On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

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].     

On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming does not exist. Namely, if is a Jónsson cardinal then , provided that there is no non-trivial elementary embedding . There are a number of related results in ZFC concerning in V and inner models, for a Jónsson or singular cardinal. Received: 8 December 1998     

On guessing generalized clubs at the successors of regulars

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.     

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

The axiomatic power of Kolmogorov complexity

The famous Gödel incompleteness theorem states that for every consistent, recursive, and sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T  . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us prove new interesting theorems. This result answers a question recently asked by Lipton. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE$\mathrm{NP}=\mathrm{PSPACE}$).     

Small embedding characterizations for large cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces

Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups, where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS]. Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case. Received: 10 February 1999 / Published online: 1 February 2002     

The approachability ideal without a maximal set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ${\omega }_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[{\omega }_2]$ does not have a maximal set modulo clubs.     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

Weakly measurable cardinals

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection $\mathcal {A}$ containing at most κ⁺ many subsets of κ, there exists a nonprincipal κ‐complete filter on κ measuring all sets in $\mathcal {A}$. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Reflexive abelian groups and measurable cardinals and full MAD families

Answering problem (DG) of [1], [2], we show that there is a reflexive group of cardinality equal to the first measurable cardinal.     

More results on congruent modules

W.R. Scott characterized the infinite abelian groups G for which H≅G for every subgroup H of G of the same cardinality as G [W.R. Scott, On infinite groups, Pacific J. Math. 5 (1955) 589-598]. In [G. Oman, On infinite modules M over a Dedekind domain for which N≅M for every submodule N of cardinality |M|, Rocky Mount. J. Math. 39 (1) (2009) 259-270], the author extends Scott’s result to infinite modules over a Dedekind domain, calling such modules congruent, and in a subsequent paper [G. Oman, On modules M for which N≅M for every submodule N of size |M|, J. Commutative Algebra (in press)] the author obtains results on congruent modules over more general classes of rings. In this paper, we continue our study.     

There are no hereditary productive γ-spaces

We show that if X is an uncountable productive γ-set [F. Jordan, Productive local properties of function spaces, Topology Appl. 154 (2007) 870-883], then there is a countable Y⊆X such that X?Y is not Hurewicz.Along the way we answer a question of A. Miller by showing that an increasing countable union of γ-spaces is again a γ-space. We will also show that λ-spaces with the Hurewicz property are precisely those spaces for which every co-countable set is Hurewicz.