DS-2009-13:
Bold, Stefan
(2009)
*Cardinals as Ultrapowers. A Canonical Measure Analysis under the Axiom of Determinacy.*
Doctoral thesis, University of Amsterdam.

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## 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

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2009-13 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2009 |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2083 |

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