Definable maximal discrete sets in forcing extensions

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $\Delta^1_2$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in [5] by showing that in the Sacks and Miller extensions there is a $\Pi^1_1$ maximal orthogonal family ("mof") of Borel probability measures on Cantor space. A similar result is also obtained for $\Pi^1_1$ mad families. By contrast, we show that if there is a Mathias real over $L$ then there are no $\Sigma^1_2$ mofs.Comment: 16 page

Σ21\Sigma^1_2 and Π11\Pi^1_1 mad families

We answer in the affirmative the following question of J\"org Brendle: If there is a $\Sigma^1_2$ mad family, is there then a $\Pi^1_1$ mad family?Comment: This will appear in the Journal of Symbolic Logic (March 2014

Projective maximal families of orthogonal measures with large continuum

We study maximal orthogonal families of Borel probability measures on $2^\omega$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a $\Delta^1_3$-definable well order of the reals, there is a $\Pi^1_2$-definable m.o. family, there are no $\mathbf{\Sigma}^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=\omega_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $\Delta^1_3$-definable well order of the reals, there is a $\Pi^1_2$-definable m.o. family, there are no $\mathbf{\Sigma}^1_2$-definable m.o. families, $\mathfrak{b}=\omega_1$ and $\mathfrak{c}=\omega_2$.Comment: 12 page

Definable maximal cofinitary groups of intermediate size

Using almost disjoint coding, we show that for each $1<M<N<\omega$ consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$, where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary group.Comment: 22 page

Transfinite inductions producing coanalytic sets

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.Comment: preliminary versio

Definable MAD families and forcing axioms

We show that under the Bounded Proper Forcing Axiom and an anti-large cardinal assumption, there is a $\mathbf{\Pi}^1_2$ MAD family.Comment: 13 page