Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

Notes on the partition property of {\mathcal{P}_\kappa\lambda}

We investigate the partition property of ${\mathcal{P}_{\kappa}\lambda}$ . Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that ${\mathcal{P}_{\kappa}\lambda}$ carries a (λ ^κ , 2)-distributive normal ideal without the partition property, then λ is ${\Pi^1_n}$ -indescribable for all n?<?ω but not ${\Pi^2_1}$ -indescribable. (2) If cf(λ) ≥?κ, then every ineffable subset of ${\mathcal{P}_{\kappa}\lambda}$ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over ${\mathcal{P}_{\kappa}\lambda}$ has the partition property.     

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.

     

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⁺).     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

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.     

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

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.     

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.     

Note on the point character of l 1-spaces

We prove that for any ordinal α, any integer t ≥ 0, the point character of the space l ₁(ω _{α + t} ) is no more than ω _α . Combined with an earlier result from [5], this yields that for any infinite cardinal κ the point character of l ₁(κ) is the largest cardinal ω _α ≤ κ where α = 0 or a limit ordinal.     

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.     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.     

Magidor-like and radin-like forcing

We force over a model M of ZF+κ→(κ)^<γ to obtain M[G] with cf(κ)=γ. The method is reminiscent of Magidor-forcing but uses no choice. Mimicing Radin-forcing, we generalize this for strong partition cardinals κ to add a subset of κ while preserving all cardinalities, cofinalities and κ's measurability. We apply these techniques to construct models of unusual partition properties, such as ω₂→[ω₂]^ω₁ but $\text{ω}{}_2\text{?[ω}{}_2\text{]}{}^{\mathrm{\omega }}$.     

Cichoń’s diagram for uncountable cardinals

We develop a version of Cichoń’s diagram for cardinal invariants on the generalized Cantor space 2 ^κ or the generalized Baire space κ ^κ , where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example, we obtain a natural generalization of the Bartoszyński–Raisonnier–Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.     

A partition theorem for a large dense linear order

Let κ be a cardinal which is measurable after generically adding ?_κ+ω many Cohen subsets to κ, and let ?_κ = (Q, ≤ _Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t _m ⁺ such that the set [Q] ^m of m-tuples from Q can be partitioned into classes 〈C _i : i < t _m ⁺ }〉 such that for any coloring a class C _i in fewer than κ colors, there is a copy ?* of ?_κ such that [?*] ^m ? C _i is monochromatic. It follows that \(\mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m \), that is, for any coloring of [?_κ] ^m with fewer than κ colors there is a copy Q′ ? Q of ?_κ such that [Q′] ^m has at most t _m ⁺ colors. On the other hand, we show that there are colorings of ?_κ such that if Q′ ? Q is any copy of ?_κ then C _i ? [Q′] ≠ ø; for all i < t _m ⁺ , and hence \(\mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m \).We characterize t _m ⁺ as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t ₂ ⁺ = 2 and for m > 2 we have t _m ⁺ > t _m , the m-th tangent number.The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ^+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.     

Pressing Down Lemma for λ-trees and its applications

For any ordinal λ of uncountable cofinality, a λ-tree is a tree T of height λ such that |T _α| < cf(λ) for each α < λ, where T _α = {x ∈ T: ht(x) = α}. In this note we get a Pressing Down Lemma for λ-trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that |T _α | ? ω for each α < η, then the tree T is collectionwise Hausdorff if and only if for each antichain C ? T and for each limit ordinal α ? η with cf(α) > ω, {ht(c): c ∈ C} ∩ α is not stationary in α. In the last part of this note, we investigate some properties of κ-trees, κ-Suslin trees and almost κ-Suslin trees, where κ is an uncountable regular cardinal.     

A counter-example in the partition calculus for an uncountable ordinal

The main theorem of the paper is a counter-example in the partition calculus introduced by P. Erdös and R. Rado: Ifκ is a regular cardinal and α=?(α,3)², then α ? (α,3)². The proof is combinatorial. Other counter-examples are produced from this one through the pinning relation which was introduced by E. Specker.     

More on regular and decomposable ultrafilters in ZFC

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (_m(λ⁺ⁿ), _m(λ⁺ⁿ))‐regular, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ 2^λ (Theorem 4.3(a^')). (b) If λ is a strong limit cardinal and D is (_m(λ⁺ⁿ), _m(λ⁺ⁿ))‐regular, then either D is (cf λ, cf λ)‐regular or there are arbitrarily large κ < λ for which D is κ ‐decomposable (Theorem 4.3(b)). (c) Suppose that λ is singular, λ < κ, cf κ ≠ cf λ and D is (λ⁺, κ)‐regular. Then: (i) D is either (cf λ, cf λ)‐regular, or (λ^', κ)‐regular for some λ^' < λ (Theorem 2.2). (ii) If κ is regular, then D is either (λ, κ)‐regular, or (ω, κ^')‐regular for every κ^' < κ (Corollary 6.4). (iii) If either (1) λ is a strong limit cardinal and λ^<λ < 2^κ, or (2) λ^<λ < κ, then D is either λ ‐decomposable, or (λ^', κ)‐regular for some λ^' < λ (Theorem 6.5). (d) If λ is singular, D is (μ, cf λ)‐regular and there are arbitrarily large ν < λ for which D is ν ‐decomposable, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ λ^<μ (Theorem 5.1; actually, our result is stronger and involves a covering number). (e) D × D^' is (λ, μ)‐regular if and only if there is a ν such that D is (ν, μ)‐regular and D^' is (λ, ν^')‐regular for all ν^～ < ν (Proposition 7.1). We also list some problems, and furnish applications to topological spaces and to extended logics (Corollar‐ies 4.6 and 4.8) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-saturation of the nonstationary ideal on P κ (λ) in case κ ≤ cf (λ) < λ

Given a regular cardinal κ > ω ₁ and a cardinal λ with κ?≤ cf (λ) < λ, we show that NS _κ,λ | T is not λ⁺-saturated, where T is the set of all ${a\in P_\kappa (\lambda)}$ such that ${| a | = | a \cap \kappa|}$ and ${{\rm cf} \big( {\rm sup} (a\cap\kappa)\big) = {\rm cf} \big({\rm sup} (a)\big) = \omega}$ .     

Characters of ultrafilters and tightness of products of fans

As applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of βN?N are not smaller than any submeasurable cardinal less or equal to ω². (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal to κ. In fact, a little more general results are proved.