Cardinals as Ultrapowers

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.<br /> By the term "measure analysis'' we understand the following procedure: given a strong partition cardinal κ and some cardinal λ > κ, we assign a measure µ on κ to λ such that the ultrapower with respect to µ equals λ . A canonical measure analysis is a measure assignment for cardinals larger than a strong partition cardinal κ and a binary operation on the measures of this assignment that corresponds to ordinal addition on indices of the cardinals.<br /> This thesis provides a canonical measure analysis up to the (ω^ω)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 Jónsson cardinals. With the canonicity results of this thesis we can state that, if κ is an odd projective ordinal, κ^(n), κ^(ωn+1), and κ^(ω^n+1), for n<ω, are Jónsson under AD.