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.
"Standard sets in nonstandard set theory." J. Symbolic Logic 69 (1) 165 - 182, March 2004. https://doi.org/10.2178/jsl/1080938835