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Cohen's first model is a model of Zermelo--Fraenkel set theory in which thereis a Dedekind-finite set of real numbers, and it is perhaps the most famousmodel where the Axiom of Choice fails. We force over this model to add afunction from this Dedekind-finite set to some infinite ordinal $\kappa$. Inthe case that we force the function to be injective, it turns out that theresulting 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-finiteset. In the case where the function is merely surjective it turns out that wedo not add any reals, sets of ordinals, or collapse any Dedekind-finite sets.This motivates the question if there is any combinatorial condition on aDedekind-finite set $A$ which characterises when a forcing will preserve itsDedekind-finiteness or not add new sets of ordinals. We answer this question inthe case of "Adding a Cohen subset" by presenting a varied list of conditionseach equivalent to the preservation of Dedekind-finiteness. For example, $2^A$is extremally disconnected, or $[A]^{<\omega}$ is Dedekind-finite.

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