Saharon Shelah
David Aspero asks on the possibility of having Forcing axiom FA_{aleph_2}(K), where K is the class of forcing notions preserving stationarity of subsets of aleph_1 and of aleph_2. We answer negatively, in fact we show the negative result for any regular lambda>aleph_1 even demanding adding no new sequence of ordinals of length<lambda.
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We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{ω_1, ω_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.
We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0}. We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced separable abelian p-group in lambda. (Note: both results fail if lambda = lambda^{aleph_0} or if lambda is strong limit, cf (mu)= aleph_0< mu).
It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .
In a countable superstable NDOP theory, the existence of a rigid aleph_epsilon-saturated model implies the existence of 2^lambda rigid aleph_epsilon-saturated models of power lambda for every lambda>2^{aleph_0}.
We study aleph_0-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of lambda-Borel completeness and prove that such theories are lambda-Borel complete. Using this, we conclude that an aleph_0-stable theory has 2^lambda pairwise non-L(infinity,aleph_0) equivalent models of size lambda for all infinite cardinals lambda if and only if T either has eni-DOP or is eni-deep.
We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $σ$-linked partial orderings. This answers an old question of the first author about the relative strength of Martin's axiom for $σ$-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can de…
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then there is an isomorphism i:M-> N which is the identity on P(M). T is relatively categorical if it is relatively lambda-categorical for every lambda. The question arises whether the relative lambda-categoricity of T for s…
Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a "partial" countable support iteration but it is c.c.c.
A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.
We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, that, for example, consistently, aleph_{omega_1} not-> (aleph_{omega_1}, omega+1)^2 and consistently, 2^{aleph_0} not-> (2^{aleph_0},omega +1)^2 .
Let K^0_lambda be the class of structures < lambda,<,A>, where A subseteq lambda is disjoint from a club, and let K^1_lambda be the class of structures < lambda,<,A>, where A subseteq lambda contains a club. We prove that if lambda = lambda^{< kappa} is regular, then no sentence of L_{lambda^+ kappa} separates K^0_lambda and K^1_lambda. On the other hand, we prove that if lambda = mu^+, mu = mu^{< mu}, and a forcing axiom holds (and aleph_1^L= aleph_1 if mu = aleph_0), then there is a sentence of L_{lambda lambda} which separates K^0_lambda and K^1_lambda .
Echeloned saturation and forcing axioms
针对Paul Larson的问题，研究结合不同力迫公理与集合论假设，证明特定条件下ℵ₁上无弱Laver理想，以及Baumgartner公理下ℵ₂上无Laver理想。
Forcing axioms and the structure theory of Aronszajn trees
本文研究Aronszajn树的结构定理对力迫公理的需求，证明MA(ℵ₁)足以保证存在2^ℵ₁棵两两非同构的相干A-树。
The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum
本文证明针对 $\aleph_1$ 规模偏序集的真力迫公理与任意大的连续统相容，并进一步构造模型同时满足 Martin's Maximum 及多种组合原则。
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…
Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to $\aleph\_1$. Later we give applications, among them the consistency of ${\rm MM}$ with $\aleph\_ω$ not being Jonsson which answers a question raised during Oberwolfach 2005.
Ordinary infinitary languages L_{lambda, kappa} satisfy the Interpolation Theorem only in the case lambda <= {aleph_1}, kappa = {aleph_0}, this include first order logic of course. There are also some pairs of such logics satifying interpolation, e.g. (L_{lambda^+,{aleph_0}}, L_{(2^lambda)^+, lambda^+}) . Does this come from an intermidiate logic satisfying it? Is it nice? unique? We define for kappa = beth_kappa a new logic L^1_kappa such that L_{kappa omega}< L^1_kappa LL_{kappa kappa} and L^1_kappa is very nice; in particular satisfies the Interpolation Theorem. Moreover, L^1_kappa has a mo…
Assume <aleph_0,aleph_1>-><lambda,lambda^+>. Assume M is a model of a first order theory T of cardinality at most lambda^+ in a vocabulary L(T) of cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be a set of first order formulas in L(T) and let D be a regular filter on lambda. Then M is Delta-embeddable into the reduced power N^lambda/D, provided that every Delta-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over lambda, M^lambda/D is lambda^{++}-universal. Our second result is as follo…
CWH, CWN stand for collectionwise Hausdorff and collectionwise normal respectively. We analyze the statement "there is a lambda-CWH not CWH first countable (Hausdorff topological) space". We prove the existence of such a space under various conditions, show its equivalence to: there is a lambda-CWN not CWN first countable space and give an equivalent set theoretic statement; the nicest version we can obtain is in \S4. The author had a flawed proof of the existence of such spaces in ZFC, for some lambda > aleph_1, in June of 1992; still we decided that there is some interest in the correct part…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
If cf(kappa) = kappa, kappa^+< cf(lambda) = λ, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: {C_delta \cap alpha: alpha \in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{<aleph_1}(lambda) of cardinality cf(S_{<aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domai…
Baumgartner's Axiom and Small Posets
本文在MA条件下证明了存在一个实数的ℵ₁密集集，其在特定扩张中不包含可数无限的序同构子集，从而加强了相关结果并提供了新证明。
We show that adding a random real number destroys a large fragment of Martin's axiom, namely Martin's axiom for partial orders that have precalibre-$\aleph_1$, thus answering an old question of J. Roitman [9]. We also answer a question of J. Steprans and S. Watson [13] by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.
We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength ``NS_lambda is aleph_1-preserving'', for lambda > aleph_2 .
Martin's Axiom, Large Continuum and Global $Σ^1_n$-Uniformization
本文研究在马丁公理、$2^{\aleph_0}=\aleph_3$、实数的轻语气$Δ^1_3$良序及所有$n\geq 2$的$Σ^1_n$-统一化条件下，构造一个满足这些条件的广义扩展。
We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l<omega}). Then we present results complementary to those of section 2 (but not close enough): if IND…
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength…
An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R^+ is multiplication by some ring-element. The existence of almost-free E-rings of cardinality greater than 2^{aleph_0} is undecidable in ZFC. While they exist in Goedel's universe, they do not exist in other models of set theory. For a regular cardinal aleph_1 <= lambda <= 2^{aleph_0} we construct E-rings of cardinality lambda in ZFC which have aleph_1-free additive structure. For lambda = aleph_1 we therefore obtain the existence of almost-free E-rings of cardinality aleph_1 in ZFC.
We continue the investigations in the author's book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S_{<= aleph_0}(kappa), subseteq) for kappa real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(lambda, lambda, theta^+,2)=pp(lambda) in more cases even when cf(lambda)= aleph_0 (Section 1), deal with bounds of pp(lambda) for lambda limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6).
Open Colorings and Baumgartner's Axiom
构造一个满足 $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ 但 Baumgartner 公理不成立的模型，并给出一个既不可逆也不递增的 $\aleph_1$-稠密集。
We prove that if cf(lambda) > aleph_0 and 2^{cf(lambda)}<lambda, then lambda->(lambda,omega+1)^2.
Martin's Axiom does not imply that all automorphisms of P(N)/[N]^{<aleph_0} are somewhere trivial.
An abelian group is said to be aleph_1-free if all its countable subgroups are free. Our main result is: If R is a ring with R^+ free and |R|<lambda <= 2^{aleph_0}, then there exists an aleph_1-free abelian group G of cardinality lambda with End(G)=R . A corollary to this theorem is: Indecomposable aleph_1-free abelian groups of cardinality aleph_1 do exist.
Uniformization of ladder system colorings and stationary precaliber forcings
本文研究了梯子系统染色均匀化变体与 Martin 公理片段的关系，提出了平稳预 caliber $\aleph_1$ 性质的力迫公理，并证明了其与 $σ$-链接偏序集力迫公理的分离。
We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $ω_1$ that don't seem to have been considered in the literature before.
Maximality Principles and Ressurection Axioms under a Laver-generic large cardinal
本文研究了 Laver-一般性公理与极大性原理及复活公理的关系，证明无参数和参数化极大性原理独立于 Laver-一般性公理，而局部极大性原理和复活公理可由足够强的 tightly Laver-一般大基数存在性推出。
In this short note we show that if lambda>aleph_1 is regular and lambda is not the successor of a singular cardinal of cofinality aleph_0, and G is a lambda-free abelian group of size lambda, then there is a free group G' subseteq G of size lambda.
Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.
We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{\aleph_0}=\aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many…
A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [Sh:24]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(mu) on an infinite cardinal mu are all of the form S_kappa(mu)= the subgroup consisting of elements whose support has cardinality <kappa for some kappa <= mu^+. A many-sorted structure M_{kappa lambda mu} is defined which, it is shown, encapsulates the first order properties of the group S_lambda (mu)/S_kappa (mu). Specifically, these two structures are (uniformly) bi-interpretable, …
Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a base of cardinality <= mu, lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has cardinality <= mu or >= 2^lambda. This is our main conclusion. First we prove it under some set theoretic assumption, which is clear when lambda = mu ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove that the simplest examples are the basic ones; they occur in every example (for lambda = aleph_0 this fulf…
Given a cardinal $λ$, category forcing axioms for $λ$-suitable classes $Γ$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_λ$, modulo generic extensions via forcing notions from $Γ$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairwise incompatible category forcing axioms for $ω_1$-suitable classes.
section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)
Let lambda be aleph_0 or a strong limit of cofinality aleph_0. Suppose that (G_m,p_{m,n}:m =< n<omega) and (H_m,p^t_{m,n}: m=< n < omega) are projective systems of groups of cardinality less than lambda and suppose that for every n<omega there is a homomorphism h:H_n-->G_n such that all the diagrams commute. If for every mu<lambda there exists (f_i in G_omega:i<mu) such that for distinct i,j we have: f_i f_j^{-1} notin h_omega(H_omega), then there exists (f_i in G_omega:i<2^lambda) such that for distinct i,j we have f_i f_j^{-1} notin h_omega(H_omega).
The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a separable (reduced, torsion-free) R-module G respectively of cardinality |G|=lambda with End_RG=A oplus Fin G.
The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M is linearly ordered by <, we will say that the linear ordering (M,<) characterizes kappa. It is known that if kappa is characterizable, then kappa plus is characterizable by a linear ordering. Also, if kappa is characterizable by a dense linear ordering with an increasing sequence of size kappa, then 2^kappa is characterizable. We show that if kappa is homo…
Reflection and Recurrence
本文研究结构性反射原则与递归公理对 ZFC 的扩展，提出其极大一致组合 Laver-generic Maximum（LGM），将连续统确定为 aleph_2 并整合多种已知独立陈述。
Extendible cardinals, and Laver-generic large cardinal axioms for extendibility
引入用于可扩展性的Laver-生成大基数公理，证明其能推导出先前已知的大多数推论，并探讨了其一致性、反射性质及相关分离结果。
Forcing Axioms, the Uniformization and the Basis Property
该数理逻辑研究证明存在满足马丁公理的模型具有粗体$Σ^1_3$-一致化性质，且有界恰当力迫公理加$ℵ_1$对实数可及可直接推出该性质，推进了力迫公理与描述集合论性质关联的研究。
An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…
Our aim is to prove that if T is a complete first order theory, which is not superstable (no knowledge on this notion is required), included in a theory T_1 then for any lambda > |T_1| there are 2^lambda models of T_1 such that for any two of them the tau(T)-reducts of one is not elementarily embeddable into the tau(T)-reduct of the other, thus completing the investigation of [Sh:a, Ch. VIII]. Note the difference with the case of unstable T: there lambda > |T_1| + aleph_0 suffices. By [Sh:E59] it suffices for every such lambda to find a complicated enough family of trees with omega + 1 levels …
We prove that for any regular kappa and mu > kappa below the first fix point (lambda = aleph_lambda) above kappa, there is a graph with chromatic number > kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic number less than or equal to kappa.
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10). In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{ω_1}$ such that, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$ (Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory…
Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$
该研究从CH和Hajnal着色出发，通过σ-中心力迫的有限支持迭代构造出满足$MA_{ω_1}$（σ-中心）等条件且$ω_1^2\nrightarrow(ω_1^2,3)^2$成立的模型，同时提出并证明了相关力迫保持原理。
Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
本文研究了在支持大小为$\aleph_1$的迭代中保持$\aleph_1$和$\aleph_2$的性质B，扩展到其他满足$κ^{<κ}=κ$的基数，并引入了修正的完美集forcing。
In [Sh:89] we, answering a question of Monk, have explicated the notion of ``a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it'' (see section 2 here) and proved the existence of one with character density aleph_0, assuming first diamondsuit_{aleph_1} and then only CH. The idea was that if h is an endomorphism of B, not among the ``trivial'' ones, then there are pairwise disjoint D_n in B with h(d_n) not subset d_n. Then we can, for some S subset omega, add an element x such that d <= x for n in S, x cap d_n=0 for n not in S while forbidding a solution for {y…
Let $\mathsf{MM}^{++}(κ)$ state that the forcing axiom $\mathsf{MM}^{++}$ can be instantiated only for stationary set preserving posets of size at most $κ$. We give a detailed account of Asperò and Schindler's proof that $\mathsf{MM}^{++}(κ)+$there are class many Woodin cardinals implies Woodin's axiom $(*)$ if $\Diamond_κ$ holds and $κ>\aleph_2$. Our presentation takes advantage of the notion of consistency property: specifically we rephrase Asperò and Schindler's forcing as a specific instantiation of the notion of ``consistency property'' used by Makkai, Keisler, Mansfield and others in the…
What model companionship can say about the Continuum problem
本文探讨集合论模型伴随概念，指出最自然的模型伴随描述$H_{κ^+}$理论，并论证$2^{\aleph_0}=\aleph_2$是连续统问题唯一可属于模型伴随的解，以极大性原理辩护此模型论方法。
This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form aleph_n where n > 1. We investigate the difficulties of doing this and survey some of the successes
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