An advanced graduate course. Some knowledge of forcing is assumed, and
some elementary Mathematical Logic, e.g. the Lowenheim-Skolem Theorem. A
student with one semester of mathematical logic and 1 of set theory
should be prepared to read these notes. The first half deals with the
general area of Borel hierarchies. What are the possible lengths of a
Borel hierarchy in a separable metric space? Lebesgue showed that in an
uncountable complete separable metric space the Borel hierarchy has
uncountably many distinct levels, but for incomplete spaces the answer
is independent. The second half includes Harrington's Theorem - it is
consistent to have sets on the second level of the projective hierarchy
of arbitrary size less than the continuum and a proof and appl- ications
of Louveau's Theorem on hyperprojective parameters.