The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some clarification of the higher-dimensional infinitary combinatorics lying at its core. In the course of this work, we describe simplicial characterizations of the ordinals $ω_n$, higher-dimensional generalizations of coherent Aronszajn trees, bases for critical inverse systems over large index sets, nontrivial $n$-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique. These constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$ framework; at their heart is a simple, finitely iterable technique of compounding $C$-sequences.
An omega-coloring is a pair <f,B> where f:[B]^{2} ---> omega. The set B is the field of f and denoted Fld(f). Let f,g be omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g) ---> Fld(f) such that for all {x,y}, {u,v} in dom(g) we have f({k(x),k(y)}) not= f({k(u),k(v)}) => g({x,y}) not= g({u,v}). We write f~g if f realizes g and g realizes f. We call the ~-classes of omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f in I. For a cardinal kappa and f:[kappa]^2 ---> omega we define I(f) to be…
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to other cardinals.
Let 0<n^*< omega and f:X-> n^*+1 be a function where X subseteq omega backslash (n^*+1) is infinite. Consider the following set S_f= {x subset aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)= aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=< aleph_omega, ... > there's a B prec A such that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)= aleph_{f(…
We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.
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…
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if kappa <= 2^{aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when kappa <= 2^{aleph_0}. In particular, we are concerned with the case aleph_1 <= kappa <= aleph_omega
Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <ω$, is consistent with an arbitrary continuum function below $\aleph_ω$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<ω$. Thus the tree property has no provable effect on the continuum function below $\aleph_ω$ except for the restriction that the tree property at $κ^{++}$ implies $2^κ>κ^+$ for every infinite $κ$.
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by declaring the G_delta sets to be open. Milovich proved that if Square_{aleph_omega} holds and (aleph_omega)^omega=aleph_{omega+1} then Nt(X)=omega_1. Answering a question of Spadaro, we show that if (aleph_omega)^omega=aleph_{omega+1} and a strong form of Chang Conjecture holds for aleph_ωthen Nt(X)>omega_1.
We show using a proof of the Global Square property in Core Models below a measurable of Mitchell order o(kappa)=kappa^++ (a result originally due to Jensen & Zeman) that Foreman and Magidor's Mutual Stationarity property MS(Aleph_n (1<n<omega), Cof(omega_1)) implies the existence of inner models with measurables of high Mitchell order. This MS property states that any sequence of independently chosen stationary subsets S_n of the Aleph_n (of fixed cofinality omega_1) is mutually stationary below aleph_omega.
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 .
After the main gap theorem was proved (see [Sh:c]), in discussion, Harrington expressed a desire for a finer structure - of finitary character (when we have a structure theorem at all). I point out that the logic L_{infty,aleph_0}(d.q.) (d.q. stands for dimension quantifier) does not suffice: e.g., for T=Th(lambda x 2^ω,E_n)_{n<omega} where (alpha,eta)E_n(beta,nu) =: eta|n=nu|n and for a subset S of 2^omega we define M_S = M | {(alpha,eta): [eta in S -> alpha<omega_1] and [eta in 2^ωbackslash S -> alpha<omega]}. Hence, it seems to me we should try L_{infty,aleph_epsilon}(d.q.) (essentially, in…
In the early 1970's S.Tennenbaum proved that all countable models of PA^- + forall_1-Th(N) are embeddable into the reduced product N^omega/F, where F is the cofinite filter. In this paper we show that if M is a model of PA^- + forall_1-Th(N), and |M|= aleph_1, then M is embeddable into N^omega/D, where D is any regular filter on omega.
We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal an…
The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…
Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B which we denote by G_{omega_1}(A,B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1 moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of cardinality <= aleph_1. We prove the following results: Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such that the game G_{omega_1}(A,B) is non-determined. Theorem B: If it is consistent that there is a measura…
We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}.
It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.
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…
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 .
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.
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.
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).
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 reflecting cardinal. We also prove that the addition of $MA$ to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of Erdos. It is also consistent that there is a graph X with Chr(X)=|X|= aleph_1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)<=aleph_2 (so the Taylor conjecture may fail).
Whenever x is a tame cardinal invariant and ZFC+large cardinals proves that x=aleph one implies WCG then ZFC+large cardinals proves that x=aleph one implies b=aleph one, and b=aleph one implies WCG. Here WCG is a certain prediction principle on omega one. This theorem is one of the many possible interpolation theorems of this kind.
For g < f in omega^omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force aleph_1 many different values for different functions (f,g). In the language of Blass: There may be aleph_1 many distinct uniform Pi^0_1 characteristics.
We show that square(theta) implies that there is a first countable <theta-collectionwise Hausdorff space that is not weakly theta-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to omega_2, first countable aleph_1-collectionwise Hausdorff spaces are weakly aleph_2-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming E^omega_theta, a certain theta-family of integer valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to omega…
A subgroup G \leq Z^omega exhibits the Specker phenomenon if every homomorphism G \to Z maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal se, the size of the smallest G \leq Z^omega exhibiting the Specker phenomenon. We also prove the consistency of b < e, where b is the unbounding number and e the evasion number introduced recently by Blass. Finally we show that e \geq \min { aleph_0--e , cov(M) }, where aleph_0--e is the aleph_0--evasion number (the uniformity of the evasion ideal) and cov(M) is the covering number of the meager ideal. All the…
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an …
A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of lambda proper subgroups. The cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove Theorem (A): Let T be any subset of omega setminus {0}. Then it is consistent that aleph_n in CF(S) if and only if n in T . One might suspect that it is consistent that…
We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a Büchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Büchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A…
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 …
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…
This paper answers three questions posed by the first author. In Theorem 2.6 we show that the family of strong measure zero subsets of {}^{omega_1}2 is 2^{aleph_1}-additive under GMA and CH. In Theorem 3.1 we prove that the generalized Borel conjecture is false in {}^{omega_1}2 assuming ZFC+CH. Next, in Theorem 4.2, we show that the family of subsets of {}^{omega_1}2 with the property of Baire is not closed under the Souslin operation.
Let chi be the minimum cardinal of a subset of 2^omega that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that s<chi is consistent. We thus answer a question by Vojtas. We give two kinds of models for the strict inequality. The first is the combination of an aleph_2-iteration of some proper forcing with adding aleph_1 random reals. The second kind of models is got by adding delta random reals to a model of MA_{< kappa} for some delta in [aleph_1,kappa). It was a conjecture of Blass that s=aleph_1<chi=kappa holds in such…
We prove (ZF+DC) e.g. : if mu =|H(mu)| then mu^+ is regular non measurable. This is in contrast with the results for mu = aleph_{omega} on measurability see Apter Magidor [ApMg]
Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraisse-game of length gamma of A and B denoted by EFG_gamma(A,B) is defined as follows: There are two players called for all and exists. First for all plays x_0 and then exists plays y_0. After this for all plays x_1, and exists plays y_1, and so on. Eventually a sequence <(x_beta,y_beta): beta<gamma> has been played. The rules of the game say that both players have to play elements of A cup B. Moreover, if for all plays his x_beta in A (B), then exists has to play his y_beta in B (A). Thus the sequ…
