Computability on the space of countable ordinals
release_j4t2mbvb2nhopl42f2t2pnfwbm
by
Arno Pauly
2016
Abstract
While there is a well-established notion of what a computable ordinal is, the
question which functions on the countable ordinals ought to be computable has
received less attention so far. We propose a notion of computability on the
space of countable ordinals via a representation in the sense of computable
analysis. The computability structure is characterized by the computability of
four specific operations, and we prove further relevant operations to be
computable. Some alternative approaches are discussed, too.
As an application in effective descriptive set theory, we can then state and
prove computable uniform versions of the Lusin separation theorem and the
Hausdorff-Kuratowski theorem. Furthermore, we introduce an operator on the
Weihrauch lattice corresponding to iteration of some principle over a countable
ordinal.
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