Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Joshua Brot; Mengyang Cao; David Fernández-Bretón

doi:10.1016/j.apal.2021.102961

Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Joshua Brot, Mengyang Cao, David Fernández-Bretón

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.

Original language	English
Article number	102961
Journal	Annals of Pure and Applied Logic
Volume	172
Issue number	6
DOIs	https://doi.org/10.1016/j.apal.2021.102961
State	Published - Jun 2021
Externally published	Yes

Keywords

Amorphous set
Axiom of choice
Dedekind-finite set
Fraenkel–Mostowski model
Hindman's theorem
Ramsey's theorem

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10.1016/j.apal.2021.102961

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abstract = "We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.",

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AU - Cao, Mengyang

AU - Fernández-Bretón, David

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AB - We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.

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KW - Axiom of choice

KW - Dedekind-finite set

KW - Fraenkel–Mostowski model

KW - Hindman's theorem

KW - Ramsey's theorem

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Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Abstract

Keywords

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