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We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses: (1) $\text{DC}(\mathbf\Pi^1_n)\land\neg\text{AC}_\omega(\varPi^1_{n+1})\,;$ (2) $\text{AC}_\omega(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_\omega(\mathbf\Pi^1_{n+1});$ (3) $\text{AC}_\omega\land\text{DC}(\mathbf\Pi^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$ (4) $\text{AC}_\omega\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbf\Pi^1_{n+1}).$ Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal P(\omega)$ in this model, such that $\langle\omega, M\rangle$ satisfies (5) PA$^0_2$ + $\text{AC}_\omega(\varSigma^1_{\infty})$ + $\text{CA}(\mathbf\Sigma^1_{n+1})$ + $\neg\text{CA}(\mathbf\Sigma^1_{n+1})$. Thus $\text{CA}(\mathbf\Sigma^1_{n+1})$ does not imply $\text{CA}(\mathbf\Sigma^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\text{AC}_\omega(\varSigma^1_{\infty}).$, Comment: A section on the Reflection principle is added in the last chapter. Using the results on DC we prove that Reflection is not finitely axiomatizable

The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $\omega$-model of $\text{PA}_2^\ast + \text{CA}(\Sigma^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $\omega$-model of $\text{PA}_2^\ast$, in which not every set has its complement, and hence the full $\text{CA}$ fails in a rather elementary way., Comment: 13 pages

In this paper, we prove the following: If $n\ge3$, there is a generic extension of $L$ -- the constructible universe -- in which it is true that the Separation principle holds for both effective (lightface) classes $\varSigma^1_n$ and $\varPi^1_n$ for sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for $n=3$; its full proof has never been presented. Our methods are based on a countable product of almost-disjoint forcing notions independent in the sense of Jensen--Solovay., Comment: 17 pages

A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe., Comment: arXiv admin note: substantial text overlap with arXiv:2111.07654

By Tzouvaras, a set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD$\subseteq$HNT$\subseteq$V holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of ZFC is presented in which HOD$\subsetneqq$HNT$\subsetneqq$V, and another model of ZFC, in which HNT does not satisfy the axiom of choice.

Using an invariant modification of Jensen's "minimal $\varPi^1_2$ singleton" forcing, we define a model of ZFC, in which, for a given $n\ge2$, there exists a lightface $\varPi^1_n$ unordered pair of non-OD (hence, OD-indiscernible) countable sets of reals, but there is no $\varSigma^1_n$ unordered pairs of this kind.

We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbf\Sigma^1_n$ and $\mathbf\Pi^1_n$, for a given $n\ge3$., Comment: arXiv admin note: text overlap with arXiv:1702.03566

The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$, or (2) $M[x]=M[a]$, or (3) $M[x]$ is a random extension of $M$ and $M[a]$ is a random extension of $M[x]$. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that $\Sigma^1_n$-Reduction holds for all $n\ge3$, in a model extending the constructible universe $L$ by $\aleph_1$-many random reals., Comment: 11 pages

In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbf\Pi^1_3$-Separation holds but $\mathbf\Sigma^1_3$-Reduction fails. The result has never appeared in a journal or book publication except for a few of old references., Comment: Revised version contains more consistent description on some results of Sami

A canonization scheme for smooth equivalence relations on $\mathbb R^\omega$ modulo restriction to infinite perfect products is proposed. It shows that given a pair of Borel smooth equivalence relations $\mathsf E,\mathsf F$ on $\mathbb R^\omega$, there is an infinite perfect product $P\subseteq\mathbb R^\omega$ such that either ${\mathsf F}\subseteq{\mathsf E}$ on $P$, or, for some $j<\omega$, the following is true for all $x,y\in P$: $x\,\mathsf E \,y$ implies $x(j)=y(j)$, and $x\restriction{(\omega\smallsetminus\{j\})}=y\restriction{(\omega\smallsetminus\{j\})}$ implies $x\,\mathsf F \,y$., Comment: A revised version

We present a model of set theory, in which, for a given $n\ge2$, there exists a non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\bf\Sigma^1_n$ sets with countable cross-sections are $\bf\Delta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level., Comment: A revised version of the originally submitted preprint

Using a non-Laver modification of Uri Abraham's minimal $\varDelta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge3$, $\{a\}$ is a lightface $\varPi^1_n$ singleton, $a$ effectively codes a cofinal map $\omega\to\omega_1^L$ minimal over $L$, while every $\varSigma^1_n$ set $X\subseteq\omega$ is still constructible., Comment: 30 pages

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals. Keywords: definability; hyperreal; superstructure; elementary embedding., Comment: 8 pages, to appear in Journal of Symbolic Logic

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily constructible, hence contains only OD reals., Comment: With TOC and Index

We make use of a finite support product of $\omega_1$ 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 $\Sigma$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation., Comment: 4 pages

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension that every countable ordinal-definable set of reals belongs to to the ground universe, Comment: 16 pages

A generic extension $L[x,y]$ of $L$ by reals $x,y$ is defined, in which the union of $\mathsf E_0$-classes of $x$ and $y$ is a $\Pi^1_2$ set, but neither of these two $\mathsf E_0$-classes is separately ordinal-definable.

We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and $\Sigma^1_2$ thin sets in the assumption that $\omega_1^{L[x]}<\omega_1$ for all reals $x$. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model., Comment: arXiv admin note: substantial text overlap with arXiv:1408.1202

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 $\varPi^1_3$., Comment: arXiv admin note: text overlap with arXiv:1408.3901

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 $\Pi^1_2$ set containing no ordinal-definable reals., Comment: arXiv admin note: substantial text overlap with arXiv:1408.3901

We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty lightface $\Pi^1_2$ set of reals containing no ordinal-definable real., Comment: Revised version. The revision includes an updated proof of Lemma 4.5 (the density-preservation lemma for the product)

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., Comment: to appear