We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: If m ≥ 1 and the ultrafilter D is , equation imagem)-regular, then D is κ -decomposable for some κ with λ ≤ κ ≤ 2λ ). If λ is a strong limit cardinal and D is , equation imagem)-regular, then either D is -regular or there are arbitrarily large κ < λ for which D is κ -decomposable ). Suppose that λ is singular, (...) λ < κ, cf κ ≠ cf λ and D is -regular. Then: D is either -regular, or -regular for some λ' < λ . If κ is regular, then D is either -regular, or -regular for every κ' < κ . If either λ is a strong limit cardinal and λ<λ < 2κ, or λ<λ < κ, then D is either λ -decomposable, or -regular for some λ' < λ . If λ is singular, D is -regular and there are arbitrarily large ν < λ for which D is ν -decomposable, then D is κ -decomposable for some κ with λ ≤ κ ≤ λ<μ . D × D' is -regular if and only if there is a ν such that D is -regular and D' is -regular for all ν∼ < ν .We also list some problems, and furnish applications to topological spaces and to extended logics. (shrink)