How to have more things by forgetting how to count them

TL;DR

This paper explores how certain forcing methods affect Dedekind-finite sets in models of set theory, revealing conditions that preserve or alter their finiteness and related set-theoretic properties.

Contribution

It characterizes when forcing preserves Dedekind-finiteness of sets and provides equivalent conditions for this preservation in the context of Cohen forcing.

Findings

Forcing with injective functions adds an enumeration of Dedekind-finite sets.

Surjective forcing does not add reals or collapse Dedekind-finite sets.

Several equivalent conditions are identified for preserving Dedekind-finiteness, such as extremal disconnectedness of $2^A$.

Abstract

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.