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.)