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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...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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The Royal Society Publishing
2020
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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 |
collection | PubMed |
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. |
format | Online Article Text |
id | pubmed-7426041 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2020 |
publisher | The Royal Society Publishing |
record_format | MEDLINE/PubMed |
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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