5 research outputs found
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
The Weihrauch lattice at the level of : the Cantor-Bendixson theorem
This paper continues the program connecting reverse mathematics and
computable analysis via the framework of Weihrauch reducibility. In particular,
we consider problems related to perfect subsets of Polish spaces, studying the
perfect set theorem, the Cantor-Bendixson theorem and various problems arising
from them. In the framework of reverse mathematics these theorems are
equivalent respectively to and
, the two strongest subsystems of second
order arithmetic among the so-called big five. As far as we know, this is the
first systematic study of problems at the level of
in the Weihrauch lattice.
We show that the strength of some of the problems we study depends on the
topological properties of the Polish space under consideration, while others
have the same strength once the space is rich enough.Comment: 35 page