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The axiom of dependent choice (DC) and the axiom of countable choice (AC_ω) are two weak forms of the axiom of choice that can be stated for a specific set: DC(X)asserts that any total binary relation on X has an infinite chain, while AC_ω(X) asserts that any countable collection of nonempty subsets of X has a choice function. It is well-known that DC implies AC_ω. We study for which sets and under which hypotheses DC(X)double right arrow AC_ω(X) and then we show it is consistent with ZF that there is a set A subset of R for which DC(A) holds, but AC_ω(A) fails

Does DC imply AC_ω, uniformly? / Andretta, Alessandro; Notaro, Lorenzo. - In: THE JOURNAL OF SYMBOLIC LOGIC. - ISSN 0022-4812. - ELETTRONICO. - (2024), pp. 1-25. [10.1017/jsl.2024.33]

Does DC imply AC_ω, uniformly?

LORENZO NOTARO
2024

Abstract

The axiom of dependent choice (DC) and the axiom of countable choice (AC_ω) are two weak forms of the axiom of choice that can be stated for a specific set: DC(X)asserts that any total binary relation on X has an infinite chain, while AC_ω(X) asserts that any countable collection of nonempty subsets of X has a choice function. It is well-known that DC implies AC_ω. We study for which sets and under which hypotheses DC(X)double right arrow AC_ω(X) and then we show it is consistent with ZF that there is a set A subset of R for which DC(A) holds, but AC_ω(A) fails
2024
THE JOURNAL OF SYMBOLIC LOGIC
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2992785