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Using a modification of the invariant Jensen forcing of [11] , we define a model of ZFC , in which, for a given n ≥ 3 , there exists a lightface Π n 1 -set of reals, which is a E 0 -equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable Σ n 1 -set of reals is constructible, hence contains only OD reals. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We make use of a finite support product of the Jensen minimal Π 2 1 singleton forcing to define a model in which Π 2 1 uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which separation fails for Π 3 1 . [ABSTRACT FROM AUTHOR]

A generic extension $${\mathbf{L}[x]}$$ by a real x is defined, in which the $${\mathsf{E}_0}$$ -class of x is a lightface $${{\it \Pi}^1_2}$$ (hence, ordinal-definable) set containing no ordinal-definable reals. [ABSTRACT FROM AUTHOR]