Descriptive Set Theory and Forcing: How to prove theorems about Borel sets the hard way

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This text is an advanced graduate course with some knowledge of forcing is assumed along with some elementary mathematical logic and set theory. The first half of the text 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 size less than on the continuum and a proof and applications of Louveau’s Theorem on hyperprojective parameters.