Loftiness is a weak notion of saturation. It was defined and studied in detail by Matt Kaufmann and Jim Schmerl in two substantial papers published in 1984 and 1987. Kaufman and Schmerl discovered that there are many shades of loftiness. I will give an overview of model theory of lofty models of arithmetic and I will talk about constructions of lofty models that are not recursively saturated.

I will discuss more work of Kaufmann and Schmerl around loftiness. In particular I will discuss how in the definition of e-loftiness we may restrict our attention to only those types that define cuts. These consideration lead to a simple proof of a theorem of Pabion’s that for kappa an uncountable cardinal a model M of PA is kappa-saturated if and only if its underlying ordering is kappa-saturated. Time permitting I will also discuss how for countable models M, being lofty is equivalent to having a recursively saturated simple extension.