A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.

A generic extension L[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{L}[x]}$$\end{document} by a real x is defined, in which the E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{E}_0}$$\end{document}-class of x is a lightface Π21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\it \Pi}^1_2}$$\end{document} set containing no ordinal-definable reals.

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.

We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product P⊆Rω such that either F⊆E on P, or, for some ℓ<ω, the following is true for all x,y∈P: xEy implies x(ℓ)=y(ℓ), and x↾(ω∖{ℓ})=y↾(ω∖{ℓ}) implies xFy.

For over a century, Haeckel's Gastraea theory remained a dominant theory to explain the origin of multicellular animals. According to this theory, the animal ancestor was a blastula‐like colony of uniform cells that gradually evolved cell differentiation. Today, however, genes that typically control metazoan development, cell differentiation, cell‐to‐cell adhesion, and cell‐to‐matrix adhesion are found in various unicellular relatives of the Metazoa, which suggests the origin of the genetic programs of cell differentiation and adhesion in the root of the Opisthokonta. Multicellular (...) stages occurring in the complex life cycles of opisthokont protists (mesomycetozoeans and choanoflagellates) never resemble a blastula. Here, we discuss a more realistic scenario of transition to multicellularity through integration of pre‐existing transient cell types into the body of an early metazoon, which possessed a complex life cycle with a differentiated sedentary filter‐feeding trophic stage and a non‐feeding blastula‐like larva, the synzoospore. Choanoflagellates are considered as forms with secondarily simplified life cycles. (shrink)