Abstract Assuming that every set is constructible, we find a ${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of $\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a ${\text{\Pi }}_1^1 $ maximal cofinitary group inL.

Abstract We consider natural ${\rm{\Sigma }}_2^1$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these ${\rm{\Sigma }}_2^1$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for ${\rm{\Sigma }}_2^1$ colourings which hold precisely when there is a non-constructible real.

We answer in the affirmative the following question of Jörg Brendle: If there is a $\\Sigma^1_2$ mad family, is there then a $\\Pi^1_1$ mad family?

Abstract We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

Abstract In this paper we show that there are “E0many” orbit inequivalent free actions of the free groups , 2 ≤n≤ ∞ by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.