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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

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., arXiv admin note: substantial text overlap with arXiv:2111.07654

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

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, Comment: A revised version

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

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., With TOC and Index

We prove that Solovay's set $��$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation., 4 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 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)

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