We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH:
THEOREM
(i) If there is a singular strong limit cardinal $κ$ such that $2^κ> kappa^+$ then there is an inner model with a cardinal $κ$ such that for all ordinals $α<κ$ there is an ordinal $ν< κ$ with $o(ν) > α$.
(ii) If there is a singular strong limit cardinal $κ$ of uncountable cofinality such that $2^κ> κ^+$ then there is an inner model with $o(κ) = κ^{++}$.
Since this paper was originally submitted, Gitik has improved this result to give exact lower bounds.
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.
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…
In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $κ\ge ω_2$, $\textsf{ISP}(κ)$ implies that $\textsf{SCH}$ holds above $κ$, and (3) forcing posets which have the $ω_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $κ$ is a singular strong limit cardinal and $2^κ>= λ$ where $λ$ is not the successor of a cardinal of cofinality at most $κ$. (i) If $\cofinality(κ)>\gw$ then $o(κ)\geλ$. (ii) If $\cofinality(κ)=\gw$ then either $o(κ)\geλ$ or $\set{\ga:K\sat o(\ga)\ge\ga^{+n}}$ is cofinal in $κ$ for each $n\in\gw$. In order to prove this theorem we give a detailed analysis of the sequences of indiscernibles which come from appl…
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 …
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 the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{ω^ω}$.
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.
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.
We prove that consistently there is a singular cardinal $κ$ of uncountable cofinality such that $2^κ$ is weakly inaccessible, and every regular cardinal strictly between $κ$ and $2^κ$ is the character of some uniform ultrafilter on $κ$.
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.
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.
We investigate the circumstances under which there exist a singular cardinal $μ$ and a short $(κ, μ)$-extender $E$ witnessing "$κ$ is $μ$-strong", such that $μ$ is singular in $\Ult(V, E)$.
The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones.
We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(μ^+,μ^+,μ^+,\cf(μ))$ for singular $μ$.
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
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 describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
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…
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.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
We deal here with colorings of the pair (mu^+, mu), when mu is a strong limit and singular cardinal. We show that there exists a coloring c, with no refinement. It follows, that the properties of identities of (mu^+, mu) when mu is singular, differ in an essential way from the case of regular mu .
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…
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.
We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…
We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.
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+…
Let $\mathrm{cof}(μ)=μ$ and $κ$ be a supercompact cardinal with $μ<κ$. Assume that there is an increasing and continuous sequence of cardinals $\langleκ_ξ\mid ξ<μ\rangle$ with $κ_0:=κ$ and such that, for each $ξ<μ$, $κ_{ξ+1}$ is supercompact. Besides, assume that $λ$ is a weakly compact cardinal with $\sup_{ξ<μ}κ_ξ<λ$. Let $Θ\geqλ$ be a cardinal with $\mathrm{cof}(Θ)>κ$. Assuming the $\mathrm{GCH}_{\geqκ}$, we construct a generic extension where $κ$ is strong limit, $\mathrm{cof}(κ)=μ$, $2^κ= Θ$ and both $\mathrm{TP}(κ^+)$ and $\mathrm{TP}(κ^{++})$ hold. Further, in this model there is a very …
For an infinite cardinal mu, MAD(mu) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over mu. Erdos and Hechler proved the consistency of [mu in MAD(mu)] for a singular cardinal mu and asked if it was ever possible for a singular mu that [mu notin MAD(mu)], and also whether 2^{cf(mu)}<mu ===> [mu in MAD(mu)] for every singular cardinal mu. We introduce a new method for controlling MAD(mu) for a singular mu and, among other new results about the structure of MAD(mu) for singular mu, settle both problems affirmatively.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether …
The $κ$-density of a cardinal $μ\geκ$ is the least cardinality of a dense collection of $κ$-subsets of $μ$ and is denoted by $\mathcal D(μ,κ)$. The Singular Density Hypothesis (SDH) for a singular cardinal $μ$ of cofinality $cfμ=κ$ is the equation $\mathcal D(μ,κ)=μ^+$. The Generalized Density Hypothesis (GDH) for $μ$ and $λ$ such that $λ\leμ$ is: $\mathcal D(μ,λ)=μ$ if $cfμ\not=cfλ$ and $\mathcal D(μ,λ)=μ^+$ if $cfμ=cfλ$. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6. If $κ=cfκ<θ=cfμ<μ$ and the set of cardinals $λ<μ$ of cofinality $κ$ that satisfy the \…
We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $κ$ is supercompact and the cardinals $θ< κ$, $λ> κ$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(κ) = θ$, the Singular Cardinal Hypothesis fails at $κ$ and there exists a universal graph in cardinality $λ\in (κ,2^κ)$.
Let $M$ denote the Merimovich's model in which for each infinite cardinal $λ, 2^λ=λ^{+3}$. We show that in $M$ the following hold: (1) Shelah's strong hypothesis fails at all singular cardinals, indeed, $\forall λ(λ$ is a singular cardinal $\Rightarrow pp(λ)=λ^{+3}).$ (2) For each singular cardinal $λ$ there is an inner model $N$ of $M$ such that $M$ and $N$ have the same bounded subsets of $λ,$ $λ$ is a singular cardinal in $N$, $(λ^{+i})^N=(λ^{+i})^M$, for $i=1,2,3,$ and $N \models 2^λ=λ^{+}$. Thus it is possible to add many new fresh subsets to $λ$ without adding any new bounded subsets to …
In this note, we relax the hypothesis of the main results in Kellner-Shelah-Tǎnasie's "Another ordering of the ten cardinal characteristics in Cichoń's diagram".
We introduce a combinatorial notion of measures called Rudin-Keisler capturing and use it to give a new construction of elementary substructures around singular cardinals. The new construction is used to establish mutual stationary results at the first successor of singular cardinals $\langle \aleph_{ωn + 1}\rangle_{n< ω}$.
We give an affirmative answer to a question of Gorelic \cite{Gorelic}, by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals $α$ with $cf(α)=ω_1$ and $α^ω> α.$
We show that some cardinal arithmetic configurations related to the negation of the Shelah Weak Hypothesis and natural from the forcing point of view are impossible.
