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Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations
Published online by Cambridge University Press: 15 November 2003
Abstract
We prove that for every countable ordinal α one cannot decide
whether a given infinitary rational relation is in the Borel class
${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$
). Furthermore
one cannot
decide whether a given infinitary rational relation is a Borel set or a
${\bf \Sigma_{1}^1}$
-complete set. We prove some recursive analogues to these
properties. In
particular one cannot decide whether an infinitary rational relation is an
arithmetical set.
We then deduce from the proof of
these results some other ones, like: one cannot decide whether the
complement of
an infinitary rational relation is also an infinitary rational relation.
Keywords
- Type
- Research Article
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- © EDP Sciences, 2003
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