Mathematical Logic Quarterly

A Remark on Ascending Chain Conditions, the Countable Axiom of Choice and the Principle of Dependent Choices

Journal Article

Abstract

It is easy to prove in ZF (= Zermelo‐Fraenkel set theory without the axioms of choice and foundation) that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well‐founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the (finite ascending) chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF anymore (if ZF is consistent). More precisely, we will prove that this statement is equivalent in ZF to the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski.

Mathematics Subject Classification: 06B05, 08A65, 08B20, 03E99.

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