Скачать или смотреть Corey Switzer: Some set theory of Kaufman models

Oxford Set Theory Seminar
http://jdh.hamkins.org/oxford-set-the...

Abstract. A Kaufmann model is an -like, recursively saturated, rather classless model of PA. Such models were shown to exist by Kaufmann under the assumption that Diamond holds, and in ZFC by Shelah via an absoluteness argument involving strong logics. They are important in the theory of models of arithmetic notably because they show that many classic results about countable, recursively saturated models of arithmetic cannot be extended to uncountable models. They are also a particularly interesting example of set theoretic incompactness at aleph_1, similar to an Aronszajn tree.

In this talk we’ll look at several set theoretic issues relating to this class of models motivated by the seemingly naïve question of whether or not such models can be killed by forcing without collapsing aleph_1. Surprisingly the answer to this question turns out to be independent: under MA_{aleph_1} no aleph_1-preserving forcing can destroy Kaufmann-ness whereas under Diamond there is a Kaufmann model M and a Souslin tree S so that forcing with S adds a satisfaction class to M (thus killing rather classlessness). The techniques involved in these proofs also yield another surprising side of Kaufmann models: it is independent of ZFC whether the class of Kaufmann models can be axiomatized in the logic L_{omega_1,omega}(Q) where Q is the quantifier “there exists uncountably many”. This is the logic used in Shelah’s aforementioned result, hence the interest in this level of expressive power.