How to have more things by forgetting how to count them

Karagila, Asaf 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 (former - to 2024)
UEA Research Groups:	Faculty of Science > Research Groups > Logic (former - to 2024)
Related URLs:	http://www.scopus.com/inward/record.url?... https://arxiv.org/abs/1910.14480
Depositing User:	LivePure Connector
Date Deposited:	26 Jun 2020 00:02
Last Modified:	30 Jun 2025 00:39
URI:	https://ueaeprints.uea.ac.uk/id/eprint/75774
DOI:	10.1098/rspa.2019.0782

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