  Motivated by ideas from the model theory of metric structures, we introduce a
metric set theory, $\mathsf{MSE}$, which takes bounded quantification as
primitive and consists of a natural metric extensionality axiom (the distance
between two sets is the Hausdorff distance between their extensions) and an
approximate, non-deterministic form of full comprehension (for any real-valued
formula $\varphi(x,y)$, tuple of parameters $a$, and $r < s$, there is a set
containing the class $\{x: \varphi(x,a) \leq r\}$ and contained in the class
$\{x:\varphi(x,a) < s\}$). We show that $\mathsf{MSE}$ is sufficient to develop
classical mathematics after the addition of an appropriate axiom of infinity.
We then construct canonical representatives of well-order types and prove that
ultrametric models of $\mathsf{MSE}$ always contain externally ill-founded
ordinals, conjecturing that this is true of all models. To establish several
independence results and, in particular, consistency, we construct a variety of
models, including pseudo-finite models and models containing arbitrarily large
standard ordinals. Finally, we discuss how to formalize $\mathsf{MSE}$ in
either continuous logic or {\L}ukasiewicz logic.
