Lindelöf models of the reals: Solution to a problem of Sikorski

We show that it is consistent with ZFC that there is a modelM of ZF + DC such that the integers ofM areω ₁-like, the reals ofM have cardinalityω ₂, and the unit interval [0, 1] ^M is Lindelöf (i.e. every open cover has a countable subcover). This answers an old question of Sikorski.     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

Some descriptive set theory and core models

We analyse the trees given by sharps for Π¹₂ sets via inner core models to give a canonical decomposition of such sets when a core model is Σ¹₃ absolute. This is by way of analogy with Solovay's analysis of Π¹₁ sets into ω₁ Borel sets — Borel in codes for wellorders. We find that Π¹₂ sets are also unions of ω₁ Borel sets — but in codes for mice and wellorders. We give an application of this technique in showing that if a core model, K, is Σ¹₃ absolute thenTheorem. Every real is in K iff every Π¹₃ set of reals contains a Π¹₃ singleton.     

On completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal not greater than 2^ω . We show that for a c. c. c. σ -ideal 𝕀 with a Borel base of subsets of an uncountable Polish space, if 𝒜 is a point-finite family of subsets from 𝕀, then there is a subfamily of 𝒜 whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong to the σ -field generated by Borel sets and the ideal 𝕀. This result is a generalization of the Four Poles Theorem (see [1]) and a result from [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

The Strengths of Some Violations of Covering

We consider two models V₁, V₂ of ZFC such that V₁ ⊆ V₂, the cofinality functions of V₁ and of V₂ coincide, V₁ and V₂ have that same hereditarily countable sets, and there is some uncountable set in V₂ that is not covered by any set in V₁ of the same cardinality. We show that under these assumptions there is an inner model of V₂ with a measurable cardinal κ of Mitchell order κ⁺⁺. This technical result allows us to show that changing cardinal characteristics without changing cofinalities or ω‐sequences (which was done for some characteristics in [13]) has consistency strength at least Mitchell order κ⁺⁺. From this we get that the changing of cardinal characteristics without changing cardinals or ω‐sequences has consistency strength Mitchell order ω₁, even in the case of characteristics that do not stem from a transitive relation. Hence the known forcing constructions for such a change have lowest possible consistency strength. We consider some stronger violations of covering which have appeared as intermediate steps in forcing constructions.     

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.     

Irreducible Components of Fixed Point Subvarieties of Flag Varieties

Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P, and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T, and a Borel subgroup, B, of G, so that T?B∩L, B ? P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let ??_e be the subset of ?? consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let ??_ω be the B-orbit in ?? containing ^ωB. We consider the intersections ??_ω ∩ ??_e. The main result is that if dim ??_ω ∩ ??_e = dim ??_e, then ??_ω ∩ ??_e is an affine space. Thus, the irreducible components of ??_e are indexed by Weyl group elements. It is also shown that if G is of type A, then this set of Weyl group elements is a right cell in W.     

A dedekind finite borel set

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if ${B\subseteq 2^\omega}$ is a G _??? -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ^?? is the countable union of countable sets, then there exists an F _??? set ${C\subseteq 2^\omega}$ such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite ${D\subseteq 2^\omega}$ which is F _??? .     

Internally club and approachable

We prove that PFA implies for all regular λ?ω₂, there are stationarily many N⊆H(λ) with size ℵ₁ which are internally club but not internally approachable.     

The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types II₁, II_∞ or IIIλ, 0?λ?1, up to isomorphism by a Borel measurable assignment of “countable structures” as invariants. In particular the isomorphism relation of type II₁ factors is not smooth. We also prove that the isomorphism relation for von Neumann II₁ factors is analytic, but is not Borel.     

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.     

The dual space of L∞ is L1

The provocative title may be supposed to hold if the axiom of choice is weakened. Even more: The dual space of any Köthe space (real-valued and over a σ-finite measure space) coincides with its associate space in the canonical way; in particular, any Köthe space with Fatou's property is re-flexive. We will show that all this is true in ZF + DC + PM_ω, where PM_ω is introduced in the text and holds e.g. in Solovay's model.     

On Internal Resonance of Two Damped Oscillators

The damped motion of two oscillators with natural frequencies ω₁ and ω₂ is studied on the assumptions that: the oscillators are uncoupled for infinitesimal displacements and quadratically coupled for finite displacements; the frequencies are approximately in the ratio 2:1, such that ω₂?2ω₁=O(εω₁), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ₁ and λ₁, are O(ε). The motion may be described by modulated sine waves with carrier frequencies ω_1,2 and slowly varying energies and phases that satisfy four first-order, nonlinear differential equations. These equations admit one invariant and may be reduced to two first-order equations if ω₂=2ω₁ and λ_1,2<0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω₂=2ω₁ and 2λ₂=λ₁. Numerical results are presented for typical parametric combinations.     

Uniqueness and Flat Core of Positive Solutions for Quasilinear Elliptic Eigenvalue Problems in General Smooth Domains

The structure of positive solutions to the quasilinear elliptic problems –div(|Du|^p–2Du = λf(u) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ R^Na bounded smooth domain, is precisely studied when λ is sufficiently large, for a class of logistic‐type nonlinearities f(u) satisfying that f(0) = f(a) = 0, a > 0, f(u) > 0 for u ∈ (0,a), , while u = a is a zero point of f with order ω. It is shown that if ω ≥ p – 1, the problem has a unique positive solution u_λ with sup _Ω u_λ < a, which develops a boundary layer near ∂Ω. It is shown that if 0 < ω < p – 1, the problem also has a unique positive solution u _λ, but the flat core {x ∈ Ω : u_λ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞.     

Decompositions of Borel bimeasurable mappings between complete metric spaces

We prove that every Borel bimeasurable mapping can be decomposed to a σ-discrete family of extended Borel isomorphisms and a mapping with a σ-discrete range. We get a new proof of a result containing the Purves and the Luzin-Novikov theorems as a by-product. Assuming an extra assumption on f, or that Fleissner's axiom (SCω₂) holds, we characterize extended Borel bimeasurable mappings as those extended Borel measurable ones which may be decomposed to countably many extended Borel isomorphisms and a mapping with a σ-discrete range.     

On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo PP-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0?α<ω₁.     

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 $\text{ω}{}_2\text{?[ω}{}_2\text{]}{}^{\mathrm{\omega }}$.