7,049 research outputs found
Many Different Uniformity Numbers of Yorioka Ideals
Using a countable support product of creature forcing posets, we show that
consistently, for uncountably many different functions the associated Yorioka
ideals' uniformity numbers can be pairwise different. In addition we show that,
in the same forcing extension, for two other types of simple cardinal
characteristics parametrised by reals (localisation and anti-localisation
cardinals), for uncountably many parameters the corresponding cardinals are
pairwise different.Comment: 29 pages, 4 figure
Decisive creatures and large continuum
For f,g\in\omega\ho let \mycfa_{f,g} be the minimal number of uniform
-splitting trees needed to cover the uniform -splitting tree, i.e. for
every branch of the -tree, one of the -trees contains .
\myc_{f,g} is the dual notion: For every branch , one of the -trees
guesses infinitely often.
It is consistent that
\myc_{f_\epsilon,g_\epsilon}=\mycfa_{f_\epsilon,g_\epsilon}=\kappa_\epsilon
for \al1 many pairwise different cardinals and suitable
pairs .
For the proof we use creatures with sufficient bigness and halving. We show
that the lim-inf creature forcing satisfies fusion and pure decision. We
introduce decisiveness and use it to construct a variant of the countable
support iteration of such forcings, which still satisfies fusion and pure
decision.Comment: major revisio
Creature forcing and large continuum: The joy of halving
For let be the minimal number of
uniform -splitting trees needed to cover the uniform -splitting tree,
i.e., for every branch of the -tree, one of the -trees contains
. Let be the dual notion: For every branch , one of
the -trees guesses infinitely often. We show that it is consistent
that
for continuum many pairwise different cardinals and suitable
pairs . For the proof we introduce a new mixed-limit
creature forcing construction
(kappa,theta)-weak normality
We characterize the situation of small cardinality for a product of cardinals
divided by an ultrafilter. We develop the notion of weak normality. We include
an application to Boolean Algebras.Comment: 13 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
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