  We introduce a generalized notion of finiteness that provides a structural
principle for the set of effective theories that can be consistently coupled to
quantum gravity. More concretely, we propose a Tameness Conjecture that states
that all valid effective theories are labelled by a definable parameter space
and must have scalar field spaces and coupling functions that are definable
using the tame geometry built from an o-minimal structure. We give a brief
introduction to tame geometry and describe how it restricts sets, manifolds,
and functions. We then collect evidence for the Tameness Conjecture by studying
various effective theories arising from string theory compactifications by
using some recent advances in tame geometry. In particular, we will exploit the
fact that coset spaces and period mappings are definable in an o-minimal
structure and argue for non-trivial tameness results in higher-supersymmetric
theories and in Calabi-Yau compactifications. As strongest evidence for the
Tameness Conjecture over a discrete parameter space, we then discuss a recent
theorem stating that the locus of self-dual flux vacua of F-theory admits a
tame geometry even if one allows for any flux choice satisfying the tadpole
constraint. This result implies the finiteness of self-dual flux vacua in
F-theory.
