How to have more things by forgetting how to count them

Asaf Karagila*, Philipp Schlicht

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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, 2^A is extremally disconnected, or [A]^<ω is Dedekind-finite.
Original languageEnglish
Article number476
Number of pages12
JournalProceedings of the Royal Society A: Mathematical and Physical Sciences
Volume476
Issue number2239
DOIs
Publication statusPublished - 29 Jul 2020

Keywords

  • Dedekind-finite sets
  • Cohen forcing
  • Cohen’s first model
  • Axiom of Choice
  • symmetric extensions

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