The full basis theorem does not imply analytic wellordering

A finite support product of ${\omega }_1$ clones of Jensen's minimal ${\Pi }_2^1$ singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals.     

A countable definable set containing no definable elements

The consistency of the existence of a countable definable set of reals, containing no definable elements, is established. The model, where such a set exists, is obtained by means of a countable product of Jensen’s forcing with finite support.     

Transfer theorems and the algebra of modal operators

A set theory ZFI′ which does not employ the Law of the Excluded Middle φ ∀ ⊥ φ, for all φ, retians the stock of expressive capacities of the classical set theory ZF, on the one hand, and has many of the features of an effective theory on the other. In the article, a broad class of formulas σ is constructed for which ZF ⊥ σ implies ZFI′ ⊥ σ. This result provides a generalization of Friedman's theorem on AE-arithmetic formulas. Besides, we prove transfer theorems of classical logic for the case of rings; in particular, Hilbert's theorem on zeros and Artin's theorem on ordered fields are extended to the case of regular f-rings, and we bring in appropriate upper bounds for them. Supported by RFFR grant No. 93-012-1027. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 282–303, May–June, 1997.     

Borel reducibility as an additive property of domains

We prove that under certain requirements, if E and F are Borel equivalence relations, , is a countable union of Borel sets, and E ↾ X _n , is Borel reducible to F for all n, then E ↾ X is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 189–198.     

Nature of weak inter-and intramolecular interactions in crystals 6. Intramolecular O—H...O bond in enol forms of (phosphoryl)acylacetonitriles

The nature of the intramolecular O—H...O bond was studied and its energy was estimated by X-ray diffraction analysis and quantum-chemical calculations (B3LYP/6-311G**) of (diphenylphosphoryl)acylacetonitriles. The influence of the nature of the substituents at the double bond in the H-bonded ring and the crystal packing effects on the hydrogen bond were investigated. Dedicated to Corresponding Member of the Russian Academy of Sciences T. A. Mastryukova. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2406–2413, November, 2005.     

Absoluteness of the Solovay Set Σ

We prove that the Solovay set Σ is absolutely definable in a sufficiently wide sense; in particular, Σ does not depend on the choice of the ground model.

     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

An effective minimal encoding of uncountable sets

We propose a method for encoding sets of the countable ordinals by generic reals which preserves cardinality and enjoys the property of minimality over the encoded set.