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\begin{center}{\huge Large Cardinals and the Determinacy of Long
Games}\end{center}
\begin{center}{\large Itay Neeman}\end{center}
\begin{center}{\large Harvard University}\end{center}

\bigskip
\large

Transfinite games on integers have been the object of substantial
research in Set Theory. Most particularly, there has been a lot
of work concerning the {\em determinacy} of games of length $\omega$;
first in establishing mathematical consequences of the determinacy
of those games (perhaps most well known --- the determinacy of all integer games
of length $\omega$ implies that all sets of reals are Lebesgue measurable,
and have the property of Baire) and more recently establishing
close connections between determinacy, and {\em large cardinal assumptions}.
Perhaps the most famous result along these lines is the Martin-Steel
theorem, which states that granted the existence of $n$ Woodin cardinals
and a measurable cardinal above them, all games with a $\Pi^1_{n+1}$ payoff
are determined.

Thus, the determinacy of length $\omega$ games with certain definable payoff
was seen to follow from the existence of large cardinals (above, from
the existence of Woodin cardinals).
It is natural to generalize this, and consider games of length greater
than $\omega$. Once we restrict to games with definable payoff (e.g.,
$\Pi^1_1$), the determinacy of our games should also follow from large
cardinals.

In my talk, I will present a hierarchy of games of different lengths
(all greater than $\omega$), and the large cardinals needed to prove
their determinacy. The following theorem is an example of the
determinacy results for relatively short games:

\vspace{0.2in}

{\bf Theorem} (Neeman): Let
$\alpha$ be a countable ordinal.
Assume that there exist $\alpha$ Woodin cardinals and
a measurable cardinal above them. Then all real games of
length $\alpha$ with $\Pi^1_1$ payoff are determined.

\vspace{0.2in}

The games we consider all have payoff sets which can be naturally
coded as sets of reals. Thus the length of the game must be
below $\omega_1$. The natural generalization of the above theorem
($\alpha>\omega_1$) seems to be excluded. We may however consider
games of {\em variable} length --- where the length of the game,
though countable, dependes on the moves the two players make.
For example,

\vspace{0.2in}

{\bf Theorem} (Neeman): Assume that there exist a cardinal which is strong
past a Woodin cardinal, and a measurable cardinal above it. Then
all games of continuously coded length, with $\Pi^1_1$ payoff,
are determined.

\vspace{0.2in}

Games of transfinite length begin to occur naturally in
the theory of Inner Models, once we try to construct inner models
for large cardinals in the region of a Woodin cardinal. The games
considered in inner model theory have a certain canonical form ---
they are all {\em iteration games}. I will attempt to persent an
intuitive survey of how those games
are used in inner model theory, and how tightly they connect large cardinals
with determinacy.

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