A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ ⁺). Then with further forcing we show that it is consistent that GCH fails at ? _ω , ? _ω strong limit, while there is a lightface definable wellorder of H(? _ω+1) (“definable failure” of the singular cardinal hypothesis at ? _ω ). The large cardinal hypothesis used is the existence of a κ ⁺⁺-strong cardinal, where κ is κ ⁺⁺-strong if there is an embedding j: V → M with critical point κ such that H(κ ⁺⁺) ? M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ⁺⁺-strong cardinal is used throughout.     

Aronszajn trees and the successors of a singular cardinal

From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ ⁺⁺ has the tree property. In particular this model has no special κ ⁺-trees.     

Superatomic Boolean algebras constructed from strongly unbounded functions

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ^{++ +} ≤ λ, κ^<κ = κ and 2^κ = κ⁺, and η is an ordinal with κ⁺ ≤ η < κ⁺⁺ and cf(η) = κ⁺. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal BUsing Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ^{++ +} ≤ λ, κ^<κ = κ and 2^κ = κ⁺, and η is an ordinal with κ⁺ ≤ η < κ⁺⁺ and cf(η) = κ⁺. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal B$ such that $\mathrm{ht}(\mathcal B) = \eta + 1$, $\mathrm{wd}_{\alpha }(\mathcal B) = \kappa$ for every α < η and $\mathrm{wd}_{\eta }(\mathcal B) = \lambda$(i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉_η^??〈λ〉). Especially, $\langle {\omega }\rangle _{{\omega }_1}{}^{\smallfrown } \langle {\omega }_3\rangle$ and $\langle {\omega }_1\rangle _{{\omega }_2}{}^{\smallfrown } \langle {\omega }_4\rangle$ can be cardinal sequences of superatomic Boolean algebras.     

On the consistency strength of ‘Accessible’ Jonsson Cardinals and of the Weak Chang Conjecture

Using the core model K we determine better lower bounds for the consistency strength of some combinatorial principles:I. Assume that λ is a Jonsson cardinal which is ‘accessible’ in the sense that at least one of (1)-(4) holds: (1) λ is a successor cardinal; (2) λ = ω_ξ and ξ<λ; (3) λ is singular of uncountable cofinality; (4) λ is a regular but not weakly hyper-Mahlo. Then 0² exists.II. For λ = ?⁺ a successor cardinal we consider the weak Chang Conjecture, wCC(λ), which is a consequence of the Chang transfer property (λ⁺, λ)?(λ, ?).III. If λ = ?⁺?ω₂, then wCC(λ) implies the existence of 0².IV. We can determine the consistency strenght of wCC(ω₁). We include a relatively simple definition of the core model which together with the results of Dodd and Jensen suffices for our proofs.     

Some Remarks on Normal Measures and Measurable Cardinals

We prove two theorems which in a certain sense show that the number of normal measures a measurable cardinal κ can carry is independent of a given fixed behavior of the continuum function on any set having measure 1 with respect to every normal measure over κ . First, starting with a model V ⊨ “ZFC + GCH + o(κ) = δ*” for δ* ≤ κ⁺ any finite or infinite cardinal, we force and construct an inner model N ⊆ V [G] so that N ⊨ “ZF + (∀δ < κ) [DC_δ] + ¬AC_κ + κ carries exactly δ* normal measures + 2^δ = δ⁺⁺ on a set having measure 1 with respect to every normal measure over κ”. There is nothing special about 2^δ = δ here, and other stated values for the continuum function will be possible as well. Then, starting with a modelV ⊨ “ZFC + GCH + κis supercompact”, we force and construct models of AC in which, roughly speaking, regardless of the specified behavior of the continuum function below κ on any set having measure 1 with respect to every normal measure over κ, κ can in essence carry any number of normal measures δ* ≥ κ⁺⁺.     

Tall cardinals

A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j (κ) > θ and M^κ ? M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2^κ as high as desired. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

On uncountable cardinal sequences for superatomic Boolean algebras

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results:

1. Ifθ = 〈κ _α :α <ω ₁〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.
2. Ifκ is an uncountable cardinal such thatκ ^<κ =κ andθ = 〈κ _α :α <κ ⁺〉 is a cardinal sequence such thatκ _α ≥κ for everyα <κ ⁺ andκ _α =κ for everyα <κ ⁺ such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.

     

Fragility and indestructibility of the tree property

We prove various theorems about the preservation and destruction of the tree property at ω ₂. Working in a model of Mitchell [9] where the tree property holds at ω ₂, we prove that ω ₂ still has the tree property after ccc forcing of size ${\aleph_1}$ or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that the tree property at ω ₂ can be indestructible under ω ₂-directed closed forcing.     

Generalized Prikry forcing and iteration of generic ultrapowers

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈M_n, j_m,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j_0,n(κ) | n ∈ ω〉 is a Prikry generic sequence over M_ω. In this paper we generalize this for normal precipitous filters. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The structure of the Mitchell order—I

We isolate here a wide class of well-founded orders called tame orders, and show that each such order of cardinality at most κ can be realized as the Mitchell order on a measurable cardinal κ, from a consistency assumption weaker than o(κ) = κ⁺.     

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ ⁺ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ ⁺ strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ ⁺, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ ⁺ strongly compact.     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 ^κ ?=?κ ⁺, another for which 2 ^κ ?=?κ ⁺⁺ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that ${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$ . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH?+?V?=?HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.     

The Consistency Strength of \aleph_{\omega} and \aleph_{{\omega}_1} Being Rowbottom Cardinals Without the Axiom of Choice

We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω ₂-Rowbottom cardinal carrying an ω ₂-Rowbottom filter and ω ₁ is regular” has the same consistency strength as the theory “ZFC + There exist ω ₁ measurable cardinals”. We also discuss some generalizations of these results.     

Resolvability vs. almost resolvability

A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

Destructibility of stationary subsets of Pκλ

For a regular cardinal κ with κ ^<κ = κ and κ ≤ λ , we construct generically (forcing by a < κ‐closed κ ⁺‐c. c. p. o.‐set ℙ₀) a subset S of {x ∈ P _κ λ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (F ^IA_κ λ ‐stationary in our terminology) but the stationarity of S can be destroyed by a κ ⁺‐c. c. forcing ℙ* (in V ^ℙ) which does not add any new element of P _κ λ . Actually ℙ* can be chosen so that ℙ* is κ‐strategically closed. However we show that such ℙ* itself cannot be κ‐strategically closed or even <κ‐strategically closed if κ is inaccessible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Absolute E-rings

A ring R with 1 is called an E-ring if EndZR is ring-isomorphic to R under the canonical homomorphism taking the value 1σ for any σ∈EndZR. Moreover R is an absolute E-ring if it remains an E-ring in every generic extension of the universe. E-rings are an important tool for algebraic topology as explained in the introduction. The existence of an E-ring R of each cardinality of the form λℵ₀ was shown by Dugas, Mader and Vinsonhaler (1987) [9]. We want to show the existence of absolute E-rings. It turns out that there is a precise cardinal-barrier κ(ω) for this problem: (The cardinal κ(ω) is the first ω-Erd?s cardinal defined in the introduction. It is a relative of measurable cardinals.) We will construct absolute E-rings of any size λ<κ(ω). But there are no absolute E-rings of cardinality ?κ(ω). The non-existence of huge, absolute E-rings ?κ(ω) follows from a recent paper by Herden and Shelah (2009) [24] and the construction of absolute E-rings R is based on an old result by Shelah (1982) [31] where families of absolute, rigid colored trees (with no automorphism between any distinct members) are constructed. We plant these trees into our potential E-rings with the aim to prevent unwanted endomorphisms of their additive group to survive. Endomorphisms will recognize the trees which will have branches infinitely often divisible by primes. Our main result provides the existence of absolute E-rings for all infinite cardinals λ<κ(ω), i.e. these E-rings remain E-rings in all generic extensions of the universe (e.g. using forcing arguments). Indeed all previously known E-rings (Dugas, Mader and Vinsonhaler, 1987 [9]; Göbel and Trlifaj, 2006 [23]) of cardinality ?ℵ₀² have a free additive group R⁺ in some extended universe, thus are no longer E-rings, as explained in the introduction. Our construction also fills all cardinal-gaps of the earlier constructions (which have only sizes λℵ₀). These E-rings are domains and as a by-product we obtain the existence of absolutely indecomposable abelian groups, compare Göbel and Shelah (2007) [22].     

Martin’s Maximum and tower forcing

There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω ₂ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\overrightarrow I $ is a tower of ideals which concentrates on the class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally club sets, then $\overrightarrow I $ is not presaturated (a set is ω ₁-guessing iff its transitive collapse has the ω ₁-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA ⁺ or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω ₂ which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin’s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω ₂ has similar implications for towers of ideals which concentrate on the wider class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally stationary sets. Finally, we show that the word “presaturated” cannot be replaced by “precipitous” in the theorems above: Martin’s Maximum (which implies SRP and the Tree Property at ω ₂) is consistent with a precipitous tower on $GI{C_{{\omega _1}}}$ .     

Strong Compactness and a Global Version of a Theorem of Ben‐David and Magidor

Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ⁺ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □_γ+ω fails but □_{γ+ω, ω} holds. This generalizes a result of Ben‐David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.