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.