Series: Penn State Logic Seminar Date: Tuesday, October 21, 2003 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Carl Mummert, Penn State, Mathematics Title: Reverse Mathematics of Maximal Filters on Countable Partial Orders Abstract: The Reverse Mathematics results covered in this talk determine which nonconstructive techniques are required to extend a filter in a countable partial order to a maximal filter. This nonconstructivity is measured using subsystems of Second Order Arithmetic, such as $\RCA_0$, $\ACA_0$ and $\Pi^1_1-\CA_0$. I will begin this talk with an introduction to these subsystems of Second Order Arithmetic. The main results to be covered are: (1) ``Every filter in a countable partial order can be extended to a maximal filter'' is equivalent to $\ACA_0$ over $\RCA_0$, and (2) ``Every directed filter in a countable partial order can be extended to a maximal directed filter'' is equivalent to $\Pi^1_1-\CA_0$ over $\RCA_0$. (For a partial order $P$, a filter $F$ is a nonempty proper subset of $P$ which is upward closed, such that every pair of elements in $F$ has a lower bound in $P$. A directed filter is a filter which contains a lower bound for each pair of its elements. A maximal filter is a filter which cannot be extended to a larger filter; a maximal directed filter is defined analogously.)