Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists
arxiv(2022)
摘要
We present three different perspectives of oracle. First, an oracle is a
blackbox; second, an oracle is a tool to change the way we access mathematical
objects; and third, an oracle is a factor that causes a change in truth values.
Formally, the second perspective advocates that an oracle is an endofunctor on
the category of coded sets (preserving underlying sets) – we associate it with
a universal closure operator. The third perspective advocates that an oracle is
an operation on the object of truth values – we associate it with a
Lawvere-Tierney topology. These three perspectives create a link between the
three fields, computability theory, synthetic descriptive set theory, and
effective topos theory.
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