Monotone Principle of Forked Points and Its Consequences

Milan Tasković

This paper presents applications of the Axiom of Infinite Choice: Given any set P, there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Ta s kov i c [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Monotone principle of forked points is a direct consequence of the Axiom of Infinite Choice, i.e., of the Lemma of Infinite Maximality! Brouwer and Schauder theorems are two direct censequences of the monotone principle od forked points.