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.
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