We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
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 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 negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
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 $κ^{++}$.
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.
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.
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 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.
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…
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…
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…
If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.
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…
Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is χ-tame. If K is categorical in some λ^+ >LS(K) then it is categorical in all μ\geq (λ+χ)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all μ\geq LS(K).
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 …
We prove that $α_M(λ)$ can be successor of a supercompact cardinal, when $λ$ is a Magidor cardinal. From this result we obtain the consistency of $α_M(λ)$ being a successor of a singular cardinal with uncountable cofinality.
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 obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using di…
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let μbe the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^μ)^+} then K is categorical in all cardinals greater than \beth_{(2^μ)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).
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.
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
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…
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, …
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/95042…
We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $κ\leq λ$, where $κ$ is the successor of a regular cardinal, there is a rigid saturated ideal on $\mathcal{P}_κλ$. We also show the consistency of some instances of rigid saturated ideals on $\mathcal{P}_κλ$ where $κ$ is the successor of a singular cardinal.
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.
Let $κ$,$λ$ be regular uncountable cardinals such that $λ> κ^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(κ) = λ$ starting from a ground model in which $o(κ) = λ$ and prove that assuming $\neg 0^¶$, $s(κ) = λ$ implies that $o(κ) \geq λ$ in the core model.
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.
We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals.
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^+ .
We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.
We establish a coloring theorem for successors of a singular cardinals, and use it prove that for any such cardinal $μ$, we have $μ^+\nrightarrow[μ^+]^2_{μ^+}$ if and only if $μ^+\nrightarrow[μ^+]^2_θ$ for arbitrarily large $θ<μ$.
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.
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a…
A cardinal $λ$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < λ$, $λ$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals can be satisfied robustly by small cardinals. We focus in particular on stationary reflection and the tree property, both of which can consistently hold but fail to be robust at small cardinals. We introduce natural strengthenings of these principles which are always robust and which hold at sufficiently large cardinals, consider the extent to which these strengthe…
Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $κ$---which we denote $\Diamond_{\text{Lav}}(κ)$---asserting the existence of a single function from $κ\to H_κ$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(ω_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(κ)$ provides a uniform, simple construc…
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.
We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.
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 GCH hold and let $j:V\longrightarrow M$ be a definable elementary embedding such that $crit(j)=κ$, $^κM\subseteq M$ and $κ^{++}=κ_{M}^{++}$. H. Woodin proved that there is a cofinality preserving generic extension in which $κ$ is measurable and GCH fails at it. This is done by using an Easton support iteration of Cohen forcings for blowing the power of every inaccessible $α\leqκ$ to $α^{++}$, and then adding another forcing on top of that. We show that it is enough to use the iterated forcing, and that the latter forcing is not needed. We will show this not only for the case where $κ^{++}=…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal $κ$ satisfies the \emph{narrow system property} if every narrow system of height $κ$ has a cofinal branch. In this paper, we study connections between the narrow system property, square principles, and forcing axioms. We prove, assuming large cardinals, both that it is consistent that $\aleph_{ω+1}$ satisfies the narrow system property and $\square_{\aleph_ω, < \aleph_ω}$ holds and that it is con…
