Saharon Shelah
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.
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We give an example of iteration of length omega of (<kappa)-complete kappa^+-cc forcing notions with the limit collapsing kappa^+. The construction is decoded from the proof of Shelah [Proper and Improper Forcing, Appendix, Theorem 3.6(1)].
This paper studies cardinal invariants b_kappa and t_kappa, the natural generalizations of the invariants b and t to a regular cardinal kappa.
We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S subseteq omega_1''. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal kappa and a stationary set S subseteq kappa with fat complement we can have uniformization for < A_delta : delta in S'>, A_delta subseteq delta = sup A_delta, cf(delta)=otp(A_delta) and a stationary non-reflecting set S' subseteq…
For a cardinal kappa and a model M of cardinality kappa let No(M) denote the number of non-isomorphic models of cardinality kappa which are L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when kappa is a weakly compact cardinal and mu<=kappa is a nonzero cardinal, there exists a model M of cardinality kappa with No(M)=mu. We prove here that if kappa is a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality kappa is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Sigma…
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.
A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed cardinal and that alpha is a successor cardinal of K with kappa < alpha < kappa^+. Then cf( alpha ) = kappa. In particular, K computes successors of countably closed singular cardinals correctly. (The hypothesis of countable closure is not required; see "Weak covering without countable closure", W. J. Mitchell and E. Schimmerling, Math. Res. Lett., Vol. 2, No. 5, …
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.
We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have stri…
Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying 2^kappa=kappa^++ and GCH below kappa. By a result of W. Mitchell and the author the assumptions are optimal.
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 > …
The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible …
Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence F and a parameter R subseteq H(kappa) such that functions f,g:kappa --> 2 are equivalent iff for some h:kappa --> 2, the structure (H(kappa),in,R,f,g,h) satisfies F, where in, R, f, g, and h are interpretations of the symbols appearing in F. All the values mu, 1 leq mu leq kappa^+ or mu=2^kappa, are possible numbers of equivalence classes for such a Sigma_1^1-equivalence relation. Additionally, the possibilities are close…
We investigate a notion called uniqueness in power kappa that is akin to categoricity in power kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is kappa-unique for one uncountable kappa, then it is kappa-unique for every uncountable kappa ; in particular, it is categorical in powers greater than the cardinality of T.
Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, kappa is theta-unfoldable when for any transitive structure M of size kappa that contains kappa as an element, there is an elementary embedding j:M-->N with critical point kappa for which j(kappa) is at least theta. Define that kappa is fully unfoldable, then, when it is theta-unfoldable for every …
In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset of kappa not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar principles formulated for many other large cardinal notions, including weakly compact, i…
If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.
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 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
We are interested in examples of a.e.c. with amalgamation having some (extreme) behaviour concerning types. Note we deal with k being sequence-local, i.e. local for increasing chains of length a regular cardinal (for types, equality of all restrictions imply equality). . For any cardinal theta>= aleph_0 we construct an a.e.c. with amalgamation k with L.S.T.(k) = theta, |tau_K| = theta such that {kappa : kappa is a regular cardinal and K is not (2^kappa, kappa)-sequence-local} is maximal. In fact we have a direct characterization of this class of cardinals: the regular kappa such that there is …
Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.
We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{k…
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
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 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.
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.
We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.
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…
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
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…
This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras), measuring prod_{i<kappa}f(i) by a family of ideals and a family of sequences <B_i:i<kappa> with |B_i|<mu_i, and finally we deal with the existence of strongly almost disjoint families.
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…
For a regular uncountable cardinal kappa, we discuss the order relationship between the unbounding and dominating numbers on kappa and cardinal invariants of the higher meager ideal M_kappa. In particular, we obtain a complete characterization of add(M_kappa) and cof(M_kappa) in terms of cov(M_kappa) and non(M_kappa) and unbounding and dominating numbers, and we provide models showing that there are no restrictions on the value of non(M_kappa) in the degenerate case 2^{<kappa} > kappa except 2^{<kappa} leq non(M_kappa) leq 2^kappa. The corresponding question for cof(M_kappa) remains open. Our …
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals…
Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal
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.
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…
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 .
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)…
If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K computes successors of weakly compact cardinals correctly. o K^c is an iterate of K. o (with Mitchell) If alpha is a cardinal > aleph_1, then K-restriction-alpha is universal for mice of height alpha. Other results in this paper, when combined with work of Woodin, imply: o If square-kappa-finite fails and kappa is a singular, strong limit cardinal, then Inductive D…
A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.
The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals
该研究通过强制选择公理，在满足ZF的最小初始段的Nairian模型中展示所有不可数基数上平方失败，并表明正则基数在HOD中为omega-强可测，从而揭示了HOD假设在ZFC中的不可证性。
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.
We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg's theorem: "there are 2^2^kappa many maximal (=precomplete) clones on a set of size kappa." The clones we construct here do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many cardinals kappa there are 2^2^kappa many such clones on a set of size kappa. Finally, we show that on a weakly compact cardinal there are e…
Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence <B_i:i<kappa> of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i<kappa}B_i/D)<|prod_{i<kappa}C(B_i)/D|. This answers a number of problems posed by Monk.
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).
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} .
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.
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.
We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular then d(X_kappa)= kappa, hence kappa is not a caliber of X_kappa, while all other uncountable regular cardinals are. We also prove it consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelof T_3 space X such that rho=cf(rho)>omega is a caliber of X exactly if rho not in A.
The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I=B or I=N, where B denotes the sigma-ideal generated by the compact subsets of the Baire space omega^omega, and N is the ideal of the null sets. For countable sets we give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular…
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 …
Square-kappa-finite, the finite family version of weak square, holds at all cardinals kappa in the Mitchell-Steel inner models.
For every regular cardinal kappa there exists a simple complete Boolean algebra with kappa generators.
By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it has kappa many branches for some kappa strictly between omega_1 and 2^{omega_1}. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist Kurepa trees and there are no Jech--Kunen trees, (2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees with omega_3 many branches exist.
We consider, for infinite cardinals kappa and alpha <= kappa^+, the group Pi(kappa,< alpha) of sequences of integers, of length kappa, with non-zero entries in fewer than alpha positions. Our main result tells when Pi(kappa,< alpha) can be embedded in Pi(lambda,< beta). The proof involves some set-theoretic results, one about familes of finite sets and one about families of ultrafilters.
One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size κsatisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.
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
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