Marcin Sabok
Bounded forcing axioms and Baumgartner's conjecture.
Mon, 02/14/2011 - 16:01 — SomNambulAuthors: Marcin Sabok, David Aspero, Sy-David Friedman, Miguel Angel Mota
Subjects: Logic
AbstractWe study the spectrum of forcing notions between the iterations of
$\sigma$-closed followed by ccc forcings and the proper forcings. This includes
the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals
as well as the Axiom A forcings. We focus on the bounded forcing axioms for the
hierarchy of $\alpha$-proper forcings and connect them to a hierarchy of weak
club guessing principles. We show that they are, in a sense, dual to each
other.Complexity of Ramsey null sets.
Thu, 04/01/2010 - 15:01 — SomNambulAuthors: Marcin Sabok
Subjects: Logic
AbstractWe show that the set of codes for Ramsey positive analytic sets is
$\mathbf{\Sigma}^1_2$-complete. This is a one projective-step higher analogue
of the Hurewicz theorem saying that the set of codes for uncountable analytic
sets is $\mathbf{\Sigma}^1_1$-complete. This shows a close resemblance between
the Sacks forcing and the Mathias forcing. In particular, we get that the
$\sigma$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem
posed by Ikegami, Pawlikowski and Zapletal.Forcing properties of ideals of closed sets.
Tue, 01/19/2010 - 16:01 — SomNambulAuthors: Marcin Sabok, Jindrich Zapletal
Subjects: Logic
AbstractWith every $\sigma$-ideal $I$ on a Polish space we associate the
$\sigma$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing
notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^*$ and
find connections between their forcing properties. To this end, we associate to
a $\sigma$-ideal on a Polish space an ideal on a countable set and show how
forcing properties of the forcing depend on combinatorial properties of the
ideal. For $\sigma$-ideals generated by closed sets we also study the degrees
of reals added in the forcing extensions.Forcing, games and families of closed sets.
Wed, 10/14/2009 - 11:03 — SomNambulAuthors: Marcin Sabok
Subjects: Logic
AbstractWe propose a new, game-theoretic, approach to the idealized forcing, in terms
of fusion games. This generalizes the classical approach to the Sacks and the
Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1)
$\sigma$-ideals we show that if a $\sigma$-ideal is generated by closed sets,
then it is generated by closed sets in all forcing extensions. We also prove an
infinite-dimensional version of the Solecki dichotomy for analytic sets.Forcing, games and families of closed sets.
Wed, 10/14/2009 - 11:03 — SomNambulAuthors: Marcin Sabok
Subjects: Logic
AbstractWe propose a new, game-theoretic, approach to the idealized forcing, in terms
of fusion games. This generalizes the classical approach to the Sacks and the
Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1)
$\sigma$-ideals we show that if a $\sigma$-ideal is generated by closed sets,
then it is generated by closed sets in all forcing extensions. We also prove an
infinite-dimensional version of the Solecki dichotomy for analytic sets.
User login
