Saharon Shelah, Lee Stanley
We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section 1, and the forcing is a ``many cardinals'' version of the forcing there.
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We prove some consistency results about b(lambda) and d(lambda), which are natural generalisations of the cardinal invariants of the continuum b and d. We also define invariants b_cl(lambda) and d_cl(lambda), and prove that almost always b(lambda) = b_cl(lambda) and d(lambda)=d_cl(lambda)
We prove that if cf(lambda) > aleph_0 and 2^{cf(lambda)}<lambda, then lambda->(lambda,omega+1)^2.
We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH around lambda) that if K is categorical in lambda, lambda^+, LS(K) <= lambda and 1 <= I(lambda^{++},K)< 2^{lambda^{++}} then K has a model in lambda^{+++} .
We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of lambda strong compactness or lambda supercompactness measures on P_kappa(lambda) can be exactly lambda+, if lambda>kappa is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.
Let kappa a regular uncountable cardinal and lambda a cardinal >kappa, and suppose lambda^{<kappa} is less than the covering number for category cov(M_{kappa,kappa}). Then (a) I_{kappa,lambda}^+ -->^kappa (I_{kappa, lambda}^+,omega +1)^2, (b) I_{kappa, lambda}^+ -->^kappa [I_{kappa, lambda}^+]_{kappa^+}^2 if kappa is a limit cardinal, and (c) I_{kappa, lambda}^+ -->^kappa (I_{kappa, lambda}^+)^2 if kappa is weakly compact.
If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.
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).
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…
We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.
In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…
We extend a transitive model V of ZFC + GCH cardinal preservingly to a model N of ZF + "GCH holds below Alef_omega" + "there is a surjection from the power set of Alef_omega onto lambda" where lambda is an arbitrarily high fixed cardinal in V.
Let kappa be a regular uncountable cardinal and lambda > kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_kappa (lambda) and use this to prove that the nonstationary ideal on P_kappa (lambda) is nowhere precipitous.
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 .
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…
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
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…
We prove that if lambda is a strong limit singular cardinal and kappa a regular uncountable cardinal < lambda, then NS_{kappa lambda}, the non-stationary ideal over P_{kappa} lambda, is nowhere precipitous. We also show that under the same hypothesis every stationary subset of P_{kappa} lambda can be partitioned into lambda^{< kappa} disjoint stationary sets.
Consider an a.e.c. (abstract elementary class), that is, a class K of models with a partial order refining inclusion (submodel) which satisfy the most basic properties of an elementary class. Our test question is trying to show that the function dot I (lambda, K), counting the number of models in K of cardinality lambda up to isomorphism, is "nice", not chaotic, even without assuming it is sometimes 1, i.e. categorical in some lambda's. We prove here that for some closed unbounded class C of cardinals we have (a), (b) or (c) where (a) for every lambda in C of cofinality aleph_0, dot I (lambda,…
1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .
Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa the principle is known to fail at some lambda. This note shows that it fails at every lambda if kappa is the successor of a regular uncountable cardinal or kappa is countably closed.
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [CK], modulo the assumption of supercompactness. In section 2 we construct several lambda-Archimedean ultrapowers of omega under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exi…
We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V[G] models ZFC + GCH in which, (a) (preservation) for kappa <= lambda regular, if V models ``kappa is lambda supercompact'', then V[G] models ``kappa is lambda supercompact'' and so that, (b) (equivalence) for kappa <= lambda regular, V[G] models ``kappa is lambda strong…
We prove that for every (infinite cardinal) lambda there is a T_3-space X with clopen basis, 2^mu points where mu = 2^lambda, such that every closed subspace of cardinality <|X| has cardinality < lambda .
An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.
We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our main result is that we can find a good lambda-frame s categorical in lambda such that K_s subseteq K_lambda. We then show how to use [Sh:705] to get categoricity in every large enough cardinality if K has cases of mu-amalgamation for enough mu and 2^mu<2^{mu^{+1}} <... < 2^{mu^{+n}}... for enough mu.
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality. Then there is sigma_* < mu such that for every chi < theta the following holds: theta > …
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda. We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.
The authors show, by means of a finitary version square^{fin}_{lambda,D} of the combinatorial principle square^{b^*}_{lambda}, the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal lambda, if M_i and N_i are elementarily equivalent models of a language of size <= lambda, then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda^+ on prod_i M_i/D and prod_i N_i/D. If in addition 2^{lambda}= lambda^+ and i<lambda implies |M_i|+|N_i| <= lambda^+ this means that the ultra…
We prove combinatorial theorems concerning the stick principle and cardinal characteristics.
We describe some (countably many) classes K^{n,e} of finite graphs and prove that if lambda^{aleph_0}= lambda then every lambda^+-chromatic graph of cardinal lambda^+ contains, for some n, e, all members of K^{n,e} as subgraphs. On the other hand, it is consistent for every regular infinite cardinal kappa that there is a kappa^+-chromatic graph on kappa^+ that contains finite subgraphs only from K^{n,e} .
A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelof regular space of density lambda. This extends an analogous earlier result of ours that only worked for regular lambda.
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 .
We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can be generalized. E.g., there is a class of regular cardinals, and we can prove cardinal inequality.
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
In this paper we present a few properties of $K$-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. Among the result proven we have general existence result that state that the existence of any $K$-partition implies the existence of $K$-partition of a metrizable space as well as existence of $K$-partition of a compact space implies the existence of $K$-partition of a completely metrizable space. We also prove some connections between existence of $K$-partitions and existence of precipitous ideals as well as measu…
We prove the consistency of $\mathfrak{r}_λ<\mathfrak{d}_λ$ and even $\mathfrak{u}_λ<\mathfrak{d}_λ$ for a singular cardinal $λ$.
We prove that the strong polarized relation of $θ$ above $ω$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.
For cardinals lambda, kappa, theta we consider the class of graphs of cardinality lambda which has no subgraph which is (kappa, theta)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove nonexistence of universals for this problem and some related ones.
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of 2^lambda needed to cover it. This answers a question of Matet.
This article continues Rosłanowski and Shelah math.LO/9906024, math.LO/0508272, math.LO/0210205, math.LO/0611131 and math.LO/0605067. We introduce here a new property of <lambda-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+ (for a strongly inaccessible cardinal lambda).
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal lambda with generating system of size less than 2^lambda . We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with lambda-supports (and we show th…
Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a …
The main result is that for lambda strong limit singular failing the continuum hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.
We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD. Nevertheless, the procedure resembles the one developed by Steel. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of lambda such that lambda is a limit of Woodin cardinals and the derived model at lambda satisfies AD the first member of the Solovay sequence is is less than Theta then there is a transitive model …
The Kalikow problem for a pair (lambda, kappa) of cardinal numbers, lambda > kappa (in particular kappa =2) is whether we can map the family of omega --sequences from lambda to the family of omega --sequences from kappa in a very continuous manner. Namely, we demand that for eta, nu in lambda^omega we have: eta, nu are almost equal if and only if their images are. We show consistency of the negative answer e.g. for aleph_omega but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.
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…
Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+, we present a ``mini-coding'' between kappa and kappa^{+ omega}. This allows us to prove that any subset of kappa^{+ omega} can be coded into a subset, W of kappa^+ which, further, ``reshapes'' the interval [kappa, kappa^+), i.e., for all kappa < delta < kappa^+, kappa = (card delta)^{L[W cap delta]}. We sketch two applications of this result, assuming 0^sharp does not exist. First, we point out that this shows that any set can be coded by a…
We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal lambda with the large cardinal property, if tightness^+(B) >= lambda^+, then depth (B) >= lambda. This improves a theorem of Dow and Monk.
We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $ω_1$ for every colouring of the finite sets of countable ordinals that is definable by a $Σ_1$-formula that only uses the cardinal $ω_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given …
We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal lambda which is lambda x omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra on the successor of beth_omega. Lastly, we give further information on guessing of clubs.
In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)= lambda there is C club in alpha such that F restriction C is strictly increasing. We prove that it is consistent to have a cardinal theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than theta .
Dedekind-finite cardinals having countable partitions
本文研究没有可数子集但在两种满射意义上非“surjectively Dedekind finite”的集合可能具有的结构。
We show that, consistently, for some regular cardinals theta<lambda, there exists a Boolean algebra B such that B=lambda^+ and for every subalgebra B' of B of size lambda^+ we have Depth(B')=theta.
The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of ^{lambda}2 needed to cover it. This answers a question of Matet.
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…
The theta--fan F_theta is the quotient space obtained by identifying the non--isolated points of the product theta \times (omega+1) with a single point \infty. (Here, theta has the discrete topology and omega+1 has the order topology.) We define several notions of forcing that allow us to manipulate the tightness t of products of fans. Some consequences include: t (F_theta \times F_omega) = theta does not imply the existence of a (theta,omega)--gap, new examples of first countable <theta--collectionwise Hausdorff (cwH) spaces that are not \leq theta--cwH for singular cardinals theta, and for c…
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality lambda is described by a sentence of L_{infty, lambda}. In fact this sentence can be chosen in the L^*_{lambda}. (L^*_{lambda} is the result of …
We investigate the unbalanced ordinary partition relations of the form $λ\rightarrow {(λ, α)}^{2}$ for various values of the cardinal $λ$ and the ordinal $α$. For example, we show that for every infinite cardinal $κ,$ the existence of a $κ^{+}-$Suslin tree implies $κ^{+} \not\rightarrow {\left( κ^{+}, {\log}_κ(κ^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, α)}^{2}$ for all $α< ω_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.
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