Slides from a talk I gave on counting the number of models of the theory of
the random graph. The main point is to show that there are really a lot of
models. This provides a segue into stability theory and the
There are many ways to define the random graph, but they all lead to the same, unique, structure. The first order theory of this structure gives a class of structures which "look like" the random graph. However, while there is a unique random graph, there are many, many look-a-likes in every uncountable cardinality. This talk will show how the many different look-a-likes come from the presence of a formula with the "independence property"; that the random graph is the simplest theory exhibiting this property; and, finally, survey other theories with this property. Also, the negation of having the independence property has its own, profound, consequences. All logic jargon will be explained, in patient, loving detail.
I would like to revise the talk to an approach focusing on approximating the (countable) random graph with finite models. This I think would also be interesting, more concrete, and moreover, it could then go into amalgamation type constructions. Another time...
Fall 2008
Notes for a talk I gave on a proof showing the theory of (Z,+) does not have a prime model, but does have a lot of distinct minimal models. This shows that the different ways of defining a "smallest" or "simplest" model are not equivalent. The abstract is:
The vagueness of the phrase ``simplest possible'' is shown through the sometimes coinciding sometimes wildly diverging notions of "prime model", a model which can be embedded inside any other model of a given theory, and a "minimal model", a model which does not contain any submodels of a given theory. Hijinks ensue. The highlight will be the exhibition of a theory with no prime model and uncluntably many minimal models along with a proof of this fact due to Baldwin, Blass, Glass, and Kueker.
Spring 2007
These are notes I made and papers I found on prime testing algorithms. The focus is more on complexity issues than implementation details.
Spring 2006
The Banach-Tarski paradox is the one where you can cut a ball up into a finite number of pieces and then rearrange these pieces to make two balls the same size as the first one. Read some notes on the construction.
Spring 2006