A classification of definable forcings on ω1
Under the assumption of the existence of sharps for reals all simply definable posets on are classified up to forcing equivalence.
Preserving P-points in definable forcing
I isolate a simple condition that is equivalent to preservation of P-points in definable proper forcing.
On the existence of a -closed dense subset
It is consistent with the axioms of set theory that there are two co-dense partial orders, one of them -closed and the other one without a -closed dense subset.
Separating equivalence classes
Given a countable Borel equivalence relation, I introduce an invariant measuring how difficult it is to find Borel sets separating its equivalence classes. I evaluate these invariants in several standard generic extensions.
Coloring triangles and rectangles
It is consistent that ZF + DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles on does not.
Forcing with ideals generated by closed sets
Consider the poset where is an arbitrary -ideal -generated by a projective collection of closed sets. Then the extension is given by a single real of an almost minimal degree: every real is Cohen-generic over or .
Proper forcings and absoluteness in
We show that in the presence of large cardinals proper forcings do not change the theory of with real and ordinal parameters and do not code any set of ordinals into the reals unless that set has already been so coded in the ground model.
Games with creatures
Many forcing notions obtained using the creature technology are naturally connected with certain integer games.
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