How to have more things by forgetting how to count them

Karagila, Asaf ORCID: https://orcid.org/0000-0003-1289-0904 and Schlicht, Philipp (2020) How to have more things by forgetting how to count them. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476 (2239). ISSN 1364-5021

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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.

Item Type: Article
Uncontrolled Keywords: axiom of choice,cohen forcing,cohen's first model,dedekind-finite sets,symmetric extensions,mathematics(all),engineering(all),physics and astronomy(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 26 Jun 2020 00:02
Last Modified: 14 Aug 2022 03:42
URI: https://ueaeprints.uea.ac.uk/id/eprint/75774
DOI: 10.1098/rspa.2019.0782

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