Tue, 25 October 2022, 1:25 pm MDT
A non-standard framework over a base set $A$ is an extension of a fragment of set theory containing $A$ that realizes many types with parameters from $A$ and preserves bounded quantifier formulas. In particular, if $A$ contains an indexing set $I$ and $U$ is an ultrafilter over $I$, then a saturated enough non-standard framework over $A$ will contain elements that appear to be principal generators for $U$. The properties of the ultrafilter are reflected in properties of their hyperprincipal generators in a non-standard framework. In this talk we will go over some background information about non-standard frameworks and then discuss how the properties of being Regular and Good (key properties related to Keisler's Order for countable first-order theories) are reflected in their hyperprincipal generators.