4 research outputs found
Lightface -indescribable cardinals
-absoluteness for ccc forcing means that for any ccc forcing ,
. "
inaccessible to reals" means that for any real ,
. To measure the exact consistency strength of
"-absoluteness for ccc forcing and is inaccessible to
reals", we introduce a weak version of a weakly compact cardinal, namely, a
(lightface) -indescribable cardinal; has this property
exactly if it is inaccessible and
Maximal sets without Choice
We show that it is consistent relative to ZF, that there is no well-ordering
of while a wide class of special sets of reals such as Hamel
bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more
precise, we can assume that every projective hypergraph on has a
maximal independent set, among a few other things. For example, we get
transversals for all projective equivalence relations. Moreover, this is
possible while either holds, or countable choice for
reals fails. Assuming the consistency of an inaccessible cardinal, "projective"
can even be replaced with "". This vastly strengthens earlier
consistency results in the literature.Comment: 16 page
Absoluteness via Resurrection
The resurrection axioms are forcing axioms introduced recently by Hamkins and
Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a
stronger form of resurrection axioms (the \emph{iterated} resurrection axioms
for a class of forcings and a given
ordinal ), and show that implies generic
absoluteness for the first-order theory of with respect to
forcings in preserving the axiom, where is a
cardinal which depends on ( if is any
among the classes of countably closed, proper, semiproper, stationary set
preserving forcings).
We also prove that the consistency strength of these axioms is below that of
a Mahlo cardinal for most forcing classes, and below that of a stationary limit
of supercompact cardinals for the class of stationary set preserving posets.
Moreover we outline that simultaneous generic absoluteness for
with respect to and for with respect to
with is in principle
possible, and we present several natural models of the Morse Kelley set theory
where this phenomenon occurs (even for all simultaneously). Finally,
we compare the iterated resurrection axioms (and the generic absoluteness
results we can draw from them) with a variety of other forcing axioms, and also
with the generic absoluteness results by Woodin and the second author.Comment: 34 page