Various kinds of Aronszajn tree with no subtree of a different kind -- On set theories in toposes -- A representation for Spector second order classes in computation theories on two types -- Precipitousness of the ideal of thin sets on a measurable cardinal -- Axiom systems of nonstandard set theory -- Semi-formal finitist proof of the transfinite induction in an initial segment of Cantor's second number class -- On the length of proofs in a formal system of recursive arithmetic -- Homogeneous formulas and definability theorems -- Boolean valued combinatorics -- Undecidability of extensions of the monadic first-order theory of successor and two-dimensional finite automata -- Sections and envelopes of type 2 objects -- Heyting valued universes of intuitionistic set theory -- Independence of a problem in elementary analysis from set theory -- Intuitionistic theories and toposes -- The Hahn-Banach theorem and a restricted inductive definition