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.
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
We compare the forcing-related properties of a complete Boolean algebra with the properties of the convergences (the algebraic convergence) and on generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that is a topological convergence iff forcing by does not produce new reals and that is weakly topological if satisfies condition (implied by the -cc). On the other hand, if is a weakly topological convergence, then is a -cc algebra...
A generalization of Shoenfield theorem on sets
A note on a question of Abe
Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.
A note on forcing with ideals and Hechler forcing
A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem
We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.
A topology generated by eventually different functions
Absolute Indiscernibles and Standard Models of ZFC
Adding a random or a Cohen real: topological consequences and the effect on Martin's axiom
An Application of Shoenfield's Absoluteness Theorem to the Theory of Uniform Distribution.
Another ⋄-like principle
A new ⋄-like principle consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that is consistent with CH and that in many models of = ω₁ the principle holds. As implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....