This week in the Logic Seminar in Keller 314, David Ross will give an easy proof of a slight extension of a result of Lagarias on the Diophantine equation

       $$ c(1/x_1+cdots+1/x_s)+b/(x_1 x_2cdots x_s)=a$$

The proof will be nonstandard, but really only require a sufficiently-saturated ordered field extension of $mathbb R$.

Mushfeq Khan will coninue to speak on amenability and symbolic dynamics.  The focus will be the Ornstein-Weiss combinatorial
lemmas from Adam Day’s paper.

Title: Universes of sets

Abstract: As is well-known, all mathematical objects can be coded as sets and thereby all of mathematics can be formally founded in set theory. What is perhaps less well-known is that there are many different models of set theory, each of which is powerful enough to function as a universe of sets and found (most) of mathematics, but these models can have very different properties.

This talk will aim to explore the question: what is a model of set theory? We will learn about Skolem’s paradox, that there are countable models of set theory, even though these countable models think they contain uncountable sets like the set of reals. We will be introduced to transitive models, usually considered to be the best behaved, but also meet ill-founded models, such as models which think ZFC is inconsistent. To conclude we will briefly discuss two positions in the philosophy of set theory: universism, the view that there is a unique maximal universe of sets, and multiversism, the view that there are many equally valid universes of sets.

This is an introductory talk, aimed to be understandable by those with little background in set theory. It is a prequel to my next talk, which is in turn a prequel to my talk after that.