399 research outputs found
Generic Large Cardinals and Systems of Filters
We introduce the notion of -system of filters, generalizing the
standard definitions of both extenders and towers of normal ideals. This
provides a framework to develop the theory of extenders and towers in a more
general and concise way. In this framework we investigate the topic of
definability of generic large cardinals properties.Comment: 36 page
Dense ideals and cardinal arithmetic
From large cardinals we show the consistency of normal, fine,
-complete -dense ideals on for
successor . We explore the interplay between dense ideals, cardinal
arithmetic, and squares, answering some open questions of Foreman
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Ideal Projections and Forcing Projections
It is well known that saturation of ideals is closely related to the âantichain-catchingâ phenomenon from Foreman-Magidor-Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If I is a normal ideal on Ï2 which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal; (2) For any n â Ï, it is consistent relative to large cardinals that there is a normal ideal I on Ïn which satisfies projective antichain catching, yet I is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foremanâs chapter in the Handbook of Set Theory ([7])
Universality properties of forcing
The purpose of this paper is to investigate forcing as a tool to construct
universal models. In particular, we look at theories of initial segments of the
universe and show that any model of a sufficiently rich fragment of those
theories can be embedded into a model constructed by forcing. Our results rely
on the model-theoretic properties of good ultrafilters, for which we provide a
new existence proof on non-necessarily complete Boolean algebras
Assessment of the environmental aspects of the DOE phosphoric acid fuel cell program
The likely facets of a nationwide phosphoric acid fuel cell (PAFC) power plant commercial system are described. The beneficial and adverse environmental impacts produced by the system are assessed. Eleven specific system activities are characterized and evaluated. Also included is a review of fuel cell technology and a description of DOE's National Fuel Cell Program. Based on current and reasonably foreseeable PAFC characteristics, no environmental or energy impact factor was identified that would significantly inhibit the commercialization of PAFC power plant technology
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