Topic Archive: recursive saturation
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.
Given a model M, a maximal automorphism is one which fixes as few points in M as possible. We begin by outlining what the correct definition of “as few points as possible” should be and then proceed to study the notion. An interesting question arises when one considers the existence of maximal automorphisms of countable recursively saturated models. In particular an interesting dichotomy arises when one asks whether for a given theory T all countable recursively saturated models of T have a maximal automorphism. Our primary goal is to determine which classes of theories T lie on the positive side of this dichotomy. We give several examples of such classes. Attacking this problem requires a detailed understanding of recursive saturation, which we will also review in this talk.
It is still unknown whether there are nonisomorphic countable recursively saturated models M and N whose automorphism groups Aut(M) and Aut(N) are isomorphic. I will discuss what has happened over the last 20 years towards showing that such models do not exist, including some very recent results.
A model $M$ of PA is full if for every definable in $(M,omega)$ set $X$, $Xcap omega$ is coded in $M$. In a recent paper, Richard Kaye proved that $M$ is full if and only if its standard system is a model of full second order comprehension. Later in the semester we will examine Kaye’s proof. In this talk I will discuss some preliminary results and I will show an example of a model that is not full, using an argument that does not depend on Kaye’s theorem