An equivalent form of the Dedekind axiom and its application. II. Also on the unity of the continuous induction, the mathematical induction and the transfinite induction (Q2716363)

This paper is a continuation of Part I [Chin. Q. J. Math. 13, No. 3, 74-80 (1998; Zbl 0952.26003))]. Continuous induction is the following statement: Assume that \(P_x\) is a condition concerning a real number \(x\) such that (1) there is \(x_0\) such that \(P_x\) holds for every \(x<x_0\); (2) if \(P_x\) holds for all \(x<y\), then there is \(\delta_y>0\) such that \(P_x\) is true for every \(x<y+\delta_y\). Then \(P_x\) holds for every real number \(x\). NEWLINENEWLINENEWLINEThis principle was introduced by \textit{Zhang Jingzhong} [Research on educational mathematics (Sichuan Educational Press, Chengdu (Chinese)) (1994)]. In this paper the authors show that continuous induction is equivalent to the Dedekind continuity axiom. They prove also a variant of continuous induction for linearly ordered sets \(S\) with the property that any union of initial segments of \(S\) is either an initial segment in \(S\), or \(S\) (e.g., if \(S\) is well-ordered).