2 research outputs found
Restrictions on Forcings That Change Cofinalities
In this paper we investigate some properties of forcing which can be
considered "nice" in the context of singularizing regular cardinals to have an
uncountable cofinality. We show that such forcing which changes cofinality of a
regular cardinal, cannot be too nice and must cause some "damage" to the
structure of cardinals and stationary sets. As a consequence there is no
analogue to the Prikry forcing, in terms of "nice" properties, when changing
cofinalities to be uncountable.Comment: 8 pages; post-refereeing versio
Equivalence of generics
Given a countable transitive model of set theory and a partial order
contained in it, there is a natural countable Borel equivalence relation on
generic filters over the model; two are equivalent if they yield the same
generic extension. We examine the complexity of this equivalence relation for
various partial orders, with particular focus on Cohen and random forcing. We
prove, amongst other results, that the former is an increasing union of
countably many hyperfinite Borel equivalence relations, while the latter is
neither amenable nor treeable.Comment: 18 pages. We have made minor stylistic changes and corrected an error
in the statement of Lemma 3.5, now appearing as Lemmas 3.5 and 3.