In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, ω-model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of Σ1 transfinite dependent choice and Π 1 2 reflection on ω-models of Σ1-DC.

A criterion is established for the transitivity of connectedness in a transfinite graph. Its proof is much shorter than a prior argument published previously for that

The bounds on the statuses of the nodes in a finite graph established by Entringer, Jackson, and Snyder are extended herein so that they apply to the statuses of the nodes in transfinite graphs of a certain kind.

We investigate a hierarchy of arithmetical structures obtained by transfinite addition of a canonic universal predicate, where the canonic universal predicate for M is defined as a minimum universal predicate for M in terms of definability. We determine the upper bound of the hierarchy and give a characterisation for the sets definable in the hierarchy.

The Furstenberg-Zimmer structure theorem for Z actions says that every measurepreserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi’s theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi’s theorem. Beleznay and Foreman showed that, in general, this tower can have...