  Cohen's first model is a model of Zermelo--Fraenkel set theory in which there
is a Dedekind-finite set of real numbers, and it is perhaps the most famous
model where the Axiom of Choice fails. We force over this model to add a
function from this Dedekind-finite set to some infinite ordinal $\kappa$. In
the case that we force the function to be injective, it turns out that the
resulting model is the same as adding $\kappa$ Cohen reals to the ground model,
and that we have just added an enumeration of the canonical Dedekind-finite
set. In the case where the function is merely surjective it turns out that we
do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets.
This motivates the question if there is any combinatorial condition on a
Dedekind-finite set $A$ which characterises when a forcing will preserve its
Dedekind-finiteness or not add new sets of ordinals. We answer this question in
the case of "Adding a Cohen subset" by presenting a varied list of conditions
each equivalent to the preservation of Dedekind-finiteness. For example, $2^A$
is extremally disconnected, or $[A]^{<\omega}$ is Dedekind-finite.
