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\begin{flushright}
  A. Miller \\
  M873        \\
  Topics in Descriptive Set Theory \\
  Fall 2000 \\
\end{flushright}

\topic Open games are determined (Gale-Stewart \cite{gs}).
\topic $G_\delta$ games are determined (Wolfe \cite{wolfe})
\topic P implies $\omega_1$ is inaccessible in L (Levy \cite{levy})
\topic AD implies P, every set of reals is countable or contains a perfect subset
 (Davis \cite{davis})
\topic Inconsistent versions of AD (Mycielski \cite{myc})
\topic DC iff Baire category theorem (Blair \cite{blair}, Goldblatt
 \cite{gold})
\topic (ZFC) $L[\R]\models$``ZF+DC'' (Takeuti \cite{tak})
\topic Equivalence of AD$_\R$ and AD for games of length $\omega^2$
 (Blass \cite{blass})
\topic Banach-Mazur games and Baire property, AD implies BP (see Oxtoby
 \cite{oxtoby})
\topic AD$_\R$ implies every set $A\subseteq [\omega]^\omega$ is
completely Ramsey (Prikry \cite{prikry} also Ellentuck \cite{ellen})
\topic AD for Banach games is equivalent to AD (Freiling \cite{freiling},
Becker \cite{becker})
\topic AD implies LM, every set is Lebesgue measurable
(Mycielski,Swierczkowski \cite{myciel})
\topic The boundedness theorem for ${\bf \Pi}^1_1$
\topic AD implies the club filter on $\omega_1$ is an ultrafilter,
(Solovay 1967, see Kechris \cite{kecp})
\topic Wadge's Lemma, well-foundedness of Wadge ordering,
(Wadge, Martin, Monk, see Van Wesep \cite{vanwesep})
\topic PWO($\Gamma$) implies Red($\Gamma$) implies Sep(dual$\Gamma$)
and not Sep($\Gamma$)
\topic PWO(${\bf\Pi^1_1}$), PWO(${\bf\Sigma^1_2}$)
\topic (V=L) PWO(${\bf\Sigma^1_{n}}$) for $n\geq 2$
(Addison \cite{addison},\cite{addison2} see also \cite{mos})
\topic Projective Determanacy PD implies PWO(${\bf\Sigma^1_{n}}$)
for $n$ even and PWO(${\bf\Pi^1_{n}}$) for $n$ odd,
(Blackwell, Martin, Addison, Moschovakis see \cite{addison3}, \cite{mos})
\topic The Separation holds on exactly one side of a nonselfdual
Wadge class, (Steel \cite{steel}, Van Wesep \cite{van2})
\topic Borel sets are determined, (Martin \cite{martin1},\cite{martin2})
\topic Existence of measurable cardinal equivalent to an elementary
embedding (Los).  Measurable implies $V\not=L$ (Scott).
\topic Existence of measurables does not decide continuum hypothesis
(Levy, Solovay). No nontrival embedding of $V$ into $V$ (Kunen).
\topic Measurables are Ramsey, limit of Ramsey's (Rowbottom).  Erdos
cardinals are strongly inaccessible.
\topic Analytic sets are determined assuming there is a measurable
cardinal (or even Erdos cardinal), (Martin \cite{martin3})
\topic The theory of $0^\sharp$, Club of indiscernibles for $L$,
$\Pi^1_2$-singleton (Silver, Solovay, Rowbottom).
\topic Existence of $0^\sharp$ equivalent to a nontrivial elementary
embedding of $L$ into $L$ (Kunen).
\topic Sharps are equivalent to Analytic Determanacy, (Harrington \cite{har}
see also Sami \cite{sami})
\topic Equivalence of sharps, Wadge's lemma for $\Pi^1_1$, Borel isomorphism,
and $<\omega^2-\Pi^1_1$-det (Harrington, Steel, Martin, see
also Dubose \cite{dubose})


\begin{thebibliography}{99}

\bibitem{addison}
Addison, J. W.,
Some consequences of the axiom of constructibility.
Fund. Math. 46 1959 337--357.

\bibitem{addison2}
Addison, J. W.,
Separation principles in the hierarchies of classical and effective
descriptive set theory. Fund. Math. 46 1959 123--135.

\bibitem{addison3}
Addison, J. W.; Moschovakis, Yiannis N.,
Some consequences of the axiom of definable determinateness.
Proc. Nat. Acad. Sci. U.S.A. 59 1968 708--712.

\bibitem{becker}
Becker, Howard,
Determinacy of Banach games.
J. Symbolic Logic 50 (1985), no. 1, 110--122.

\bibitem{blair}
Blair, Charles E.,
The Baire category theorem implies the principle of dependent choices.
Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom.
Phys. 25 (1977), no. 10, 933--934.

\bibitem{blass}
Blass, Andreas,
Equivalence of two strong forms of determinacy.
Proc. Amer. Math. Soc. 52 (1975), 373--376.

\bibitem{davis}
Davis, Morton,
Infinite games of perfect information. 1964
Advances in game theory 85--101
Princeton Univ. Press, Princeton, N.J.

\bibitem{dubose}
Dubose, Derrick Albert, The equivalence of determinancy and iterated sharps.
J. Symbolic Logic 55 (1990), 502--525.


\bibitem{ellen}
Ellentuck, Erik,
A new proof that analytic sets are Ramsey.
J. Symbolic Logic 39 (1974), 163--165.

\bibitem{freiling}
Freiling, Chris,
Banach games.
J. Symbolic Logic 49 (1984), no. 2, 343--375.


\bibitem{gs}
Gale, David; Stewart, F. M.
Infinite games with perfect information,
Contributions to the theory of games, vol. 2, pp. 245--266,
Annals of Mathematics Studies, no. 28,
Princeton University Press, Princeton, N. J., 1953.

\bibitem{gold}
Goldblatt, Robert,
On the role of the Baire category theorem and dependent choice in the
foundations of logic. J. Symbolic Logic 50 (1985), no. 2, 412--422.

\bibitem{har}
Harrington, Leo,
Analytic determinacy and $0\sp{\sharp }$.
J. Symbolic Logic 43 (1978), no. 4, 685--693.


\bibitem{kecp}
Kechris, Alexander S.,
${\rm AD}$ and projective ordinals.
Cabal Seminar 76--77 (Proc. Caltech-UCLA Logic Sem., 1976--77), pp. 91--132,
Lecture Notes in Math., 689,
Springer, Berlin, 1978.

\bibitem{levy}
Levy, Azriel,
Definability in axiomatic set theory. II. 1970
Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq.,
Jerusalem, 1968) pp. 129--145 North-Holland, Amsterdam

\bibitem{martin3}
Martin, Donald A.,
Measurable cardinals and analytic games.
Fund. Math. 66 1969/1970 287--291.

\bibitem{martin1}
Martin, Donald A.,
Borel determinacy.
Ann. of Math. (2) 102 (1975), no. 2, 363--371.

\bibitem{martin2}
Martin, Donald A.,
A purely inductive proof of Borel determinacy.
Recursion theory (Ithaca, N.Y., 1982), 303--308,
Proc. Sympos. Pure Math., 42,
Amer. Math. Soc., Providence, R.I., 1985.

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

\bibitem{myc}
Mycielski, Jan,
On the axiom of determinateness.
Fund. Math. 53 1963/1964 205--224.


\bibitem{myciel}
Mycielski, Jan; \'Swierczkowski, S.,
On the Lebesgue measurability and the axiom of determinateness.
Fund. Math. 54 1964 67--71.

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

\bibitem{prikry}
Prikry, Karel,
Determinateness and partitions.
Proc. Amer. Math. Soc. 54 (1976), 303--306.

\bibitem{sami}
Sami, Ramez L., Analytic determinacy and $0\sp {\sharp}$. A forcing-free
proof of Harrington's theorem. Fund. Math. 160 (1999), 153--159.

\bibitem{steel}
Steel, John R.,
Determinateness and the separation property.
J. Symbolic Logic 46 (1981), no. 1, 41--44.

\bibitem{tak}
Takeuti, Gaisi, Hypotheses on power set. Axiomatic set theory (Proc. Sympos.
Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967),
pp. 439--446. Amer. Math. Soc., Providence, R.I., 1971.

\bibitem{vanwesep}
Van Wesep, Robert,
Wadge degrees and descriptive set theory.
Cabal Seminar 76--77 (Proc. Caltech-UCLA Logic Sem., 1976--77), pp. 151--170,
Lecture Notes in Math., 689,
Springer, Berlin, 1978.

\bibitem{van2}
Van Wesep, Robert A.,
Separation principles and the axiom of determinateness.
J. Symbolic Logic 43 (1978), no. 1, 77--81.

\bibitem{wolfe}
Wolfe, Philip,
The strict determinateness of certain infinite games.
Pacific J. Math. 5 (1955), 841--847.

\end{thebibliography}
\end{document}

