622 research outputs found
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
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
The measure algebra does not always embed
The Open Colouring Axiom implies that the measure algebra cannot be embedded
into P(N)/fin. We also discuss the errors in previous results on the
embeddability of the measure algebra.Comment: 10 pages 2000/01/24: final version, 11 page
Measure Recognition Problem
This is an article in mathematics, specifically in set theory. On the example
of the Measure Recognition Problem (MRP) the article highlights the phenomenon
of the utility of a multidisciplinary mathematical approach to a single
mathematical problem, in particular the value of a set-theoretic analysis. MRP
asks if for a given Boolean algebra \algB and a property of measures
one can recognize by purely combinatorial means if \algB supports a strictly
positive measure with property . The most famous instance of this problem
is MRP(countable additivity), and in the first part of the article we survey
the known results on this and some other problems. We show how these results
naturally lead to asking about two other specific instances of the problem MRP,
namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v
zamonja and Plebanek (2006) gives an easy solution to the former of these
problems, and gives some partial information about the latter. The long term
goal of this line of research is to obtain a structure theory of Boolean
algebras that support a finitely additive strictly positive measure, along the
lines of Maharam theorem which gives such a structure theorem for measure
algebras
Some Banach spaces added by a Cohen real
We study certain Banach spaces that are added in the extension by one Cohen
real. Specifically, we show that adding just one Cohen real to any model adds a
Banach space of density which does not embed into any such space in
the ground model such a Banach space can be chosen to be UG This has
consequences on the the isomorphic universality number for Banach spaces of
density , which is hence equal to in the standard Cohen
model and the same is true for UG spaces. Analogous universality results for
Banach spaces are true for other cardinals, by a different proof.Comment: The version to appear in Topology and Its Applications arXiv admin
note: substantial text overlap with arXiv:1308.364
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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