  This article expands our work in [Ca16]. By its reliance on Turing
computability, the classical theory of effectivity, along with effective
reducibility and Weihrauch reducibility, is only applicable to objects that are
either countable or can be encoded by countable objects. We propose a notion of
effectivity based on Koepke's Ordinal Turing Machines (OTMs) that applies to
arbitrary set-theoretical $\Pi_{2}$-statements, along with according variants
of effective reducibility and Weihrauch reducibility. As a sample application,
we compare various choice principles with respect to effectivity.
  We also propose a generalization to set-theoretical formulas of arbitrary
quantifier complexity.
