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How to have more things by forgetting how to count them(†)

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...

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Autores principales: Karagila, Asaf, Schlicht, Philipp
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society Publishing 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7426041/
https://www.ncbi.nlm.nih.gov/pubmed/32821239
http://dx.doi.org/10.1098/rspa.2019.0782
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author Karagila, Asaf
Schlicht, Philipp
author_facet Karagila, Asaf
Schlicht, Philipp
author_sort Karagila, Asaf
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description 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.
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spelling pubmed-74260412020-08-18 How to have more things by forgetting how to count them(†) Karagila, Asaf Schlicht, Philipp Proc Math Phys Eng Sci Research Article 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. The Royal Society Publishing 2020-07 2020-07-29 /pmc/articles/PMC7426041/ /pubmed/32821239 http://dx.doi.org/10.1098/rspa.2019.0782 Text en © 2020 The Authors. http://creativecommons.org/licenses/by/4.0/ http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Research Article
Karagila, Asaf
Schlicht, Philipp
How to have more things by forgetting how to count them(†)
title How to have more things by forgetting how to count them(†)
title_full How to have more things by forgetting how to count them(†)
title_fullStr How to have more things by forgetting how to count them(†)
title_full_unstemmed How to have more things by forgetting how to count them(†)
title_short How to have more things by forgetting how to count them(†)
title_sort how to have more things by forgetting how to count them(†)
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7426041/
https://www.ncbi.nlm.nih.gov/pubmed/32821239
http://dx.doi.org/10.1098/rspa.2019.0782
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