7,254 research outputs found
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 Îș of uncountable cofinality, while Îș^+ 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 Îș of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{Îș^+} of graphs on Îș^+ such that any graph on Îș^+ embeds into one of the graphs in the collection
On constructions with -cardinals
We propose developing the theory of consequences of morasses relevant in
mathematical applications in the language alternative to the usual one,
replacing commonly used structures by families of sets originating with
Velleman's neat simplified morasses called -cardinals. The theory of related
trees, gaps, colorings of pairs and forcing notions is reformulated and
sketched from a unifying point of view with the focus on the applicability to
constructions of mathematical structures like Boolean algebras, Banach spaces
or compact spaces.
A new result which we obtain as a side product is the consistency of the
existence of a function
with the
appropriate -version of property for regular
satisfying .Comment: Minor correction
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
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
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
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