Successor Cardinals Research Articles

Compactness versus hugeness at successor cardinals

If [Formula: see text] is regular and [Formula: see text], then the existence of a weakly presaturated ideal on [Formula: see text] implies [Formula: see text]. This partially answers a question of Foreman and Magidor about the approachability ideal on [Formula: see text]. As a corollary, we show that if there is a presaturated ideal [Formula: see text] on [Formula: see text] such that [Formula: see text] 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.

Controlling the number of normal measures at successor cardinals

AbstractWe examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.

Easton collapses and a strongly saturated filter

We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses.

The super tree property at the successor of a singular

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 ℵω+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 large cardinal-type properties at smaller cardinals.

Singularizing successor cardinals by forcing

We exhibit models of set theory, using large cardinals and forcing, in which successor cardinals can be made singular by some “Namba-like” further set forcing, in most cases without collapsing cardinals below that successor cardinal. For successors of regular cardinals, we work from consistency-wise optimal assumptions in the ground model. Successors of singular cardinals require stronger hypotheses. Our partial orderings are different from Woodin’s stationary tower forcing, which requires much stronger hypotheses when singularizing successors of regular cardinals, and collapses cardinals above the cardinal whose cofinality is changed when singularizing successors of singular cardinals.

ZF + DC + AX4

We consider mainly the following version of set theory:"ZF + DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a sequence $\bar\delta = \langle \delta_s:s \in Y\rangle,\textrm{cf}(\delta_s)$ large enough compared to $Y$, we can prove the pcf theorem with minor changes (using true cofinalities not the pseudo one). We then deduce the existence of covering numbers and define and prove existence of truly successor cardinals. From this we show that some diagonalization arguments (more specifically some black boxes and consequence) on Abelian groups. We end by showing some such consequences hold in ZF above.

Indestructibility of generically strong cardinals

Foreman (For13) proved the Duality Theorem, which gives an al- gebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of !1 is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (Cla10)). As an application we prove that if !1 is generically strong, then it remains so after adding any number of Cohen subsets of !1; however many other !1-closed posets—such as Colp!1,!2q—can destroy the generic strength of !1. This generalizes some results of Gitik-Shelah (GS89a) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than !1.

Δ1-Definability of the non-stationary ideal at successor cardinals

Δ₁-Definability of the non-stationary ideal at successor cardinals

THE TREE PROPERTY UP TO אω+1

AbstractAssuming ω supercompact cardinals we force to obtain a model where the tree property holds both at אω+1, and at אn for all 2 ≤ n < ω. A model with the former was obtained by Magidor–Shelah from a large cardinal assumption above a huge cardinal, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals.

The combinatorial essence of supercompactness

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal. Utilizing the failure of a weak version of a square, we show that the best currently known lower bounds for the consistency strength of these principles can be applied.

Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

We provide upper and lower bounds in consistency strength for the theories “ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω1”. In particular, our models for both of these theories satisfy “ZF + ¬ACω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”. There are many instances in the literature where choiceless large cardinal patterns are initially forced from strong hypotheses which one later sees can be weakened somewhat. For example, it is shown in [4] that the models constructed in [11] from an almost huge cardinal can actually be ∗2000 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55. †

Perfect trees and elementary embeddings

AbstractAn important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j* : M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0#.

Making all cardinals almost Ramsey

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + \({\neg {\rm AC}_\omega}\) in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”.

CARDINAL INVARIANTS AND EVENTUALLY DIFFERENT FUNCTIONS

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ<κ = κ and 2κ = κ+, the following is not provable in Zermelo–Fraenkel set theory. There is a κ+-cc poset which does not collapse κ and which forces a(κ) = κ+ < ae(κ) = κ++ = 2κ. We also apply the ideas from the proofs of these results to study a = a(ω) and non(M). 2000 Mathematics Subject Classification 03E17 (primary), 03E05 (secondary).

CARDINAL INVARIANTS AND EVENTUALLY DIFFERENT FUNCTIONS

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ<κ = κ and 2κ = κ+, the following is not provable in Zermelo–Fraenkel set theory. There is a κ+-cc poset which does not collapse κ and which forces a(κ) = κ+ < ae(κ) = κ++ = 2κ. We also apply the ideas from the proofs of these results to study a = a(ω) and non(M). 2000 Mathematics Subject Classification 03E17 (primary), 03E05 (secondary).

Almost free groups and Ehrenfeucht–Fraı̈ssé games for successors of singular cardinals

We strengthen nonstructure theorems for almost free Abelian groups by studying long Ehrenfeucht–Fraı̈ssé games between a fixed group of cardinality λ and a free Abelian group. A group is called ε-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length ε∈ λ. We prove for a large set of successor cardinals λ= μ + the existence of nonfree ( μ· ω 1)-game-free groups of cardinality λ. We concentrate on successors of singular cardinals.

Generic saturation

Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one definable in a set-generic extension of L[O#]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be almost codable in the sense that it is definable from a real which is generic for an L-forcing (and which belongs to a set-generic extension of L[0#]). This result is best possible in the sense that for any countable ordinal α there is an L-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“O# exists” is actually necessary for these results.Let P denote a class forcing definable over an amenable ground model 〈L, A〉 and assume that O# exists.Definition. P is relevant if P has a generic definable in L[0#]. P is almost relevant if P has a generic definable in a set-generic extension of L[0#].Remark. The reverse Easton product of Cohen forcings 2&lt;κ, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in L) is almost relevant.Definition. κ is α-Erdös if whenever C is CUB in κ and f: [C]&lt;ω → κ is regressive (i.e., f(a) &lt; min(a)) then f has a homogeneous set of ordertype α.

The essentially free spectrum of a variety

We partially prove a conjecture from [MeSh] which says that the spectrum of almost free, essentially free non-free algebras in a variety is either empty or consists of the class of all successor cardinals.

$\omega\sb 3\omega\sb 1\to (\omega\sb 3\omega\sb 1,3)\sp 2$ requires an inaccessible

We show that if there is a simplified $({\omega _2},1)$-morass with linear limits and ${2^{{\aleph _1}}} = {\aleph _2}$, then ${\omega _3}{\omega _1} \nrightarrow {({\omega _3}{\omega _1},3)^2}$. Thus, assuming ${2^{{\aleph _1}}} = {\aleph _2}$, this negative relation holds in $V$ if both ${\aleph _2}$ and ${\aleph _3}$ are (successor cardinals)$^{L}$, since in this case, well-known arguments show there is a simplified $({\omega _2},1)$-morass with linear limits. The contrapositive is that, assuming ${2^{{\aleph _1}}} = {\aleph _2}$, the positive relation holds only if either ${\aleph _2}$ or ${\aleph _3}$ is (inaccessible)$^{L}$.

Partition relations for successor cardinals

This paper investigates the relations κ + → ( α) 2 2 and its variants for uncountable cardinals κ. First of all, the extensive literature in this area is reviewed. Then, some possibilities afforded by large cardinal hypotheses are derived, for example, if κ is measurable, then κ + → ( κ + κ + 1, α) 2 2 for every α < κ +. Finally, the limitations imposed on provability in ZFC by L and by relative consistency via forcing are considered, primarily the consistency of: if κ is not weakly compact, then κ + ⇅ [κ : κ] 2 κ .