\(AD+DC_{\aleph_ 1}\) is incompatible (Q2367868)

In Lect. Notes Math. 689, 171-183 (1978; Zbl 0388.03022), \textit{R. M. Solovay} first supposed that \(\text{ZF}+\text{DA}_ R\) is consistent, and then he proved the independence of DC from AD by using inner models for \(\text{AD}_ R\). In this note, we further investigate the relation between the generalized version of the principle of dependent choices and the axiom of determinateness. We know that AD implies AC and AC is equivalent to \((\forall k) \text{DC}_ k\), where \(k\) is an Aleph. Since \(\neg(\forall k) \text{DC}_ k\) iff \((\exists k)(\neg\text{DC}_ k)\), we prove that AD implies \((\exists k)(\neg\text{DC}_ k)\). Then, what is \(k\)? This note shows that \(k\) is just \(\aleph_ 1\), i.e. \(\text{AD}+\text{DC}_{\aleph_ 1}\) is not a consistent system. From this we can deduce that AD and \(\text{DC}_ k\) are incompatible, where \(k>\aleph_ 1\). The relation between the axiom of determinateness and the generalized version of the principle of dependent choices is thoroughly solved.