How to have more things by forgetting how to count them†.

Asaf Karagila,Philipp Schlicht

doi:10.1098/rspa.2019.0782

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 κ. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ 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, 2A is extremally disconnected, or [A]<ω is Dedekind-finite.

Highlights

Cohen developed the method of forcing to prove that Cantor’s Continuum Hypothesis is not provable from the axioms of Zermelo–Fraenkel and the Axiom of Choice
We show that forcing over Cohen’s first model can have some counterintuitive results
Our two main results to that effect are theorem 4.2, which shows that we can introduce an arbitrary enumeration of the canonical Dedekind-finite set and the resulting model is itself the appropriate Cohen extension of the ground model; and theorem 5.1 where we show that an analogue of the Levy collapse adds a surjection from the canonical Dedekind-finite set onto any fixed ordinal, but does not add new sets of ordinals

Summary

Introduction

Cohen developed the method of forcing to prove that Cantor’s Continuum Hypothesis is not provable from the axioms of Zermelo–Fraenkel and the Axiom of Choice. Monro [4] (later developed by the first author in [5] and by Shani in [6]), where adding Cohen subsets to Dedekind-finite sets is used iteratively to prove the independence of certain weak choice principles from one another. In these works, it is crucial that no new sets of ordinals are added, and in particular no real numbers. We give 10 different equivalent conditions for this preservation, and we show that if the Dedekind-finite set is a set of real numbers, like in Cohen’s first model, these conditions are satisfied

Preliminaries

Cohen’s first model

Injective collapse