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  A. Miller \\
  M873        \\
  Descriptive Set Theory \\
  Fall 2000 \\
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Descriptive set theory goes back to the work of the French mathematicians
Borel, Baire, and Lebesgue around the turn of the century who studied the
basic properties of the Borel subsets of the real line, functions with the
Baire property, and Lebesgue measure. In his most famous monograph Lebesgue
made the error of thinking that the projection of a Borel set is Borel. 
This error was caught by the Russian mathematician Suslin resulting in the
definition of analytic sets and leading to work by Luzin and Sierpinski,
e.g., every analytic set is either countable of size continuum.


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Some possible topics in Descriptive Set Theory \\
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\topic (Gale-Stewart) Open subsets of $\om^\om$ are determaned.

\topic (Wolfe) $G_\delta$ subsets of $\om^\om$ are determaned.

\topic (Davis) Axiom of Determanacy AD$\to$P$\to\om_1$ is inacc in $L$

\topic Banach-Mazur games, AD$\to$BP Baire Property

\topic Becker on Banach games, Blass on long games, Prikry on Ramsey

\topic (Mycielski-Swierczkowski) AD$\to$LM Lebesgue Measure

\topic (Solovay) AD$\to\om_1$ is a measurable cardinal  (club filter)

\topic Wadge's Lemma and well-foundedness of Wadge ordering

\topic (Van Wesep-Steel) Separation

\topic (Kechris) AD$\to$SP superperfect trees

\topic (Kunen-Miller) compactly-Borel

\topic Classical Desc Set Thy: Prewellordering, boundedness, union of $\om_1$
Borel sets, absoluteness, MA+notCH+$\om_1=\om_1^L$, regularity of analytic
sets

\topic (Martin) Borel Det

\topic (Friedman) Borel Det not provable in Z

\topic (Martin) Analytic Det and measurable cardinals

\topic Equivalents of $0^\sharp$

\topic Largest thin coanalytic set, coanalytic sets under V=L

\topic (Miller-Steel-Van Engelen) Borel sets not rigid

\topic Solovay model for P+LM+BP, (Mathias) Ramsey, Galvin-Prikry Thm

\topic Invariant Descriptive Set theory : Vaught transform and Lopez-Escobar
Thm

\topic (Morley) Weak Vaught's Conjecture for PC(L$_{\om_1,\om}$) sentences


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  References
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\bigskip{\bf Classical descriptive set theory},  Alexander S. Kechris,
Springer-Verlag, 1995.

\bigskip{\bf Recursive aspects of descriptive set theory}, Richard Mansfield,
Galen Weitkamp ; with a chapter by Stephen Simpson, Oxford University Press,
Clarendon Press, 1985.

\bigskip{\bf Descriptive set theory and forcing : how to prove theorems about
Borel sets the hard way}, Arnold W. Miller, Lecture notes in logic 4.
Springer-Verlag, 1995.

\bigskip{\bf Descriptive set theory},  Yiannis N. Moschovakis,
Elsevier-North Holland, 1980.

\bigskip{\bf Measure and category}, J.Oxtoby, Springer-Verlag, 1971.

\bigskip{\bf Analytic sets},  edited by C. A. Rogers ... [et al.],
Academic Press, 1980.

\bigskip {\bf Cabal Seminar} 76-77,77-79,79-81,81-85, Lecture Notes in
Mathematics, 689, 839, 1019, 1333, Springer-Verlag.


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