### Mathematical logic seminar - September 14 2009

 Time: 12:00 - 13:20 Room: Doherty Hall 4303 Speaker: Spencer Unger    CMU Title: PCF theory I Abstract: This week I will give the first of two talks on Shelah's PCF Theory. PCF Theory has many applications in set theory. The most well known is Shelah's result in cardinal arithmetic that if $\aleph_\omega$ is strong limit, then $2^{\aleph_\omega}<\aleph_{\omega_4}$. On can also use PCF concepts to construct a Jonsson Algebra of size $\aleph_{\omega+1}$. This talk will be devoted to some of the basic PCF theoretic machinery. Those who have studied some PCF theory will know that building a sequence of functions with an exact upper bound is a key technical point. Much hinges on whether you are working modulo an ultrafilter or working modulo an ideal. In this series I will present some theorems of PCF illustrating this point. In the first talk, I will state the relevant definitions and proceed to a proof of Shelah's Dichotomy theorem. Finally, I will state some applications of the dichotomy theorem for obtaining exact upper bounds. Small background knowledge of set theory will be presumed. Familiarity with PCF theory is not required. Eventually the notes for the talk will be posted at my logic web page.