arXiv subjects
Publications and source records attributed to Vassily Lyubetsky.
We make use of a finite support product of $ω_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the continuum.
We prove that Solovay's set $Σ$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.
It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.
We make use of a finite support product of the Jensen minimal $\varPi^1_2$ singleton forcing to define a model in which $\varPi^1_2$ Uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which Separation fails for $\varPi^1_3$.
A generic extension $L[x]$ of $L$ by a real $x$ is defined, in which the $\mathsf E_0$-class of $x$ is a lightface $Π^1_2$ set containing no ordinal-definable reals.
We prove that in some cases definable chains of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic chains, ROD chains in the Solovay model, and $Σ^1_2$ chains in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$.
For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.
The classical attenuation regulation of gene expression in bacteria is considered. We propose to represent the secondary RNA structure in the leader region of a gene or an operon by a term, and we give a probabilistic term rewriting system modeling the whole process of such a regulation.