Cardinals as Ultrapowers. A Canonical Measure Analysis under the Axiom of Determinacy.
Stefan Bold
Abstract:
This thesis is in the field of Descriptive Set Theory and examines
some consequences of the Axiom of Determinacy concerning partition
properties that define large cardinals. The Axiom of Determinacy (AD)
is a game-theoretic statement expressing that all infinite two-player
perfect information games with a countable set of possible moves are
determined, i.e., admit a winning strategy for one of the players.
By the term "measure analysis" we understand the following procedure:
given a strong partition cardinal \kappa and some cardinal \lambda >
\kappa, we assign a measure \mu on \kappa to \lambda such that
\kappa^\kappa/\mu = \lambda. A canonical measure analysis is a measure
assignment for cardinals larger than a strong partition cardinal
\kappa and a binary operation \oplus on the measures of this
assignment that corresponds to ordinal addition on indices of the
cardinals.
This thesis provides a canonical measure analysis up to the \omega
\omega th cardinal after an odd projective cardinal. Using this
canonical measure analysis we show that all cardinals that are
ultrapowers with respect to basic order measures are Jonsson
cardinals. With the canonicity results of this thesis we can state
that, if \kappa is an odd projective ordinal, \kappa^(n) ,
\kappa^(\omega.n+1), and \kappa^(\omega^n+1), for n<\omega, are
Jonsson under AD.
2000 Mathematics Subject Classification: 03E15, 03E60, 03E55, 03E02