Set Theory and Analysis Program
Saharon Shelah, Hebrew University and Rutgers University
October 1st - 3rd, 2002 at 3:30 pm
Adding measures to normed forcing
Normed forcing first established its usefulness in Saharon Shelah's
seminal proof that the splitting number can be greater than the unboundedness
number. The essential idea was to use measures (often on finite sets)
to control the splitting nodes of trees used as forcing conditions.
These ideas later evolved into a sophisticated theory of "creature
forcing" which is described in a short book by A. Roslanowski and
S. Shelah. Lately, in more joint work with A. Roslanowski, a simple
form of integration was added to the mix and has allowed the solution
of several longstanding problems in the theory of measurable functions.
For example, it has been shown to be consistent with set theory that
all real valued functions are continuous on a non-measurable set of
reals. The same techniques have also been applied to questions dealing
with superposition measurable functions; namely those function from
the plane to the reals whose restriction to the graph of any measurable
function is measurable. The study of a certain class of solutions to
the Cauchy problem leads to the question of whether every superposition
measurable function is actually measurable. Measured creatures can be
used to show that the answer is consistently positive.
The first lecture will provide an introduction and overview of the
main results, while the last two lectures will provide details for experts
in the field.