12,925 research outputs found
Even more simple cardinal invariants
Using GCH, we force the following: There are continuum many simple cardinal
characteristics with pairwise different values.Comment: a few changes (minor corrections) from first versio
The Higher Cicho\'n Diagram
For a strongly inacessible cardinal , we investigate the
relationships between the following ideals:
- the ideal of meager sets in the -box product topology
- the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799)
- the ideal of nowhere stationary subsets of a (naturally defined) stationary
set .
In particular, we analyse the provable inequalities between the cardinal
characteristics for these ideals, and we give consistency results showing that
certain inequalities are unprovable.
While some results from the classical case () can be easily
generalized to our setting, some key results (such as a Fubini property for the
ideal of null sets) do not hold; this leads to the surprising inequality
cov(null)non(null). Also, concepts that did not exist in the classical
case (in particular, the notion of stationary sets) will turn out to be
relevant.
We construct several models to distinguish the various cardinal
characteristics; the main tools are iterations with -support
(and a strong "Knaster" version of -cc) and one iteration with
-support (and a version of -properness).Comment: 84 page
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
Preserving levels of projective determinacy by tree forcings
We prove that various classical tree forcings -- for instance Sacks forcing,
Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve
the statement that every real has a sharp and hence analytic determinacy. We
then lift this result via methods of inner model theory to obtain
level-by-level preservation of projective determinacy (PD). Assuming PD, we
further prove that projective generic absoluteness holds and no new equivalence
classes classes are added to thin projective transitive relations by these
forcings.Comment: 3 figure
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