93 research outputs found
A general tool for consistency results related to I1
In this paper we provide a general tool to prove the consistency of
with various combinatorial properties at typical at
settings with , that does not need a profound knowledge of
the forcing notions involved. Examples of such properties are the first failure
of GCH, a very good scale and the negation of the approachability property, or
the tree property at and
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
Subcompact cardinals, squares, and stationary reflection
We generalise Jensen's result on the incompatibility of subcompactness with
square. We show that alpha^+-subcompactness of some cardinal less than or equal
to alpha precludes square_alpha, but also that square may be forced to hold
everywhere where this obstruction is not present. The forcing also preserves
other strong large cardinals. Similar results are also given for stationary
reflection, with a corresponding strengthening of the large cardinal assumption
involved. Finally, we refine the analysis by considering Schimmerling's
hierarchy of weak squares, showing which cases are precluded by
alpha^+-subcompactness, and again we demonstrate the optimality of our results
by forcing the strongest possible squares under these restrictions to hold.Comment: 18 pages. Corrections and improvements from referee's report mad
Stationary reflection principles and two cardinal tree properties
We study consequences of stationary and semi-stationary set reflection. We
show that the semi stationary reflection principle implies the Singular
Cardinal Hypothesis, the failure of weak square principle, etc. We also
consider two cardinal tree properties introduced recently by Weiss and prove
that they follow from stationary and semi stationary set reflection augmented
with a weak form of Martin's Axiom. We also show that there are some
differences between the two reflection principles which suggest that stationary
set reflection is analogous to supercompactness whereas semi-stationary set
reflection is analogous to strong compactness.Comment: 19 page
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
A framework for forcing constructions at successors of singular cardinals
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ^+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection
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