Nonstandard and relative set theories.
Karel Hrbacek
City College of New York, CUNY
These theories axiomatize a universe of sets that can have nonstandard elements such as infinitesimals. The nonstandard set theory BST [respectively, the relative set theory GRIST] extends the language of ZFC by a unary predicate “x is standard” [respectively, by a binary predicate “x is standard relative to y”].
Theorem. Every model M of ZFC has an extension to a model of BST [respectively, GRIST] in which M is the universe of standard sets. If M is countable, then the extension is unique, modulo an isomorphism that fixes standard sets.
Corollary. BST [respectively, GRIST] is conservative and complete over ZFC.
I will describe some ideas used to prove these results, in particular, the technique of internally iterated ultrapowers.
Professor Hrbacek undertakes research in the area of mathematical logic, particularly in set theory, with a focus on non-standard analysis and nonstandard set theory. He wrote in co-authorship with Thomas Jech the highly regarded book Introduction to Set Theory.