UNDECIDABILITY OF TOPOLOGICAL AND ARITHMETICAL PROPERTIES OF INFINITARY RATIONAL RELATIONS

摘要

We prove that for every countable ordinal a one cannot decide whether a given infinitary rational relation is in the Borel class Σ_α~0 (respectively Π_α~0). Furthermore one cannot decide whether a given infmitary rational relation is a Borel set or a Σ_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.