Blog Archives
Topic Archive: nonstandard set theory
Nonstandard and relative set theories.
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.
Some problems motivated by nonstandard set theory
Nonstandard set theory enriches the usual set theory by a unary “standardness” predicate. Investigations of its foundations raise a number of questions that can be formulated in ZFC or GB and appear open. I will discuss several such problems concerning elementary embeddings, ultraproducts, ultrafilters and large cardinals.