A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC.
"An axiomatic presentation of the nonstandard methods in mathematics." J. Symbolic Logic 67 (1) 315 - 325, March 2002. https://doi.org/10.2178/jsl/1190150046