# 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.