 3.6 Corollary (Shelah [79]). If κ is a regular cardinal and if a forcing P makes cf κ cf κ, thenp collapses κ +. Proof. Assume that κ + is not collapsed; thus in V [G], (κ + ) V = λ + where λ = κ. In V there is a s.a.d. family {X ξ : ξ<(κ + ) V }, and it remains a s.a.d. family in V [G], of size λ +. Since cf κ cfλ, inv [G], this contradicts Lemma Theorem (Shelah). If κ = λ +,ifν cf λ is regular and if I is a normal κ-complete κ + -saturated ideal on κ, thene κ ν I. Proof. If not, then forcing with I-positive subsets of E κ ν well as cf λ, andmakescfκ = ν; a contradicton. preserves κ + as Theorem 3.4 leaves open the following problem: If λ is a regular cardinal, can I NS E λ+ λ be λ ++ -saturated? (For instance can I NS E ℵ2 ℵ 1 be ℵ 3 - saturated?) Let us also mention that for all regular ν and κ not excluded by Corollary 3.7, it is consistent that I NS S is κ + -saturated for some S E κ ν (see [33]).

20 20 I. Stationary Sets If κ is a large cardinal, then I NS Reg can be κ + -saturated, as the following theorem shows. Of course, κ cannot be too large: if κ is greatly Mahlo, then the canonical stationary sets E α κ α<κ + witness nonsaturation. 3.8 Theorem (Jech-Woodin [58]). For any α < κ +, the following are equiconsistent: (i) κ is measurable of order α, (ii) κ is α-mahlo and the ideal I NS Reg on κ is κ + -saturated Precipitousness An important property of saturated ideals is that the generic ultrapower is well-founded. It has been recognized that this property is important enough to single out and study the class of ideals that have it. The ideals for which the generic ultrapower is well founded are called precipitous. They are described in detail in Foreman s chapter in this volume; here we address the question of when the nonstationary ideal is precipitous. Precipitous ideals were introduced by Jech and Prikry in [51]. There are several equivalent formulations of precipitousness. Let I be an ideal on some set E. AnI-partition is a maximal family of I-positive sets such that the intersection of any two of them is in I. LetG I denote the infinite game of two players who alternately pick I-positive sets S n such that S 1 S 2 S 3. The first player wins if n=1 S n =. 3.9 Theorem (Jech-Prikry [51, 45, 46, 29]). Let I be an ideal on a set E. The following are equivalent: (i) forcing with P (E)/I makes the generic ultrapower well-founded, (ii) for every sequence {W n } n=1 of I-partitions there exists a sequence {X n } n=1 such that X n W n for each n, and n=1 X n, (iii) the first player does not have a winning strategy in the game G I. The problem of whether the nonstationary ideal on κ can be precipitous involves large cardinals. For κ = ℵ 1 the exact consistency strength is the existence of a measurable cardinal: 3.10 Theorem (Jech-Magidor-Mitchell-Prikry [50]). The following are equiconsistent: (i) there exists a measurable cardinal, (ii) the nonstationary ideal on ℵ 1 is precipitous.

x < y iff x < y, or x and y are incomparable and x χ(x,y) < y χ(x,y).

x < y iff x < y, or x and y are incomparable and x χ(x,y) < y χ(x,y). 12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory currently. There are many provably different kinds of large cardinals whose descriptions

More information

Chang s Conjecture and weak square

Chang s Conjecture and weak square Chang s Conjecture and weak square Hiroshi Sakai Graduate School of System Informatics, Kobe University hsakai@people.kobe-u.ac.jp Abstract We investigate how weak square principles are denied by Chang

More information

Continuous tree-like scales

Continuous tree-like scales Carnegie Mellon University Research Showcase @ CMU Department of Mathematical Sciences Mellon College of Science 4-2010 Continuous tree-like scales James Cummings Carnegie Mellon University, jcumming@andrew.cmu.edu

More information

The Power Set Function

The Power Set Function The Power Set Function Moti Gitik School of Mathematical Sciences Tel Aviv University Tel Aviv, Israel e-mail: gitik@post.tau.ac.il Abstract We survey old and recent results on the problem of finding a

More information

EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE DEGREES

EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE DEGREES EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE ENUMERATION DEGREES AND THE ω-enumeration DEGREES MARIYA I. SOSKOVA AND IVAN N. SOSKOV 1. Introduction One of the most basic measures of the complexity of a

More information

COFINAL MAXIMAL CHAINS IN THE TURING DEGREES

COFINAL MAXIMAL CHAINS IN THE TURING DEGREES COFINA MAXIMA CHAINS IN THE TURING DEGREES WEI WANG, IUZHEN WU, AND IANG YU Abstract. Assuming ZF C, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω 1 in the

More information

SETS OF GOOD INDISCERNIBLES AND CHANG CONJECTURES WITHOUT CHOICE

SETS OF GOOD INDISCERNIBLES AND CHANG CONJECTURES WITHOUT CHOICE SETS OF GOOD INDISCERNIBLES AND CHANG CONJECTURES WITHOUT CHOICE IOANNA M. DIMITRIOU Abstract. With the help of sets of good indiscernibles above a certain height, we show that Chang conjectures involving

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

GENTLY KILLING S SPACES TODD EISWORTH, PETER NYIKOS, AND SAHARON SHELAH

GENTLY KILLING S SPACES TODD EISWORTH, PETER NYIKOS, AND SAHARON SHELAH GENTLY KILLING S SPACES TODD EISWORTH, PETER NYIKOS, AND SAHARON SHELAH Abstract. We produce a model of ZFC in which there are no locally compact first countable S spaces, and in which 2 ℵ 0 < 2 ℵ 1. A

More information

MODELS OF SET THEORY

MODELS OF SET THEORY MODELS OF SET THEORY STEFAN GESCHKE Contents 1. First order logic and the axioms of set theory 2 1.1. Syntax 2 1.2. Semantics 2 1.3. Completeness, compactness and consistency 3 1.4. Foundations of mathematics

More information

SET-THEORETIC CONSTRUCTIONS OF TWO-POINT SETS

SET-THEORETIC CONSTRUCTIONS OF TWO-POINT SETS SET-THEORETIC CONSTRUCTIONS OF TWO-POINT SETS BEN CHAD AND ROBIN KNIGHT AND ROLF SUABEDISSEN Abstract. A two-point set is a subset of the plane which meets every line in exactly two points. By working

More information

arxiv:math/0510680v3 [math.gn] 31 Oct 2010

arxiv:math/0510680v3 [math.gn] 31 Oct 2010 arxiv:math/0510680v3 [math.gn] 31 Oct 2010 MENGER S COVERING PROPERTY AND GROUPWISE DENSITY BOAZ TSABAN AND LYUBOMYR ZDOMSKYY Abstract. We establish a surprising connection between Menger s classical covering

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

A NEW CONDENSATION PRINCIPLE

A NEW CONDENSATION PRINCIPLE A NEW CONDENSATION PRINCIPLE THORALF RÄSCH AND RALF SCHINDLER Abstract. We generalize (A), which was introduced in [Sch ], to larger cardinals. For a regular cardinal κ > ℵ 0 we denote by κ (A) the statement

More information

The Banach-Tarski Paradox

The Banach-Tarski Paradox University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES

THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES ANDREW E.M. LEWIS 1. Introduction This will be a course on the Turing degrees. We shall assume very little background knowledge: familiarity with

More information

SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

The Markov-Zariski topology of an infinite group

The Markov-Zariski topology of an infinite group Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

AN INTRODUCTION TO SET THEORY. Professor William A. R. Weiss

AN INTRODUCTION TO SET THEORY. Professor William A. R. Weiss AN INTRODUCTION TO SET THEORY Professor William A. R. Weiss October 2, 2008 2 Contents 0 Introduction 7 1 LOST 11 2 FOUND 19 3 The Axioms of Set Theory 23 4 The Natural Numbers 31 5 The Ordinal Numbers

More information

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

More information

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005 Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

More information

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements. 3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but

More information

Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

More information

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

More information

Low upper bound of ideals, coding into rich Π 0 1 classes

Low upper bound of ideals, coding into rich Π 0 1 classes Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low

More information

There is no degree invariant half-jump

There is no degree invariant half-jump There is no degree invariant half-jump Rod Downey Mathematics Department Victoria University of Wellington P O Box 600 Wellington New Zealand Richard A. Shore Mathematics Department Cornell University

More information

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

More information

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in

More information

Cellular objects and Shelah s singular compactness theorem

Cellular objects and Shelah s singular compactness theorem Cellular objects and Shelah s singular compactness theorem Logic Colloquium 2015 Helsinki Tibor Beke 1 Jiří Rosický 2 1 University of Massachusetts tibor beke@uml.edu 2 Masaryk University Brno rosicky@math.muni.cz

More information

Embeddability and Decidability in the Turing Degrees

Embeddability and Decidability in the Turing Degrees ASL Summer Meeting Logic Colloquium 06. Embeddability and Decidability in the Turing Degrees Antonio Montalbán. University of Chicago Nijmegen, Netherlands, 27 July- 2 Aug. of 2006 1 Jump upper semilattice

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

Finite Projective demorgan Algebras. honoring Jorge Martínez

Finite Projective demorgan Algebras. honoring Jorge Martínez Finite Projective demorgan Algebras Simone Bova Vanderbilt University (Nashville TN, USA) joint work with Leonardo Cabrer March 11-13, 2011 Vanderbilt University (Nashville TN, USA) honoring Jorge Martínez

More information

Logic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic

Logic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic Logic, Algebra and Truth Degrees 2008 September 8-11, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,

More information

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

More information

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3

FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3 FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Minimal R 1, minimal regular and minimal presober topologies

Minimal R 1, minimal regular and minimal presober topologies Revista Notas de Matemática Vol.5(1), No. 275, 2009, pp.73-84 http://www.saber.ula.ve/notasdematematica/ Comisión de Publicaciones Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold: Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),

More information

Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

More information

Separation Properties for Locally Convex Cones

Separation Properties for Locally Convex Cones Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Extension of measure

Extension of measure 1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

More information

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

More information

Group Theory. Contents

Group Theory. Contents Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

More information

On The Existence Of Flips

On The Existence Of Flips On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry

More information

On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

More information

Adaptive Online Gradient Descent

Adaptive Online Gradient Descent Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

More information

Row Ideals and Fibers of Morphisms

Row Ideals and Fibers of Morphisms Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

More information

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

More information

A domain of spacetime intervals in general relativity

A domain of spacetime intervals in general relativity A domain of spacetime intervals in general relativity Keye Martin Department of Mathematics Tulane University New Orleans, LA 70118 United States of America martin@math.tulane.edu Prakash Panangaden School

More information

A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES THEODORE A. SLAMAN AND ANDREA SORBI Abstract. We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: In fact, below

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

More information

Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.

Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true

More information

Notes on Richard Dedekind s Was sind und was sollen die Zahlen?

Notes on Richard Dedekind s Was sind und was sollen die Zahlen? Notes on Richard Dedekind s Was sind und was sollen die Zahlen? David E. Joyce, Clark University December 2005 Contents Introduction 2 I. Sets and their elements. 2 II. Functions on a set. 5 III. One-to-one

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385 mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

More information

Ri and. i=1. S i N. and. R R i

Ri and. i=1. S i N. and. R R i The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

More information

The Ideal Class Group

The Ideal Class Group Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

More information

Conjectures and Questions from Gerald Sacks s Degrees of Unsolvability

Conjectures and Questions from Gerald Sacks s Degrees of Unsolvability Conjectures and Questions from Gerald Sacks s Degrees of Unsolvability Richard A. Shore Department of Mathematics Cornell University Ithaca NY 14853 Abstract We describe the important role that the conjectures

More information

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

Finite dimensional topological vector spaces

Finite dimensional topological vector spaces Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

More information

R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

More information

Global Properties of the Turing Degrees and the Turing Jump

Global Properties of the Turing Degrees and the Turing Jump Global Properties of the Turing Degrees and the Turing Jump Theodore A. Slaman Department of Mathematics University of California, Berkeley Berkeley, CA 94720-3840, USA slaman@math.berkeley.edu Abstract

More information

This chapter is all about cardinality of sets. At first this looks like a

This chapter is all about cardinality of sets. At first this looks like a CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

More information

Introduction to finite fields

Introduction to finite fields Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

More information

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

More information

Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

More information

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

More information

9 More on differentiation

9 More on differentiation Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

More information

Point Set Topology. A. Topological Spaces and Continuous Maps

Point Set Topology. A. Topological Spaces and Continuous Maps Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

On an anti-ramsey type result

On an anti-ramsey type result On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

More information