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=lambda^+ and lambda is a strong limit singular, then the corresponding guessing holds.
Both results are also connected to an earlier result of Dzamonja and Shelah in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that a result of math.LO/9601219 can to a certain extent be extended to the successor of a regular. The negative result here gives limitations to the extent to which one can hope to extend this result.
We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.
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 $o(μ)=μ^{++}$.
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
We define a generic Vopěnka cardinal to be an inaccessible cardinal $κ$ such that for every first-order language $\mathcal{L}$ of cardinality less than $κ$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if $|\mathscr{B}| = κ$ and every structure in $\mathscr{B}$ has cardinality less than $κ$, then an elementary embedding between two structures in $\mathscr{B}$ exists in some generic extension of $V$. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of $\aleph_1$-Suslin sets of reals in models of ZF. In particular, we show tha…
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
It is known that the assumption that ``GCH first fails at \aleph_ω'' leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that this is not so in ZF: namely there is a generic cardinal-preserving extension of L (or any universe of ZFC + GCH in which all ZF axioms hold, the axiom of choice fails, GCH holds for all cardinals \aleph_n, but there is a surjection from PowerSet(\aleph_ω) onto λ, where λ is any previously chosen cardinal in L greater than \aleph_ω, for instance, \aleph_{ω+17}. In other words, in such an extension GCH holds in proper sense for all cardinals \aleph_n but …
Suppose that there is a measurable cardinal. If \aleph_ωis a strong limit cardinal, but the power of \aleph_ωis bigger than \aleph_{ω_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the measurable cardinal this answers a question of Gitik and Mitchell.
An inaccessible cardinal $κ$ is supercompact when $(κ, λ)$-ITP holds for all $λ\geq κ.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where simultaneously $(\aleph_2, μ)$-ITP and $(\aleph_3, μ')$-ITP hold, for all $μ\geq \aleph_2$ and $μ'\geq \aleph_3.$
Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
We discuss the generalized Kurepa hypothesis $KH_λ$ at singular cardinals $λ$. In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $KH_{\aleph_ω}$ nor the existence of a family $ \mathcal{F} \subseteq [\aleph_ω]^{\aleph_0}$ of size $\aleph_{ω+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.
Starting from a supercompact cardinal we build a model in which $2^{\aleph_{ω_1}}=2^{\aleph_{ω_1+1}}=\aleph_{ω_1+3}$ but there is a jointly universal family of size $\aleph_{ω_1+2}$ of graphs on $\aleph_{ω_1+1}$. The same technique will work for any uncountable cardinal in place of $ω_1$.
Starting from a stationary set of supercompact cardinals we find a generic extension in which the tree property holds at every regular cardinal between $\aleph_2$ and $\aleph_{ω^2}$.
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.
The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $κ$ is singular strong limit, then $2^κ=κ^+$. We prove that given a singular cardinal $κ$ of {\em cofinality} $η$ in the ground model, which is a limit of suitable large cardinals, and $η^+=\aleph_γ$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $η$, such that $κ$ becomes $\aleph_{γ+η}$, and SCH fails at $κ$. Furthermore, if $η$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_η$. Our large c…
Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\] Under the same assumption, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
In this paper it is introduced a generic large cardinal akin to I0, and its consequences are analyzed in the case that $\aleph_ω$ is such a generic large cardinal. In this case $\aleph_ω$ is Jónsson, and in a choiceless inner model many properties hold that are in contrast with PCF in ZFC.
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.
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 $κ$.
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 use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
We show that from a supercompact cardinal κ, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, κ and κ^+ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF + not AC_ω in which either (1) aleph_1 and aleph_2 are both singular and the continuum function at aleph_1 can be precisely controlled, or (2) aleph_ω and aleph_{ω+…
Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
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.
Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than $\aleph_1$, we produce a model in which the approachability property fails (hence there are no special Aronszajn trees) at all regular cardinals in the interval $[\aleph_2, \aleph_{ω^2+3}]$ and $\aleph_{ω^2}$ is strong limit.
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.
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 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 .
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.
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 …
We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$ when $κ$ is supercompact. The actual proofs show that $ω\_1$-regressive Kurepa-trees are consistent above a supercompact cardinal even though ${\rm MM}$ destroys them on all regular cardinals. This rather paradoxical fact contradicts the common intuition.
A ZFC Dowker space is constructed which has cardinality $\aleph_{ω+1}$. This provides a bound in ZFC to the first cardinal in which there is a ZFC Dowker space. The space we construct is a closed and cofinal subspace of M.~E.~Rudin's Dowker space from 1971. A theorem from pcf theory used in the proof, but otherwise the proof is elementary.
We give Woodin's original proof that if there exists a $(κ+2)-$strong cardinal $κ,$ then there is a generic extension of the universe in which $κ=\aleph_ω,$ $GCH$ holds below $\aleph_ω$ and $2^{\aleph_ω}=\aleph_{ω+2}.$
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.
It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on ${\aleph}_{ω+1}$ which is generated by fewer than ${2}^{{\aleph}_{ω+1}}$ sets.
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 .
Assuming $κ$ is a supercompact cardinal and $λ$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $κ=\aleph_ω$ and $λ=\aleph_{ω+1}.$
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of …
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 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.
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…
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…
One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality aleph_0, aleph_1 and aleph_2 while on aleph_2 club, the ``small'' brother of diamond, holds. As a consequence we get the consistency of ``there is no Gross space''. Another aim is to present a case of ``Historic iteration''.
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and S…
This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is …
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.
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).
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 prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
