Guessing and non-guessing of canonical functions

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Abstract

It is possible to control to a large extent, via semiproper forcing, the parameters (ß, ß) measuring the guessing density of the members of any given antichain of stationary subsets of ? (assuming the existence of an inaccessible limit of measurable cardinals). Here, given a pair (ß, ß) of ordinals, we will say that a stationary set S ? ? has guessing density (ß, ß) if ß = ? (S) and ß = sup {? (S) : S ? S, S stationary}, where ? (S) is, for every stationary S ? ?, the infimum of the set of ordinals t = ? + 1 for which there is a function F : S {long rightwards arrow} P (?) with o t (F (?)) <t for all ? ? S and with {? ? S : g (?) ? F (?)} stationary for every a <? and every canonical function g for a. This work involves an analysis of iterations of models of set theory relative to sequences of measures on possibly distinct measurable cardinals. As an application of these techniques I show how to force, from the existence of a supercompact cardinal, a model of PFA in which there is a well-order of H (?) definable, over <H (?), ? >, by a formula without parameters.
Original languageEnglish
Pages (from-to)150-179
Number of pages30
JournalAnnals of Pure and Applied Logic
Volume146
Issue number2-3
DOIs
Publication statusPublished - 1 May 2007

Keywords

  • Guessing canonical functions
  • Iterations relative to sequences of measures on cardinals
  • PFA++
  • Definable well-orders of H(ω2)

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